A Bayesian nonparametric chi-squared goodness-of-fit test

TL;DR

This paper introduces a Bayesian nonparametric chi-squared goodness-of-fit test using the Dirichlet process and Kullback-Leibler distance, providing a flexible alternative to traditional tests with asymptotic equivalence.

Contribution

It proposes a novel Bayesian nonparametric test based on the Dirichlet process and KL divergence, extending chi-squared testing to a Bayesian framework.

Findings

Asymptotic convergence to chi-squared distribution.

Method for selecting Dirichlet process concentration parameter.

Extension to independence testing in contingency tables.

Abstract

The Bayesian nonparametric inference and Dirichlet process are popular tools in statistical methodologies. In this paper, we employ the Dirichlet process in hypothesis testing to propose a Bayesian nonparametric chi-squared goodness-of-fit test. In our Bayesian nonparametric approach, we consider the Dirichlet process as the prior for the distribution of data and carry out the test based on the Kullback-Leibler distance between the updated Dirichlet process and the hypothesized distribution F0. We prove that this distance asymptotically converges to the same chi-squared distribution as the chi-squared test does. Similarly, a Bayesian nonparametric chi-squared test of independence for a contingency table is provided. Also, by computing the Kullback-Leibler distance between the Dirichlet process and the hypothesized distribution, a method to obtain an appropriate concentration parameter for the Dirichlet process is suggested.