*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.

## Construction

The ‘s are defined as the posterior mean predictive density, when the posterior is conditioned on . That is,

and

Clearly

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).

[…] We assume, without too much loss of generality, that our priors are discrete. When dealing with Hellinger separable density spaces, it is possible to discretize posterior distributions to study consistency (see this post about it). […]

[…] me consider well-specified discrete priors for simplicity. More generally, the discretization trick could possibly yield similar […]