We use martingales to study Bayesian consistency. We derive sufficient conditions for both Hellinger and Kullback–Leibler consistency, which do not rely on the use of a sieve. Alternative sufficient conditions for Hellinger consistency are also found and demonstrated on examples.
"New approaches to Bayesian consistency." Ann. Statist. 32 (5) 2028 - 2043, October 2004. https://doi.org/10.1214/009053604000000409