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D-分离

2021-07-21T02:00:35Z

Chancychen：

'''d-分离'''是一套决定准则：对于给定的因果图，决定在给定变量集合Z的情况下，变量集合X和变量集合Y是否独立<ref>Koller D, Friedman N. Probabilistic graphical models: principles and techniques[M]. MIT press, 2009.</ref>。

== d-分离的基本思想 ==
它的基本想法是将统计意义上的“独立性”与图论中的“分离性”（“非连通性”）联系起来。这个基本想法首先需要我们去定义在给定的有向图中给定结点集Z取值下的“激活路径”。d-分离中的“d”实际上指示了我们讨论的目标为有向图。

== d-分离的定义 ==
''令<math> \mathbf{X} </math>，<math> \mathbf{Y} </math>，是图<math> \mathcal{G} </math>中的三个结点集合。如果在给定Z的条件下，任意<math> X \in \mathbf{X} </math>与<math> Y \in \mathbf{Y} </math>两个结点间没有激活路径，则称<math> \mathbf{X} </math>和<math> \mathbf{Y} </math>是给定<math>\mathbf{Z} </math>下d-分离的，记作<math> d-sep_{\mathcal{G}(\mathbf{X}l\mathbf{Y} \mid \mathbf{Z}) } </math>。''

''将结点间的一系列独立性关系记作<math> \mathcal{I}(\mathcal{G}) </math>，从而可将独立性和d-分离联系起来：''

''<math> \mathcal{I}(\mathcal{G}) = \left\{ (\mathbf{X} \perp \!\!\! \perp \mathbf{Y} \mid \mathbf{Z}) : d-sep_{\mathcal{G}(\mathbf{X}l\mathbf{Y} \mid \mathbf{Z}) } \right\} </math>''

== d-分离的解释 ==

=== 激活路径的定义 ===
''假设<math> \mathcal{G} </math>是一个贝叶斯网络，<math> \mathbf{Z} </math>是给定的已观测变量集，我们称一条路径<math> \mathbf{X}_1 \leftrightarrow \cdots \leftrightarrow \mathbf{X}_n </math>是给定<math> \mathbf{Z} </math>''下激活的，当且仅当：

* 当路径中出现如此形式的V-结构：''<math> \mathbf{X}_{i-1} \rightarrow \mathbf{X}_{i} \leftarrow \mathbf{X}_{i+1} </math>''时，''<math> \mathbf{X}_{i} </math>或它的某个子孙结点在<math> \mathbf{Z} </math>''中。
* 其它任何路径中的结点都不在''<math> \mathbf{Z} </math>中。''

=== 根据激活路径的解释 ===
由d-分离的定义可知，<math> d-sep_{\mathcal{G}(\mathbf{X}l\mathbf{Y} \mid \mathbf{Z}) } </math>实际上指的就是将所有<math> \mathbf{X} </math>和<math> \mathbf{Y} </math>间的激活路径以及相应激活结点集找出，只要保证<math> \mathbf{Z} </math>中没有激活节点集中的结点，且不含<math> \mathbf{X} </math>与<math> \mathbf{Y} </math>中结点即可做到d-分离。

=== 根据阻断路径的解释<ref>Pearl J, Glymour M, Jewell N P. Causal inference in statistics: A primer[M]. John Wiley & Sons, 2016.</ref> ===
类似激活路径的定义，我们还可以定义阻断路径：

''一条路径被一组结点<math> \mathbf{Z} </math>阻断，当且仅当：''

* ''路径中包含链结构或分叉结构，则中间结点包含在<math> \mathbf{Z} </math>中。''
* ''路径中包含对撞结构，则对撞结点及其子孙结点都不在<math> \mathbf{Z} </math>中。''

有了阻断路径的定义，我们可以重新定义d-分离为：

''如果<math> \mathbf{Z} </math>阻断了<math> \mathbf{X} </math>和<math> \mathbf{Y} </math>之间的每一条路径，则称<math> \mathbf{X} </math>和<math> \mathbf{Y} </math>是给定<math>\mathbf{Z} </math>下d-分离的，<math> \mathbf{X} </math>和<math> \mathbf{Y} </math>在给定<math>\mathbf{Z} </math>下条件独立。''

=== d-分离实例<ref>Spirtes P. Introduction to causal inference[J]. Journal of Machine Learning Research, 2010, 11(5).</ref> ===
[[文件:D-sep ex1.png|缩略图|实例]]
在此图中，T和Y是给定W下d-分离的。但当Z取空集、{W，X1}、{W，X2}等情况下，T和Y不满足d-分离条件。

== d-分离算法 ==

=== 算法 ===
d-分离算法的基本思路如下：

# 构造概率表达式中提到的所有变量的“有向图”，即对应的贝叶斯网络。
# 对于具有公共子代的每对变量，在它们之间绘制无向边（线）。（如果一个变量有两个以上的父母，则在每对父母之间画一条线）
# 将所有边变成无向边。
# 删除所有给定变量的节点，以及他们的边。
# 解释得到的图。

具体的算法可参考文献<ref>Geiger D, Verma T, Pearl J. d-separation: From theorems to algorithms[M]//Machine Intelligence and Pattern Recognition. North-Holland, 1990, 10: 139-148.</ref>。

=== 代码实现 ===
Python开源包NetworkX中包含[https://networkx.org/documentation/stable/reference/algorithms/d_separation.html D-Separation]算法的实现。

== d-分离与因果推断 ==

=== 相关性与因果性 ===
d-分离实际上暗含了对相关性与因果性的区分。假设我们观测到了两个变量X、Y之间有统计相关性，为了鉴别这种相关性是否有两个变量间的直接因果关系导致，我们可以在变量对应的因果图中搜索可将X与Y作d-分离的集合Z，如果存在这样的Z，则说明X与Y可d-分离，即X与Y之间无直接因果关系，它们的相关性由共同原因导致。

=== Faithfulness假设 ===
Faithfulness假设指的是从概率分布中的独立性中，我们可以推断出图中的d-分离性。Faithfulness假设常作为因果推断与因果发现算法的前提假设，更多关于Faithfulness假设的背景与讨论可见参考文献<ref>Uhler C, Raskutti G, Bühlmann P, et al. Geometry of the faithfulness assumption in causal inference[J]. The Annals of Statistics, 2013: 436-463.</ref>。

=== d-分离与后门调整 ===
后门调整是[[Do演算|do-演算]]的一种特例，找出满足后门准则的变量集W，给定W即可将干预下的条件分布识别为统计意义下的含W的条件分布，从而来计算变量间的因果效应<ref>Pearl J. Causality[M]. Cambridge university press, 2009.</ref>。后门准则中寻找的变量集，实际上就是对结果变量Y和干预变量T之间寻找d-分离变量集Z。

== 参考文献 ==
<references>''<math>\mathbf{Z} </math>''</references>

D-分离

2021-07-21T01:56:41Z

Chancychen：

'''d-分离'''是一套决定准则：对于给定的因果图，决定在给定变量集合Z的情况下，变量集合X和变量集合Y是否独立<ref>Koller D, Friedman N. Probabilistic graphical models: principles and techniques[M]. MIT press, 2009.</ref>。

== d-分离的基本思想 ==
它的基本想法是将统计意义上的“独立性”与图论中的“分离性”（“非连通性”）联系起来。这个基本想法首先需要我们去定义在给定的有向图中给定结点集Z取值下的“激活路径”。d-分离中的“d”实际上指示了我们讨论的目标为有向图。

== d-分离的定义 ==
''令<math> \mathbf{X} </math>，<math> \mathbf{Y} </math>，是图<math> \mathcal{G} </math>中的三个结点集合。如果在给定Z的条件下，任意<math> X \in \mathbf{X} </math>与<math> Y \in \mathbf{Y} </math>两个结点间没有激活路径，则称<math> \mathbf{X} </math>和<math> \mathbf{Y} </math>是给定<math>\mathbf{Z} </math>下d-分离的，记作<math> d-sep_{\mathcal{G}(\mathbf{X}l\mathbf{Y} \mid \mathbf{Z}) } </math>。''

''将结点间的一系列独立性关系记作<math> \mathcal{I}(\mathcal{G}) </math>，从而可将独立性和d-分离联系起来：''

''<math> \mathcal{I}(\mathcal{G}) = \left\{ (\mathbf{X} \perp \!\!\! \perp \mathbf{Y} \mid \mathbf{Z}) : d-sep_{\mathcal{G}(\mathbf{X}l\mathbf{Y} \mid \mathbf{Z}) } \right\} </math>''

== d-分离的解释 ==

=== 激活路径的定义 ===
''假设<math> \mathcal{G} </math>是一个贝叶斯网络，<math> \mathbf{Z} </math>是给定的已观测变量集，我们称一条路径<math> \mathbf{X}_1 \leftrightarrow \cdots \leftrightarrow \mathbf{X}_n </math>是给定<math> \mathbf{Z} </math>''下激活的，当且仅当：

* 当路径中出现如此形式的V-结构：''<math> \mathbf{X}_{i-1} \rightarrow \mathbf{X}_{i} \leftarrow \mathbf{X}_{i+1} </math>''时，''<math> \mathbf{X}_{i} </math>或它的某个子孙结点在<math> \mathbf{Z} </math>''中。
* 其它任何路径中的结点都不在''<math> \mathbf{Z} </math>中。''

=== 根据激活路径的解释 ===
由d-分离的定义可知，<math> d-sep_{\mathcal{G}(\mathbf{X}l\mathbf{Y} \mid \mathbf{Z}) } </math>实际上指的就是将所有<math> \mathbf{X} </math>和<math> \mathbf{Y} </math>间的激活路径以及相应激活结点集找出，只要保证<math> \mathbf{Z} </math>中没有激活节点集中的结点，且不含<math> \mathbf{X} </math>与<math> \mathbf{Y} </math>中结点即可做到d-分离。

=== 根据阻断路径的解释<ref>Pearl J, Glymour M, Jewell N P. Causal inference in statistics: A primer[M]. John Wiley & Sons, 2016.</ref> ===
类似激活路径的定义，我们还可以定义阻断路径：

''一条路径被一组结点<math> \mathbf{Z} </math>阻断，当且仅当：''

* ''路径中包含链结构或分叉结构，则中间结点包含在<math> \mathbf{Z} </math>中。''
* ''路径中包含对撞结构，则对撞结点及其子孙结点都不在<math> \mathbf{Z} </math>中。''

有了阻断路径的定义，我们可以重新定义d-分离为：

''如果<math> \mathbf{Z} </math>阻断了<math> \mathbf{X} </math>和<math> \mathbf{Y} </math>之间的每一条路径，则称<math> \mathbf{X} </math>和<math> \mathbf{Y} </math>是给定<math>\mathbf{Z} </math>下d-分离的，<math> \mathbf{X} </math>和<math> \mathbf{Y} </math>在给定<math>\mathbf{Z} </math>下条件独立。''

=== d-分离实例<ref>Spirtes P. Introduction to causal inference[J]. Journal of Machine Learning Research, 2010, 11(5).</ref> ===
[[文件:D-sep ex1.png|缩略图|实例]]
在此图中，T和Y是给定W下d-分离的。但当Z取空集、{W，X1}、{W，X2}等情况下，T和Y不满足d-分离条件。

== d-分离算法 ==
d-分离算法的基本思路如下：

# 构造概率表达式中提到的所有变量的“有向图”，即对应的贝叶斯网络。
# 对于具有公共子代的每对变量，在它们之间绘制无向边（线）。（如果一个变量有两个以上的父母，则在每对父母之间画一条线）
# 将所有边变成无向边。
# 删除所有给定变量的节点，以及他们的边。
# 解释得到的图。

具体的算法可参考文献<ref>Geiger D, Verma T, Pearl J. d-separation: From theorems to algorithms[M]//Machine Intelligence and Pattern Recognition. North-Holland, 1990, 10: 139-148.</ref>。

== d-分离与因果推断 ==

=== 相关性与因果性 ===
d-分离实际上暗含了对相关性与因果性的区分。假设我们观测到了两个变量X、Y之间有统计相关性，为了鉴别这种相关性是否有两个变量间的直接因果关系导致，我们可以在变量对应的因果图中搜索可将X与Y作d-分离的集合Z，如果存在这样的Z，则说明X与Y可d-分离，即X与Y之间无直接因果关系，它们的相关性由共同原因导致。

=== Faithfulness假设 ===
Faithfulness假设指的是从概率分布中的独立性中，我们可以推断出图中的d-分离性。Faithfulness假设常作为因果推断与因果发现算法的前提假设，更多关于Faithfulness假设的背景与讨论可见参考文献<ref>Uhler C, Raskutti G, Bühlmann P, et al. Geometry of the faithfulness assumption in causal inference[J]. The Annals of Statistics, 2013: 436-463.</ref>。

=== d-分离与后门调整 ===
后门调整是[[Do演算|do-演算]]的一种特例，找出满足后门准则的变量集W，给定W即可将干预下的条件分布识别为统计意义下的含W的条件分布，从而来计算变量间的因果效应<ref>Pearl J. Causality[M]. Cambridge university press, 2009.</ref>。后门准则中寻找的变量集，实际上就是对结果变量Y和干预变量T之间寻找d-分离变量集Z。

<references>''<math>\mathbf{Z} </math>''</references>

D-分离

2021-07-21T01:19:15Z

Chancychen：/* 根据阻断路径的解释 */

文件:D-sep ex1.png

2021-07-21T01:15:10Z

2021-07-20T23:58:09Z

Chancychen：建立内容为“'''d-分离'''是一套决定准则：对于给定的因果图，决定在给定变量集合Z的情况下，变量集合X和变量集合Y是否独立<ref>@book{ko…”的新页面

'''d-分离'''是一套决定准则：对于给定的因果图，决定在给定变量集合Z的情况下，变量集合X和变量集合Y是否独立<ref>@book{koller2009probabilistic,
title={Probabilistic graphical models: principles and techniques},
author={Koller, Daphne and Friedman, Nir},
year={2009},
publisher={MIT press}
}</ref>。

鲁宾因果模型

2021-06-09T15:16:56Z

Chancychen：/* 书籍推荐 */

The Rubin causal model (RCM), also known as the Neyman–Rubin causal model,[1] is an approach to the statistical analysis of cause and effect based on the framework of potential outcomes, named after Donald Rubin. The name "Rubin causal model" was first coined by Paul W. Holland.[2] The potential outcomes framework was first proposed by Jerzy Neyman in his 1923 Master's thesis,[3] though he discussed it only in the context of completely randomized experiments.[4] Rubin extended it into a general framework for thinking about causation in both observational and experimental studies.[1]

''' 鲁宾因果模型 Rubin Causal Model (RCM)''' ，也称为 Neyman-Rubin 因果模型，<ref name="sekhon">{{cite book |last=Sekhon |first=Jasjeet |chapter=The Neyman–Rubin Model of Causal Inference and Estimation via Matching Methods |title=The Oxford Handbook of Political Methodology |year=2007 |chapter-url=http://sekhon.berkeley.edu/papers/SekhonOxfordHandbook.pdf }}</ref>是一种基于潜在结果框架的因果统计分析方法，以Donald Rubin的名字命名。“鲁宾因果模型”这个名字最早是由 Paul W. Holland 创造的。 <ref name="holland:causal86">{{cite journal |last=Holland |first=Paul W. |title=Statistics and Causal Inference |journal=[[Journal of the American Statistical Association|J. Amer. Statist. Assoc.]] |volume=81 |issue=396 |year=1986 |pages=945–960 |jstor=2289064 |doi=10.1080/01621459.1986.10478354}}</ref> ''' 潜在结果框架 Potential Outcomes Framework'''最初是由 Jerzy Neyman 在他 1923 年的硕士论文中提出的，<ref name="neyman:masters">Neyman, Jerzy. ''Sur les applications de la theorie des probabilites aux experiences agricoles: Essai des principes.'' Master's Thesis (1923). Excerpts reprinted in English, Statistical Science, Vol. 5, pp. 463–472. ([[Dorota Dabrowska|D. M. Dabrowska]], and T. P. Speed, Translators.)</ref>尽管他只在完全随机实验的背景下讨论了它。 <ref name="Jasa1">{{cite journal |last=Rubin |first=Donald |year=2005 |title=Causal Inference Using Potential Outcomes |journal=[[Journal of the American Statistical Association|J. Amer. Statist. Assoc.]] |volume=100 |issue=469 |pages=322–331 |doi=10.1198/016214504000001880 }}</ref>鲁宾将其扩展为在观察性和实验性研究中思考因果关系的一般框架。<ref name="sekhon"/>

==介绍==

The Rubin causal model is based on the idea of potential outcomes. For example, a person would have a particular income at age 40 if he had attended college, whereas he would have a different income at age 40 if he had not attended college. To measure the causal effect of going to college for this person, we need to compare the outcome for the same individual in both alternative futures. Since it is impossible to see both potential outcomes at once, one of the potential outcomes is always missing. This dilemma is the "fundamental problem of causal inference".

Because of the fundamental problem of causal inference, unit-level causal effects cannot be directly observed. However, randomized experiments allow for the estimation of population-level causal effects.[5] A randomized experiment assigns people randomly to treatments: college or no college. Because of this random assignment, the groups are (on average) equivalent, and the difference in income at age 40 can be attributed to the college assignment since that was the only difference between the groups. An estimate of the average causal effect (also referred to as the average treatment effect) can then be obtained by computing the difference in means between the treated (college-attending) and control (not-college-attending) samples.

In many circumstances, however, randomized experiments are not possible due to ethical or practical concerns. In such scenarios there is a non-random assignment mechanism. This is the case for the example of college attendance: people are not randomly assigned to attend college. Rather, people may choose to attend college based on their financial situation, parents' education, and so on. Many statistical methods have been developed for causal inference, such as propensity score matching. These methods attempt to correct for the assignment mechanism by finding control units similar to treatment units.

鲁宾因果模型基于潜在结果的想法。例如，如果一个人上过大学，他在 40 岁时会有特定的收入，而如果他没有上过大学，他在 40 岁时会有不同的收入。为了衡量这个人上大学的因果效应，我们需要比较同一个人在两种不同的未来中的结果。由于不可能同时看到两种潜在结果，因此总是缺少其中一种潜在结果。这种困境就是“因果推理的基本问题”。

由于因果推理的根本问题，无法直接观察到单元级别的因果效应。然而，随机实验允许估计人口水平的因果效应。<ref name=":01">{{cite journal |last=Rubin |first=Donald |title=Estimating Causal Effects of Treatments in Randomized and Nonrandomized Studies |journal=[[Journal of Educational Psychology|J. Educ. Psychol.]] |volume=66 |issue=5 |year=1974 |pages=688–701 [p. 689] |doi=10.1037/h0037350 }}</ref>随机实验将人们随机分配到对照组：大学或非大学。由于这种随机分配，各组（平均）相等，40 岁时的收入差异可归因于大学分配，因为这是各组之间的唯一差异。然后可以通过计算处理（上大学）和对照（非上大学）样本之间的平均值差异来获得平均因果效应（也称为平均处理效应）的估计值。

然而，在许多情况下，由于伦理或实际问题，随机实验是不可能的。在这种情况下，存在非随机分配机制。上大学的例子就是这种情况：人们不是随机分配上大学的。相反，人们可能会根据他们的经济状况、父母的教育等来选择上大学。已经开发了许多用于因果推断的统计方法，例如倾向得分匹配。这些方法试图通过寻找类似于处理单元的控制单元来纠正分配机制。

==一个扩展案例==

Rubin defines a causal effect:

Intuitively, the causal effect of one treatment, E, over another, C, for a particular unit and an interval of time from {\displaystyle t_{1}}t_{1} to {\displaystyle t_{2}}t_{2} is the difference between what would have happened at time {\displaystyle t_{2}}t_{2} if the unit had been exposed to E initiated at {\displaystyle t_{1}}t_{1} and what would have happened at {\displaystyle t_{2}}t_{2} if the unit had been exposed to C initiated at {\displaystyle t_{1}}t_{1}: 'If an hour ago I had taken two aspirins instead of just a glass of water, my headache would now be gone,' or 'because an hour ago I took two aspirins instead of just a glass of water, my headache is now gone.' Our definition of the causal effect of the E versus C treatment will reflect this intuitive meaning."[5]

According to the RCM, the causal effect of your taking or not taking aspirin one hour ago is the difference between how your head would have felt in case 1 (taking the aspirin) and case 2 (not taking the aspirin). If your headache would remain without aspirin but disappear if you took aspirin, then the causal effect of taking aspirin is headache relief. In most circumstances, we are interested in comparing two futures, one generally termed "treatment" and the other "control". These labels are somewhat arbitrary.

鲁宾定义了一个因果效应：

<blockquote>
直观地说，一种处理 E 对另一种处理 C 的因果效应对于特定单位和从 <math>t_{1}</math> 至 <math>t_{2}</math> 是当时会发生的事情之间的差异 <math>t_{2}</math> 如果装置暴露于 E 开始于 <math>t_{1}</math> 以及会发生什么 <math>t_{2}</math> 如果该装置已暴露于 C 开始于<math>t_{1}</math>：“如果一个小时前我服用了两个阿司匹林而不是一杯水，我的头痛现在就会消失”，或者“因为一个小时前我服用了两个阿司匹林而不是一杯水，我的头痛现在消失了.' 我们对 E 与 C 处理的因果效应的定义将反映这种直观的含义。<ref name=":01"/>”
</blockquote>

根据 RCM，您在一小时前服用或不服用阿司匹林的因果效应是您的头部在情况 1（服用阿司匹林）和情况 2（未服用阿司匹林）中的感受的差异。如果没有阿司匹林你的头痛仍然存在，但如果服用阿司匹林头痛就会消失，那么服用阿司匹林的因果效应是头痛缓解。在大多数情况下，我们对比较两种特性感兴趣，一种通常称为“处理”，另一种称为“控制”。这些标签有些随意。

===潜在结果===

Potential outcomes

Suppose that Joe is participating in an FDA test for a new hypertension drug. If we were omniscient, we would know the outcomes for Joe under both treatment (the new drug) and control (either no treatment or the current standard treatment). The causal effect, or treatment effect, is the difference between these two potential outcomes.

假设 Joe 正在参加 FDA 对一种新的高血压药物的测试。如果我们无所不知，我们就会知道乔在处理t（新药）和控制c（未处理或当前标准处理）下的结果。因果效应或处理效应是这两种潜在结果之间的差异。

{| class="wikitable" align="center"

!主题!!<math>Y_{t}(u)</math>!!<math>Y_{c}(u)</math>!!<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||135||−5

|}

Y_{t}(u) is Joe's blood pressure if he takes the new pill. In general, this notation expresses the potential outcome which results from a treatment, t, on a unit, u. Similarly, {\displaystyle Y_{c}(u)}Y_{c}(u) is the effect of a different treatment, c or control, on a unit, u. In this case, {\displaystyle Y_{c}(u)}Y_{c}(u) is Joe's blood pressure if he doesn't take the pill. {\displaystyle Y_{t}(u)-Y_{c}(u)}Y_{t}(u)-Y_{c}(u) is the causal effect of taking the new drug.

From this table we only know the causal effect on Joe. Everyone else in the study might have an increase in blood pressure if they take the pill. However, regardless of what the causal effect is for the other subjects, the causal effect for Joe is lower blood pressure, relative to what his blood pressure would have been if he had not taken the pill.

Consider a larger sample of patients:

如果乔服用新药丸，<math>Y_{t}(u)</math>就是他的血压。通常，该符号表示对单位u进行处理t 所产生的潜在结果。类似的，<math>Y_{c}(u)</math>是不同处理t或控制c对单元u的影响。在这种情况下，如果乔不服药，<math>Y_{c}(u)</math>就是他的血压。<math>Y_{t}(u)-Y_{c}(u)</math> 是服用新药的因果效应。

从这张表中我们只知道对 Joe 的因果效应。研究中的其他人如果服用药物，血压可能会升高。然而，不管对其他受试者的因果效应是什么，乔的因果效应是血压降低（相对于他没有服用避孕药时的血压）。

考虑更大的患者样本：

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||135||−5
|-
!玛丽||140||150||−10
|-
!莎莉||135||125||10
|-
!鲍勃||135||150||−15

|}

The causal effect is different for every subject, but the drug works for Joe, Mary and Bob because the causal effect is negative. Their blood pressure is lower with the drug than it would have been if each did not take the drug. For Sally, on the other hand, the drug causes an increase in blood pressure.

In order for a potential outcome to make sense, it must be possible, at least a priori. For example, if there is no way for Joe, under any circumstance, to obtain the new drug, then {\displaystyle Y_{t}(u)}Y_{t}(u) is impossible for him. It can never happen. And if {\displaystyle Y_{t}(u)}Y_{t}(u) can never be observed, even in theory, then the causal effect of treatment on Joe's blood pressure is not defined.

因果效应对于每一个主题是不同的，但药物对乔，玛丽和鲍勃都有影响，因为因果效应是负的。他们服用药物后的血压低于每个人不服用药物时的血压。另一方面，对于 Sally 来说，这种药物会导致血压升高。

为了使潜在结果有意义，它必须是可能的，至少是先验已知可能的。例如，如果乔在任何情况下都无法获得新药，那么<math>Y_{t}(u)</math>对他来说是不可能的。它永远不会发生。而如果<math>Y_{t}(u)</math>永远无法观察到（即使在理论上），那么对乔的血压的因果效应是不确定的。

===没有操纵就没有因果关系===

No causation without manipulation

The causal effect of new drug is well defined because it is the simple difference of two potential outcomes, both of which might happen. In this case, we (or something else) can manipulate the world, at least conceptually, so that it is possible that one thing or a different thing might happen.

This definition of causal effects becomes much more problematic if there is no way for one of the potential outcomes to happen, ever. For example, what is the causal effect of Joe's height on his weight? Naively, this seems similar to our other examples. We just need to compare two potential outcomes: what would Joe's weight be under the treatment (where treatment is defined as being 3 inches taller) and what would Joe's weight be under the control (where control is defined as his current height).

A moment's reflection highlights the problem: we can't increase Joe's height. There is no way to observe, even conceptually, what Joe's weight would be if he were taller because there is no way to make him taller. We can't manipulate Joe's height, so it makes no sense to investigate the causal effect of height on weight. Hence the slogan: No causation without manipulation.

新药的因果效应是明确定义的，因为它是两种可能发生的潜在结果的简单差异。在这种情况下，我们（或其他事物）可以干预世界，至少在概念上是这样，因此可能会发生不同的事。

如果永远不可能发生其中一种潜在结果，那么这种因果效应的定义就会变得更加棘手。例如，乔的身高对他的体重有什么因果关系？这似乎与我们的其他示例相似。我们只需要比较两个潜在的结果：Joe 在处理下的体重（处理被定义为高3英寸）和 Joe 在控制下的体重（控制被定义为他当前的身高）。

问题在于：我们无法增加乔的身高。没有办法观察如果乔更高，他的体重会是多少，因为没有办法让他更高。我们无法操纵乔的身高，因此调查身高对体重的因果关系毫无意义。因此有一个口号：没有操纵就没有因果关系。

===稳定单元处理值假设 (SUTVA)===

Stable unit treatment value assumption (SUTVA)

See also: Spillover (experiment)

We require that "the [potential outcome] observation on one unit should be unaffected by the particular assignment of treatments to the other units" (Cox 1958, §2.4). This is called the stable unit treatment value assumption (SUTVA), which goes beyond the concept of independence.

In the context of our example, Joe's blood pressure should not depend on whether or not Mary receives the drug. But what if it does? Suppose that Joe and Mary live in the same house and Mary always cooks. The drug causes Mary to crave salty foods, so if she takes the drug she will cook with more salt than she would have otherwise. A high salt diet increases Joe's blood pressure. Therefore, his outcome will depend on both which treatment he received and which treatment Mary receives.

SUTVA violation makes causal inference more difficult. We can account for dependent observations by considering more treatments. We create 4 treatments by taking into account whether or not Mary receives treatment.

我们要求“对一个单元的 [潜在结果] 观察不应受到其他单元的特定处理分配的影响”（Cox 1958，第 2.4 节）。这被称为稳定单元处理值假设（SUTVA），它超越了独立性的概念。

在我们的例子中，Joe 的血压不应该取决于 Mary 是否接受了药物。但如果真的发生了呢？假设乔和玛丽住在同一所房子里，玛丽总是做饭。这种药物会导致玛丽渴望咸的食物，所以如果她服用这种药物，她会用比其他情况下更多的盐来烹饪。高盐饮食会增加乔的血压。因此，他的结果将取决于他接受的处理和玛丽接受的处理。

在不满足SUTVA的情况下，因果推断会更加困难。我们可以通过考虑更多的处理来解释相关的观察结果。我们通过考虑 Mary 是否接受处理来创建 4 个处理。

{| class="wikitable" align="center"
!主题||乔 = c，玛丽 = t||乔 = t，玛丽 = t||乔 = c，玛丽 = c||乔 = t，玛丽 = c
|-
!乔||140||130||125||120
|}

Recall that a causal effect is defined as the difference between two potential outcomes. In this case, there are multiple causal effects because there are more than two potential outcomes. One is the causal effect of the drug on Joe when Mary receives treatment and is calculated, {\displaystyle 130-140}{\displaystyle 130-140}. Another is the causal effect on Joe when Mary does not receive treatment and is calculated {\displaystyle 120-125}{\displaystyle 120-125}. The third is the causal effect of Mary's treatment on Joe when Joe is not treated. This is calculated as {\displaystyle 140-125}{\displaystyle 140-125}. The treatment Mary receives has a greater causal effect on Joe than the treatment which Joe received has on Joe, and it is in the opposite direction.

By considering more potential outcomes in this way, we can cause SUTVA to hold. However, if any units other than Joe are dependent on Mary, then we must consider further potential outcomes. The greater the number of dependent units, the more potential outcomes we must consider and the more complex the calculations become (consider an experiment with 20 different people, each of whom's treatment status can effect outcomes for every one else). In order to (easily) estimate the causal effect of a single treatment relative to a control, SUTVA should hold.

回想一下，因果效应被定义为两个潜在结果之间的差异。在这种情况下，存在多种因果效应，因为存在两个以上的潜在结果。一是玛丽接受处理时药物对乔的因果效应<math>130-140</math>。另一个是当玛丽没有接受处理时对乔的因果效应<math>120-125</math>。第三是在乔没有得到处理的情况下，玛丽的处理对乔的因果效应<math>140-125</math>。Mary 接受的处理对 Joe 的因果影响比 Joe 接受的处理对 Joe 的影响更大，而且是相反的方向。

通过以这种方式考虑更多潜在结果，我们可以使SUTVA成立。但是，如果 Joe 以外的任何单位都依赖于 Mary，那么我们必须考虑进一步的潜在结果。依赖单位的数量越多，我们必须考虑的潜在结果就越多，计算也变得越复杂（考虑对不同的20个人进行的实验，每个人的处理状态都会影响其他人的结果）。为了（轻松）估计单一处理相对于对照的因果效应，SUTVA 应该成立。

===平均因果效应（Average Causal Effect, ACE）===

考虑：

{| class="wikitable" align="center"
!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||135||−5
|-
!玛丽||130||145||−15
|-
!莎莉||130||145||−15
|-
!鲍勃||140||150||−10
|-
!詹姆士||145||140||+5
|-
!平均||135||143||−8
|}

One may calculate the average causal effect by taking the mean of all the causal effects.

How we measure the response affects what inferences we draw. Suppose that we measure changes in blood pressure as a percentage change rather than in absolute values. Then, depending in the exact numbers, the average causal effect might be an increase in blood pressure. For example, assume that George's blood pressure would be 154 under control and 140 with treatment. The absolute size of the causal effect is −14, but the percentage difference (in terms of the treatment level of 140) is −10%. If Sarah's blood pressure is 200 under treatment and 184 under control, then the causal effect in 16 in absolute terms but 8% in terms of the treatment value. A smaller absolute change in blood pressure (−14 versus 16) yields a larger percentage change (−10% versus 8%) for George. Even though the average causal effect for George and Sarah is +1 in absolute terms, it is −1 in percentage terms.

人们可以通过取所有因果效应的平均值来计算平均因果效应。

我们如何测量反馈效应会影响我们得出的推论。假设我们以百分比变化而不是绝对值来测量血压的变化。然后，根据确切的数字，平均因果效应可能是血压升高。例如，假设乔治的血压在控制下为 154，在处理后为 140。因果效应的绝对大小为 -14，但百分比差异（就 140 的处理水平而言）为 -10%。如果莎拉的血压在治疗下为 200，在控制下为 184，那么因果效应的绝对值是 16，而治疗值是 8%。乔治的血压绝对变化较小（-14 对 16）会产生较大的百分比变化（-10% 对 8%）。

===因果推理的基本问题===

The fundamental problem of causal inference

The results we have seen up to this point would never be measured in practice. It is impossible, by definition, to observe the effect of more than one treatment on a subject over a specific time period. Joe cannot both take the pill and not take the pill at the same time. Therefore, the data would look something like this:

到目前为止，我们所看到的结果永远无法在实践中衡量。根据定义，不可能在特定时间段内观察多种处理对受试者的影响。乔不能同时服用药物和不服用药物。因此，数据看起来像这样：

{| class="wikitable" align="center"
!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||?||?
|}

Question marks are responses that could not be observed. The Fundamental Problem of Causal Inference[2] is that directly observing causal effects is impossible. However, this does not make causal inference impossible. Certain techniques and assumptions allow the fundamental problem to be overcome.

Assume that we have the following data:

问号是无法观察到的反馈。因果推断的基本问题<ref name="holland:causal86"/>是不可能直接观测因果效应。然而，这并不是说因果推断是不可能的。某些技术和假设可以克服基本问题。

假设我们有以下数据：

{| class="wikitable" align="center"
!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||?||?
|-
!玛丽||?||125||?
|-
!莎莉||100||?||?
|-
!鲍勃||?||130||?
|-
!詹姆士||?||120||?
|-
!平均||115||125||−10
|}

We can infer what Joe's potential outcome under control would have been if we make an assumption of constant effect:

如果我们假设效应恒定，我们可以推断出乔在控制下的潜在结果是什么：

<math>Y_{t}(u)=T+Y_{c}(u)</math>

和

<math>Y_{t}(u)-T=Y_{c}(u)</math>。

If we wanted to infer the unobserved values we could assume a constant effect. The following tables illustrates data consistent with the assumption of a constant effect.

如果我们想推断未观察到的值，我们可以假设一个恒定的影响。下表说明了与恒定效应假设一致的数据。

{| class="wikitable" align="center"
!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||140||−10
|-
!玛丽||115||125||−10
|-
!莎莉||100||110||−10
|-
!鲍勃||120||130||−10
|-
!詹姆士||110||120||−10
|-
!平均||115||125||−10
|}

All of the subjects have the same causal effect even though they have different outcomes under the treatment.

所有受试者即使在治疗下有不同的结果，也具有相同的因果效应。

===分配机制（Assignment Mechanism）===

The assignment mechanism

The assignment mechanism, the method by which units are assigned treatment, affects the calculation of the average causal effect. One such assignment mechanism is randomization. For each subject we could flip a coin to determine if she receives treatment. If we wanted five subjects to receive treatment, we could assign treatment to the first five names we pick out of a hat. When we randomly assign treatments we may get different answers.

Assume that this data is the truth:

分配机制，即分配单位处理的方法，影响平均因果效应的计算。一种分配机制是随机化。对于每个受试者，我们可以抛硬币来确定她是否接受处理。如果我们希望五个受试者接受处理，我们可以将处理分配给我们从帽子里挑选出来的前五个名字。当我们随机分配处理时，我们可能会得到不同的答案。

假设这个数据是真实的：

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||115||15
|-
!玛丽||120||125||−5
|-
!莎莉||100||125||−25
|-
!鲍勃||110||130||−20
|-
!詹姆士||115||120||−5
|-
!平均||115||123||−8

|}

The true average causal effect is −8. But the causal effect for these individuals is never equal to this average. The causal effect varies, as it generally (always?) does in real life. After assigning treatments randomly, we might estimate the causal effect as:

真正的平均因果效应是 -8。但是对这些人的因果效应永远不会等于这个平均值。因果效应各不相同，因为它通常（总是？）在现实生活中也是如此。在随机分配处理后，我们可以估计因果效应为：

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>

|-
!乔||130||?||?
|-
!玛丽||120||?||?
|-
!莎莉||?||125||?
|-
!鲍勃||?||130||?
|-
!詹姆士||115||?||?
|-
!平均||121.66||127.5||−5.83

|}

A different random assignment of treatments yields a different estimate of the average causal effect.

处理的不同随机分配产生对平均因果效应的不同估计。

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>

|-
!乔||130||?||?
|-
!玛丽||120||?||?
|-
!莎莉||100||?||?
|-
!鲍勃||?||130||?
|-
!詹姆士||?||120||?
|-
!平均||116.67||125||−8.33

|}

The average causal effect varies because our sample is small and the responses have a large variance. If the sample were larger and the variance were less, the average causal effect would be closer to the true average causal effect regardless of the specific units randomly assigned to treatment.

Alternatively, suppose the mechanism assigns the treatment to all men and only to them.

平均因果效应会有所不同，因为我们的样本很小并且反馈效应的方差很大。如果样本较大且方差较小，则无论随机分配给处理的特定单位如何，平均因果效应将更接近真实的平均因果效应。

或者，假设该机制将处理分配给所有男性且仅分配给他们。

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||?||?
|-
!鲍勃||110||?||?
|-
!詹姆士||105||?||?
|-
!玛丽||?||130||?
|-
!莎莉||?||125||?
|-
!苏茜||?||135||?
|-
!平均||115||130||−15

|}

Under this assignment mechanism, it is impossible for women to receive treatment and therefore impossible to determine the average causal effect on female subjects. In order to make any inferences of causal effect on a subject, the probability that the subject receive treatment must be greater than 0 and less than 1.

在这种分配机制下，女性不可能接受处理，因此无法确定对女性受试者的平均因果效应。为了对受试者做出因果效应的任何推断，受试者接受治疗的概率必须大于 0 且小于 1。

===完美的医生(Perfect Doctor Mechanism)===

The perfect doctor

Consider the use of the perfect doctor as an assignment mechanism. The perfect doctor knows how each subject will respond to the drug or the control and assigns each subject to the treatment that will most benefit her. The perfect doctor knows this information about a sample of patients:

考虑使用完美医生作为分配机制。完美的医生知道每个受试者对药物或对照的反应如何，并为每个受试者分配对她最有益的处理。完美的医生知道有关患者样本的以下信息：

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||115||15
|-
!鲍勃||120||125||−5
|-
!詹姆士||100||150||−50
|-
!玛丽||115||125||−10
|-
!莎莉||120||130||−10
|-
!苏茜||135||105||30
|-
!平均||120||125||−5

|}

Based on this knowledge she would make the following treatment assignments:

基于这些知识，她将进行以下处理分配：

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||?||115||?
|-
!鲍勃||120||?||?
|-
!詹姆士||100||?||?
|-
!玛丽||115||?||?
|-
!莎莉||120||?||?
|-
!苏茜||?||105||?
|-
!平均||113.75||110||3.75
|}

The perfect doctor distorts both averages by filtering out poor responses to both the treatment and control. The difference between means, which is the supposed average causal effect, is distorted in a direction that depends on the details. For instance, a subject like Susie who is harmed by taking the drug would be assigned to the control group by the perfect doctor and thus the negative effect of the drug would be masked.

完美的医生通过过滤掉对处理和控制的不良反应来扭曲这两个平均值。均值之间的差异，即假定的平均因果效应，在取决于细节的方向上发生扭曲。例如，像Susie这样因服药而受到伤害的受试者会被完美的医生分配到对照组，从而掩盖了药物的负面影响。

==结论==

Conclusion

The causal effect of a treatment on a single unit at a point in time is the difference between the outcome variable with the treatment and without the treatment. The Fundamental Problem of Causal Inference is that it is impossible to observe the causal effect on a single unit. You either take the aspirin now or you don't. As a consequence, assumptions must be made in order to estimate the missing counterfactuals.

The Rubin causal model has also been connected to instrumental variables (Angrist, Imbens, and Rubin, 1996)[6] and other techniques for causal inference. For more on the connections between the Rubin causal model, structural equation modeling, and other statistical methods for causal inference, see Morgan and Winship (2007).[7]

在某个时间点对单个单位的处理的因果效应是经过处理和未经过处理的结果变量之间的差异。因果推断的基本问题是不可能观察到对单个单元的因果效应。你要么现在服用阿司匹林，要么不服用。因此，必须做出假设以估计缺失的反事实。

Rubin 因果模型还与工具变量（Angrist、Imbens 和 Rubin，1996 年）<ref name="">Angrist J.,Imbens G.,Rubin D. (1996) [https://pattern.swarma.org/paper?id=86ab704a-c3ad-11eb-ab25-0242ac170007 Identification of Causal effects Using Instrumental Variables].J. Amer. Statist. Assoc.91.434:(444–455)</ref>和其他因果推断技术相关联。有关 Rubin 因果模型、结构方程建模和其他因果推断统计方法之间联系的更多信息，请参见 Morgan 和 Winship (2007)。<ref name="">Morgan S.,Winship C. (2007) [https://pattern.swarma.org/paper?id=b87f164e-c3ad-11eb-ba86-0242ac170007 Counterfactuals and Causal Inference: Methods and Principles for Social Research].</ref>

==另见==

- [https://en.wikipedia.org/wiki/Causality 因果关系]
- [https://en.wikipedia.org/wiki/Principal_stratification 主要分层]
- [https://en.wikipedia.org/wiki/Propensity_score_matching 倾向得分匹配]
<noinclude>

==编者推荐==
===书籍推荐===

*[[Causal Inference: What If]]
作者：Miguel A. Hernán, James M. Robins
[[File:preface.png|200px|thumb|right|《Causal Inference: What If》封面]]

==参考文献==
</noinclude>

文件:Preface.png

2021-06-09T15:15:53Z

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鲁宾因果模型

2021-06-09T15:14:57Z

Chancychen：/* 书籍推荐 */

The Rubin causal model (RCM), also known as the Neyman–Rubin causal model,[1] is an approach to the statistical analysis of cause and effect based on the framework of potential outcomes, named after Donald Rubin. The name "Rubin causal model" was first coined by Paul W. Holland.[2] The potential outcomes framework was first proposed by Jerzy Neyman in his 1923 Master's thesis,[3] though he discussed it only in the context of completely randomized experiments.[4] Rubin extended it into a general framework for thinking about causation in both observational and experimental studies.[1]

''' 鲁宾因果模型 Rubin Causal Model (RCM)''' ，也称为 Neyman-Rubin 因果模型，<ref name="sekhon">{{cite book |last=Sekhon |first=Jasjeet |chapter=The Neyman–Rubin Model of Causal Inference and Estimation via Matching Methods |title=The Oxford Handbook of Political Methodology |year=2007 |chapter-url=http://sekhon.berkeley.edu/papers/SekhonOxfordHandbook.pdf }}</ref>是一种基于潜在结果框架的因果统计分析方法，以Donald Rubin的名字命名。“鲁宾因果模型”这个名字最早是由 Paul W. Holland 创造的。 <ref name="holland:causal86">{{cite journal |last=Holland |first=Paul W. |title=Statistics and Causal Inference |journal=[[Journal of the American Statistical Association|J. Amer. Statist. Assoc.]] |volume=81 |issue=396 |year=1986 |pages=945–960 |jstor=2289064 |doi=10.1080/01621459.1986.10478354}}</ref> ''' 潜在结果框架 Potential Outcomes Framework'''最初是由 Jerzy Neyman 在他 1923 年的硕士论文中提出的，<ref name="neyman:masters">Neyman, Jerzy. ''Sur les applications de la theorie des probabilites aux experiences agricoles: Essai des principes.'' Master's Thesis (1923). Excerpts reprinted in English, Statistical Science, Vol. 5, pp. 463–472. ([[Dorota Dabrowska|D. M. Dabrowska]], and T. P. Speed, Translators.)</ref>尽管他只在完全随机实验的背景下讨论了它。 <ref name="Jasa1">{{cite journal |last=Rubin |first=Donald |year=2005 |title=Causal Inference Using Potential Outcomes |journal=[[Journal of the American Statistical Association|J. Amer. Statist. Assoc.]] |volume=100 |issue=469 |pages=322–331 |doi=10.1198/016214504000001880 }}</ref>鲁宾将其扩展为在观察性和实验性研究中思考因果关系的一般框架。<ref name="sekhon"/>

==介绍==

The Rubin causal model is based on the idea of potential outcomes. For example, a person would have a particular income at age 40 if he had attended college, whereas he would have a different income at age 40 if he had not attended college. To measure the causal effect of going to college for this person, we need to compare the outcome for the same individual in both alternative futures. Since it is impossible to see both potential outcomes at once, one of the potential outcomes is always missing. This dilemma is the "fundamental problem of causal inference".

Because of the fundamental problem of causal inference, unit-level causal effects cannot be directly observed. However, randomized experiments allow for the estimation of population-level causal effects.[5] A randomized experiment assigns people randomly to treatments: college or no college. Because of this random assignment, the groups are (on average) equivalent, and the difference in income at age 40 can be attributed to the college assignment since that was the only difference between the groups. An estimate of the average causal effect (also referred to as the average treatment effect) can then be obtained by computing the difference in means between the treated (college-attending) and control (not-college-attending) samples.

In many circumstances, however, randomized experiments are not possible due to ethical or practical concerns. In such scenarios there is a non-random assignment mechanism. This is the case for the example of college attendance: people are not randomly assigned to attend college. Rather, people may choose to attend college based on their financial situation, parents' education, and so on. Many statistical methods have been developed for causal inference, such as propensity score matching. These methods attempt to correct for the assignment mechanism by finding control units similar to treatment units.

鲁宾因果模型基于潜在结果的想法。例如，如果一个人上过大学，他在 40 岁时会有特定的收入，而如果他没有上过大学，他在 40 岁时会有不同的收入。为了衡量这个人上大学的因果效应，我们需要比较同一个人在两种不同的未来中的结果。由于不可能同时看到两种潜在结果，因此总是缺少其中一种潜在结果。这种困境就是“因果推理的基本问题”。

由于因果推理的根本问题，无法直接观察到单元级别的因果效应。然而，随机实验允许估计人口水平的因果效应。<ref name=":01">{{cite journal |last=Rubin |first=Donald |title=Estimating Causal Effects of Treatments in Randomized and Nonrandomized Studies |journal=[[Journal of Educational Psychology|J. Educ. Psychol.]] |volume=66 |issue=5 |year=1974 |pages=688–701 [p. 689] |doi=10.1037/h0037350 }}</ref>随机实验将人们随机分配到对照组：大学或非大学。由于这种随机分配，各组（平均）相等，40 岁时的收入差异可归因于大学分配，因为这是各组之间的唯一差异。然后可以通过计算处理（上大学）和对照（非上大学）样本之间的平均值差异来获得平均因果效应（也称为平均处理效应）的估计值。

然而，在许多情况下，由于伦理或实际问题，随机实验是不可能的。在这种情况下，存在非随机分配机制。上大学的例子就是这种情况：人们不是随机分配上大学的。相反，人们可能会根据他们的经济状况、父母的教育等来选择上大学。已经开发了许多用于因果推断的统计方法，例如倾向得分匹配。这些方法试图通过寻找类似于处理单元的控制单元来纠正分配机制。

==一个扩展案例==

Rubin defines a causal effect:

Intuitively, the causal effect of one treatment, E, over another, C, for a particular unit and an interval of time from {\displaystyle t_{1}}t_{1} to {\displaystyle t_{2}}t_{2} is the difference between what would have happened at time {\displaystyle t_{2}}t_{2} if the unit had been exposed to E initiated at {\displaystyle t_{1}}t_{1} and what would have happened at {\displaystyle t_{2}}t_{2} if the unit had been exposed to C initiated at {\displaystyle t_{1}}t_{1}: 'If an hour ago I had taken two aspirins instead of just a glass of water, my headache would now be gone,' or 'because an hour ago I took two aspirins instead of just a glass of water, my headache is now gone.' Our definition of the causal effect of the E versus C treatment will reflect this intuitive meaning."[5]

According to the RCM, the causal effect of your taking or not taking aspirin one hour ago is the difference between how your head would have felt in case 1 (taking the aspirin) and case 2 (not taking the aspirin). If your headache would remain without aspirin but disappear if you took aspirin, then the causal effect of taking aspirin is headache relief. In most circumstances, we are interested in comparing two futures, one generally termed "treatment" and the other "control". These labels are somewhat arbitrary.

鲁宾定义了一个因果效应：

<blockquote>
直观地说，一种处理 E 对另一种处理 C 的因果效应对于特定单位和从 <math>t_{1}</math> 至 <math>t_{2}</math> 是当时会发生的事情之间的差异 <math>t_{2}</math> 如果装置暴露于 E 开始于 <math>t_{1}</math> 以及会发生什么 <math>t_{2}</math> 如果该装置已暴露于 C 开始于<math>t_{1}</math>：“如果一个小时前我服用了两个阿司匹林而不是一杯水，我的头痛现在就会消失”，或者“因为一个小时前我服用了两个阿司匹林而不是一杯水，我的头痛现在消失了.' 我们对 E 与 C 处理的因果效应的定义将反映这种直观的含义。<ref name=":01"/>”
</blockquote>

根据 RCM，您在一小时前服用或不服用阿司匹林的因果效应是您的头部在情况 1（服用阿司匹林）和情况 2（未服用阿司匹林）中的感受的差异。如果没有阿司匹林你的头痛仍然存在，但如果服用阿司匹林头痛就会消失，那么服用阿司匹林的因果效应是头痛缓解。在大多数情况下，我们对比较两种特性感兴趣，一种通常称为“处理”，另一种称为“控制”。这些标签有些随意。

===潜在结果===

Potential outcomes

Suppose that Joe is participating in an FDA test for a new hypertension drug. If we were omniscient, we would know the outcomes for Joe under both treatment (the new drug) and control (either no treatment or the current standard treatment). The causal effect, or treatment effect, is the difference between these two potential outcomes.

假设 Joe 正在参加 FDA 对一种新的高血压药物的测试。如果我们无所不知，我们就会知道乔在处理t（新药）和控制c（未处理或当前标准处理）下的结果。因果效应或处理效应是这两种潜在结果之间的差异。

{| class="wikitable" align="center"

!主题!!<math>Y_{t}(u)</math>!!<math>Y_{c}(u)</math>!!<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||135||−5

|}

Y_{t}(u) is Joe's blood pressure if he takes the new pill. In general, this notation expresses the potential outcome which results from a treatment, t, on a unit, u. Similarly, {\displaystyle Y_{c}(u)}Y_{c}(u) is the effect of a different treatment, c or control, on a unit, u. In this case, {\displaystyle Y_{c}(u)}Y_{c}(u) is Joe's blood pressure if he doesn't take the pill. {\displaystyle Y_{t}(u)-Y_{c}(u)}Y_{t}(u)-Y_{c}(u) is the causal effect of taking the new drug.

From this table we only know the causal effect on Joe. Everyone else in the study might have an increase in blood pressure if they take the pill. However, regardless of what the causal effect is for the other subjects, the causal effect for Joe is lower blood pressure, relative to what his blood pressure would have been if he had not taken the pill.

Consider a larger sample of patients:

如果乔服用新药丸，<math>Y_{t}(u)</math>就是他的血压。通常，该符号表示对单位u进行处理t 所产生的潜在结果。类似的，<math>Y_{c}(u)</math>是不同处理t或控制c对单元u的影响。在这种情况下，如果乔不服药，<math>Y_{c}(u)</math>就是他的血压。<math>Y_{t}(u)-Y_{c}(u)</math> 是服用新药的因果效应。

从这张表中我们只知道对 Joe 的因果效应。研究中的其他人如果服用药物，血压可能会升高。然而，不管对其他受试者的因果效应是什么，乔的因果效应是血压降低（相对于他没有服用避孕药时的血压）。

考虑更大的患者样本：

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||135||−5
|-
!玛丽||140||150||−10
|-
!莎莉||135||125||10
|-
!鲍勃||135||150||−15

|}

The causal effect is different for every subject, but the drug works for Joe, Mary and Bob because the causal effect is negative. Their blood pressure is lower with the drug than it would have been if each did not take the drug. For Sally, on the other hand, the drug causes an increase in blood pressure.

In order for a potential outcome to make sense, it must be possible, at least a priori. For example, if there is no way for Joe, under any circumstance, to obtain the new drug, then {\displaystyle Y_{t}(u)}Y_{t}(u) is impossible for him. It can never happen. And if {\displaystyle Y_{t}(u)}Y_{t}(u) can never be observed, even in theory, then the causal effect of treatment on Joe's blood pressure is not defined.

因果效应对于每一个主题是不同的，但药物对乔，玛丽和鲍勃都有影响，因为因果效应是负的。他们服用药物后的血压低于每个人不服用药物时的血压。另一方面，对于 Sally 来说，这种药物会导致血压升高。

为了使潜在结果有意义，它必须是可能的，至少是先验已知可能的。例如，如果乔在任何情况下都无法获得新药，那么<math>Y_{t}(u)</math>对他来说是不可能的。它永远不会发生。而如果<math>Y_{t}(u)</math>永远无法观察到（即使在理论上），那么对乔的血压的因果效应是不确定的。

===没有操纵就没有因果关系===

No causation without manipulation

The causal effect of new drug is well defined because it is the simple difference of two potential outcomes, both of which might happen. In this case, we (or something else) can manipulate the world, at least conceptually, so that it is possible that one thing or a different thing might happen.

This definition of causal effects becomes much more problematic if there is no way for one of the potential outcomes to happen, ever. For example, what is the causal effect of Joe's height on his weight? Naively, this seems similar to our other examples. We just need to compare two potential outcomes: what would Joe's weight be under the treatment (where treatment is defined as being 3 inches taller) and what would Joe's weight be under the control (where control is defined as his current height).

A moment's reflection highlights the problem: we can't increase Joe's height. There is no way to observe, even conceptually, what Joe's weight would be if he were taller because there is no way to make him taller. We can't manipulate Joe's height, so it makes no sense to investigate the causal effect of height on weight. Hence the slogan: No causation without manipulation.

新药的因果效应是明确定义的，因为它是两种可能发生的潜在结果的简单差异。在这种情况下，我们（或其他事物）可以干预世界，至少在概念上是这样，因此可能会发生不同的事。

如果永远不可能发生其中一种潜在结果，那么这种因果效应的定义就会变得更加棘手。例如，乔的身高对他的体重有什么因果关系？这似乎与我们的其他示例相似。我们只需要比较两个潜在的结果：Joe 在处理下的体重（处理被定义为高3英寸）和 Joe 在控制下的体重（控制被定义为他当前的身高）。

问题在于：我们无法增加乔的身高。没有办法观察如果乔更高，他的体重会是多少，因为没有办法让他更高。我们无法操纵乔的身高，因此调查身高对体重的因果关系毫无意义。因此有一个口号：没有操纵就没有因果关系。

===稳定单元处理值假设 (SUTVA)===

Stable unit treatment value assumption (SUTVA)

See also: Spillover (experiment)

We require that "the [potential outcome] observation on one unit should be unaffected by the particular assignment of treatments to the other units" (Cox 1958, §2.4). This is called the stable unit treatment value assumption (SUTVA), which goes beyond the concept of independence.

In the context of our example, Joe's blood pressure should not depend on whether or not Mary receives the drug. But what if it does? Suppose that Joe and Mary live in the same house and Mary always cooks. The drug causes Mary to crave salty foods, so if she takes the drug she will cook with more salt than she would have otherwise. A high salt diet increases Joe's blood pressure. Therefore, his outcome will depend on both which treatment he received and which treatment Mary receives.

SUTVA violation makes causal inference more difficult. We can account for dependent observations by considering more treatments. We create 4 treatments by taking into account whether or not Mary receives treatment.

我们要求“对一个单元的 [潜在结果] 观察不应受到其他单元的特定处理分配的影响”（Cox 1958，第 2.4 节）。这被称为稳定单元处理值假设（SUTVA），它超越了独立性的概念。

在我们的例子中，Joe 的血压不应该取决于 Mary 是否接受了药物。但如果真的发生了呢？假设乔和玛丽住在同一所房子里，玛丽总是做饭。这种药物会导致玛丽渴望咸的食物，所以如果她服用这种药物，她会用比其他情况下更多的盐来烹饪。高盐饮食会增加乔的血压。因此，他的结果将取决于他接受的处理和玛丽接受的处理。

在不满足SUTVA的情况下，因果推断会更加困难。我们可以通过考虑更多的处理来解释相关的观察结果。我们通过考虑 Mary 是否接受处理来创建 4 个处理。

{| class="wikitable" align="center"
!主题||乔 = c，玛丽 = t||乔 = t，玛丽 = t||乔 = c，玛丽 = c||乔 = t，玛丽 = c
|-
!乔||140||130||125||120
|}

Recall that a causal effect is defined as the difference between two potential outcomes. In this case, there are multiple causal effects because there are more than two potential outcomes. One is the causal effect of the drug on Joe when Mary receives treatment and is calculated, {\displaystyle 130-140}{\displaystyle 130-140}. Another is the causal effect on Joe when Mary does not receive treatment and is calculated {\displaystyle 120-125}{\displaystyle 120-125}. The third is the causal effect of Mary's treatment on Joe when Joe is not treated. This is calculated as {\displaystyle 140-125}{\displaystyle 140-125}. The treatment Mary receives has a greater causal effect on Joe than the treatment which Joe received has on Joe, and it is in the opposite direction.

By considering more potential outcomes in this way, we can cause SUTVA to hold. However, if any units other than Joe are dependent on Mary, then we must consider further potential outcomes. The greater the number of dependent units, the more potential outcomes we must consider and the more complex the calculations become (consider an experiment with 20 different people, each of whom's treatment status can effect outcomes for every one else). In order to (easily) estimate the causal effect of a single treatment relative to a control, SUTVA should hold.

回想一下，因果效应被定义为两个潜在结果之间的差异。在这种情况下，存在多种因果效应，因为存在两个以上的潜在结果。一是玛丽接受处理时药物对乔的因果效应<math>130-140</math>。另一个是当玛丽没有接受处理时对乔的因果效应<math>120-125</math>。第三是在乔没有得到处理的情况下，玛丽的处理对乔的因果效应<math>140-125</math>。Mary 接受的处理对 Joe 的因果影响比 Joe 接受的处理对 Joe 的影响更大，而且是相反的方向。

通过以这种方式考虑更多潜在结果，我们可以使SUTVA成立。但是，如果 Joe 以外的任何单位都依赖于 Mary，那么我们必须考虑进一步的潜在结果。依赖单位的数量越多，我们必须考虑的潜在结果就越多，计算也变得越复杂（考虑对不同的20个人进行的实验，每个人的处理状态都会影响其他人的结果）。为了（轻松）估计单一处理相对于对照的因果效应，SUTVA 应该成立。

===平均因果效应（Average Causal Effect, ACE）===

考虑：

{| class="wikitable" align="center"
!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||135||−5
|-
!玛丽||130||145||−15
|-
!莎莉||130||145||−15
|-
!鲍勃||140||150||−10
|-
!詹姆士||145||140||+5
|-
!平均||135||143||−8
|}

One may calculate the average causal effect by taking the mean of all the causal effects.

How we measure the response affects what inferences we draw. Suppose that we measure changes in blood pressure as a percentage change rather than in absolute values. Then, depending in the exact numbers, the average causal effect might be an increase in blood pressure. For example, assume that George's blood pressure would be 154 under control and 140 with treatment. The absolute size of the causal effect is −14, but the percentage difference (in terms of the treatment level of 140) is −10%. If Sarah's blood pressure is 200 under treatment and 184 under control, then the causal effect in 16 in absolute terms but 8% in terms of the treatment value. A smaller absolute change in blood pressure (−14 versus 16) yields a larger percentage change (−10% versus 8%) for George. Even though the average causal effect for George and Sarah is +1 in absolute terms, it is −1 in percentage terms.

人们可以通过取所有因果效应的平均值来计算平均因果效应。

我们如何测量反馈效应会影响我们得出的推论。假设我们以百分比变化而不是绝对值来测量血压的变化。然后，根据确切的数字，平均因果效应可能是血压升高。例如，假设乔治的血压在控制下为 154，在处理后为 140。因果效应的绝对大小为 -14，但百分比差异（就 140 的处理水平而言）为 -10%。如果莎拉的血压在治疗下为 200，在控制下为 184，那么因果效应的绝对值是 16，而治疗值是 8%。乔治的血压绝对变化较小（-14 对 16）会产生较大的百分比变化（-10% 对 8%）。

===因果推理的基本问题===

The fundamental problem of causal inference

The results we have seen up to this point would never be measured in practice. It is impossible, by definition, to observe the effect of more than one treatment on a subject over a specific time period. Joe cannot both take the pill and not take the pill at the same time. Therefore, the data would look something like this:

到目前为止，我们所看到的结果永远无法在实践中衡量。根据定义，不可能在特定时间段内观察多种处理对受试者的影响。乔不能同时服用药物和不服用药物。因此，数据看起来像这样：

{| class="wikitable" align="center"
!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||?||?
|}

Question marks are responses that could not be observed. The Fundamental Problem of Causal Inference[2] is that directly observing causal effects is impossible. However, this does not make causal inference impossible. Certain techniques and assumptions allow the fundamental problem to be overcome.

Assume that we have the following data:

问号是无法观察到的反馈。因果推断的基本问题<ref name="holland:causal86"/>是不可能直接观测因果效应。然而，这并不是说因果推断是不可能的。某些技术和假设可以克服基本问题。

假设我们有以下数据：

{| class="wikitable" align="center"
!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||?||?
|-
!玛丽||?||125||?
|-
!莎莉||100||?||?
|-
!鲍勃||?||130||?
|-
!詹姆士||?||120||?
|-
!平均||115||125||−10
|}

We can infer what Joe's potential outcome under control would have been if we make an assumption of constant effect:

如果我们假设效应恒定，我们可以推断出乔在控制下的潜在结果是什么：

<math>Y_{t}(u)=T+Y_{c}(u)</math>

和

<math>Y_{t}(u)-T=Y_{c}(u)</math>。

If we wanted to infer the unobserved values we could assume a constant effect. The following tables illustrates data consistent with the assumption of a constant effect.

如果我们想推断未观察到的值，我们可以假设一个恒定的影响。下表说明了与恒定效应假设一致的数据。

{| class="wikitable" align="center"
!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||140||−10
|-
!玛丽||115||125||−10
|-
!莎莉||100||110||−10
|-
!鲍勃||120||130||−10
|-
!詹姆士||110||120||−10
|-
!平均||115||125||−10
|}

All of the subjects have the same causal effect even though they have different outcomes under the treatment.

所有受试者即使在治疗下有不同的结果，也具有相同的因果效应。

===分配机制（Assignment Mechanism）===

The assignment mechanism

The assignment mechanism, the method by which units are assigned treatment, affects the calculation of the average causal effect. One such assignment mechanism is randomization. For each subject we could flip a coin to determine if she receives treatment. If we wanted five subjects to receive treatment, we could assign treatment to the first five names we pick out of a hat. When we randomly assign treatments we may get different answers.

Assume that this data is the truth:

分配机制，即分配单位处理的方法，影响平均因果效应的计算。一种分配机制是随机化。对于每个受试者，我们可以抛硬币来确定她是否接受处理。如果我们希望五个受试者接受处理，我们可以将处理分配给我们从帽子里挑选出来的前五个名字。当我们随机分配处理时，我们可能会得到不同的答案。

假设这个数据是真实的：

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||115||15
|-
!玛丽||120||125||−5
|-
!莎莉||100||125||−25
|-
!鲍勃||110||130||−20
|-
!詹姆士||115||120||−5
|-
!平均||115||123||−8

|}

The true average causal effect is −8. But the causal effect for these individuals is never equal to this average. The causal effect varies, as it generally (always?) does in real life. After assigning treatments randomly, we might estimate the causal effect as:

真正的平均因果效应是 -8。但是对这些人的因果效应永远不会等于这个平均值。因果效应各不相同，因为它通常（总是？）在现实生活中也是如此。在随机分配处理后，我们可以估计因果效应为：

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>

|-
!乔||130||?||?
|-
!玛丽||120||?||?
|-
!莎莉||?||125||?
|-
!鲍勃||?||130||?
|-
!詹姆士||115||?||?
|-
!平均||121.66||127.5||−5.83

|}

A different random assignment of treatments yields a different estimate of the average causal effect.

处理的不同随机分配产生对平均因果效应的不同估计。

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>

|-
!乔||130||?||?
|-
!玛丽||120||?||?
|-
!莎莉||100||?||?
|-
!鲍勃||?||130||?
|-
!詹姆士||?||120||?
|-
!平均||116.67||125||−8.33

|}

The average causal effect varies because our sample is small and the responses have a large variance. If the sample were larger and the variance were less, the average causal effect would be closer to the true average causal effect regardless of the specific units randomly assigned to treatment.

Alternatively, suppose the mechanism assigns the treatment to all men and only to them.

平均因果效应会有所不同，因为我们的样本很小并且反馈效应的方差很大。如果样本较大且方差较小，则无论随机分配给处理的特定单位如何，平均因果效应将更接近真实的平均因果效应。

或者，假设该机制将处理分配给所有男性且仅分配给他们。

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||?||?
|-
!鲍勃||110||?||?
|-
!詹姆士||105||?||?
|-
!玛丽||?||130||?
|-
!莎莉||?||125||?
|-
!苏茜||?||135||?
|-
!平均||115||130||−15

|}

Under this assignment mechanism, it is impossible for women to receive treatment and therefore impossible to determine the average causal effect on female subjects. In order to make any inferences of causal effect on a subject, the probability that the subject receive treatment must be greater than 0 and less than 1.

在这种分配机制下，女性不可能接受处理，因此无法确定对女性受试者的平均因果效应。为了对受试者做出因果效应的任何推断，受试者接受治疗的概率必须大于 0 且小于 1。

===完美的医生(Perfect Doctor Mechanism)===

The perfect doctor

Consider the use of the perfect doctor as an assignment mechanism. The perfect doctor knows how each subject will respond to the drug or the control and assigns each subject to the treatment that will most benefit her. The perfect doctor knows this information about a sample of patients:

考虑使用完美医生作为分配机制。完美的医生知道每个受试者对药物或对照的反应如何，并为每个受试者分配对她最有益的处理。完美的医生知道有关患者样本的以下信息：

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||115||15
|-
!鲍勃||120||125||−5
|-
!詹姆士||100||150||−50
|-
!玛丽||115||125||−10
|-
!莎莉||120||130||−10
|-
!苏茜||135||105||30
|-
!平均||120||125||−5

|}

Based on this knowledge she would make the following treatment assignments:

基于这些知识，她将进行以下处理分配：

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||?||115||?
|-
!鲍勃||120||?||?
|-
!詹姆士||100||?||?
|-
!玛丽||115||?||?
|-
!莎莉||120||?||?
|-
!苏茜||?||105||?
|-
!平均||113.75||110||3.75
|}

The perfect doctor distorts both averages by filtering out poor responses to both the treatment and control. The difference between means, which is the supposed average causal effect, is distorted in a direction that depends on the details. For instance, a subject like Susie who is harmed by taking the drug would be assigned to the control group by the perfect doctor and thus the negative effect of the drug would be masked.

完美的医生通过过滤掉对处理和控制的不良反应来扭曲这两个平均值。均值之间的差异，即假定的平均因果效应，在取决于细节的方向上发生扭曲。例如，像Susie这样因服药而受到伤害的受试者会被完美的医生分配到对照组，从而掩盖了药物的负面影响。

==结论==

Conclusion

The causal effect of a treatment on a single unit at a point in time is the difference between the outcome variable with the treatment and without the treatment. The Fundamental Problem of Causal Inference is that it is impossible to observe the causal effect on a single unit. You either take the aspirin now or you don't. As a consequence, assumptions must be made in order to estimate the missing counterfactuals.

The Rubin causal model has also been connected to instrumental variables (Angrist, Imbens, and Rubin, 1996)[6] and other techniques for causal inference. For more on the connections between the Rubin causal model, structural equation modeling, and other statistical methods for causal inference, see Morgan and Winship (2007).[7]

在某个时间点对单个单位的处理的因果效应是经过处理和未经过处理的结果变量之间的差异。因果推断的基本问题是不可能观察到对单个单元的因果效应。你要么现在服用阿司匹林，要么不服用。因此，必须做出假设以估计缺失的反事实。

Rubin 因果模型还与工具变量（Angrist、Imbens 和 Rubin，1996 年）<ref name="">Angrist J.,Imbens G.,Rubin D. (1996) [https://pattern.swarma.org/paper?id=86ab704a-c3ad-11eb-ab25-0242ac170007 Identification of Causal effects Using Instrumental Variables].J. Amer. Statist. Assoc.91.434:(444–455)</ref>和其他因果推断技术相关联。有关 Rubin 因果模型、结构方程建模和其他因果推断统计方法之间联系的更多信息，请参见 Morgan 和 Winship (2007)。<ref name="">Morgan S.,Winship C. (2007) [https://pattern.swarma.org/paper?id=b87f164e-c3ad-11eb-ba86-0242ac170007 Counterfactuals and Causal Inference: Methods and Principles for Social Research].</ref>

==另见==

- [https://en.wikipedia.org/wiki/Causality 因果关系]
- [https://en.wikipedia.org/wiki/Principal_stratification 主要分层]
- [https://en.wikipedia.org/wiki/Propensity_score_matching 倾向得分匹配]
<noinclude>

==编者推荐==
===书籍推荐===

*[[Causal Inference: What If]]
作者：Miguel A. Hernán, James M. Robins
[[File:preface.jpg|200px|thumb|right|《统计因果推理入门》封面]]

==参考文献==
</noinclude>

鲁宾因果模型

2021-06-09T15:14:21Z

Chancychen：/* 书籍推荐 */

The Rubin causal model (RCM), also known as the Neyman–Rubin causal model,[1] is an approach to the statistical analysis of cause and effect based on the framework of potential outcomes, named after Donald Rubin. The name "Rubin causal model" was first coined by Paul W. Holland.[2] The potential outcomes framework was first proposed by Jerzy Neyman in his 1923 Master's thesis,[3] though he discussed it only in the context of completely randomized experiments.[4] Rubin extended it into a general framework for thinking about causation in both observational and experimental studies.[1]

''' 鲁宾因果模型 Rubin Causal Model (RCM)''' ，也称为 Neyman-Rubin 因果模型，<ref name="sekhon">{{cite book |last=Sekhon |first=Jasjeet |chapter=The Neyman–Rubin Model of Causal Inference and Estimation via Matching Methods |title=The Oxford Handbook of Political Methodology |year=2007 |chapter-url=http://sekhon.berkeley.edu/papers/SekhonOxfordHandbook.pdf }}</ref>是一种基于潜在结果框架的因果统计分析方法，以Donald Rubin的名字命名。“鲁宾因果模型”这个名字最早是由 Paul W. Holland 创造的。 <ref name="holland:causal86">{{cite journal |last=Holland |first=Paul W. |title=Statistics and Causal Inference |journal=[[Journal of the American Statistical Association|J. Amer. Statist. Assoc.]] |volume=81 |issue=396 |year=1986 |pages=945–960 |jstor=2289064 |doi=10.1080/01621459.1986.10478354}}</ref> ''' 潜在结果框架 Potential Outcomes Framework'''最初是由 Jerzy Neyman 在他 1923 年的硕士论文中提出的，<ref name="neyman:masters">Neyman, Jerzy. ''Sur les applications de la theorie des probabilites aux experiences agricoles: Essai des principes.'' Master's Thesis (1923). Excerpts reprinted in English, Statistical Science, Vol. 5, pp. 463–472. ([[Dorota Dabrowska|D. M. Dabrowska]], and T. P. Speed, Translators.)</ref>尽管他只在完全随机实验的背景下讨论了它。 <ref name="Jasa1">{{cite journal |last=Rubin |first=Donald |year=2005 |title=Causal Inference Using Potential Outcomes |journal=[[Journal of the American Statistical Association|J. Amer. Statist. Assoc.]] |volume=100 |issue=469 |pages=322–331 |doi=10.1198/016214504000001880 }}</ref>鲁宾将其扩展为在观察性和实验性研究中思考因果关系的一般框架。<ref name="sekhon"/>

==介绍==

The Rubin causal model is based on the idea of potential outcomes. For example, a person would have a particular income at age 40 if he had attended college, whereas he would have a different income at age 40 if he had not attended college. To measure the causal effect of going to college for this person, we need to compare the outcome for the same individual in both alternative futures. Since it is impossible to see both potential outcomes at once, one of the potential outcomes is always missing. This dilemma is the "fundamental problem of causal inference".

Because of the fundamental problem of causal inference, unit-level causal effects cannot be directly observed. However, randomized experiments allow for the estimation of population-level causal effects.[5] A randomized experiment assigns people randomly to treatments: college or no college. Because of this random assignment, the groups are (on average) equivalent, and the difference in income at age 40 can be attributed to the college assignment since that was the only difference between the groups. An estimate of the average causal effect (also referred to as the average treatment effect) can then be obtained by computing the difference in means between the treated (college-attending) and control (not-college-attending) samples.

In many circumstances, however, randomized experiments are not possible due to ethical or practical concerns. In such scenarios there is a non-random assignment mechanism. This is the case for the example of college attendance: people are not randomly assigned to attend college. Rather, people may choose to attend college based on their financial situation, parents' education, and so on. Many statistical methods have been developed for causal inference, such as propensity score matching. These methods attempt to correct for the assignment mechanism by finding control units similar to treatment units.

鲁宾因果模型基于潜在结果的想法。例如，如果一个人上过大学，他在 40 岁时会有特定的收入，而如果他没有上过大学，他在 40 岁时会有不同的收入。为了衡量这个人上大学的因果效应，我们需要比较同一个人在两种不同的未来中的结果。由于不可能同时看到两种潜在结果，因此总是缺少其中一种潜在结果。这种困境就是“因果推理的基本问题”。

由于因果推理的根本问题，无法直接观察到单元级别的因果效应。然而，随机实验允许估计人口水平的因果效应。<ref name=":01">{{cite journal |last=Rubin |first=Donald |title=Estimating Causal Effects of Treatments in Randomized and Nonrandomized Studies |journal=[[Journal of Educational Psychology|J. Educ. Psychol.]] |volume=66 |issue=5 |year=1974 |pages=688–701 [p. 689] |doi=10.1037/h0037350 }}</ref>随机实验将人们随机分配到对照组：大学或非大学。由于这种随机分配，各组（平均）相等，40 岁时的收入差异可归因于大学分配，因为这是各组之间的唯一差异。然后可以通过计算处理（上大学）和对照（非上大学）样本之间的平均值差异来获得平均因果效应（也称为平均处理效应）的估计值。

然而，在许多情况下，由于伦理或实际问题，随机实验是不可能的。在这种情况下，存在非随机分配机制。上大学的例子就是这种情况：人们不是随机分配上大学的。相反，人们可能会根据他们的经济状况、父母的教育等来选择上大学。已经开发了许多用于因果推断的统计方法，例如倾向得分匹配。这些方法试图通过寻找类似于处理单元的控制单元来纠正分配机制。

==一个扩展案例==

Rubin defines a causal effect:

Intuitively, the causal effect of one treatment, E, over another, C, for a particular unit and an interval of time from {\displaystyle t_{1}}t_{1} to {\displaystyle t_{2}}t_{2} is the difference between what would have happened at time {\displaystyle t_{2}}t_{2} if the unit had been exposed to E initiated at {\displaystyle t_{1}}t_{1} and what would have happened at {\displaystyle t_{2}}t_{2} if the unit had been exposed to C initiated at {\displaystyle t_{1}}t_{1}: 'If an hour ago I had taken two aspirins instead of just a glass of water, my headache would now be gone,' or 'because an hour ago I took two aspirins instead of just a glass of water, my headache is now gone.' Our definition of the causal effect of the E versus C treatment will reflect this intuitive meaning."[5]

According to the RCM, the causal effect of your taking or not taking aspirin one hour ago is the difference between how your head would have felt in case 1 (taking the aspirin) and case 2 (not taking the aspirin). If your headache would remain without aspirin but disappear if you took aspirin, then the causal effect of taking aspirin is headache relief. In most circumstances, we are interested in comparing two futures, one generally termed "treatment" and the other "control". These labels are somewhat arbitrary.

鲁宾定义了一个因果效应：

<blockquote>
直观地说，一种处理 E 对另一种处理 C 的因果效应对于特定单位和从 <math>t_{1}</math> 至 <math>t_{2}</math> 是当时会发生的事情之间的差异 <math>t_{2}</math> 如果装置暴露于 E 开始于 <math>t_{1}</math> 以及会发生什么 <math>t_{2}</math> 如果该装置已暴露于 C 开始于<math>t_{1}</math>：“如果一个小时前我服用了两个阿司匹林而不是一杯水，我的头痛现在就会消失”，或者“因为一个小时前我服用了两个阿司匹林而不是一杯水，我的头痛现在消失了.' 我们对 E 与 C 处理的因果效应的定义将反映这种直观的含义。<ref name=":01"/>”
</blockquote>

根据 RCM，您在一小时前服用或不服用阿司匹林的因果效应是您的头部在情况 1（服用阿司匹林）和情况 2（未服用阿司匹林）中的感受的差异。如果没有阿司匹林你的头痛仍然存在，但如果服用阿司匹林头痛就会消失，那么服用阿司匹林的因果效应是头痛缓解。在大多数情况下，我们对比较两种特性感兴趣，一种通常称为“处理”，另一种称为“控制”。这些标签有些随意。

===潜在结果===

Potential outcomes

Suppose that Joe is participating in an FDA test for a new hypertension drug. If we were omniscient, we would know the outcomes for Joe under both treatment (the new drug) and control (either no treatment or the current standard treatment). The causal effect, or treatment effect, is the difference between these two potential outcomes.

假设 Joe 正在参加 FDA 对一种新的高血压药物的测试。如果我们无所不知，我们就会知道乔在处理t（新药）和控制c（未处理或当前标准处理）下的结果。因果效应或处理效应是这两种潜在结果之间的差异。

{| class="wikitable" align="center"

!主题!!<math>Y_{t}(u)</math>!!<math>Y_{c}(u)</math>!!<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||135||−5

|}

Y_{t}(u) is Joe's blood pressure if he takes the new pill. In general, this notation expresses the potential outcome which results from a treatment, t, on a unit, u. Similarly, {\displaystyle Y_{c}(u)}Y_{c}(u) is the effect of a different treatment, c or control, on a unit, u. In this case, {\displaystyle Y_{c}(u)}Y_{c}(u) is Joe's blood pressure if he doesn't take the pill. {\displaystyle Y_{t}(u)-Y_{c}(u)}Y_{t}(u)-Y_{c}(u) is the causal effect of taking the new drug.

From this table we only know the causal effect on Joe. Everyone else in the study might have an increase in blood pressure if they take the pill. However, regardless of what the causal effect is for the other subjects, the causal effect for Joe is lower blood pressure, relative to what his blood pressure would have been if he had not taken the pill.

Consider a larger sample of patients:

如果乔服用新药丸，<math>Y_{t}(u)</math>就是他的血压。通常，该符号表示对单位u进行处理t 所产生的潜在结果。类似的，<math>Y_{c}(u)</math>是不同处理t或控制c对单元u的影响。在这种情况下，如果乔不服药，<math>Y_{c}(u)</math>就是他的血压。<math>Y_{t}(u)-Y_{c}(u)</math> 是服用新药的因果效应。

从这张表中我们只知道对 Joe 的因果效应。研究中的其他人如果服用药物，血压可能会升高。然而，不管对其他受试者的因果效应是什么，乔的因果效应是血压降低（相对于他没有服用避孕药时的血压）。

考虑更大的患者样本：

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||135||−5
|-
!玛丽||140||150||−10
|-
!莎莉||135||125||10
|-
!鲍勃||135||150||−15

|}

The causal effect is different for every subject, but the drug works for Joe, Mary and Bob because the causal effect is negative. Their blood pressure is lower with the drug than it would have been if each did not take the drug. For Sally, on the other hand, the drug causes an increase in blood pressure.

In order for a potential outcome to make sense, it must be possible, at least a priori. For example, if there is no way for Joe, under any circumstance, to obtain the new drug, then {\displaystyle Y_{t}(u)}Y_{t}(u) is impossible for him. It can never happen. And if {\displaystyle Y_{t}(u)}Y_{t}(u) can never be observed, even in theory, then the causal effect of treatment on Joe's blood pressure is not defined.

因果效应对于每一个主题是不同的，但药物对乔，玛丽和鲍勃都有影响，因为因果效应是负的。他们服用药物后的血压低于每个人不服用药物时的血压。另一方面，对于 Sally 来说，这种药物会导致血压升高。

为了使潜在结果有意义，它必须是可能的，至少是先验已知可能的。例如，如果乔在任何情况下都无法获得新药，那么<math>Y_{t}(u)</math>对他来说是不可能的。它永远不会发生。而如果<math>Y_{t}(u)</math>永远无法观察到（即使在理论上），那么对乔的血压的因果效应是不确定的。

===没有操纵就没有因果关系===

No causation without manipulation

The causal effect of new drug is well defined because it is the simple difference of two potential outcomes, both of which might happen. In this case, we (or something else) can manipulate the world, at least conceptually, so that it is possible that one thing or a different thing might happen.

This definition of causal effects becomes much more problematic if there is no way for one of the potential outcomes to happen, ever. For example, what is the causal effect of Joe's height on his weight? Naively, this seems similar to our other examples. We just need to compare two potential outcomes: what would Joe's weight be under the treatment (where treatment is defined as being 3 inches taller) and what would Joe's weight be under the control (where control is defined as his current height).

A moment's reflection highlights the problem: we can't increase Joe's height. There is no way to observe, even conceptually, what Joe's weight would be if he were taller because there is no way to make him taller. We can't manipulate Joe's height, so it makes no sense to investigate the causal effect of height on weight. Hence the slogan: No causation without manipulation.

新药的因果效应是明确定义的，因为它是两种可能发生的潜在结果的简单差异。在这种情况下，我们（或其他事物）可以干预世界，至少在概念上是这样，因此可能会发生不同的事。

如果永远不可能发生其中一种潜在结果，那么这种因果效应的定义就会变得更加棘手。例如，乔的身高对他的体重有什么因果关系？这似乎与我们的其他示例相似。我们只需要比较两个潜在的结果：Joe 在处理下的体重（处理被定义为高3英寸）和 Joe 在控制下的体重（控制被定义为他当前的身高）。

问题在于：我们无法增加乔的身高。没有办法观察如果乔更高，他的体重会是多少，因为没有办法让他更高。我们无法操纵乔的身高，因此调查身高对体重的因果关系毫无意义。因此有一个口号：没有操纵就没有因果关系。

===稳定单元处理值假设 (SUTVA)===

Stable unit treatment value assumption (SUTVA)

See also: Spillover (experiment)

We require that "the [potential outcome] observation on one unit should be unaffected by the particular assignment of treatments to the other units" (Cox 1958, §2.4). This is called the stable unit treatment value assumption (SUTVA), which goes beyond the concept of independence.

In the context of our example, Joe's blood pressure should not depend on whether or not Mary receives the drug. But what if it does? Suppose that Joe and Mary live in the same house and Mary always cooks. The drug causes Mary to crave salty foods, so if she takes the drug she will cook with more salt than she would have otherwise. A high salt diet increases Joe's blood pressure. Therefore, his outcome will depend on both which treatment he received and which treatment Mary receives.

SUTVA violation makes causal inference more difficult. We can account for dependent observations by considering more treatments. We create 4 treatments by taking into account whether or not Mary receives treatment.

我们要求“对一个单元的 [潜在结果] 观察不应受到其他单元的特定处理分配的影响”（Cox 1958，第 2.4 节）。这被称为稳定单元处理值假设（SUTVA），它超越了独立性的概念。

在我们的例子中，Joe 的血压不应该取决于 Mary 是否接受了药物。但如果真的发生了呢？假设乔和玛丽住在同一所房子里，玛丽总是做饭。这种药物会导致玛丽渴望咸的食物，所以如果她服用这种药物，她会用比其他情况下更多的盐来烹饪。高盐饮食会增加乔的血压。因此，他的结果将取决于他接受的处理和玛丽接受的处理。

在不满足SUTVA的情况下，因果推断会更加困难。我们可以通过考虑更多的处理来解释相关的观察结果。我们通过考虑 Mary 是否接受处理来创建 4 个处理。

{| class="wikitable" align="center"
!主题||乔 = c，玛丽 = t||乔 = t，玛丽 = t||乔 = c，玛丽 = c||乔 = t，玛丽 = c
|-
!乔||140||130||125||120
|}

Recall that a causal effect is defined as the difference between two potential outcomes. In this case, there are multiple causal effects because there are more than two potential outcomes. One is the causal effect of the drug on Joe when Mary receives treatment and is calculated, {\displaystyle 130-140}{\displaystyle 130-140}. Another is the causal effect on Joe when Mary does not receive treatment and is calculated {\displaystyle 120-125}{\displaystyle 120-125}. The third is the causal effect of Mary's treatment on Joe when Joe is not treated. This is calculated as {\displaystyle 140-125}{\displaystyle 140-125}. The treatment Mary receives has a greater causal effect on Joe than the treatment which Joe received has on Joe, and it is in the opposite direction.

By considering more potential outcomes in this way, we can cause SUTVA to hold. However, if any units other than Joe are dependent on Mary, then we must consider further potential outcomes. The greater the number of dependent units, the more potential outcomes we must consider and the more complex the calculations become (consider an experiment with 20 different people, each of whom's treatment status can effect outcomes for every one else). In order to (easily) estimate the causal effect of a single treatment relative to a control, SUTVA should hold.

回想一下，因果效应被定义为两个潜在结果之间的差异。在这种情况下，存在多种因果效应，因为存在两个以上的潜在结果。一是玛丽接受处理时药物对乔的因果效应<math>130-140</math>。另一个是当玛丽没有接受处理时对乔的因果效应<math>120-125</math>。第三是在乔没有得到处理的情况下，玛丽的处理对乔的因果效应<math>140-125</math>。Mary 接受的处理对 Joe 的因果影响比 Joe 接受的处理对 Joe 的影响更大，而且是相反的方向。

通过以这种方式考虑更多潜在结果，我们可以使SUTVA成立。但是，如果 Joe 以外的任何单位都依赖于 Mary，那么我们必须考虑进一步的潜在结果。依赖单位的数量越多，我们必须考虑的潜在结果就越多，计算也变得越复杂（考虑对不同的20个人进行的实验，每个人的处理状态都会影响其他人的结果）。为了（轻松）估计单一处理相对于对照的因果效应，SUTVA 应该成立。

===平均因果效应（Average Causal Effect, ACE）===

考虑：

{| class="wikitable" align="center"
!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||135||−5
|-
!玛丽||130||145||−15
|-
!莎莉||130||145||−15
|-
!鲍勃||140||150||−10
|-
!詹姆士||145||140||+5
|-
!平均||135||143||−8
|}

One may calculate the average causal effect by taking the mean of all the causal effects.

How we measure the response affects what inferences we draw. Suppose that we measure changes in blood pressure as a percentage change rather than in absolute values. Then, depending in the exact numbers, the average causal effect might be an increase in blood pressure. For example, assume that George's blood pressure would be 154 under control and 140 with treatment. The absolute size of the causal effect is −14, but the percentage difference (in terms of the treatment level of 140) is −10%. If Sarah's blood pressure is 200 under treatment and 184 under control, then the causal effect in 16 in absolute terms but 8% in terms of the treatment value. A smaller absolute change in blood pressure (−14 versus 16) yields a larger percentage change (−10% versus 8%) for George. Even though the average causal effect for George and Sarah is +1 in absolute terms, it is −1 in percentage terms.

人们可以通过取所有因果效应的平均值来计算平均因果效应。

我们如何测量反馈效应会影响我们得出的推论。假设我们以百分比变化而不是绝对值来测量血压的变化。然后，根据确切的数字，平均因果效应可能是血压升高。例如，假设乔治的血压在控制下为 154，在处理后为 140。因果效应的绝对大小为 -14，但百分比差异（就 140 的处理水平而言）为 -10%。如果莎拉的血压在治疗下为 200，在控制下为 184，那么因果效应的绝对值是 16，而治疗值是 8%。乔治的血压绝对变化较小（-14 对 16）会产生较大的百分比变化（-10% 对 8%）。

===因果推理的基本问题===

The fundamental problem of causal inference

The results we have seen up to this point would never be measured in practice. It is impossible, by definition, to observe the effect of more than one treatment on a subject over a specific time period. Joe cannot both take the pill and not take the pill at the same time. Therefore, the data would look something like this:

到目前为止，我们所看到的结果永远无法在实践中衡量。根据定义，不可能在特定时间段内观察多种处理对受试者的影响。乔不能同时服用药物和不服用药物。因此，数据看起来像这样：

{| class="wikitable" align="center"
!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||?||?
|}

Question marks are responses that could not be observed. The Fundamental Problem of Causal Inference[2] is that directly observing causal effects is impossible. However, this does not make causal inference impossible. Certain techniques and assumptions allow the fundamental problem to be overcome.

Assume that we have the following data:

问号是无法观察到的反馈。因果推断的基本问题<ref name="holland:causal86"/>是不可能直接观测因果效应。然而，这并不是说因果推断是不可能的。某些技术和假设可以克服基本问题。

假设我们有以下数据：

{| class="wikitable" align="center"
!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||?||?
|-
!玛丽||?||125||?
|-
!莎莉||100||?||?
|-
!鲍勃||?||130||?
|-
!詹姆士||?||120||?
|-
!平均||115||125||−10
|}

We can infer what Joe's potential outcome under control would have been if we make an assumption of constant effect:

如果我们假设效应恒定，我们可以推断出乔在控制下的潜在结果是什么：

<math>Y_{t}(u)=T+Y_{c}(u)</math>

和

<math>Y_{t}(u)-T=Y_{c}(u)</math>。

If we wanted to infer the unobserved values we could assume a constant effect. The following tables illustrates data consistent with the assumption of a constant effect.

如果我们想推断未观察到的值，我们可以假设一个恒定的影响。下表说明了与恒定效应假设一致的数据。

{| class="wikitable" align="center"
!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||140||−10
|-
!玛丽||115||125||−10
|-
!莎莉||100||110||−10
|-
!鲍勃||120||130||−10
|-
!詹姆士||110||120||−10
|-
!平均||115||125||−10
|}

All of the subjects have the same causal effect even though they have different outcomes under the treatment.

所有受试者即使在治疗下有不同的结果，也具有相同的因果效应。

===分配机制（Assignment Mechanism）===

The assignment mechanism

The assignment mechanism, the method by which units are assigned treatment, affects the calculation of the average causal effect. One such assignment mechanism is randomization. For each subject we could flip a coin to determine if she receives treatment. If we wanted five subjects to receive treatment, we could assign treatment to the first five names we pick out of a hat. When we randomly assign treatments we may get different answers.

Assume that this data is the truth:

分配机制，即分配单位处理的方法，影响平均因果效应的计算。一种分配机制是随机化。对于每个受试者，我们可以抛硬币来确定她是否接受处理。如果我们希望五个受试者接受处理，我们可以将处理分配给我们从帽子里挑选出来的前五个名字。当我们随机分配处理时，我们可能会得到不同的答案。

假设这个数据是真实的：

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||115||15
|-
!玛丽||120||125||−5
|-
!莎莉||100||125||−25
|-
!鲍勃||110||130||−20
|-
!詹姆士||115||120||−5
|-
!平均||115||123||−8

|}

The true average causal effect is −8. But the causal effect for these individuals is never equal to this average. The causal effect varies, as it generally (always?) does in real life. After assigning treatments randomly, we might estimate the causal effect as:

真正的平均因果效应是 -8。但是对这些人的因果效应永远不会等于这个平均值。因果效应各不相同，因为它通常（总是？）在现实生活中也是如此。在随机分配处理后，我们可以估计因果效应为：

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>

|-
!乔||130||?||?
|-
!玛丽||120||?||?
|-
!莎莉||?||125||?
|-
!鲍勃||?||130||?
|-
!詹姆士||115||?||?
|-
!平均||121.66||127.5||−5.83

|}

A different random assignment of treatments yields a different estimate of the average causal effect.

处理的不同随机分配产生对平均因果效应的不同估计。

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>

|-
!乔||130||?||?
|-
!玛丽||120||?||?
|-
!莎莉||100||?||?
|-
!鲍勃||?||130||?
|-
!詹姆士||?||120||?
|-
!平均||116.67||125||−8.33

|}

The average causal effect varies because our sample is small and the responses have a large variance. If the sample were larger and the variance were less, the average causal effect would be closer to the true average causal effect regardless of the specific units randomly assigned to treatment.

Alternatively, suppose the mechanism assigns the treatment to all men and only to them.

平均因果效应会有所不同，因为我们的样本很小并且反馈效应的方差很大。如果样本较大且方差较小，则无论随机分配给处理的特定单位如何，平均因果效应将更接近真实的平均因果效应。

或者，假设该机制将处理分配给所有男性且仅分配给他们。

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||?||?
|-
!鲍勃||110||?||?
|-
!詹姆士||105||?||?
|-
!玛丽||?||130||?
|-
!莎莉||?||125||?
|-
!苏茜||?||135||?
|-
!平均||115||130||−15

|}

Under this assignment mechanism, it is impossible for women to receive treatment and therefore impossible to determine the average causal effect on female subjects. In order to make any inferences of causal effect on a subject, the probability that the subject receive treatment must be greater than 0 and less than 1.

在这种分配机制下，女性不可能接受处理，因此无法确定对女性受试者的平均因果效应。为了对受试者做出因果效应的任何推断，受试者接受治疗的概率必须大于 0 且小于 1。

===完美的医生(Perfect Doctor Mechanism)===

The perfect doctor

Consider the use of the perfect doctor as an assignment mechanism. The perfect doctor knows how each subject will respond to the drug or the control and assigns each subject to the treatment that will most benefit her. The perfect doctor knows this information about a sample of patients:

考虑使用完美医生作为分配机制。完美的医生知道每个受试者对药物或对照的反应如何，并为每个受试者分配对她最有益的处理。完美的医生知道有关患者样本的以下信息：

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||115||15
|-
!鲍勃||120||125||−5
|-
!詹姆士||100||150||−50
|-
!玛丽||115||125||−10
|-
!莎莉||120||130||−10
|-
!苏茜||135||105||30
|-
!平均||120||125||−5

|}

Based on this knowledge she would make the following treatment assignments:

基于这些知识，她将进行以下处理分配：

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||?||115||?
|-
!鲍勃||120||?||?
|-
!詹姆士||100||?||?
|-
!玛丽||115||?||?
|-
!莎莉||120||?||?
|-
!苏茜||?||105||?
|-
!平均||113.75||110||3.75
|}

The perfect doctor distorts both averages by filtering out poor responses to both the treatment and control. The difference between means, which is the supposed average causal effect, is distorted in a direction that depends on the details. For instance, a subject like Susie who is harmed by taking the drug would be assigned to the control group by the perfect doctor and thus the negative effect of the drug would be masked.

完美的医生通过过滤掉对处理和控制的不良反应来扭曲这两个平均值。均值之间的差异，即假定的平均因果效应，在取决于细节的方向上发生扭曲。例如，像Susie这样因服药而受到伤害的受试者会被完美的医生分配到对照组，从而掩盖了药物的负面影响。

==结论==

Conclusion

The causal effect of a treatment on a single unit at a point in time is the difference between the outcome variable with the treatment and without the treatment. The Fundamental Problem of Causal Inference is that it is impossible to observe the causal effect on a single unit. You either take the aspirin now or you don't. As a consequence, assumptions must be made in order to estimate the missing counterfactuals.

The Rubin causal model has also been connected to instrumental variables (Angrist, Imbens, and Rubin, 1996)[6] and other techniques for causal inference. For more on the connections between the Rubin causal model, structural equation modeling, and other statistical methods for causal inference, see Morgan and Winship (2007).[7]

在某个时间点对单个单位的处理的因果效应是经过处理和未经过处理的结果变量之间的差异。因果推断的基本问题是不可能观察到对单个单元的因果效应。你要么现在服用阿司匹林，要么不服用。因此，必须做出假设以估计缺失的反事实。

Rubin 因果模型还与工具变量（Angrist、Imbens 和 Rubin，1996 年）<ref name="">Angrist J.,Imbens G.,Rubin D. (1996) [https://pattern.swarma.org/paper?id=86ab704a-c3ad-11eb-ab25-0242ac170007 Identification of Causal effects Using Instrumental Variables].J. Amer. Statist. Assoc.91.434:(444–455)</ref>和其他因果推断技术相关联。有关 Rubin 因果模型、结构方程建模和其他因果推断统计方法之间联系的更多信息，请参见 Morgan 和 Winship (2007)。<ref name="">Morgan S.,Winship C. (2007) [https://pattern.swarma.org/paper?id=b87f164e-c3ad-11eb-ba86-0242ac170007 Counterfactuals and Causal Inference: Methods and Principles for Social Research].</ref>

==另见==

- [https://en.wikipedia.org/wiki/Causality 因果关系]
- [https://en.wikipedia.org/wiki/Principal_stratification 主要分层]
- [https://en.wikipedia.org/wiki/Propensity_score_matching 倾向得分匹配]
<noinclude>

==编者推荐==
===书籍推荐===

*[[Causal Inference: What If]]
作者：Miguel A. Hernán, James M. Robins
<gallery>
preface.jpg|
</gallery>

==参考文献==
</noinclude>

鲁宾因果模型

2021-06-09T15:10:21Z

Chancychen：

鲁宾因果模型

2021-06-09T15:09:26Z

Chancychen：

The Rubin causal model (RCM), also known as the Neyman–Rubin causal model,[1] is an approach to the statistical analysis of cause and effect based on the framework of potential outcomes, named after Donald Rubin. The name "Rubin causal model" was first coined by Paul W. Holland.[2] The potential outcomes framework was first proposed by Jerzy Neyman in his 1923 Master's thesis,[3] though he discussed it only in the context of completely randomized experiments.[4] Rubin extended it into a general framework for thinking about causation in both observational and experimental studies.[1]

''' 鲁宾因果模型 Rubin Causal Model (RCM)''' ，也称为 Neyman-Rubin 因果模型，<ref name="sekhon">{{cite book |last=Sekhon |first=Jasjeet |chapter=The Neyman–Rubin Model of Causal Inference and Estimation via Matching Methods |title=The Oxford Handbook of Political Methodology |year=2007 |chapter-url=http://sekhon.berkeley.edu/papers/SekhonOxfordHandbook.pdf }}</ref>是一种基于潜在结果框架的因果统计分析方法，以Donald Rubin的名字命名。“鲁宾因果模型”这个名字最早是由 Paul W. Holland 创造的。 <ref name="holland:causal86">{{cite journal |last=Holland |first=Paul W. |title=Statistics and Causal Inference |journal=[[Journal of the American Statistical Association|J. Amer. Statist. Assoc.]] |volume=81 |issue=396 |year=1986 |pages=945–960 |jstor=2289064 |doi=10.1080/01621459.1986.10478354}}</ref> ''' 潜在结果框架 Potential Outcomes Framework'''最初是由 Jerzy Neyman 在他 1923 年的硕士论文中提出的，<ref name="neyman:masters">Neyman, Jerzy. ''Sur les applications de la theorie des probabilites aux experiences agricoles: Essai des principes.'' Master's Thesis (1923). Excerpts reprinted in English, Statistical Science, Vol. 5, pp. 463–472. ([[Dorota Dabrowska|D. M. Dabrowska]], and T. P. Speed, Translators.)</ref>尽管他只在完全随机实验的背景下讨论了它。 <ref name="Jasa1">{{cite journal |last=Rubin |first=Donald |year=2005 |title=Causal Inference Using Potential Outcomes |journal=[[Journal of the American Statistical Association|J. Amer. Statist. Assoc.]] |volume=100 |issue=469 |pages=322–331 |doi=10.1198/016214504000001880 }}</ref>鲁宾将其扩展为在观察性和实验性研究中思考因果关系的一般框架。<ref name="sekhon"/>

==介绍==

The Rubin causal model is based on the idea of potential outcomes. For example, a person would have a particular income at age 40 if he had attended college, whereas he would have a different income at age 40 if he had not attended college. To measure the causal effect of going to college for this person, we need to compare the outcome for the same individual in both alternative futures. Since it is impossible to see both potential outcomes at once, one of the potential outcomes is always missing. This dilemma is the "fundamental problem of causal inference".

Because of the fundamental problem of causal inference, unit-level causal effects cannot be directly observed. However, randomized experiments allow for the estimation of population-level causal effects.[5] A randomized experiment assigns people randomly to treatments: college or no college. Because of this random assignment, the groups are (on average) equivalent, and the difference in income at age 40 can be attributed to the college assignment since that was the only difference between the groups. An estimate of the average causal effect (also referred to as the average treatment effect) can then be obtained by computing the difference in means between the treated (college-attending) and control (not-college-attending) samples.

In many circumstances, however, randomized experiments are not possible due to ethical or practical concerns. In such scenarios there is a non-random assignment mechanism. This is the case for the example of college attendance: people are not randomly assigned to attend college. Rather, people may choose to attend college based on their financial situation, parents' education, and so on. Many statistical methods have been developed for causal inference, such as propensity score matching. These methods attempt to correct for the assignment mechanism by finding control units similar to treatment units.

鲁宾因果模型基于潜在结果的想法。例如，如果一个人上过大学，他在 40 岁时会有特定的收入，而如果他没有上过大学，他在 40 岁时会有不同的收入。为了衡量这个人上大学的因果效应，我们需要比较同一个人在两种不同的未来中的结果。由于不可能同时看到两种潜在结果，因此总是缺少其中一种潜在结果。这种困境就是“因果推理的基本问题”。

由于因果推理的根本问题，无法直接观察到单元级别的因果效应。然而，随机实验允许估计人口水平的因果效应。<ref name=":01">{{cite journal |last=Rubin |first=Donald |title=Estimating Causal Effects of Treatments in Randomized and Nonrandomized Studies |journal=[[Journal of Educational Psychology|J. Educ. Psychol.]] |volume=66 |issue=5 |year=1974 |pages=688–701 [p. 689] |doi=10.1037/h0037350 }}</ref>随机实验将人们随机分配到对照组：大学或非大学。由于这种随机分配，各组（平均）相等，40 岁时的收入差异可归因于大学分配，因为这是各组之间的唯一差异。然后可以通过计算处理（上大学）和对照（非上大学）样本之间的平均值差异来获得平均因果效应（也称为平均处理效应）的估计值。

然而，在许多情况下，由于伦理或实际问题，随机实验是不可能的。在这种情况下，存在非随机分配机制。上大学的例子就是这种情况：人们不是随机分配上大学的。相反，人们可能会根据他们的经济状况、父母的教育等来选择上大学。已经开发了许多用于因果推断的统计方法，例如倾向得分匹配。这些方法试图通过寻找类似于处理单元的控制单元来纠正分配机制。

==一个扩展案例==

Rubin defines a causal effect:

Intuitively, the causal effect of one treatment, E, over another, C, for a particular unit and an interval of time from {\displaystyle t_{1}}t_{1} to {\displaystyle t_{2}}t_{2} is the difference between what would have happened at time {\displaystyle t_{2}}t_{2} if the unit had been exposed to E initiated at {\displaystyle t_{1}}t_{1} and what would have happened at {\displaystyle t_{2}}t_{2} if the unit had been exposed to C initiated at {\displaystyle t_{1}}t_{1}: 'If an hour ago I had taken two aspirins instead of just a glass of water, my headache would now be gone,' or 'because an hour ago I took two aspirins instead of just a glass of water, my headache is now gone.' Our definition of the causal effect of the E versus C treatment will reflect this intuitive meaning."[5]

According to the RCM, the causal effect of your taking or not taking aspirin one hour ago is the difference between how your head would have felt in case 1 (taking the aspirin) and case 2 (not taking the aspirin). If your headache would remain without aspirin but disappear if you took aspirin, then the causal effect of taking aspirin is headache relief. In most circumstances, we are interested in comparing two futures, one generally termed "treatment" and the other "control". These labels are somewhat arbitrary.

鲁宾定义了一个因果效应：

<blockquote>
直观地说，一种处理 E 对另一种处理 C 的因果效应对于特定单位和从 <math>t_{1}</math> 至 <math>t_{2}</math> 是当时会发生的事情之间的差异 <math>t_{2}</math> 如果装置暴露于 E 开始于 <math>t_{1}</math> 以及会发生什么 <math>t_{2}</math> 如果该装置已暴露于 C 开始于<math>t_{1}</math>：“如果一个小时前我服用了两个阿司匹林而不是一杯水，我的头痛现在就会消失”，或者“因为一个小时前我服用了两个阿司匹林而不是一杯水，我的头痛现在消失了.' 我们对 E 与 C 处理的因果效应的定义将反映这种直观的含义。<ref name=":01"/>”
</blockquote>

根据 RCM，您在一小时前服用或不服用阿司匹林的因果效应是您的头部在情况 1（服用阿司匹林）和情况 2（未服用阿司匹林）中的感受的差异。如果没有阿司匹林你的头痛仍然存在，但如果服用阿司匹林头痛就会消失，那么服用阿司匹林的因果效应是头痛缓解。在大多数情况下，我们对比较两种特性感兴趣，一种通常称为“处理”，另一种称为“控制”。这些标签有些随意。

===潜在结果===

Potential outcomes

Suppose that Joe is participating in an FDA test for a new hypertension drug. If we were omniscient, we would know the outcomes for Joe under both treatment (the new drug) and control (either no treatment or the current standard treatment). The causal effect, or treatment effect, is the difference between these two potential outcomes.

假设 Joe 正在参加 FDA 对一种新的高血压药物的测试。如果我们无所不知，我们就会知道乔在处理t（新药）和控制c（未处理或当前标准处理）下的结果。因果效应或处理效应是这两种潜在结果之间的差异。

{| class="wikitable" align="center"

!主题!!<math>Y_{t}(u)</math>!!<math>Y_{c}(u)</math>!!<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||135||−5

|}

Y_{t}(u) is Joe's blood pressure if he takes the new pill. In general, this notation expresses the potential outcome which results from a treatment, t, on a unit, u. Similarly, {\displaystyle Y_{c}(u)}Y_{c}(u) is the effect of a different treatment, c or control, on a unit, u. In this case, {\displaystyle Y_{c}(u)}Y_{c}(u) is Joe's blood pressure if he doesn't take the pill. {\displaystyle Y_{t}(u)-Y_{c}(u)}Y_{t}(u)-Y_{c}(u) is the causal effect of taking the new drug.

From this table we only know the causal effect on Joe. Everyone else in the study might have an increase in blood pressure if they take the pill. However, regardless of what the causal effect is for the other subjects, the causal effect for Joe is lower blood pressure, relative to what his blood pressure would have been if he had not taken the pill.

Consider a larger sample of patients:

如果乔服用新药丸，<math>Y_{t}(u)</math>就是他的血压。通常，该符号表示对单位u进行处理t 所产生的潜在结果。类似的，<math>Y_{c}(u)</math>是不同处理t或控制c对单元u的影响。在这种情况下，如果乔不服药，<math>Y_{c}(u)</math>就是他的血压。<math>Y_{t}(u)-Y_{c}(u)</math> 是服用新药的因果效应。

从这张表中我们只知道对 Joe 的因果效应。研究中的其他人如果服用药物，血压可能会升高。然而，不管对其他受试者的因果效应是什么，乔的因果效应是血压降低（相对于他没有服用避孕药时的血压）。

考虑更大的患者样本：

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||135||−5
|-
!玛丽||140||150||−10
|-
!莎莉||135||125||10
|-
!鲍勃||135||150||−15

|}

The causal effect is different for every subject, but the drug works for Joe, Mary and Bob because the causal effect is negative. Their blood pressure is lower with the drug than it would have been if each did not take the drug. For Sally, on the other hand, the drug causes an increase in blood pressure.

In order for a potential outcome to make sense, it must be possible, at least a priori. For example, if there is no way for Joe, under any circumstance, to obtain the new drug, then {\displaystyle Y_{t}(u)}Y_{t}(u) is impossible for him. It can never happen. And if {\displaystyle Y_{t}(u)}Y_{t}(u) can never be observed, even in theory, then the causal effect of treatment on Joe's blood pressure is not defined.

因果效应对于每一个主题是不同的，但药物对乔，玛丽和鲍勃都有影响，因为因果效应是负的。他们服用药物后的血压低于每个人不服用药物时的血压。另一方面，对于 Sally 来说，这种药物会导致血压升高。

为了使潜在结果有意义，它必须是可能的，至少是先验已知可能的。例如，如果乔在任何情况下都无法获得新药，那么<math>Y_{t}(u)</math>对他来说是不可能的。它永远不会发生。而如果<math>Y_{t}(u)</math>永远无法观察到（即使在理论上），那么对乔的血压的因果效应是不确定的。

===没有操纵就没有因果关系===

No causation without manipulation

The causal effect of new drug is well defined because it is the simple difference of two potential outcomes, both of which might happen. In this case, we (or something else) can manipulate the world, at least conceptually, so that it is possible that one thing or a different thing might happen.

This definition of causal effects becomes much more problematic if there is no way for one of the potential outcomes to happen, ever. For example, what is the causal effect of Joe's height on his weight? Naively, this seems similar to our other examples. We just need to compare two potential outcomes: what would Joe's weight be under the treatment (where treatment is defined as being 3 inches taller) and what would Joe's weight be under the control (where control is defined as his current height).

A moment's reflection highlights the problem: we can't increase Joe's height. There is no way to observe, even conceptually, what Joe's weight would be if he were taller because there is no way to make him taller. We can't manipulate Joe's height, so it makes no sense to investigate the causal effect of height on weight. Hence the slogan: No causation without manipulation.

新药的因果效应是明确定义的，因为它是两种可能发生的潜在结果的简单差异。在这种情况下，我们（或其他事物）可以干预世界，至少在概念上是这样，因此可能会发生不同的事。

如果永远不可能发生其中一种潜在结果，那么这种因果效应的定义就会变得更加棘手。例如，乔的身高对他的体重有什么因果关系？这似乎与我们的其他示例相似。我们只需要比较两个潜在的结果：Joe 在处理下的体重（处理被定义为高3英寸）和 Joe 在控制下的体重（控制被定义为他当前的身高）。

问题在于：我们无法增加乔的身高。没有办法观察如果乔更高，他的体重会是多少，因为没有办法让他更高。我们无法操纵乔的身高，因此调查身高对体重的因果关系毫无意义。因此有一个口号：没有操纵就没有因果关系。

===稳定单元处理值假设 (SUTVA)===

Stable unit treatment value assumption (SUTVA)

See also: Spillover (experiment)

We require that "the [potential outcome] observation on one unit should be unaffected by the particular assignment of treatments to the other units" (Cox 1958, §2.4). This is called the stable unit treatment value assumption (SUTVA), which goes beyond the concept of independence.

In the context of our example, Joe's blood pressure should not depend on whether or not Mary receives the drug. But what if it does? Suppose that Joe and Mary live in the same house and Mary always cooks. The drug causes Mary to crave salty foods, so if she takes the drug she will cook with more salt than she would have otherwise. A high salt diet increases Joe's blood pressure. Therefore, his outcome will depend on both which treatment he received and which treatment Mary receives.

SUTVA violation makes causal inference more difficult. We can account for dependent observations by considering more treatments. We create 4 treatments by taking into account whether or not Mary receives treatment.

我们要求“对一个单元的 [潜在结果] 观察不应受到其他单元的特定处理分配的影响”（Cox 1958，第 2.4 节）。这被称为稳定单元处理值假设（SUTVA），它超越了独立性的概念。

在我们的例子中，Joe 的血压不应该取决于 Mary 是否接受了药物。但如果真的发生了呢？假设乔和玛丽住在同一所房子里，玛丽总是做饭。这种药物会导致玛丽渴望咸的食物，所以如果她服用这种药物，她会用比其他情况下更多的盐来烹饪。高盐饮食会增加乔的血压。因此，他的结果将取决于他接受的处理和玛丽接受的处理。

在不满足SUTVA的情况下，因果推断会更加困难。我们可以通过考虑更多的处理来解释相关的观察结果。我们通过考虑 Mary 是否接受处理来创建 4 个处理。

{| class="wikitable" align="center"
!主题||乔 = c，玛丽 = t||乔 = t，玛丽 = t||乔 = c，玛丽 = c||乔 = t，玛丽 = c
|-
!乔||140||130||125||120
|}

Recall that a causal effect is defined as the difference between two potential outcomes. In this case, there are multiple causal effects because there are more than two potential outcomes. One is the causal effect of the drug on Joe when Mary receives treatment and is calculated, {\displaystyle 130-140}{\displaystyle 130-140}. Another is the causal effect on Joe when Mary does not receive treatment and is calculated {\displaystyle 120-125}{\displaystyle 120-125}. The third is the causal effect of Mary's treatment on Joe when Joe is not treated. This is calculated as {\displaystyle 140-125}{\displaystyle 140-125}. The treatment Mary receives has a greater causal effect on Joe than the treatment which Joe received has on Joe, and it is in the opposite direction.

By considering more potential outcomes in this way, we can cause SUTVA to hold. However, if any units other than Joe are dependent on Mary, then we must consider further potential outcomes. The greater the number of dependent units, the more potential outcomes we must consider and the more complex the calculations become (consider an experiment with 20 different people, each of whom's treatment status can effect outcomes for every one else). In order to (easily) estimate the causal effect of a single treatment relative to a control, SUTVA should hold.

回想一下，因果效应被定义为两个潜在结果之间的差异。在这种情况下，存在多种因果效应，因为存在两个以上的潜在结果。一是玛丽接受处理时药物对乔的因果效应<math>130-140</math>。另一个是当玛丽没有接受处理时对乔的因果效应<math>120-125</math>。第三是在乔没有得到处理的情况下，玛丽的处理对乔的因果效应<math>140-125</math>。Mary 接受的处理对 Joe 的因果影响比 Joe 接受的处理对 Joe 的影响更大，而且是相反的方向。

通过以这种方式考虑更多潜在结果，我们可以使SUTVA成立。但是，如果 Joe 以外的任何单位都依赖于 Mary，那么我们必须考虑进一步的潜在结果。依赖单位的数量越多，我们必须考虑的潜在结果就越多，计算也变得越复杂（考虑对不同的20个人进行的实验，每个人的处理状态都会影响其他人的结果）。为了（轻松）估计单一处理相对于对照的因果效应，SUTVA 应该成立。

===平均因果效应（Average Causal Effect, ACE）===

考虑：

{| class="wikitable" align="center"
!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||135||−5
|-
!玛丽||130||145||−15
|-
!莎莉||130||145||−15
|-
!鲍勃||140||150||−10
|-
!詹姆士||145||140||+5
|-
!平均||135||143||−8
|}

One may calculate the average causal effect by taking the mean of all the causal effects.

How we measure the response affects what inferences we draw. Suppose that we measure changes in blood pressure as a percentage change rather than in absolute values. Then, depending in the exact numbers, the average causal effect might be an increase in blood pressure. For example, assume that George's blood pressure would be 154 under control and 140 with treatment. The absolute size of the causal effect is −14, but the percentage difference (in terms of the treatment level of 140) is −10%. If Sarah's blood pressure is 200 under treatment and 184 under control, then the causal effect in 16 in absolute terms but 8% in terms of the treatment value. A smaller absolute change in blood pressure (−14 versus 16) yields a larger percentage change (−10% versus 8%) for George. Even though the average causal effect for George and Sarah is +1 in absolute terms, it is −1 in percentage terms.

人们可以通过取所有因果效应的平均值来计算平均因果效应。

我们如何测量反馈效应会影响我们得出的推论。假设我们以百分比变化而不是绝对值来测量血压的变化。然后，根据确切的数字，平均因果效应可能是血压升高。例如，假设乔治的血压在控制下为 154，在处理后为 140。因果效应的绝对大小为 -14，但百分比差异（就 140 的处理水平而言）为 -10%。如果莎拉的血压在治疗下为 200，在控制下为 184，那么因果效应的绝对值是 16，而治疗值是 8%。乔治的血压绝对变化较小（-14 对 16）会产生较大的百分比变化（-10% 对 8%）。

===因果推理的基本问题===

The fundamental problem of causal inference

The results we have seen up to this point would never be measured in practice. It is impossible, by definition, to observe the effect of more than one treatment on a subject over a specific time period. Joe cannot both take the pill and not take the pill at the same time. Therefore, the data would look something like this:

到目前为止，我们所看到的结果永远无法在实践中衡量。根据定义，不可能在特定时间段内观察多种处理对受试者的影响。乔不能同时服用药物和不服用药物。因此，数据看起来像这样：

{| class="wikitable" align="center"
!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||?||?
|}

Question marks are responses that could not be observed. The Fundamental Problem of Causal Inference[2] is that directly observing causal effects is impossible. However, this does not make causal inference impossible. Certain techniques and assumptions allow the fundamental problem to be overcome.

Assume that we have the following data:

问号是无法观察到的反馈。因果推断的基本问题<ref name="holland:causal86"/>是不可能直接观测因果效应。然而，这并不是说因果推断是不可能的。某些技术和假设可以克服基本问题。

假设我们有以下数据：

{| class="wikitable" align="center"
!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||?||?
|-
!玛丽||?||125||?
|-
!莎莉||100||?||?
|-
!鲍勃||?||130||?
|-
!詹姆士||?||120||?
|-
!平均||115||125||−10
|}

We can infer what Joe's potential outcome under control would have been if we make an assumption of constant effect:

如果我们假设效应恒定，我们可以推断出乔在控制下的潜在结果是什么：

<math>Y_{t}(u)=T+Y_{c}(u)</math>

和

<math>Y_{t}(u)-T=Y_{c}(u)</math>。

If we wanted to infer the unobserved values we could assume a constant effect. The following tables illustrates data consistent with the assumption of a constant effect.

如果我们想推断未观察到的值，我们可以假设一个恒定的影响。下表说明了与恒定效应假设一致的数据。

{| class="wikitable" align="center"
!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||140||−10
|-
!玛丽||115||125||−10
|-
!莎莉||100||110||−10
|-
!鲍勃||120||130||−10
|-
!詹姆士||110||120||−10
|-
!平均||115||125||−10
|}

All of the subjects have the same causal effect even though they have different outcomes under the treatment.

所有受试者即使在治疗下有不同的结果，也具有相同的因果效应。

===分配机制（Assignment Mechanism）===

The assignment mechanism

The assignment mechanism, the method by which units are assigned treatment, affects the calculation of the average causal effect. One such assignment mechanism is randomization. For each subject we could flip a coin to determine if she receives treatment. If we wanted five subjects to receive treatment, we could assign treatment to the first five names we pick out of a hat. When we randomly assign treatments we may get different answers.

Assume that this data is the truth:

分配机制，即分配单位处理的方法，影响平均因果效应的计算。一种分配机制是随机化。对于每个受试者，我们可以抛硬币来确定她是否接受处理。如果我们希望五个受试者接受处理，我们可以将处理分配给我们从帽子里挑选出来的前五个名字。当我们随机分配处理时，我们可能会得到不同的答案。

假设这个数据是真实的：

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||115||15
|-
!玛丽||120||125||−5
|-
!莎莉||100||125||−25
|-
!鲍勃||110||130||−20
|-
!詹姆士||115||120||−5
|-
!平均||115||123||−8

|}

The true average causal effect is −8. But the causal effect for these individuals is never equal to this average. The causal effect varies, as it generally (always?) does in real life. After assigning treatments randomly, we might estimate the causal effect as:

真正的平均因果效应是 -8。但是对这些人的因果效应永远不会等于这个平均值。因果效应各不相同，因为它通常（总是？）在现实生活中也是如此。在随机分配处理后，我们可以估计因果效应为：

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>

|-
!乔||130||?||?
|-
!玛丽||120||?||?
|-
!莎莉||?||125||?
|-
!鲍勃||?||130||?
|-
!詹姆士||115||?||?
|-
!平均||121.66||127.5||−5.83

|}

A different random assignment of treatments yields a different estimate of the average causal effect.

处理的不同随机分配产生对平均因果效应的不同估计。

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>

|-
!乔||130||?||?
|-
!玛丽||120||?||?
|-
!莎莉||100||?||?
|-
!鲍勃||?||130||?
|-
!詹姆士||?||120||?
|-
!平均||116.67||125||−8.33

|}

The average causal effect varies because our sample is small and the responses have a large variance. If the sample were larger and the variance were less, the average causal effect would be closer to the true average causal effect regardless of the specific units randomly assigned to treatment.

Alternatively, suppose the mechanism assigns the treatment to all men and only to them.

平均因果效应会有所不同，因为我们的样本很小并且反馈效应的方差很大。如果样本较大且方差较小，则无论随机分配给处理的特定单位如何，平均因果效应将更接近真实的平均因果效应。

或者，假设该机制将处理分配给所有男性且仅分配给他们。

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||?||?
|-
!鲍勃||110||?||?
|-
!詹姆士||105||?||?
|-
!玛丽||?||130||?
|-
!莎莉||?||125||?
|-
!苏茜||?||135||?
|-
!平均||115||130||−15

|}

Under this assignment mechanism, it is impossible for women to receive treatment and therefore impossible to determine the average causal effect on female subjects. In order to make any inferences of causal effect on a subject, the probability that the subject receive treatment must be greater than 0 and less than 1.

在这种分配机制下，女性不可能接受处理，因此无法确定对女性受试者的平均因果效应。为了对受试者做出因果效应的任何推断，受试者接受治疗的概率必须大于 0 且小于 1。

===完美的医生(Perfect Doctor Mechanism)===

The perfect doctor

Consider the use of the perfect doctor as an assignment mechanism. The perfect doctor knows how each subject will respond to the drug or the control and assigns each subject to the treatment that will most benefit her. The perfect doctor knows this information about a sample of patients:

考虑使用完美医生作为分配机制。完美的医生知道每个受试者对药物或对照的反应如何，并为每个受试者分配对她最有益的处理。完美的医生知道有关患者样本的以下信息：

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||130||115||15
|-
!鲍勃||120||125||−5
|-
!詹姆士||100||150||−50
|-
!玛丽||115||125||−10
|-
!莎莉||120||130||−10
|-
!苏茜||135||105||30
|-
!平均||120||125||−5

|}

Based on this knowledge she would make the following treatment assignments:

基于这些知识，她将进行以下处理分配：

{| class="wikitable" align="center"

!主题||<math>Y_{t}(u)</math>||<math>Y_{c}(u)</math>||<math>Y_{t}(u)-Y_{c}(u)</math>
|-
!乔||?||115||?
|-
!鲍勃||120||?||?
|-
!詹姆士||100||?||?
|-
!玛丽||115||?||?
|-
!莎莉||120||?||?
|-
!苏茜||?||105||?
|-
!平均||113.75||110||3.75
|}

The perfect doctor distorts both averages by filtering out poor responses to both the treatment and control. The difference between means, which is the supposed average causal effect, is distorted in a direction that depends on the details. For instance, a subject like Susie who is harmed by taking the drug would be assigned to the control group by the perfect doctor and thus the negative effect of the drug would be masked.

完美的医生通过过滤掉对处理和控制的不良反应来扭曲这两个平均值。均值之间的差异，即假定的平均因果效应，在取决于细节的方向上发生扭曲。例如，像Susie这样因服药而受到伤害的受试者会被完美的医生分配到对照组，从而掩盖了药物的负面影响。

==结论==

Conclusion

The causal effect of a treatment on a single unit at a point in time is the difference between the outcome variable with the treatment and without the treatment. The Fundamental Problem of Causal Inference is that it is impossible to observe the causal effect on a single unit. You either take the aspirin now or you don't. As a consequence, assumptions must be made in order to estimate the missing counterfactuals.

The Rubin causal model has also been connected to instrumental variables (Angrist, Imbens, and Rubin, 1996)[6] and other techniques for causal inference. For more on the connections between the Rubin causal model, structural equation modeling, and other statistical methods for causal inference, see Morgan and Winship (2007).[7]

在某个时间点对单个单位的处理的因果效应是经过处理和未经过处理的结果变量之间的差异。因果推断的基本问题是不可能观察到对单个单元的因果效应。你要么现在服用阿司匹林，要么不服用。因此，必须做出假设以估计缺失的反事实。

Rubin 因果模型还与工具变量（Angrist、Imbens 和 Rubin，1996 年）<ref name="">Angrist J.,Imbens G.,Rubin D. (1996) [https://pattern.swarma.org/paper?id=86ab704a-c3ad-11eb-ab25-0242ac170007 Identification of Causal effects Using Instrumental Variables].J. Amer. Statist. Assoc.91.434:(444–455)</ref>和其他因果推断技术相关联。有关 Rubin 因果模型、结构方程建模和其他因果推断统计方法之间联系的更多信息，请参见 Morgan 和 Winship (2007)。<ref name="">Morgan S.,Winship C. (2007) [https://pattern.swarma.org/paper?id=b87f164e-c3ad-11eb-ba86-0242ac170007 Counterfactuals and Causal Inference: Methods and Principles for Social Research].</ref>

==另见==

- [https://en.wikipedia.org/wiki/Causality 因果关系]
- [https://en.wikipedia.org/wiki/Principal_stratification 主要分层]
- [https://en.wikipedia.org/wiki/Propensity_score_matching 倾向得分匹配]
<noinclude>

==参考文献==
</noinclude>

==编者推荐==
===书籍推荐===

2021-06-01T01:18:19Z

Chancychen：

{{Short description|Conditionals that discuss what would have been if things were otherwise}}

{{Redirect|Counterfactual}}

'''Counterfactual conditionals''' (also ''subjunctive'' or ''X-marked'') are [[conditional sentence]]s which discuss what would have been true under different circumstances, e.g.  "If Peter believed in ghosts, he would be afraid to be here." Counterfactuals are contrasted with [[indicative conditionals|indicatives]], which are generally restricted to discussing open possibilities. Counterfactuals are characterized grammatically by their use of [[Counterfactual conditional#Fake tense|fake tense morphology]], which some languages use in combination with other kinds of [[Morphology (linguistics)|morphology]] including [[Counterfactual conditional#Fake aspect|aspect]] and [[Counterfactual conditional#mood|mood]].

''' 反事实条件句 Counterfactual conditionals'''（虚拟条件或X标记的）是用来讨论在不同情况下什么为真的的条件句。例如：如果彼得相信鬼魂的存在，他就会害怕来到这里。反事实句与指示句形成对比，后者一般只限于讨论开放的可能性。反事实动词的语法特征是使用虚拟时态语法，这种虚拟时态语法与时态和语态等其他语法结合使用。

Counterfactuals are one of the most studied phenomena in [[philosophical logic]], [[formal semantics (natural language)|formal semantics]], and [[philosophy of language]]. They were first discussed as a problem for the [[material conditional]] analysis of conditionals, which treats them all as trivially true. Starting in the 1960s, philosophers and linguists developed the now-classic [[possible world]] approach, in which a counterfactual's truth hinges on its consequent holding at certain possible worlds where its antecedent holds. More recent formal analyses have treated them using tools such as [[causal model]]s and [[dynamic semantics]]. Other research has addressed their metaphysical, psychological, and grammatical underpinnings, while applying some of the resultant insights to fields including history, marketing, and epidemiology.

反事实是哲学逻辑、形式语义学和语言哲学中研究最多的现象之一。它们首先作为条件句的实质条件分析的问题被讨论，条件句把它们都当作是微不足道的真实。从20世纪60年代开始，哲学家和语言学家发展出现在经典的可能世界方法，在这种方法中，反事实的真理取决于它在某些可能世界中的后果，而这些可能世界的前因是成立的。最近的形式化分析使用因果模型和动态语义等工具对它们进行了处理。其他研究已经解决了他们的形而上学、心理学和语法基础，同时将一些结果的见解应用到包括历史，市场营销和流行病学领域。

{{TOC limit|3}}

==Overview==

=== Examples ===

The difference between [[indicative conditional|indicative]] and counterfactual conditionals can be illustrated by the following [[minimal pair]]:

指示条件句和反事实条件句之间的区别可以用下面的简单例子来说明：

# '''Indicative Conditional''': If it ''is'' raining right now, then Sally ''is'' inside.

# '''指示条件句'''：如果现在正在下雨，那么Sally 就在里面（If it ''is'' raining right now, then Sally ''is'' inside）。

# '''Simple Past Counterfactual''': If it ''was raining''  right now, then Sally ''would be'' inside.<ref>{{cite journal |last1=von Prince |first1=Kilu |date=2019 |title=Counterfactuality and past |url=https://link.springer.com/content/pdf/10.1007/s10988-019-09259-6.pdf |journal=Linguistics and Philosophy |volume=42 |issue=6|pages=577–615 |doi=10.1007/s10988-019-09259-6 |s2cid=181778834 |doi-access=free }}</ref><ref>{{cite thesis |last=Karawani |first=Hadil |date=2014 |title=The Real, the Fake, and the Fake Fake in Counterfactual Conditionals, Crosslinguistically |page=186 |publisher=Universiteit van Amsterdam |url=https://pure.uva.nl/ws/files/1695453/142017_thesis.pdf}}</ref><ref name="Linguistic Society of America">{{cite conference |url=https://journals.linguisticsociety.org/proceedings/index.php/SALT/article/view/27.547 |title=Fake Perfect in X-Marked Conditionals |last1=Schulz |first1=Katrin |date=2017 |publisher=Linguistic Society of America |book-title=Proceedings from Semantics and Linguistic Theory. |pages=547–570 |conference= Semantics and Linguistic Theory.|doi=10.3765/salt.v27i0.4149|doi-access=free }}</ref><ref>{{cite book |last1=Huddleston |first1=Rodney | last2=Pullum |first2=Geoff |date=2002 |title= The Cambridge Grammar of the English Language |publisher=Cambridge University Press |isbn=978-0521431460|pages=85–86}}</ref>

# '''一般过去时的反事实'''：如果现在正在下雨，那么Sally就会在里面（If it was raining right now, then Sally would be inside）。<ref>{{cite journal |last1=von Prince |first1=Kilu |date=2019 |title=Counterfactuality and past |url=https://link.springer.com/content/pdf/10.1007/s10988-019-09259-6.pdf |journal=Linguistics and Philosophy |volume=42 |issue=6|pages=577–615 |doi=10.1007/s10988-019-09259-6 |s2cid=181778834 |doi-access=free }}</ref><ref>{{cite thesis |last=Karawani |first=Hadil |date=2014 |title=The Real, the Fake, and the Fake Fake in Counterfactual Conditionals, Crosslinguistically |page=186 |publisher=Universiteit van Amsterdam |url=https://pure.uva.nl/ws/files/1695453/142017_thesis.pdf}}</ref><ref name="Linguistic Society of America">{{cite conference |url=https://journals.linguisticsociety.org/proceedings/index.php/SALT/article/view/27.547 |title=Fake Perfect in X-Marked Conditionals |last1=Schulz |first1=Katrin |date=2017 |publisher=Linguistic Society of America |book-title=Proceedings from Semantics and Linguistic Theory. |pages=547–570 |conference= Semantics and Linguistic Theory.|doi=10.3765/salt.v27i0.4149|doi-access=free }}</ref><ref>{{cite book |last1=Huddleston |first1=Rodney | last2=Pullum |first2=Geoff |date=2002 |title= The Cambridge Grammar of the English Language |publisher=Cambridge University Press |isbn=978-0521431460|pages=85–86}}</ref>

These conditionals differ in both form and meaning. The indicative conditional uses the present tense form "is" in both the "if" clause and the "then" clause. As a result, it conveys that the speaker is agnostic about whether it is raining. The counterfactual example uses the [[fake tense]] form "was" in the "if" clause and the [[modal verb|modal]] "would" in the "then" clause. As a result, it conveys that the speaker does not believe that it is raining.

这些条件句在形式和意义上都不同。指示条件在“如果(if)”和 “那么(then)”两个从句中都使用现在时态is。因此，它传达了这样一种信息：说话人对是否下雨是不可知的。反事实的例句在“如果”从句中使用虚拟时态“was”，在“ then”句中使用情态“ would”。因此，它传达的信息是，说话人并不相信天在下雨。

English has several other grammatical forms whose meanings are sometimes included under the umbrella of counterfactuality. One is the [[pluperfect|past perfect]] counterfactual, which contrasts with indicatives and simple past counterfactuals in its use of pluperfect morphology:<ref>{{cite book |last1=Huddleston |first1=Rodney | last2=Pullum |first2=Geoff |date=2002 |title= The Cambridge Grammar of the English Language |publisher=Cambridge University Press |page=150 |isbn=978-0521431460}}</ref>

英语还有其他几种语法形式，其含义有时被包括在反事实的范畴内。其中一个是过去完成时的反事实，在使用过去完成时态时，它与指示词和一般完成时的反事实形成对比：<ref>{{cite book |last1=Huddleston |first1=Rodney | last2=Pullum |first2=Geoff |date=2002 |title= The Cambridge Grammar of the English Language |publisher=Cambridge University Press |page=150 |isbn=978-0521431460}}</ref>

# '''Past Perfect Counterfactual''': If it ''had been raining'' yesterday, then Sally ''would have been'' inside.

# '''过去完成时的反事实'''：如果昨天下了雨，那么Sally就会在里面(If it ''had been raining'' yesterday, then Sally ''would have been'' inside)。

Another kind of conditional uses the form "were", generally referred to as the irrealis or subjunctive form.<ref>There is no standard system of terminology for these grammatical forms in English. Pullum and Huddleston (2002, pp. 85-86) adopt the term "irrealis" for this morphological form, reserving the term "subjunctive" for the English clause type whose distribution more closely parallels that of morphological subjunctives in languages that have such a form.</ref>

Another kind of conditional uses the form "were", generally referred to as the irrealis or subjunctive form.

另一种条件句的用法是“were”，一般称为“非现实（irrealis）”或“虚拟（subjunctive）”。

# '''Irrealis Counterfactual''': If it ''were raining'' right now, then Sally ''would be'' inside.

# '''非现实反事实（Irrealis Counterfactual）'''：如果现在正在下雨，那么Sally应该在里面(If it ''were raining'' right now, then Sally ''would be'' inside)。

Past perfect and irrealis counterfactuals can undergo ''conditional inversion'':<ref>{{cite encyclopedia |last1=Bhatt |first=Rajesh|last2=Pancheva|first2=Roumyana |editor-last1=Everaert |editor-first1=Martin | editor-last2=van Riemsdijk |editor-first2=Henk |encyclopedia= |title=The Wiley Blackwell Companion to Syntax |url=https://people.umass.edu/bhatt/papers/bhatt-pancheva-cond.pdf |year=2006 |publisher=Wiley Blackwell |doi=10.1002/9780470996591.ch16}}</ref>

过去完成时和非现实的反事实可以进行条件倒置：<ref>{{cite encyclopedia |last1=Bhatt |first=Rajesh|last2=Pancheva|first2=Roumyana |editor-last1=Everaert |editor-first1=Martin | editor-last2=van Riemsdijk |editor-first2=Henk |encyclopedia= |title=The Wiley Blackwell Companion to Syntax |url=https://people.umass.edu/bhatt/papers/bhatt-pancheva-cond.pdf |year=2006 |publisher=Wiley Blackwell |doi=10.1002/9780470996591.ch16}}</ref>

# Were it raining, Sally would be inside.

# 如果下雨的话，Sally就会在里面（Were it raining, Sally would be inside）。

# Had it rained, Sally would be inside.

# 那时如果下雨的话，Sally就会在里面（Had it rained, Sally would be inside）。

=== Terminology ===





<！——鉴于术语上的巨大但往往是细微的差异，本节必须经过仔细的编辑。在点击“发布更改”之前，请考虑结果文本是否有助于读者理解这些术语是如何使用的。如果结果文本读起来像是“热狗是三明治的辩论吗?”删除所有字符提示后，请不要点击”发布更改”。特别是，请确保(1)明确区分事实主张和术语定义(2)记住，不同的来源可以以不同的方式使用单一术语(3)对术语的每个术语或用法进行不偏不倚的框架性解释。-->

The term ''counterfactual conditional'' is widely used as an umbrella term for the kinds of sentences shown above. However, not all conditionals of this sort express contrary-to-fact meanings. For instance, the classic example known as the "Anderson Case" has the characteristic grammatical form of a counterfactual conditional, but does not convey that its antecedent is false or unlikely.<ref name = "vonfintel98" >{{cite encyclopedia |last1=von Fintel |first1=Kai |editor-last1=Sauerland |editor-first1=Uli |editor-last2=Percus |editor-first2=Oren |encyclopedia=The Interpretive Tract |title=The Presupposition of Subjunctive Conditionals |year=1998 |publisher=Cambridge University Press |pages=29–44|url=http://web.mit.edu/fintel/fintel-1998-subjunctive.pdf}}</ref><ref name="Conditionals">{{cite encyclopedia |last1=Egré |first1=Paul | last2=Cozic |first2=Mikaël |editor-last1=Aloni |editor-first1=Maria |editor-last2=Dekker |editor-first2=Paul |encyclopedia=Cambridge Handbook of Formal Semantics |title=Conditionals |year=2016 |publisher=Cambridge University Press |isbn=978-1-107-02839-5 |pages=515}}</ref>

The term counterfactual conditional is widely used as an umbrella term for the kinds of sentences shown above. However, not all conditionals of this sort express contrary-to-fact meanings. For instance, the classic example known as the "Anderson Case" has the characteristic grammatical form of a counterfactual conditional, but does not convey that its antecedent is false or unlikely.

“反事实条件（counterfactual conditional）”这一术语被广泛用作上述各类句子的总称。然而，并非所有这类条件句都表达与事实相反的意思。例如，被称为“ Anderson 案例”的经典例子具有反事实条件的典型语法形式，但是并不表明它的先行词是假的或不可能的。

# '''Anderson Case''': If the patient had taken arsenic, he would have blue spots.<ref>{{cite journal |last1=Anderson |first1=Alan |date=1951 |title=A Note on Subjunctive and Counterfactual Conditionals |journal=Analysis |volume=12 |issue = 2|pages=35–38|doi=10.1093/analys/12.2.35 }}</ref>

Anderson Case: If the patient had taken arsenic, he would have blue spots.

# '''Anderson案例'''：如果病人服用了砒霜，他会长出蓝斑（If the patient had taken arsenic, he would have blue spots）。

Such conditionals are also widely referred to as ''subjunctive conditionals'', though this term is likewise acknowledged as a misnomer even by those who use it.<ref>See for instance [https://pdfs.semanticscholar.org/e68d/612d7a93c98956e7314e0d131d90244c31f2.pdf Ippolito (2002)]: "Because ''subjunctive'' and ''indicative'' are the terms used in the philosophical literature on conditionals and because we will refer to that literature in the course of this paper, I have decided to keep these terms in the present discussion... however, it would be wrong to believe that mood choice is a necessary component of the semantic contrast between indicative and subjunctive conditionals." Also, [http://web.mit.edu/fintel/fintel-2011-hsk-conditionals.pdf von Fintel (2011)] "The terminology is of course linguistically inept ([since] the morphological marking is one of tense and aspect, not of indicative vs. subjunctive mood), but it is so deeply entrenched that it would be foolish not to use it."</ref> Many languages do not have a morphological [[subjunctive]] (e.g. [[Danish grammar|Danish]] and [[Dutch grammar|Dutch]]) and many that do have it don’t use it for this sort of conditional (e.g. [[French grammar|French]], [[Swahili grammar|Swahili]], all [[Indo-Aryan languages]] that have a subjunctive). Moreover, languages that do use the subjunctive for such conditionals only do so if they have a specific past subjunctive form. Thus, subjunctive marking is neither necessary nor sufficient for membership in this class of conditionals.<ref>{{cite journal |last1=Iatridou |first1=Sabine |date=2000 |title=The grammatical ingredients of counterfactuality |journal= Linguistic Inquiry |volume=31 |issue = 2|pages=231–270|doi=10.1162/002438900554352 |s2cid=57570935 |url=http://lingphil.mit.edu/papers/iatridou/counterfactuality.pdf}}</ref><ref>{{cite journal |last1= Kaufmann |first1= Stefan |s2cid= 60598513 |date=2005 |title=Conditional predictions |journal= Linguistics and Philosophy |volume=28 |issue = 2|doi= 10.1007/s10988-005-3731-9 |at=183-184}}</ref><ref name="Conditionals"/>

Such conditionals are also widely referred to as subjunctive conditionals, though this term is likewise acknowledged as a misnomer even by those who use it. Many languages do not have a morphological subjunctive (e.g. Danish and Dutch) and many that do have it don’t use it for this sort of conditional (e.g. French, Swahili, all Indo-Aryan languages that have a subjunctive). Moreover, languages that do use the subjunctive for such conditionals only do so if they have a specific past subjunctive form. Thus, subjunctive marking is neither necessary nor sufficient for membership in this class of conditionals.

这种条件句也被广泛地称为''虚拟条件句（subjunctive conditionals）''，尽管这个术语同样被使用者认为是用词不当<ref>See for instance [https://pdfs.semanticscholar.org/e68d/612d7a93c98956e7314e0d131d90244c31f2.pdf Ippolito (2002)]: "Because ''subjunctive'' and ''indicative'' are the terms used in the philosophical literature on conditionals and because we will refer to that literature in the course of this paper, I have decided to keep these terms in the present discussion... however, it would be wrong to believe that mood choice is a necessary component of the semantic contrast between indicative and subjunctive conditionals." Also, [http://web.mit.edu/fintel/fintel-2011-hsk-conditionals.pdf von Fintel (2011)] "The terminology is of course linguistically inept ([since] the morphological marking is one of tense and aspect, not of indicative vs. subjunctive mood), but it is so deeply entrenched that it would be foolish not to use it."</ref>。许多语言都没有虚拟语气（如丹麦语和荷兰语)，许多有从句的语言也不把它用于这种条件句（如法语、斯瓦希里语、所有有从句的印度-雅利安语）。此外，只有将虚拟语气用于此类条件的语言才具有特定的过去虚拟语气形式。因此，虚拟标记既不是必要的，也不是充分的。

The terms ''counterfactual'' and ''subjunctive'' have sometimes been repurposed for more specific uses. For instance, the term "counterfactual" is sometimes applied to conditionals that express a contrary-to-fact meaning, regardless of their grammatical structure.<ref name = "lewis73" >{{cite book |last=Lewis |first=David |date=1973 |title= Counterfactuals |location=Cambridge, MA |publisher=Harvard University Press|isbn= 9780631224952}}</ref><ref name = "vonfintel98" /> Along similar lines, the term "subjunctive" is sometimes used to refer to conditionals that bear fake past or irrealis marking, regardless of the meaning they convey.<ref name = "lewis73" >{{cite book |last=Lewis |first=David |date=1973 |title= Counterfactuals |location=Cambridge, MA |publisher=Harvard University Press|isbn= 9780631224952}}</ref><ref>{{cite journal |last1=Khoo |first1=Justin |date=2015 |title=On Indicative and Subjunctive Conditionals |url=https://quod.lib.umich.edu/cgi/p/pod/dod-idx/on-indicative-and-subjunctive-conditionals.pdf?c=phimp;idno=3521354.0015.032;format=pdf |journal=Philosophers' Imprint |volume=15 |issue=32}}</ref>

Recently the term X-Marked has been proposed as a replacement, evoking the extra marking that these conditionals bear. Those adopting this terminology refer to indicative conditionals as O-Marked conditionals, reflecting their ordinary marking.

''反事实（counterfactual）'' and ''从句（subjunctive）''这两个术语有时被重新用于更具体的用途。例如，不管其语法结构如何，"反事实"这个术语有时被用于表达与事实相反的意思的条件语<ref name = "lewis73" >{{cite book |last=Lewis |first=David |date=1973 |title= Counterfactuals |location=Cambridge, MA |publisher=Harvard University Press|isbn= 9780631224952}}</ref><ref name = "vonfintel98" />。按照类似的思路，不管其表达的意思如何，"从句"这个术语有时被用于指带有虚拟过去或非现实标记的条件语。

最近有人提出用术语 X-Marked这个词来替代，以概括这些条件语所带有的额外标记。采用这个术语的人把指示性条件语称为O-Marked条件语，反映了它们的普通标记。

Recently the term ''X-Marked'' has been proposed as a replacement, evoking the ''ex''tra marking that these conditionals bear. Those adopting this terminology refer to indicative conditionals as ''O-Marked'' conditionals, reflecting their ''o''rdinary marking.<ref>von Fintel, Kai; Iatridou, Sabine. [http://web.mit.edu/fintel/fintel-iatridou-2019-x-slides.pdf Prolegomena to a theory of X-marking ] Unpublished lecture slides.</ref><ref>von Fintel, Kai; Iatridou, Sabine. [https://web.mit.edu/fintel/ks-x-phlip-slides.pdf X-marked desires or: What wanting and wishing crosslinguistically can tell us about the ingredients of counterfactuality ] Unpublished lecture slides.</ref><ref name="Linguistic Society of America"/>

The ''antecedent'' of a conditional is sometimes referred to as its ''"if"-clause'' or ''protasis''. The ''consequent'' of a conditional is sometimes referred to as a ''"then"''-clause or as an apodosis.

一个条件的 ''前件（antecedent）''有时被称为 "如果"从句或条件子句。条件的结果有时被称为"那么"子句或结论子句。

==Logic and semantics==

===经典问题（Classic puzzles）===

====反事实的问题（The problem of counterfactuals）====

According to the material conditional analysis, a natural language conditional, a statement of the form ‘if P then Q’, is true whenever its antecedent, P, is false. Since counterfactual conditionals are those whose antecedents are false, this analysis would wrongly predict that all counterfactuals are vacuously true. Goodman illustrates this point using the following pair in a context where it is understood that the piece of butter under discussion had not been heated.

根据实质条件的分析，自然语言条件句即“如果p，那么q（if P then Q）”的陈述，只要其前件p为假就是真。由于反事实条件句是那些前置假设的条件句，这种分析会错误地预测所有反事实条件句都是虚假的。Goodman在理解到正在讨论的那块黄油没有被加热的情况下，用下面的一对例子来说明这一点。

If that piece of butter had been heated to 150º, it would have melted.

如果那块黄油被加热到150度，它就会融化。

Counterfactuals were first discussed by [[Nelson Goodman]] as a problem for the [[material conditional]] used in [[classical logic]]. Because of these problems, early work such as that of [[W.V. Quine]] held that counterfactuals aren't strictly logical, and do not make true or false claims about the world. However, in the 1970s, [[David Lewis (philosopher)|David Lewis]] showed that these problems are surmountable given an appropriate logical framework. Work since then in [[formal semantics (linguistics)|formal semantics]], [[philosophical logic]], [[philosophy of language]], and [[cognitive science]] has built on Lewis's insight, taking it in a variety of different directions.<ref name="Counterfactuals">{{cite encyclopedia |last1=Starr |first1=Will |editor-last1=Zalta |editor-first1=Edward N.|encyclopedia=The Stanford Encyclopedia of Philosophy|title=Counterfactuals|year=2019 |url=https://plato.stanford.edu/archives/fall2019/entries/counterfactuals}}</ref>

If that piece of butter had been heated to 150º, it would not have melted.

如果那块黄油被加热到150度，它就不会融化。

More generally, such examples show that counterfactuals are not truth-functional. In other words, knowing whether the antecedent and consequent are actually true is not sufficient to determine whether the counterfactual itself is true.

更一般地说，这些例子表明反事实不具备真理功能。换句话说，知道前件和结果是否为真并不足以确定反事实本身是否为真。

====上下文依赖和含糊不清（Context dependence and vagueness）====

Counterfactuals are ''context dependent'' and ''[[vague]]''. For example, either of the following statements can be reasonably held true, though not at the same time:<ref>{{Cite journal |last=Lewis |first=David |date=1979 |title=Counterfactual dependence and time's arrow |journal=Noûs |volume=13 |issue=4 |pages=455–476 |doi=10.2307/2215339 |jstor=2215339 |quote=Counterfactuals are infected with vagueness, as everyone agrees.|url=http://pdfs.semanticscholar.org/b736/55909cf4d6a54cd36c4ed449afbe71f3c44b.pdf }}</ref>
反事实是依赖于上下文且含糊不清的。例如，以下任一陈述都可以合理地成立，但不能同时成立：

If Caesar had been in command in Korea, he would have used the atom bomb.

# 如果凯撒（Caesar）当时在朝鲜指挥，他会使用原子弹。

If Caesar had been in command in Korea, he would have used catapults.

# 如果凯撒在朝鲜指挥，他会使用弹弓。

====非单调性（Non-monotonicity）====

Counterfactuals are ''non-monotonic'' in the sense that their truth values can be changed by adding extra material to their antecedents. This fact is illustrated by ''[[Jordan Howard Sobel|Sobel sequences]]'' such as the following:<ref name="jstor.org"/><ref>{{cite journal |last1=Lewis |first1=David |date=1973 |title= Counterfactuals and Comparative Possibility |journal=Journal of Philosophical Logic |volume=2 |issue=4 |doi=10.2307/2215339|jstor=2215339 }}</ref><ref>{{cite book |last=Lewis |first=David |date=1973 |title= Counterfactuals |location=Cambridge, MA |publisher=Harvard University Press|isbn= 9780631224952}}</ref>

反事实是非单调的，因为它们的真值可以通过在其前件中添加额外的信息而改变。这一事实可以通过 Sobel 序列得到说明，例如：

If Hannah had drunk coffee, she would be happy.
If Hannah had drunk coffee and the coffee had gasoline in it, she would be sad.
If Hannah had drunk coffee and the coffee had gasoline in it and Hannah was a gasoline-drinking robot, she would be happy.

# 如果汉娜喝了咖啡，她会很高兴。
# 如果汉娜喝了咖啡，而且咖啡里有汽油，她会很伤心。
# 如果汉娜喝了咖啡，咖啡里有汽油，而汉娜是一个喝汽油的机器人，她会很高兴。

One way of formalizing this fact is to say that the principle of ''Antecedent Strengthening'' should '''not''' hold for any connective > intended as a formalization of natural language conditionals.

对此事实进行形式化的一种方法是说，''前件增强（Antecedent Strengthening）''原则不适用于任何旨在作为自然语言条件句形式化的连接词>。

* '''Antecedent Strengthening''': <math> P > Q \models (P \land R) > Q </math>

* '''前件增强''': <math> P > Q \models (P \land R) > Q </math>

===考虑可能存在的世界Possible worlds accounts===

The most common logical accounts of counterfactuals are couched in the [[possible world semantics]]. Broadly speaking, these approaches have in common that they treat a counterfactual ''A'' > ''B'' as true if ''B'' holds across some set of possible worlds where A is true. They vary mainly in how they identify the set of relevant A-worlds.

反事实的最常见的逻辑解释是可能世界语义学。一般来说，这些方法的共同点是，如果B在A成立的某些可能世界中成立，那么它们就认为反事实 A > B为真。它们的主要区别在于如何确定相关A世界集的方式。

David Lewis's variably strict conditional is considered the classic analysis within philosophy. The closely related premise semantics proposed by Angelika Kratzer is often taken as the standard within linguistics. However, there are numerous possible worlds approaches on the market, including dynamic variants of the strict conditional analysis originally dismissed by Lewis.

大卫·刘易斯（David Lewis）严格可变的条件被认为是哲学中的经典分析。安吉利卡·克拉策（Angelika Kratzer）提出的紧密相关的前提语义常常被视为语言学中的标准。然而，市场上有许多可能世界的方法，包括最初被Lewis摒弃的严格条件分析的动态变体。

====严格的条件Strict conditional====

The [[strict conditional]] analysis treats natural language counterfactuals as being equivalent to the [[modal logic]] formula <math>\Box(P \rightarrow Q)</math>. In this formula, <math>\Box</math> expresses necessity and <math>\rightarrow</math> is understood as [[material conditional|material implication]]. This approach was first proposed in 1912 by [[C.I. Lewis]] as part of his [[Axiomatic system|axiomatic approach]] to modal logic.<ref name="Counterfactuals"/> In modern [[relational semantics]], this means that the strict conditional is true at ''w'' iff the corresponding material conditional is true throughout the worlds accessible from ''w''. More formally:

严格条件分析将自然语言反事实视为等同于模态逻辑公式<math>\Box(P \rightarrow Q)</math>。在这个公式中， <math>\Box</math>表示必要性，<math>\rightarrow</math>被理解为实质条件。这种方法最早是在1912年由C.I. Lewis提出的，作为他对模态逻辑的公理化方法的一部分。

* Given a model <math>M = \langle W,R,V \rangle</math>, we have that <math> M,w \models \Box(P \rightarrow Q) </math> iff <math>M, v \models P \rightarrow Q </math> for all <math>v</math> such that <math>Rwv</math>
* 给定一个模型 <math>M = \langle W,R,V \rangle</math>, 对于所有 <math>v</math> 使得 <math>Rwv</math>，当且仅当<math>M, v \models P \rightarrow Q </math> ，我们有 <math> M,w \models \Box(P \rightarrow Q) </math>。

Unlike the material conditional, the strict conditional is not vacuously true when its antecedent is false. To see why, observe that both <math>P</math> and <math>\Box(P \rightarrow Q)</math> will be false at <math>w</math> if there is some accessible world <math>v</math> where <math>P</math> is true and <math>Q</math> is not. The strict conditional is also context-dependent, at least when given a relational semantics (or something similar). In the relational framework, accessibility relations are parameters of evaluation which encode the range of possibilities which are treated as "live" in the context. Since the truth of a strict conditional can depend on the accessibility relation used to evaluate it, this feature of the strict conditional can be used to capture context-dependence.

与实质条件不同，严格条件在其前件为假时严格为真。要知道为什么，请观察，如果有一些可能世界<math>v</math>，其中<math>P</math>为真，<math>Q</math>为假，那么<math>P</math>和 <math>\Box(P \rightarrow Q)</math>在<math>w</math>处都为假。严格条件也是依赖于上下文的，至少在给定关系语义（或类似的东西）时是如此。在关系框架中，可及性关系是评价的参数，它编码了在上下文中被视为 "活 "的可能性范围。由于严格条件的真实性可能取决于用来评价它的可及性关系，所以严格条件的这一特征可以用来捕捉上下文的依赖性。

The strict conditional analysis encounters many known problems, notably monotonicity. In the classical relational framework, when using a standard notion of entailment, the strict conditional is monotonic, i.e. it validates ''Antecedent Strengthening''. To see why, observe that if <math>P \rightarrow Q</math> holds at every world accessible from <math>w</math>, the monotonicity of the material conditional guarantees that <math>P \land R \rightarrow Q</math> will be too. Thus, we will have that <math> \Box(P \rightarrow Q) \models \Box(P \land R \rightarrow Q) </math>.

严格条件分析遇到了许多已知的问题，特别是单调性。在经典的关系框架中，当使用标准的蕴涵概念时，严格条件是单调的，也就是说，它验证了''前件增强''。要知道为什么，观察一下，如果<math>P \rightarrow Q</math>在每个来自<math>w</math>的世界上成立。那么物质条件的单调性保证了 <math>P \land R \rightarrow Q</math> 也将是如此。因此，我们将有<math> \Box(P \rightarrow Q) \models \Box(P \land R \rightarrow Q) </math>。

This fact led to widespread abandonment of the strict conditional, in particular in favor of Lewis's [[counterfactual conditional#Variably strict conditional|variably strict analysis]]. However, subsequent work has revived the strict conditional analysis by appealing to context sensitivity. This approach was pioneered by Warmbrōd (1981), who argued that ''Sobel sequences'' don't demand a ''non-monotonic'' logic, but in fact can rather be explained by speakers switching to more permissive accessibility relations as the sequence proceeds. In his system, a counterfactual like "If Hannah had drunk coffee, she would be happy" would normally be evaluated using a model where Hannah's coffee is gasoline-free in all accessible worlds. If this same model were used to evaluate a subsequent utterance of "If Hannah had drunk coffee and the coffee had gasoline in it...", this second conditional would come out as trivially true, since there are no accessible worlds where its antecedent holds. Warmbrōd's idea was that speakers will switch to a model with a more permissive accessibility relation in order to avoid this triviality.

这一事实导致了对严格条件的广泛放弃，特别是支持刘易斯的可变严格分析。然而，随后的工作通过对语境敏感性的诉求恢复了严格条件分析。这种方法是由Warmbrōd（1981）开创的，他认为''Sobel序列'' 并不要求非单调逻辑，而事实上，随着序列的进行，说话人可以切换到更宽松的可及性关系来解释。在他的系统中，像“如果Hannah喝了咖啡，她会很高兴”这样的反事实，通常会用Hannah的咖啡在所有可及世界中不含汽油的模型进行评价。如果这个模型被用来评估随后的“如果汉娜喝了咖啡，而咖啡里有汽油……”的话语，这个第二个条件就会被认为是微不足道的真实，因为没有任何可访问的世界的前件是成立的。Warmbrōd的想法是，说话人将转向一个具有更宽松的可及性关系的模型，以避免这种琐碎性。

Subsequent work by Kai von Fintel (2001), Thony Gillies (2007), and Malte Willer (2019) has formalized this idea in the framework of [[dynamic semantics]], and given a number of linguistic arguments in favor. One argument is that conditional antecedents license [[Polarity item#Determination of licensing contexts|negative polarity items]], which are thought to be licensed only by monotonic operators.

Kai von Fintel（2001）、Thony Gillies（2007）和Malte Willer（2019）的后续工作在动态语义学的框架内将这一想法正式化，并给出了一些支持的语言学论据。其中一个论点是，条件前置词许可否定性词语，而这些词被认为只能由单调性运算符许可。

If Natalia leaves tomorrow, she will arrive on time.

# 如果Natalia明天离开，她会准时到达。

Another argument in favor of the strict conditional comes from [[Irene Heim|Irene Heim's]] observation that Sobel Sequences are generally [[Felicity (pragmatics)|infelicitous]] (i.e. sound strange) in reverse.

If Hannah had drunk coffee with gasoline in it, she would not be happy. But if she had drunk coffee, she would be happy.

# 如果Hannah喝了含有汽油的咖啡，她就不会高兴。但如果她喝了咖啡，她就会高兴。

Sarah Moss (2012) and Karen Lewis (2018) have responded to these arguments, showing that a version of the variably strict analysis can account for these patterns, and arguing that such an account is preferable since it can also account for apparent exceptions. As of 2020, this debate continues in the literature, with accounts such as Willer (2019) arguing that a strict conditional account can cover these exceptions as well.<ref name="Counterfactuals"/>

Sarah Moss（2012）和Karen Lewis（2018）对这些论点做出了回应，表明一个版本的可变严格分析可以解释这些模式，并认为这样的解释是可取的，因为它也可以解释明显的例外情况。截至2020年，这一争论在文献中仍在继续，Willer（2019）等人认为，严格条件账户也可以涵盖这些例外情况。

====可变严格条件 Variably strict conditional====

In the variably strict approach, the semantics of a conditional ''A'' > ''B'' is given by some function on the relative closeness of worlds where A is true and B is true, on the one hand, and worlds where A is true but B is not, on the other.

在可变严格方法中，条件''A'' > ''B''的语义是由一些函数给出的，一方面是A为真、B为真的世界，另一方面是A为真、B为假的世界的相对接近程度。

On Lewis's account, A > C is (a) vacuously true if and only if there are no worlds where A is true (for example, if A is logically or metaphysically impossible); (b) non-vacuously true if and only if, among the worlds where A is true, some worlds where C is true are closer to the actual world than any world where C is not true; or (c) false otherwise. Although in Lewis's ''Counterfactuals'' it was unclear what he meant by 'closeness', in later writings, Lewis made it clear that he did ''not'' intend the metric of 'closeness' to be simply our ordinary notion of [[Similarity (philosophy)#Respective and overall similarity|overall similarity]].

在刘易斯的论述中，A > C 是(a)空洞的真实，只有在没有A为真的世界时（例如，如果A在逻辑上或形而上学上是不可能的）；(b)非空洞的真实，只有在A为真的世界中，一些C为真的世界比任何C不为真的世界更接近实际世界；或者(c)虚假，在其他世界里。尽管在刘易斯的《反事实》中，他对“接近性（closeness）”的意思并不清晰，但在后来的著作中，刘易斯明确表示，他并不打算将“接近性”的尺度简单地作为我们对整体相似性的普通概念。

Example:

:If he had eaten more at breakfast, he would not have been hungry at 11 am.
例子：
:如果他在早餐时吃多一点，他在上午11点就不会饿。

On Lewis's account, the truth of this statement consists in the fact that, among possible worlds where he ate more for breakfast, there is at least one world where he is not hungry at 11 am and which is closer to our world than any world where he ate more for breakfast but is still hungry at 11 am.

根据刘易斯的说法，这个陈述的真理在于：在他早餐吃得更多的可能世界中，至少有一个他在上午11点不饿的世界比任何他早餐吃得更多但在上午11点仍然饿的世界更接近我们的世界。

In the past as modal approach, the denotation of the past tense is not fundamentally about time. Rather, it is an underspecified skeleton which can apply either to modal or temporal content. For instance, the particular past as modal proposal of Iatridou (2000), the past tense's core meaning is what's shown schematically below:

过去式作为情态动词的方法，过去时的外延从根本上说不是关于时间的。相反，它是一个未指定的框架，既可以应用于模态内容，也可以应用于时态内容。例如，Iatridou (2000)的特殊过去式作为情态提议，过去式的核心含义是下面的图示:

Stalnaker's account differs from Lewis's most notably in his acceptance of the ''limit'' and ''uniqueness assumptions''. The uniqueness assumption is the thesis that, for any antecedent A, among the possible worlds where A is true, there is a single (''unique'') one that is ''closest'' to the actual world. The limit assumption is the thesis that, for a given antecedent A, if there is a chain of possible worlds where A is true, each closer to the actual world than its predecessor, then the chain has a ''limit'': a possible world where A is true that is closer to the actual worlds than all worlds in the chain. (The uniqueness assumption [[logical consequence|entails]] the limit assumption, but the limit assumption does not entail the uniqueness assumption.) On Stalnaker's account, A > C is non-vacuously true if and only if, at the closest world where A is true, C is true. So, the above example is true just in case at the single, closest world where he ate more breakfast, he does not feel hungry at 11 am. Although it is controversial, Lewis rejected the limit assumption (and therefore the uniqueness assumption) because it rules out the possibility that there might be worlds that get closer and closer to the actual world without limit. For example, there might be an infinite series of worlds, each with a coffee cup a smaller fraction of an inch to the left of its actual position, but none of which is uniquely the closest. (See Lewis 1973: 20.)

Stalnaker的论述与Lewis的论述最明显的不同在于，他接受了“极限（limit）”和“唯一性假设（uniqueness assumptions）”。唯一性假设的论点是：对于任何前件A，在A为真的可能世界中，有一个最接近实际世界的单一（唯一）世界。极限假设的论点是，对于一个给定的前件A，如果存在一个A为真的可能世界链，每个世界都比它的前一个世界更接近实际世界，那么这个链就有一个极限：一个A为真的可能世界比这个链中的所有世界更接近实际世界。（唯一性假设包含了极限假设，但极限假设并不包含唯一性假设）。根据Stalnaker的观点，当且仅当在最接近A为真的世界中，C为真时，A>C才是非空洞的真。因此，上面的例子是真的，只是在他吃了更多早餐的唯一最接近的世界中，他在上午11点不觉得饿。虽然有争议，但Lewis拒绝了极限假设（因此也拒绝了唯一性假设），因为它排除了这样一种可能性，即可能存在着越来越接近实际世界的世界，而没有极限。例如，可能会有一系列无限的世界，每个世界的咖啡杯都在其实际位置的左边小几分之一英寸，但其中没有一个是唯一最接近的。（见Lewis 1973: 20）。

One consequence of Stalnaker's acceptance of the uniqueness assumption is that, if the [[law of excluded middle]] is true, then all instances of the formula (A > C) ∨ (A > ¬C) are true. The law of excluded middle is the thesis that for all propositions p, p ∨ ¬p is true. If the uniqueness assumption is true, then for every antecedent A, there is a uniquely closest world where A is true. If the law of excluded middle is true, any consequent C is either true or false at that world where A is true. So for every counterfactual A > C, either A > C or A > ¬C is true. This is called conditional excluded middle (CEM). Example:

Stalnaker接受唯一性假设的一个结果是，如果排除中间律是真的，那么公式(A>C)∨(A>¬C)的所有实例都是真的。排他性中间律的论题是：对于所有命题p，p∨¬p都是真的。如果唯一性假设为真，那么对于每一个前件A，都有一个唯一最接近的世界，其中A为真。如果排除中间法则是真的，任何结果C在A为真的那个世界里要么是真，要么是假。所以对于每一个反事实A>C，要么A>C，要么A>¬C为真。这就是所谓的条件排除中间法（CEM）。例子：

:(1) If the fair coin had been flipped, it would have landed heads.
:(2) If the fair coin had been flipped, it would have landed tails (i.e. not heads).

:(1) 如果公平的硬币被抛出，它将会正面朝上。
:(2) 如果公平的硬币被抛出，它将会反面朝上（即不是正面朝上）。

On Stalnaker's analysis, there is a closest world where the fair coin mentioned in (1) and (2) is flipped and at that world either it lands heads or it lands tails. So either (1) is true and (2) is false or (1) is false and (2) true. On Lewis's analysis, however, both (1) and (2) are false, for the worlds where the fair coin lands heads are no more or less close than the worlds where they land tails. For Lewis, "If the coin had been flipped, it would have landed heads or tails" is true, but this does not entail that "If the coin had been flipped, it would have landed heads, or: If the coin had been flipped it would have landed tails."

根据Stalnaker的分析，存在一个最接近的世界，在这个世界里，(1)和(2)中提到的公平的硬币被抛出，硬币要么正面朝上，要么反面朝上。因此，要么（1）是真，（2）是假，要么（1）是假，（2）是真。然而，根据Lewis的分析，(1)和(2)都是假的，因为公平的硬币正面朝上的世界并不比反面朝上的世界更接近或更远离。对Lewis来说，“如果硬币被抛出，它将正面朝上或反面朝上”是真的，但这并不意味着“如果硬币被抛出，它将落在正面”，或“如果硬币被抛出，它就会反面朝上”。

=== 其他考虑 Other accounts ===

====因果模型 Causal models====

The ''causal models framework'' analyzes counterfactuals in terms of systems of [[structural equation model|structural equations]]. In a system of equations, each variable is assigned a value that is an explicit function of other variables in the system. Given such a model, the sentence "''Y'' would be ''y'' had ''X'' been ''x''" (formally, ''X = x'' > ''Y = y'' ) is defined as the assertion: If we replace the equation currently determining ''X'' with a constant ''X = x'', and solve the set of equations for variable ''Y'', the solution obtained will be ''Y = y''. This definition has been shown to be compatible with the axioms of possible world semantics and forms the basis for causal inference in the natural and social sciences, since each structural equation in those domains corresponds to a familiar causal mechanism that can be meaningfully reasoned about by investigators. This approach was developed by [[Judea Pearl]] (2000) as a means of encoding fine-grained intuitions about causal relations which are difficult to capture in other proposed systems.<ref name="Pearl2000">{{Cite book |last=Pearl |first=Judea |title=Causality |publisher=Cambridge University Press |year=2000 }}</ref>

''因果模型框架''从结构方程（structural equations）系统的角度分析反事实。在一个方程系统中，每个变量都被分配了一个值，这个值是系统中其他变量的显式函数。给定这样一个模型，“如果X是X，Y就会是Y（''Y'' would be ''y'' had ''X'' been ''x''）”这个句子 (形式上为 ''X = x'' > ''Y = y'' )被定义为断言。如果我们用一个常数''X = x''取代当前决定 ''X''的方程，并求解变量''Y''的方程组，得到的解将是''Y = y''。这个定义已被证明与可能世界语义学的公理兼容，并构成自然科学和社会科学中因果推理的基础。因为这些领域的每个结构方程都对应于一个熟悉的因果机制，这个因果机制可以被研究者进行有意义地推理。这种方法是由Judea Pearl(2000)提出的，作为编码关于因果关系的细粒度直觉的手段，这些直觉在其他提议的系统中难以捕捉。<ref name="Pearl2000">{{Cite book |last=Pearl |first=Judea |title=Causality |publisher=Cambridge University Press |year=2000 }}</ref>

====信念修正 Belief revision====

In the [[belief revision]] framework, counterfactuals are treated using a formal implementation of the ''Ramsey test''. In these systems, a counterfactual ''A'' > ''B'' holds if and only if the addition of ''A'' to the current body of knowledge has ''B'' as a consequence. This condition relates counterfactual conditionals to [[belief revision]], as the evaluation of ''A'' > ''B'' can be done by first revising the current knowledge with ''A'' and then checking whether ''B'' is true in what results. Revising is easy when ''A'' is consistent with the current beliefs, but can be hard otherwise. Every semantics for belief revision can be used for evaluating conditional statements. Conversely, every method for evaluating conditionals can be seen as a way for performing revision.

在信念修正框架中，反事实是用 ''Ramsey检验''的形式化实现来处理的。在这些系统中，当且仅当在当前的知识体系添加 ''A''后得到的结果是B时，反事实''A'' > ''B''成立。这个条件将反事实条件与信念修正联系起来，因为对''A'' > ''B''的评价可以通过首先用''A''修正当前的知识，然后检查''B''在什么结果中是否为真。当''A''与当前的信念一致时，修正是很容易的，但在其他情况下可能会很难。每一个用于信念修正的语义都可以用于评价条件语句。反过来说，每一种评价条件语句的方法都可以被看作是一种执行修正的方法。

====Ginsberg====

Ginsberg (1986) has proposed a semantics for conditionals which assumes that the current beliefs form a set of [[propositional formula]]e, considering the maximal sets of these formulae that are consistent with ''A'', and adding ''A'' to each. The rationale is that each of these maximal sets represents a possible state of belief in which ''A'' is true that is as similar as possible to the original one. The conditional statement ''A'' > ''B'' therefore holds if and only if ''B'' is true in all such sets.<ref name="rev. no. 03011">{{Citation |title=Review of the paper: M. L. Ginsberg, "Counterfactuals," Artificial Intelligence 30 (1986), pp. 35–79 |url=https://zbmath.org/?q=an:0655.03011&format=complete |work=Zentralblatt für Mathematik |pages=13–14 |year=1989 |publisher=FIZ Karlsruhe – Leibniz Institute for Information Infrastructure GmbH |zbl=0655.03011}}.</ref>

Ginsberg(1986)提出了一种条件句的语义，它假定当前的信念形成了一组命题公式，考虑这些公式中与''A''一致的最大集合，并在每个集合中加入''A''。其理由是，这些最大集合中的每一个都代表了一种可能的信念状态，在这种状态下，''A''为真，且与原始状态尽可能相似。因此，当且仅当''B''在所有这些集合中都为真时，条件陈述句''A'' > ''B''才成立。