更改

Formally it has been written as [Y_i1, Y_i0] ⊥ Tx_i, or in words the potential Y outcome of person ''i'' had they been treated or not does not depend on whether they have really been (observable) treated or not. We can ignore in other words how people ended up in one vs. the other condition, and treat their potential outcomes as exchangeable. While this seems thick, it becomes clear if we add subscripts for the ‘realized’ and superscripts for the ‘ideal’ (potential) worlds (notation suggested by [https://www.cambridge.org/core/books/statistical-models-and-causal-inference/7CE8D4957FF6E9615AAAC4128FA8246E David Freedman]; a visual can help here: [https://drive.google.com/open?id=1nLHHH0il225LIy33nRiH3ZfgoX1_-_V9 potential outcomes simplified]).

Formally it has been written as [Y_i1, Y_i0] ⊥ Tx_i, or in words the potential Y outcome of person ''i'' had they been treated or not does not depend on whether they have really been (observable) treated or not. We can ignore in other words how people ended up in one vs. the other condition, and treat their potential outcomes as exchangeable. While this seems thick, it becomes clear if we add subscripts for the ‘realized’ and superscripts for the ‘ideal’ (potential) worlds (notation suggested by [https://www.cambridge.org/core/books/statistical-models-and-causal-inference/7CE8D4957FF6E9615AAAC4128FA8246E David Freedman]; a visual can help here: [https://drive.google.com/open?id=1nLHHH0il225LIy33nRiH3ZfgoX1_-_V9 potential outcomes simplified]).

数学形式上，它被写成<math>[Y_i1, Y_i0] ⊥ Tx_i</math> ，或者用文字来说，人们的潜在结果Y我已经治疗或不治疗不取决于他们是否真的被(可观察的)治疗。换句话说，我们可以忽略人们是如何在一种情况下和另一种情况下结束生命的，而把他们的潜在结果看作是可以交换的。虽然这看起来很厚，但是如果我们为“理想”(潜在)世界添加“已实现”的下标和上标就变得很清楚了(由 [https://www.cambridge.org/core/books/statistical-models-and-causal-inference/7CE8D4957FF6E9615AAAC4128FA8246E David Freedman]提出的符号; 一个视觉可以在这里帮助:[https://drive.google.com/open?id=1nLHHH0il225LIy33nRiH3ZfgoX1_-_V9 potential outcomes simplified]).

其数学形式可记为[Y_i1, Y_i0] ⊥ Tx_i ，或者用文字表述为，个体“i”是否接受处理的潜在结果Y并不取决于他们是否真的（可观测到的）接受处理。换句话说，个体最终是通过什么方式处于一种与另一种处理状态我们是可忽略的，并将其潜在结果视为等价可交换的。 虽然这看起来很复杂，但如果我们为“已实现”的真实处理结果添加下标，为“理想”（潜在）世界的处理结果添加上标，就会变得很清楚。

换句话说，我们可以忽略人们是如何在一种情况下和另一种情况下结束生命的，而把他们的潜在结果看作是可以交换的。虽然这看起来很厚，但是如果我们为“理想”(潜在)世界添加“已实现”的下标和上标就变得很清楚了(符号的提出可参考[https://www.cambridge.org/core/books/statistical-models-and-causal-inference/7CE8D4957FF6E9615AAAC4128FA8246E David Freedman]; 可视化帮助可参考:[https://drive.google.com/open?id=1nLHHH0il225LIy33nRiH3ZfgoX1_-_V9 potential outcomes simplified]).

So: Y₁¹/*Y₀¹ are potential Y outcomes had the person been treated (superscript ¹), when in reality they have actually been (Y₁¹, subscript ₁), or not (*Y₀¹: the * signals this quantity can never be realized or observed, or is ''fully'' contrary-to-fact or counterfactual, CF).

So: Y₁¹/*Y₀¹ are potential Y outcomes had the person been treated (superscript ¹), when in reality they have actually been (Y₁¹, subscript ₁), or not (*Y₀¹: the * signals this quantity can never be realized or observed, or is ''fully'' contrary-to-fact or counterfactual, CF).

所以:Y₁¹/*Y₀¹是潜在结果Y，如果个体被处理(superscript ¹) ，那么实际上它们是(Y₁¹, subscript ₁) ，而不是(*Y₀¹:: * 表示这个值是无法实现或不可观测的，或''完全''与事实或事实相反，CF)。

所以:Y₁¹/*Y₀¹是潜在结果Y，如果个体被处理(上角标为 ¹) ，那么实际上它们是(Y₁¹, 下角标也为 ₁) ，而不是(*Y₀¹:: * 表示这个值是无法实现或不可观测的，即''完全''与事实相反或称为反事实（counterfactual, CF）)。

 

Similarly, *Y₁⁰/Y₀⁰ are potential Y outcomes had the person not been treated (superscript ⁰), when in reality they have been (*Y₁⁰, subscript ₁), or not actually (Y₀⁰).

Similarly, *Y₁⁰/Y₀⁰ are potential Y outcomes had the person not been treated (superscript ⁰), when in reality they have been (*Y₁⁰, subscript ₁), or not actually (Y₀⁰).

 

同样，*Y₁⁰/Y₀⁰是个体未被处理 (上角标为 ⁰)的潜在结果Y，当现实中它们是(*Y₁⁰, 下角标为 ₁),或实际上不是 (Y₀⁰).

同样，*Y₁⁰/Y₀⁰是个体未被处理 (上角标0)的潜在结果Y，当现实中它们是(*Y₁⁰, subscript ₁),或实际上不是 (Y₀⁰).