The price is failure on a class of inductive inference problems that are easily solved, in contrast, by nonBayesian mechanical learners. By “mechanical” is meant “simulable by Turing machine”.
One of the central tenets of Bayesianism, which is common to the heterogeneous collection of views which fall under this rubric, is that hypothesis change proceeds via conditionalization on accumulated evidence, the posterior probability of a given hypothesis on the evidence being computed using Bayes's theorem. We show that this strategy for hypothesis change precludes the solution of certain problems of inductive inference by mechanical means—problems which are solvable by mechanical means when the restriction to this Bayesian strategy is lifted. Our discussion proceeds as follows. After some technical preliminaries, the concept of (formal) learner is introduced along with a criterion of inferential success. Next we specify a class of inductive inference problems, and then define the notion of “Bayesian behavior” on those problems. Finally, we exhibit an inductive inference problem from the specified class such that (a) some nonmechanical Bayesian learner solves the problem, (b) some nonBayesian mechanical learner solves the problem, (c) some mechanical learner manifests Bayesian behavior on the problem, but (d) no mechanical Bayesian learner solves the problem.
Insofar as possible terminology and notation are drawn from Osherson, Stob, and Weinstein .