This paper establishes a correspondence in large samples between classical hypothesis tests and Bayesian posterior odds tests for models without trends. More speciﬁcally, tests of point null hypotheses and one- or two-sided alternatives are considered (where nuisance parameters may be present under both hypotheses). It is shown that for certain priors the Bayesian posterior odds test is equivalent in large samples to classical Wald, Lagrange multiplier, and likelihood ratio tests for some signiﬁcance level and vice versa.
Andrews, Donald W.K., "The Large Sample Correspondence Between Classical Hypothesis Tests and Bayesian Posterior Odds Tests" (1992). Cowles Foundation Discussion Papers. 1278.