Pathwise concentration bounds for Bayesian beliefs
证明了即使先验分布不完全支持，贝叶斯后验也会集中到近似最小化经验分布与真实分布之间KL散度的结果分布上，并给出了先验设定错误时后验收敛速度的界。
We show that Bayesian posteriors concentrate on the outcome distributions that approximately minimize the Kullback–Leibler divergence from the empirical distribution, uniformly over sample paths, even when the prior does not have full support. This generalizes Diaconis and Freedman's (1990) uniform convergence result to, e.g., priors that have finite support, are constrained by independence assumptions, or have a parametric form that cannot match some probability distributions. The concentration result lets us provide a rate of convergence for Berk's (1966) result on the limiting behavior of posterior beliefs when the prior is misspecified. We provide a bound on approximation errors in “anticipated‐utility” models, and extend our analysis to outcomes that are perceived to follow a Markov process.