simple geometric arguments, the information that is gained from increasing restrictions on the law of the mediator. In the online Appendix, we illustrate the map by recovering the efficient influence function of van der Laan and Petersen in the case of a singleton model (i.e., a known conditional density) for the mediator and in the case of a parametric model for the mediator.
8. A semiparametric sensitivity analysis. We describe a semiparametric sensitivity analysis framework to assess the extent to which a violation of the ignorability assumption for the mediator might alter inferences about natural direct and indirect effects. Although only results for the natural direct effect are given here, the extension for the indirect effect is easily deduced from the presentation. Let
then
that is, a violation of the ignorability assumption for the mediator variable, generally implies that
Thus, we proceed as in Robins, Rotnitzky and Scharfstein (2000), and propose to recover inferences by assuming the selection bias function $t(e, m, x)$ is known, which encodes the magnitude and direction of the unmeasured confounding for the mediator. In the following, the support of $M$, $S$ is assumed to be finite. To motivate the proposed approach, suppose for the moment that $f_{M|E,X}(M|E, X)$ is known; then under the assumption that the exposure is ignorable given $X$, we show in the Appendix that
and therefore the M-functional is identified by