which is coded so that higher values indicate more favorable clinical outcomes. An ITR $g$ is a function mapping from $\mathcal{X}$ to $\mathcal{A}$.

We first review the potential outcome framework [@neyman1923applications; @rubin1974estimating; @rubin1986comment]. The potential outcome $Y^*(a)$ is the outcome for an arbitrary individual has s/he received treatment $a$. In actuality, at most one of the potential outcomes can be observed for any individual. The optimal ITR under mean regression, which maximizes the expected outcome, is $g^{\mathrm{opt}}_\mu={\mathrm{argmax}}_{g\in\mathcal{G}}{\mbox{E}}[Y^*\{g({{\bm X}})\}]$. Define the propensity score $\pi({{\bm X}})\triangleq P(A=1|{{\bm X}})$. Following [@rubin1974estimating] and [@rubin1986comment], we can compute the expectation of the potential outcome under the following two key assumptions.

-   **Stable Unit Treatment Value Assumption (SUTVA):** a patient’s observed outcome is the same as the potential outcome for the treatment that s/he actually received. Based on [@rubin1986comment], the SUTVA assumption implies that the value of the potential outcome for a subject does not depend on what treatments other subject receive. Specifically, we can write the SUTVA assumption as $$Y_i=Y_i^*(1)A_i+Y_i^*(0)(1-A_i),\;i=1,\ldots,n.$$ This is also referred as consistency assumption.

-   **Strong Ignorability Assumption:** the treatment assignment $A$ for an individual is independent of the potential outcomes conditional on the covariates ${{\bm X}}$, i.e., $A\bot \{Y^*(a)\}_{a\in\mathcal{A}}|{{\bm X}}$. For 