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.
Shaun McDonald has been supported by an NSERC Alexander Graham Bell Canada Graduate Scholarship. David Campbell has been supported by an NSERC Discovery Grant (RGPIN-2019-05115).
The authors wish to thank the following people for clarifying some ideas in personal correspondence: Eric Cator, Zhong Guan, Maria Lomeli, Omiros Papaspiliopoulos, Yushi Shi, Richard Nickl, and Bin Wang. They also wish to thank the anonymous referees for their insights and improvements to this paper.
"A review of uncertainty quantification for density estimation." Statist. Surv. 15 1 - 71, 2021. https://doi.org/10.1214/21-SS130