I explain the discretization trick that I mentioned in my previous post (Posterior consistency under possible misspecification). I think it was introduced by Walker (New approaches to Bayesian consistency (2004)).
Let be a set of densities and let be a prior on . If follows some distribution having density , then the posterior distribution of can be written as
The discretization trick is to find densities in the convex hull of (taken in the space of all densities) such that
For example, suppose , and that is a partition of of diameter at most . If there exists such that
then for some we have that
almost surely. This is because, with the -affinity defined here, we have that
goes exponentially fast towards 0 when is sufficiently small. Hence the Borel-Cantelli lemma applies, yielding the claim.
The ‘s are defined as the posterior mean predictive density, when the posterior is conditioned on . That is,
Furthermore, if is contained in a Hellinger ball of center and of radius , then also
This follows form the convexity of the Hellinger balls (an important remark for the generalization of this trick).