Potential Outcomes

You’re used to seeing $Y$. What happens. That’s very nice. For example, you give someone Aspirin and say:

$$Y = \begin{cases} 1 & \text{headache disappears} \\ 0 & \text{headache remains} \end{cases}$$

Now in Causal Inference, you deal with something subtler: Potential Outcomes. Remember the “What would have happened” part? This is how you talk about then.

• $Y_i^1$ is the outcome if person $i$ had received the treatment.
• $Y_i^0$ is the outcome if person $i$ had not received the treatment.
• $A=\{0,1\}$ is whether or not treatment was assigned. In formulae this is used as $Y_i^A$. That’s how you get the notation above.

I.e., $Y_i^A$ denotes the Potential Outcome for a treatment assignment $A$.

Words again

How do counterfactuals and potential outcomes relate?

• Potential Outcomes: What would happen under treatment and no treatment.
• Counterfactual: The potential outcome corresponding to the treatment not received

You can maybe say “A counterfactual is a potential outcome under an alternate unrealized treatment or action.” I don’t know about this.

Sidenotes

Notation

Note that $Y_i^1$ is not just an outcome! It is the outcome if you did something (administer treatment in this case.)

Next, distinguish between the individual and population-level outcomes!

• $Y_{\text{i}}^1 = Y_{\text{Alice}}^1$ denotes individual.
• $Y^1$ denotes population.

See also

Permutations

$Y_{\text{Bob}}^0 = 1$ is eminently possible: you did not give Bob Aspirin and he got a headache!

For completeness,

• $Y_i^1 = 1$ means you gave them Aspirin and their headache went away.
• $Y_i^1 = 0$ means you gave them Aspirin and their headache did not go away.
• $Y_i^0 = 0$ means you did not give them Aspirin and their headache did not go away.

What do you really know bro?

Consider what we’re saying up there. In the given experiment can you:

• Both give and not give Alice her Aspirin? No.
• Deny Alice her Aspirin but know what would have happened if she took Aspirin? No.

The Fundamental Problem

So where do the $Y_{\text{Alice}^{\text{Aspirin}}}$ and $Y_{\text{Alice}^{\text{No Aspirin}}}$ come from? You can only the latter!

Welcome to the fundamental problem: the information of what would have happened if you had given her Aspirin is lost. Poof.

Now zoom out to the population. When you say $Y^1$ this means what would have happened if everyone got the treatment. $Y^0$ is the outcome if everyone did not.

Can you have access to both? Nope.

Causal Inference is a Missing Data Problem.

But we will stay for a little in hypothetical space which is used to build up the problem and tackle it using various assumptions.