Bayesian Updating

Bayesian updating is the process of revising belief probabilities in response to new evidence according to Bayes' theorem. In the idealized form, an agent begins with a prior probability distribution over hypotheses, observes data, and computes a posterior distribution that optimally combines prior belief with likelihood of the evidence. The theorem is mathematically trivial; its philosophical weight comes from the claim that rational belief revision is Bayesian updating.

The challenge is that real agents — humans, institutions, algorithms — do not update in the Bayesian manner. They exhibit confirmation bias, anchoring, and motivated reasoning, systematically deviating from the normative prescription. The standard reply — that humans are 'bounded rationality' approximations of Bayesians — is itself a Bayesian interpretation of non-Bayesian behavior. It treats deviation as noise around a true signal rather than as evidence that the Bayesian framework mischaracterizes what belief revision actually is.

In machine learning, Bayesian updating appears as variational inference and posterior sampling, where the computational cost of exact Bayesian updating forces approximations that are no longer guaranteed to be optimal. The gap between normative Bayesianism and computational reality mirrors the gap between idealized rationality and human cognition. Whether the gap is a problem for the theory or for the practitioners depends on whether one treats Bayesianism as a descriptive claim or a prescriptive framework.

Bayesian updating is either a theorem, a methodology, or a mythology — and the field has not been clear about which it is claiming.