Counterfactuals and expected utility theory

How do counterfactuals relate to the expected reward or expected “utility” used in medical decision analysis?

To compute the counterfactual expectation, EY(t), where T is some binary treatment, Y is continuous, and X contains all confounders, we write EY(t) = E (E (Y | t,X)) = \int_x \int_y y p(y|t,x)p(x) dx dy.

In decision analysis we want to compute ER. Recall that we have A as a binary treatment, S' as a continuous post-treatment vector of covariates, and S as a continuous pre-treatment vector of covariates.

We write E_{\pi} R(S,A,S') = \int_s \int_{s'} \sum_a R(s,a,s') p(s'|a,s)\pi(a|s)p(s)ds'ds.

Now suppose R(S,A,S') = S', and we care about the expected reward when fixing A=t. In other words, we care about the expected reward under a policy \pi(A=a|S=s) = I(A=t), where we draw action t with probability 1.

We get then

E_{\pi} R = \int_s \int_{s'} \sum_a s'p(s'|a,s)I(A=t)p(s) ds'ds = \int_s \int_{s'}s' p(s'|t,s)p(s)ds'ds

Rename s' as y and s as x.

Another way: The expected counterfactual is

EY(t) = E(E(Y|t,X)) = \int_x \int_y y p(y|t,x) p(x) dxdy.

With the expected utility all we are doing is

E_{\pi} Y = \int_x \int_y \sum_a y p(y|a,x)\pi(A=a|x)p(x) dxdy.

Now what happens if \pi(A=a|x) = I(A=t)? Then

E_{I(A=t)} Y = \int_x \int_y \sum_a y p(y|a,x)I(A=t)p(x) dxdy = \int_x \int_y y p(y|t, x)p(x) dxdy

which is EY(t).

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