## Propensity scores and Bayesian inference

Zhiqiang Tan (Biostatistics, Johns Hopkins) writes, regarding my blog entry on regression and matching.

I wrote:

I’m imagining a unification of matching and regression methods, following the Cochran and Rubin approach: (1) matching, (2) keeping the treated and control units but discarding the information on who was matched with whom, (3) regression including treatment interactions. I’m still confused about exactly how the propensity score fits in.

Zhiqiang writes:

In fact, I’m also working on “causal inference”. As I understand, there is
a fundamental gap between the idea of propensity score and the likelihood
principle or Bayesian inference. The likelihood is factorized in terms of
the outcome regression and the propensity score, so that any (parametric)
likelihood or Bayesian inference would necessarily ignore the propensity
score! One way to reconcile the two “ideas” is to look at the joint
distribution of covariates and outcome, as in my paper “Efficient and
Robust Causal Inference: A Distributional Approach”
.

As you can see, the idea is connected to the likelihood formulation for Monte Carlo integration. Here I worked on propensity score weighting as opposed to matching, and followed maximum likelihood/frequentist instead of Bayesian.

My response: I agree that propensity score methods don’t tie directly to likelihood or Bayesian inference. I think the appropriate link is through poststratification. But actually carrying out this modeling in a reasonable way is a challenge–an important research problem, I think.

My quick and lazy comments on Zhiqiang’s paper: The tables should be graphs. Figure 1 could use a caption explaining what models 1-4 are, and what the two graphs are. The graphs in Figures 2 and 3 can be made smaller, and they should be rotated 90 degrees.

OK, now I have to read the paper for real.