counterfactual

Let \(T\) be the random variable that denotes treatment. Let \(Y\) denote the outcome. Let \(Y_j\) denote the outcome if treatment 1 were given.

The average treatment effect is \(E[Y_1 - Y_0] = \sum_{i} y_{1i} - y_{0i}\) where \(y_{ji}\) is the outcome of individual \(i\) who received treatment \(j\). Note that \(i\) cannot actually receive both treatments, so we need to try to account for this using causal inference. If, in the data \(i\) receives treatment 1, then \(y_{0i}\) is the counterfactual outcome.

One thing that confused me in the beginning was trying to understand \(p(Y_1 \mid T=0)\). I thought "how can we talk about an outcome for treatment 1, when treatment 0 has been given?", I realized I should read it like this: "take all the individuals who have been given treatment 0: if they had instead been given treatment 1, what would the outcome be?" The thing to remember: imagine that there are properties that look like "counterfactual outcome for this individual" that we can assign to each individual.