Marginal structural models were introduced in order to provide estimates of causal effects from interventions based on observational studies in epidemiological research. The key point is that this can be understood in terms of Girsanov’s change of measure. This offers a mathematical interpretation of marginal structural models that has not been available before. We consider both a model of an observational study and a model of a hypothetical randomized trial. These models correspond to different martingale measures – the observational measure and the randomized trial measure – on some underlying space. We describe situations where the randomized trial measure is absolutely continuous with respect to the observational measure. The resulting continuous-time likelihood ratio process with respect to these two probability measures corresponds to the weights in discrete-time marginal structural models. In order to do inference for the hypothetical randomized trial, we can simulate samples using observational data weighted by this likelihood ratio.
"A martingale approach to continuous-time marginal structural models." Bernoulli 17 (3) 895 - 915, August 2011. https://doi.org/10.3150/10-BEJ303