Towards a practicable Bayesian nonparametric density estimator

• Published in: Biometrika 1991-09, Vol.78 (3), p.531-543
• Main Author: Lenk, Peter J.
• Format: Article
• Language: English
• Subjects: Approximation; Covariance; Data smoothing; Density; Density estimation; Estimators; Exact sciences and technology; Hierarchical Bayes; Histograms; Logistic normal process; Logistics; Mathematical independent variables; Mathematics; Probability and statistics; Sciences and techniques of general use; Smoothing; Statistical variance; Statistics
• ISSN: 0006-3444; 1464-3510
• DOI: 10.1093/biomet/78.3.531

SUMMARY Nonparametric density estimators smooth the empirical distribution function and are sensitive to the choice of smoothing parameters. This paper develops an hierarchical Bayes formulation for the smoothing problem. The prior distribution for the density function is the logistic normal process, which is a logistic transform of a Gaussian process. The covariance of the Gaussian process is a smoothing kernel and has parameters that control the degree of smoothness. The likelihood function for the smoothing parameters and their posterior density are computed from an approximation of the joint moments of the logistic normal process. The marginal predictive density mixes the conditional predictive density given the smoothing parameters with their posterior distribution. This hierarchical Bayes analysis provides a fully automated, data-dependent method for smoothing and selects the amount of smoothing that is coherent with its prior specification.