On the Bernstein-von Mises theorem for the Dirichlet process

Kolyan Ray,Aad Van Der Vaart

doi:10.1214/21-ejs1821

Abstract

We establish that Laplace transforms of the posterior Dirichlet process converge to those of the limiting Brownian bridge process in a neighbourhood about zero, uniformly over Glivenko-Cantelli function classes. For real-valued random variables and functions of bounded variation, we strengthen this result to hold for all real numbers. This last result is proved via an explicit strong approximation coupling inequality.

Highlights

Zn from a distribution P0 on some measurable space (X, A), and given Z1, . . . , Zn let Pn be a draw from the Dirichlet process with base measure ν + nPn
Ν is a finite measure on the sample space and Pn| Z1, . . . , Zn ∼ DP(ν + nPn) for all n, which is the posterior distribution obtained when equipping the distribution of the observations Z1, Z2, . . . , Zn with a Dirichlet process prior with base measure ν
The case ν = 0 is allowed; the process Pn is known as the Bayesian bootstrap

Summary

Results

ΔZi be the empirical distribution of an i.i.d. , Zn from a distribution P0 on some measurable space (X , A), and given Z1, . . . , Zn let Pn be a draw from the Dirichlet process with base measure ν + nPn. Since bounded variation balls are universal Donsker classes, this is a significantly stronger requirement than G being P0-Glivenko-Cantelli in Theorem 1 We prove this result by exploiting a strong approximation, which establishes a rate of convergence for representations of these random variables defined on a common probability space and has various applications in probability and statistics, for instance√studying distributional approximations of transformed random variables ψn( n(Pn − Pn)), where the functions ψn depend on n. ≤ C2e−C3x, for all x > 0 and n ≥ 2, where C1 − C3 are universal constants This result says one can couple the Dirichlet process posteriors to a sequence of Brownian bridges independent of the underlying data. A similar, if more complicated, expression can be proved with the Brownian bridges (Bn) replaced by the Kiefer process K, in particular yielding an almost sure rate O(n−1/4(log n)5/4), for P0∞-almost every sequence Z1, Z2,

Proof of Theorem 1

Proof of Proposition 1

Proofs of strong approximation results

Some weak convergence facts