  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.
