Bayesian approaches to non- and semiparametric density estimation [with a rejoinder to my discussants]

TL;DR

This paper reviews Bayesian methods for non- and semiparametric density estimation, discussing priors, basis expansions, and local parametric models, with a focus on robustness and flexibility.

Contribution

It introduces and discusses various Bayesian approaches for density estimation, including Dirichlet process priors, basis function expansions, and local parametric modeling.

Findings

Dirichlet process priors provide flexible nonparametric density models.

Orthogonal basis expansions, especially Hermite, offer robust density approximations.

Local likelihood methods enable adaptive, locally parametric density estimation.

Abstract

This invited paper proposes and discusses several Bayesian attempts at nonparametric and semiparametric density estimation. The main categories of these ideas are as follows: 1) Build a nonparametric prior around a given parametric model. We look at cases where the nonparametric part of the construction is a Dirichlet process or relatives thereof. (2) Express the density as an additive expansion of orthogonal basis functions, and place priors on the coefficients. Here attention is given to a certain robust Hermite expansion around the normal distribution. Multiplicative expansions are also considered. (3) Express the unknown density as locally being of a certain parametric form, then construct suitable local likelihood functions to express information content, and place local priors on the local parameters.