The Logistic Normal Distribution for Bayesian, Nonparametric, Predictive Densities

Abstract

This article models the common density of an exchangeable sequence of observations by a generalization of the process derived from a logistic transform of a Gaussian process. The support of the logistic normal includes all distributions that are absolutely continuous with respect to the dominating measure of the observations. The logistic-normal family is closed in the prior to posterior Bayes analysis, with the observations entering the posterior distribution through the covariance function of the Gaussian process. The covariance of the Gaussian process plays the role of a smoothing kernel. Three features of the model provide a flexible structure for computing the predictive density: (a) The mean of the Gaussian process corresponds to the prior mean of the random density: (b) The prior variance of the Gaussian process controls the influence of the data in the posterior process. As the variance increases, the predictive density has greater fidelity to the data, (c) The prior covariance of the Gaussian process controls the smoothness of its sample paths and the amount of pooling of the sample information. For iid observations the empirical distribution function (edf) is a sufficient statistic for all inference. Since the human eye finds it difficult to distinguish important features of distribution functions, their densities often are plotted instead. Unfortunately, the edf does not possess a density function with respect to Lebesgue measure; consequently, many techniques have been proposed to smooth the edf so that its modification does possess a proper density. From the subjective, Bayesian perspective, the data are iid given a common but unspecified density. Beliefs about the unknown density are modeled through probability statements. When the density has a known functional form, the prior distribution concerns the density's parameters, which describe important, unknown features of the density. In this article the density is not constrained to a functional form, so it becomes the parameter of interest. Its posterior distribution becomes the mechanism for smoothing the edf so that the density estimator (the posterior mean) evaluated at a point can use nearby data.

Key Words:

• Lognormal distribution
• Nonparametric density estimation
• Smoothing