We give conditions that guarantee that the posterior probability of every Hellinger neighborhood of the true distribution tends to 1 almost surely. The conditions are (1) a requirement that the prior not put high mass near distributions with very rough densities and (2) a requirement that the prior put positive mass in Kullback-Leibler neighborhoods of the true distribution. The results are based on the idea of approximating the set of distributions with a finite-dimensional set of distributions with sufficiently small Hellinger bracketing metric entropy. We apply the results to some examples.
"The consistency of posterior distributions in nonparametric problems." Ann. Statist. 27 (2) 536 - 561, April 1999. https://doi.org/10.1214/aos/1018031206