  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) Bayesian decision picks the
wrong model, (5) Bayes factors fail in the presence of flat or weak priors, (6)
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.
