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 »