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[[File:Diagram of Dynamic Causal Modelling - Causal Modelling and Brain Connectivity in Functional Magnetic Resonance Imaging by Karl Friston.png|thumb|300px|Comparison of two competing causal models (DCM, GCM) used for interpretation of [[fMRI]] images<ref>{{cite journal | doi=10.1371/journal.pbio.1000033 | pmid=19226186 | pmc=2642881 | author=Karl Friston | title=Causal Modelling and Brain Connectivity in Functional Magnetic Resonance Imaging | journal=[[PLOS Biology]] | volume=7 | number=2 | pages=e1000033 | date=Feb 2009 | author-link=Karl Friston }}</ref>]]

[[File:Diagram of Dynamic Causal Modelling - Causal Modelling and Brain Connectivity in Functional Magnetic Resonance Imaging by Karl Friston.png|thumb|300px|比较两个竞争的因果模型(DCM，GCM)用于解释[[fMRI 图像]]<ref>{{cite journal | doi=10.1371/journal.pbio.1000033 | pmid=19226186 | pmc=2642881 | author=Karl Friston | title=Causal Modelling and Brain Connectivity in Functional Magnetic Resonance Imaging | journal=PLOS Biology| volume=7 | number=2 | pages=e1000033 | date=Feb 2009}}</ref>]]

 

 

 

{{Quote|text=Causal models are mathematical models representing causal relationships within an individual system or population. They facilitate inferences about causal relationships from statistical data. They can teach us a good deal about the epistemology of causation, and about the relationship between causation and probability. They have also been applied to topics of interest to philosophers, such as the logic of counterfactuals, decision theory, and the analysis of actual causation.<ref>{{Citation|last=Hitchcock|first=Christopher|title=Causal Models|date=2018|url=https://plato.stanford.edu/archives/fall2018/entries/causal-models/|encyclopedia=The Stanford Encyclopedia of Philosophy|editor-last=Zalta|editor-first=Edward N.|edition=Fall 2018|publisher=Metaphysics Research Lab, Stanford University|access-date=2018-09-08}}</ref>|sign=|source=Stanford Encyclopedia of Philosophy}} [[Judea Pearl]] defines a causal model as an ordered triple <math>\langle U, V, E\rangle</math>, where U is a set of [[exogenous variable]]s whose values are determined by factors outside the model; V is a set of endogenous variables whose values are determined by factors within the model; and E is a set of [[structural equation]]s that express the value of each endogenous variable as a function of the values of the other variables in U and V.<ref name=":0" />

[[Judea Pearl]] defines a causal model as an ordered triple <math>\langle U, V, E\rangle</math>, where U is a set of [[exogenous variable]]s whose values are determined by factors outside the model; V is a set of endogenous variables whose values are determined by factors within the model; and E is a set of [[structural equation]]s that express the value of each endogenous variable as a function of the values of the other variables in U and V.<ref name=":0" />

  Judea Pearl defines a causal model as an ordered triple <math>\langle U, V, E\rangle</math>, where U is a set of exogenous variables whose values are determined by factors outside the model; V is a set of endogenous variables whose values are determined by factors within the model; and E is a set of structural equations that express the value of each endogenous variable as a function of the values of the other variables in U and V.

  Judea Pearl defines a causal model as an ordered triple <math>\langle U, V, E\rangle</math>, where U is a set of exogenous variables whose values are determined by factors outside the model; V is a set of endogenous variables whose values are determined by factors within the model; and E is a set of structural equations that express the value of each endogenous variable as a function of the values of the other variables in U and V.

Judea Pearl 将因果模型定义为一个有序的三元组 < math > langle u，v，e rangle </math > ，其中 u 是一组外生变量，其值由模型外部的因素决定; v 是一组内生变量，其值由模型内部的因素决定; e 是一组结构方程，表示每个内生变量的值为 u 和 v 中其他变量值的函数。

Judea Pearl 将因果模型定义为一个有序的三元组<math>\langle U, V, E\rangle</math> ，其中 u 是一组外生变量，其值由模型外部的因素决定; v 是一组内生变量，其值由模型内部的因素决定; e 是一组结构方程，表示每个内生变量的值为 u 和 v 中其他变量值的函数。

[[Aristotle]] defined a taxonomy of causality, including material, formal, efficient and final causes. Hume rejected Aristotle's taxonomy in favor of [[Counterfactual conditional|counterfactuals]]. At one point, he denied that objects have "powers" that make one a cause and another an effect.<ref name=":1" />{{rp|[{{google books|plainurl=y|id=9H0dDQAAQBAJ|page=264}} 264]}} Later he adopted "if the first object had not been, the second had never existed" ("[[Sine qua non|but-for]]" causation).<ref name=":1" />{{rp|[{{google books|plainurl=y|id=9H0dDQAAQBAJ|page=265}} 265]}}

[[Aristotle]] defined a taxonomy of causality, including material, formal, efficient and final causes. Hume rejected Aristotle's taxonomy in favor of [[Counterfactual conditional|counterfactuals]]. At one point, he denied that objects have "powers" that make one a cause and another an effect.<ref name=":1" />Later he adopted "if the first object had not been, the second had never existed" ("[[Sine qua non|but-for]]" causation).<ref name=":1" />

In 1921 [[Sewall Wright|Wright]]'s [[Path analysis (statistics)|path analysis]] became the theoretical ancestor of causal modeling and causal graphs.<ref>{{Cite book|url={{google books |plainurl=y |id=yWWEIvNgUQ4C|page=707}} |title=The Oxford Handbook of Causation |volume=1 |editor-last=Beebee |editor-first=Helen|editor-last2=Hitchcock|editor-first2=Christopher|editor-last3=Menzies|editor-first3=Peter|date=2012-01-12|publisher=OUP Oxford|isbn=9780191629464|language=en|first=Samir |last=Okasha |chapter=Causation in Biology|chapter-url=http://www.oxfordhandbooks.com/view/10.1093/oxfordhb/9780199279739.001.0001/oxfordhb-9780199279739-e-0036|doi=10.1093/oxfordhb/9780199279739.001.0001 }}</ref> He developed this approach while attempting to untangle the relative impacts of [[heredity]], development and environment on [[guinea pig]] coat patterns. He backed up his then-heretical claims by showing how such analyses could explain the relationship between guinea pig birth weight, ''[[Uterus|in utero]]'' time and litter size. Opposition to these ideas by prominent statisticians led them to be ignored for the following 40 years (except among animal breeders). Instead scientists relied on correlations, partly at the behest of Wright's critic (and leading statistician), [[Ronald Fisher|Fisher]].<ref name=":1" /> One exception was Burks, a student who in 1926 was the first to apply path diagrams to represent a mediating influence (''mediator'') and to assert that holding a mediator constant induces errors. She may have invented path diagrams independently.<ref name=":1" />{{rp|[{{google books|plainurl=y|id=9H0dDQAAQBAJ|page=304}} 304]}}

In 1921 [[Sewall Wright|Wright]]'s [[Path analysis (statistics)|path analysis]] became the theoretical ancestor of causal modeling and causal graphs.<ref>{{Cite book|url={{google books |plainurl=y |id=yWWEIvNgUQ4C|page=707}} |title=The Oxford Handbook of Causation |volume=1 |editor-last=Beebee |editor-first=Helen|editor-last2=Hitchcock|editor-first2=Christopher|editor-last3=Menzies|editor-first3=Peter|date=2012-01-12|publisher=OUP Oxford|isbn=9780191629464|language=en|first=Samir |last=Okasha |chapter=Causation in Biology|chapter-url=http://www.oxfordhandbooks.com/view/10.1093/oxfordhb/9780199279739.001.0001/oxfordhb-9780199279739-e-0036|doi=10.1093/oxfordhb/9780199279739.001.0001 }}</ref> He developed this approach while attempting to untangle the relative impacts of [[heredity]], development and environment on [[guinea pig]] coat patterns. He backed up his then-heretical claims by showing how such analyses could explain the relationship between guinea pig birth weight, ''[[Uterus|in utero]]'' time and litter size. Opposition to these ideas by prominent statisticians led them to be ignored for the following 40 years (except among animal breeders). Instead scientists relied on correlations, partly at the behest of Wright's critic (and leading statistician), [[Ronald Fisher|Fisher]].<ref name=":1" /> One exception was Burks, a student who in 1926 was the first to apply path diagrams to represent a mediating influence (''mediator'') and to assert that holding a mediator constant induces errors. She may have invented path diagrams independently.<ref name=":1" />

<math>P (floss \vline toothpaste, price*2) </math>

<math>P (floss \vline toothpaste, price*2) </math>

In 1958 [[David Cox (statistician)|Cox]] warned that controlling for a variable Z is valid only if it is highly unlikely to be affected by independent variables.<ref name=":1" />{{rp|[{{google books|plainurl=y|id=9H0dDQAAQBAJ|page=154}} 154]}}

In 1958 [[David Cox (statistician)|Cox]] warned that controlling for a variable Z is valid only if it is highly unlikely to be affected by independent variables.<ref name=":1" />

In the 1960s, [[Otis Dudley Duncan|Duncan]], [[Hubert M. Blalock Jr.|Blalock]], [[Arthur Goldberger|Goldberger]] and others rediscovered path analysis. While reading Blalock's work on path diagrams, Duncan remembered a lecture by [[William Fielding Ogburn|Ogburn]] twenty years earlier that mentioned a paper by Wright that in turn mentioned Burks.<ref name=":1" />{{rp|[{{google books|plainurl=y|id=9H0dDQAAQBAJ|page=308}} 308]}}

In the 1960s, [[Otis Dudley Duncan|Duncan]], [[Hubert M. Blalock Jr.|Blalock]], [[Arthur Goldberger|Goldberger]] and others rediscovered path analysis. While reading Blalock's work on path diagrams, Duncan remembered a lecture by [[William Fielding Ogburn|Ogburn]] twenty years earlier that mentioned a paper by Wright that in turn mentioned Burks.<ref name=":1" />

For x to be a necessary cause of y, the presence of y must imply the prior occurrence of x. The presence of x, however, does not imply that y will occur. Necessary causes are also known as "but-for" causes, as in y would not have occurred but for the occurrence of x.

For x to be a necessary cause of y, the presence of y must imply the prior occurrence of x. The presence of x, however, does not imply that y will occur. Necessary causes are also known as "but-for" causes, as in y would not have occurred but for the occurrence of x.

因果图是一个有向图，它显示了因果模型中变量之间的因果关系。因果关系图包括一组变量(或节点)。每个节点通过一个箭头连接到一个或多个其他节点，对这些节点有因果影响。箭头描述了因果关系的方向，例如，一个箭头连接变量 a 和 b 与箭头在 b 表示 a 的变化导致 b 的变化(与一个相关的概率)。路径是在两个节点之间按照因果箭头进行的图的遍历。

因果图是一个有向图，它显示了因果模型中变量之间的因果关系。因果关系图包括一组变量(或节点)。每个节点通过一个箭头连接到一个或多个其他节点，对这些节点有因果影响。箭头描述了因果关系的方向，例如，一个箭头连接变量 a 和 b 与箭头在 b 表示 a 的变化导致 b 的变化(与一个相关的概率)。路径是在两个节点之间按照因果箭头进行的图的遍历。

In 1973 [[David Lewis (philosopher)|Lewis]] advocated replacing correlation with but-for causality (counterfactuals). He referred to humans' ability to envision alternative worlds in which a cause did or not occur and in which effect an appeared only following its cause.<ref name=":1" />{{rp|[{{google books|plainurl=y|id=9H0dDQAAQBAJ|page=266}} 266]}} In 1974 [[Donald Rubin|Rubin]] introduced the notion of "potential outcomes" as a language for asking causal questions.<ref name=":1" />{{rp|[{{google books|plainurl=y|id=9H0dDQAAQBAJ|page=269}} 269]}}

In 1973 [[David Lewis (philosopher)|Lewis]] advocated replacing correlation with but-for causality (counterfactuals). He referred to humans' ability to envision alternative worlds in which a cause did or not occur and in which effect an appeared only following its cause.<ref name=":1" />{{rp|[{{google books|plainurl=y|id=9H0dDQAAQBAJ|page=266}} 266]}} In 1974 [[Donald Rubin|Rubin]] introduced the notion of "potential outcomes" as a language for asking causal questions.<ref name=":1" />

因为基因在不同人群中随机变化，基因的存在通常被认为是工具变量，这意味着在许多情况下，因果关系可以用观察性研究回归来量化。

因为基因在不同人群中随机变化，基因的存在通常被认为是工具变量，这意味着在许多情况下，因果关系可以用观察性研究回归来量化。

In 1983 [[Nancy Cartwright (philosopher)|Cartwright]] proposed that any factor that is "causally relevant" to an effect be conditioned on, moving beyond simple probability as the only guide.<ref name=":1" />{{rp|[{{google books|plainurl=y|id=9H0dDQAAQBAJ|page=48}} 48]}}

In 1983 [[Nancy Cartwright (philosopher)|Cartwright]] proposed that any factor that is "causally relevant" to an effect be conditioned on, moving beyond simple probability as the only guide.<ref name=":1" />

 

<math>P(Y|do(X)) = \textstyle \sum_{z} \displaystyle P(Y|X, Z=z) P(Z=z)</math>

<math>P(Y|do(X)) = \textstyle \sum_{z} \displaystyle P(Y|X, Z=z) P(Z=z)</math>

In 1986 Baron and Kenny introduced principles for detecting and evaluating mediation in a system of linear equations. As of 2014 their paper was the 33rd most-cited of all time.<ref name=":1" />{{rp|[{{google books|plainurl=y|id=9H0dDQAAQBAJ|page=324}} 324]}} That year [[Sander Greenland|Greenland]] and [[James Robins|Robins]] introduced the "exchangeability" approach to handling confounding by considering a counterfactual. They proposed assessing what would have happened to the treatment group if they had not received the treatment and comparing that outcome to that of the control group. If they matched, confounding was said to be absent.<ref name=":1" />{{rp|[{{google books|plainurl=y|id=9H0dDQAAQBAJ|page=154}} 154]}}

In 1986 Baron and Kenny introduced principles for detecting and evaluating mediation in a system of linear equations. As of 2014 their paper was the 33rd most-cited of all time.<ref name=":1" />That year [[Sander Greenland|Greenland]] and [[James Robins|Robins]] introduced the "exchangeability" approach to handling confounding by considering a counterfactual. They proposed assessing what would have happened to the treatment group if they had not received the treatment and comparing that outcome to that of the control group. If they matched, confounding was said to be absent.<ref name=":1" />