A review of uncertainty quantification for density estimation

Shaun Mcdonald,David Campbell

doi:10.1214/21-ss130

Abstract

It is often useful to conduct inference for probability densities by constructing “plausible” sets in which the unknown density of given data may lie. Examples of such sets include pointwise intervals, simultaneous bands, or balls in a function space, and they may be frequentist or Bayesian in interpretation. For almost any density estimator, there are multiple approaches to inference available in the literature. Here we review such literature, providing a thorough overview of existing methods for density uncertainty quantification. The literature considered here comprises a spectrum from theoretical to practical ideas, and for some methods there is little commonality between these two extremes. After detailing some of the key concepts of nonparametric inference – the different types of “plausible” sets, and their interpretation and behaviour – we list the most prominent density estimators and the corresponding uncertainty quantification methods for each.

Highlights

Density estimation is one of the seminal examples of nonparametric statistical modelling
Not as abundant as other areas in nonparametric statistics, there is a sizeable body of literature on uncertainty quantification (UQ) for density estimation, ranging from rigorously theoretical to extremely practical
Some combinations of estimation and inference ideas are not represented in the literature, in principle, one could always obtain some kind of uncertainty bounds on a density estimate, either by bootstrapping a frequentist method or taking quantiles of MCMC output for a Bayesian one

Summary

Introduction

Density estimation is one of the seminal examples of nonparametric statistical modelling. Some combinations of estimation and inference ideas are not represented in the literature (in particular, a substantial gap exists between theoretical and practical UQ developments in many cases), in principle, one could always obtain some kind of uncertainty bounds on a density estimate, either by bootstrapping a frequentist method or taking quantiles of MCMC output for a Bayesian one. Whether or not such bounds have suitable coverage properties or otherwise perform adequately is another question for which the answers are not always known. Despite some of these limitations, this paper presents a comprehensive review of the work done far in unknown density UQ, and suggests promising areas to extend the research or “fill in the gaps”

Methods

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Discussion

Conclusion