Every philosophy has holes, and it is the responsibility of proponents of a philosophy to point out these problems. Here are a few holes in Bayesian data analysis: (1) the usual rules of conditional probability fail in the quantum realm, (2) flat or weak priors lead to terrible inferences about things we care about, (3) subjective priors are incoherent, (4) Bayes factors fail in the presence of flat or weak priors, (5) for Cantorian reasons we need to check our models, but this destroys the coherence of Bayesian inference.
Some of the problems of Bayesian statistics arise from people trying to do things they shouldn’t be trying to do, but other holes are not so easily patched. In particular, it may be a good idea to avoid flat, weak, or conventional priors, but such advice, if followed, would go against the vast majority of Bayesian practice and requires us to confront the fundamental incoherence of Bayesian inference.
This does not mean that we think Bayesian inference is a bad idea, but it does mean that there is a tension between Bayesian logic and Bayesian workflow which we believe can only be resolved by considering Bayesian logic as a tool, a way of revealing inevitable misfits and incoherences in our model assumptions, rather than as an end in itself.
This paper came from a talk I gave a few months ago at a physics conference. For more on Bayesian inference and the two-slit experiment, see this post by Yuling and this blog discussion from several years ago. But quantum probability is just a small part of this paper. Our main concern is to wrestle with the larger issues of incoherence in Bayesian data analysis. I think there’s more to be said on the topic, but it was helpful to write down what we could now. Also I want to make clear that these are real holes. This is different from my article, “Objections to Bayesian statistics,” which discusses some issues that non-Bayesians or anti-Bayesians have had, which I do not think are serious problems with Bayesian inference. In contrast, the “holes” discussed in this new article are real concerns to me.