Progress! (on the understanding of the role of randomization in Bayesian inference)

Leading theoretical statistician Larry Wassserman in 2008:

Some of the greatest contributions of statistics to science involve adding additional randomness and leveraging that randomness. Examples are randomized experiments, permutation tests, cross-validation and data-splitting. These are unabashedly frequentist ideas and, while one can strain to fit them into a Bayesian framework, they don’t really have a place in Bayesian inference. The fact that Bayesian methods do not naturally accommodate such a powerful set of statistical ideas seems like a serious deficiency.

To which I responded on the second-to-last paragraph of page 8 here.

Larry Wasserman in 2013:

Some people say that there is no role for randomization in Bayesian inference. In other words, the randomization mechanism plays no role in Bayes’ theorem. But this is not really true. Without randomization, we can indeed derive a posterior for theta but it is highly sensitive to the prior. This is just a restatement of the non-identifiability of theta. With randomization, the posterior is much less sensitive to the prior. And I think most practical Bayesians would consider it valuable to increase the robustness of the posterior.

Exactly! I completely agree with 2013 Larry (and it’s what we say in our Bayesian book, following the ideas of Rubin and others).

I’m happy to see this development. Much of my recent work has involved Bayesian analysis of sample surveys. And, indeed, our models typically assume simple random sampling within poststratification cells. Such models are never correct (even if the survey is conducted by a probability sampling design, nonresponse will not be random) but it’s a useful starting point that we try to approximate in many of our designs. In other settings, we simply don’t have random sampling or random assignment, and then, indeed, our inferences can be more sensitive to our assumptions. The only place I’d disagree with Larry is when he writes “sensitive to the prior,” I’d say, “sensitive to the model,” because the data model comes into play too, not just the prior distribution (that is, the model for the parameters).

P.S. Beyond appreciating Larry’s recognition of this particular issue, I find his larger point interesting, that we add noise in different ways to achieve robustness or computational efficiency.

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