Nonparametric predictive inference for discrete data via Metropolis-adjusted Dirichlet sequences

TL;DR

This paper introduces a new Bayesian predictive method using Metropolis-adjusted Dirichlet sequences for discrete data, offering computational advantages and flexible inference for count and related data types.

Contribution

It develops a novel MAD sequence model that simplifies Bayesian inference for discrete distributions without relying on traditional mixture models.

Findings

Asymptotic exchangeability of MAD sequences

Efficient algorithms for count data inference

Successful applications to multivariate and regression data

Abstract

This article is motivated by challenges in conducting Bayesian inferences on unknown discrete distributions, with a particular focus on count data. To avoid the computational disadvantages of traditional mixture models, we develop a novel Bayesian predictive approach. In particular, our Metropolis-adjusted Dirichlet (MAD) sequence model characterizes the predictive measure as a mixture of a base measure and Metropolis-Hastings kernels centered on previous data points. The resulting MAD sequence is asymptotically exchangeable and the posterior on the data generator takes the form of a martingale posterior. This structure leads to straightforward algorithms for inference on count distributions, with easy extensions to multivariate, regression, and binary data cases. We obtain a useful asymptotic Gaussian approximation and illustrate the methodology on a variety of applications.