Misspecified Bayesianism

An agent is a misspecified Bayesian if she updates her belief using Bayes' rule given a subjective, possibly misspecified model of her signals. This paper shows that a belief sequence is consistent with misspecified Bayesianism if the prior contains a grain of the average posterior, i.e., is a mixture of the average posterior and another distribution. A partition-based variant of the grain condition is both necessary and sufficient. Under correct specification, the grain condition reduces to the usual Bayes plausibility. The condition imposes no restriction on the posterior given a full-support prior over a finite or compact state space. However, it rules out posteriors that have heavier tails than the prior on unbounded state spaces. The results cast doubt on the feasibility of testing Bayesian updating in many environments. They also suggest that many seemingly non-Bayesian updating rules are observationally equivalent to Bayesian updating under misspecified beliefs.