The Dependent Dirichlet Process and Related Models

TL;DR

This paper reviews Bayesian nonparametric regression models that allow response distributions to vary flexibly with predictors, focusing on dependent Dirichlet processes and their extensions.

Contribution

It provides a comprehensive review of the development and application of dependent Dirichlet process models for flexible nonparametric regression.

Findings

Dependent Dirichlet processes enable flexible modeling of changing response distributions.

Extensions of these models address complex predictor-response relationships.

Bayesian approaches improve inference in nonparametric regression scenarios.

Abstract

Standard regression approaches assume that some finite number of the response distribution characteristics, such as location and scale, change as a (parametric or nonparametric) function of predictors. However, it is not always appropriate to assume a location/scale representation, where the error distribution has unchanging shape over the predictor space. In fact, it often happens in applied research that the distribution of responses under study changes with predictors in ways that cannot be reasonably represented by a finite dimensional functional form. This can seriously affect the answers to the scientific questions of interest, and therefore more general approaches are indeed needed. This gives rise to the study of fully nonparametric regression models. We review some of the main Bayesian approaches that have been employed to define probability models where the complete response distribution may vary flexibly with predictors. We focus on developments based on modifications of the Dirichlet process, historically termed dependent Dirichlet processes, and some of the extensions that have been proposed to tackle this general problem using nonparametric approaches.