On Rates of Convergence for Posterior Distributions Under Misspecification

We extend the approach of Walker (2003 Walker , S. ( 2003 ). On sufficient conditions for Bayesian consistency . Biometrika 90 ( 2 ): 482 – 488 .[Crossref], [Web of Science ®] , [Google Scholar]); (2004 Walker , S. ( 2004 ). New approaches to Bayesian consistency . Ann. Statist. 32 ( 5 ): 2028 – 2043 .[Crossref], [Web of Science ®] , [Google Scholar]) to the case of misspecified models. A sufficient condition for establishing rates of convergence is given based on a key identity involving martingales, which does not require construction of tests. We also show roughly that the result obtained by using tests can also be obtained by our approach, which demonstrates the potential wider applicability of this method.