# Idea **Potential** outcomes describe what would happen if a treatment were or were not administered. They're different from **observed** outcomes. They're the outcomes we **would** see under each possible treatment option. They're about [[counterfactuals]]. If treatment $A = 1$ is receiving vaccine and $A = 0$ is not receiving vaccine, then $Y^1$ is the outcome (e.g., probability of getting COVID) we would observe if $A = 1$ and $Y^0$ is the outcome (e.g., probability of getting COVID) we would observe if $A = 0$. Thus, each person/unit has two potential outcomes $Y^0$, $Y^1$. General notation for the outcome would be observed if treatment was set to $A = a$: $Y^a \text{ or } Y_a$ - the potential outcome that happened: **factual** - the potential outcome that didn't happen: **counterfactual** We often use the following notation to denote the unit and potential outcome for that unit. - $Y_{0i}$: potential outcome for unit $i$ **without treatment** - $Y_{1i}$: potential outcome for the same unit $i$ **with treatment** ## Before and after treatment/exposure: potential outcomes and counterfactual Before any treatment decision is made, any outcome is a **potential** outcome: $Y^0$, $Y^1$. Both outcomes can **potentially** happen. Thus, the term "**potential**" outcomes. After exposure to treatment, there is an observed outcome, $Y = Y^A$, and [[counterfactuals|counterfactual outcome]] $Y^{1-A}$. The $Y_1$ of the **treated** is observed. The $Y_0$ of the **untreated** is observed. The $Y_0$ of the **treated** is [[counterfactuals|counterfactual]]—often the most important counterfactual that contributes to [[bias]]. It answers the question of "if the treated had not received treatment, what their outcome would have been." If this value isn't equivalent to $Y_0$ of the untreated (i.e., observed outcome for the untreated), then we have [[bias]]. The $Y_1$ of the **untreated** is [[counterfactuals|counterfactual]]. ## Counterfactuals In many real-world situations, we can only observe one outcome for each person. A person can either receive the vaccine or not—but rarely both outcomes—the [[fundamental problem of causal inference]]. If I receive the vaccine, my observed outcome is $Y^1$. And my [[counterfactuals|counterfactual outcome]] is $Y^0$—what my outcome would be had I received $Y^0$ instead. The unobserved outcome is known as the [[counterfactuals|counterfactual outcome]]. # References - [Potential outcomes and counterfactuals - Welcome and Introduction to Causal Effects | Coursera](https://www.coursera.org/learn/crash-course-in-causality/lecture/0XWFc/potential-outcomes-and-counterfactuals) - [01 - Introduction To Causality — Causal Inference for the Brave and True](https://matheusfacure.github.io/python-causality-handbook/01-Introduction-To-Causality.html)