Mixture representations and Bayesian nonparametric inference for likelihood ratio ordered distributions

TL;DR

This paper introduces mixture representations for likelihood ratio ordered distributions and develops a Bayesian nonparametric method for density estimation and hypothesis testing under these order constraints.

Contribution

It provides the first posterior consistency results for likelihood ratio order constrained inference and proposes a practical MCMC algorithm for Bayesian analysis.

Findings

Posterior consistency is established under reasonable conditions.

The method effectively tests for likelihood ratio ordering.

Demonstrated successful application in a biomedical case study.

Abstract

In this article, we introduce mixture representations for likelihood ratio ordered distributions. Essentially, the ratio of two probability densities, or mass functions, is monotone if and only if one can be expressed as a mixture of one-sided truncations of the other. To illustrate the practical value of the mixture representations, we address the problem of density estimation for likelihood ratio ordered distributions. In particular, we propose a nonparametric Bayesian solution which takes advantage of the mixture representations. The prior distribution is constructed from Dirichlet process mixtures and has large support on the space of pairs of densities satisfying the monotone ratio constraint. Posterior consistency holds under reasonable conditions on the prior specification and the true unknown densities. To our knowledge, this is the first posterior consistency result in the literature on order constrained inference. With a simple modification to the prior distribution, we can test the equality of two distributions against the alternative of likelihood ratio ordering. We develop a Markov chain Monte Carlo algorithm for posterior inference and demonstrate the method in a biomedical application.