# The discretization trick

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 $\mathbb{F}$ be a set of densities and let $\Pi$ be a prior on $\mathbb{F}$. If $x_1, x_2, x_3, \dots$ follows some distribution $P_0$ having density $f_0$, then the posterior distribution of $\Pi$ can be written as

$\Pi(A | x_1, \dots, x_n) \propto \int_A \prod_{i=1}^n f(x_i) \Pi(df).$

The discretization trick is to find densities $f_{1}, f_2, f_3, \dots$ in the convex hull of $A$ (taken in the space of all densities) such that

$\int_A \prod_{i=1}^n f(x_i) = \prod_{i=1}^n f_i(x_i) \Pi(A).$Read More »