On Bayesian estimation of marginal structural models 1

Saarela et al. are concerned with integrating propensity scores into a Bayesian framework. Some of us have previously written (Robins and Ritov, 1997; Robins and Wasserman, 2000; http://normaldeviate.wordpress.com/2012/08/28/robins-and-wasserman-respond-to-a-nobelprize-winner/; posted 28 Aug 2012, accessed 1 Oct 2014) about this topic, every time making much the same argument. Here we present a simplified version that captures the main points. A simple setting Though our argument applies to the complex observational data considered by Saarela et al, it is easier to understand it in the simpler setting of a double-blind, placebo-controlled randomized clinical trial of a non-time-varying treatment and under complete compliance. In the spirit of the authors, we assume the trial subjects are representative of a much larger population and the trial results will guide treatment decisions in the population. Let V = { ;;  = 1  } denote the data on the  trial subjects, where  is the binary treatment arm indicator, is the binary outcome, and  is a high-dimensional vector of baseline covariates. The randomization probabilities  [ = 1|] are chosen by a randomizer. By de Finetti’s theorem (e.g., Bernardo and Smith, 1994), a Bayesian can write the marginal density  (V) of 