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).

For example, suppose \varepsilon > 2\delta > 0, A = A_\varepsilon = \{f \,|\, D_{\frac{1}{2}}(f_0, f) > \varepsilon\} and that A_i is a partition of A of diameter at most \delta. If there exists 1 > \alpha > 0 such that

\sum_i \Pi(A_i)^\alpha < \infty,

then for some \beta > 0 we have that

e^{n \beta} \left(\int_{A_{\varepsilon}} \prod_{i=1}^n \frac{f(x_i)}{f_0(x_i)} \Pi (df) \right)^\alpha \le e^{n \beta} \sum_i \prod_{j=1}^n \left(\frac{f_{i,j}(x_j)}{f_0(x_j)}\right)^\alpha \Pi(A_i)^\alpha  \rightarrow 0

almost surely. This is because, with A_{\alpha} the \alpha-affinity defined here, we have that

eq_5

goes exponentially fast towards 0 when \beta is sufficiently small.  Hence the Borel-Cantelli lemma applies.

Construction

The f_{i,j}‘s are defined as the posterior mean predictive density, when the posterior is conditioned on A_i. That is,

f_{i,j} : x \mapsto \frac{\int_{A_i} f(x) \prod_{k=1}^{j-1}f(x_k) \Pi(df)}{\int_{A_i} \prod_{k=1}^{j-1}f(x_k) \Pi(df)}

and

f_{i, 1} : x \mapsto \frac{\int_{A_i} f(x) \Pi(df)}{\Pi(A_i)}.

Clearly

\int_{A_i} \prod_{i=1}^n f(x_i) \Pi(df) = \prod_{j=1}^n f_{i,j}(x_j) \Pi(A_i).

Furthermore, if A_i is contained in a Hellinger ball of center g_i and of radius \delta, then also

H(f_{i,j}, g_i) < \delta.

This follows form the convexity of the Hellinger balls (an important remark for the generalization of this trick).

One thought on “The discretization trick

  1. Pingback: Posterior consistency under (possible) misspecification – Math. Stat. Notes

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