Standardization for a causal mean

Let $Q$ be a probability law for an observed discrete or atomic treatment $A$, a covariate random element $L$, and a real-valued observed outcome $Y$. Fix an atom $a$ with $Q(A=a)>0$, let $\mathcal{G}=\sigma (L)$, and define $I_a={\mathbf{1}}_{\{A=a\}},e_a(L)=E_Q[I_a\mid \mathcal{G}]\mathrm{.}$ Assume $e_a(L)>0$ $Q$-almost surely and $I_{a}Y\in L^1(Q)$. Define $q_a(L)=E_Q[I_{a}Y\mid \mathcal{G}],m_a(L)=\frac{q_a(L)}{e_a(L)}\mathrm{.}$ The standardization functional at $a$ is ${\mathrm{\Psi }}_{\mathrm{s}\mathrm{t}\mathrm{d}}(Q;a)=E_Q[m_a(L)],$ provided $m_a(L)$ is integrable. The numerator and denominator are conditional-expectation classes; the ratio is defined almost surely because the denominator is positive. In conventional notation $m_a(L)$ is $E_Q[Y\mid A=a,L]$, and the displayed expectation averages that treatment-specific conditional mean over the target law of $L$.

When $L$ has discrete support, this is ${\sum }_{\mathrm{\ell }}{E}_Q[Y\mid A=a,L=\mathrm{\ell }]Q(L=\mathrm{\ell })$. For general $L$ we use the version-free conditional-expectation class above and integrate it under $Q$; no value of a regression function is chosen at a null covariate stratum.

This is a population functional of the observed law and the true conditional treatment probability. It is not an algorithm for estimating a propensity from a finite sample.