Information measures of Dirichlet distribution with applications

Abstract

We explore properties of information measures of the Dirichlet family and related distributions. Representations of the information measures of the Dirichlet family in terms of the information measures of the gamma family reflect the characterization of Dirichlet distribution in terms of the ratios of independent gamma distributions to their sum. We present measures of information provided by a multinomial vector about the multinomial parameters under the Dirichlet prior and measures of information loss due to aggregation of Dirichlet components and the multinomial categories. New information characterizations of the bivariate Dirichlet distribution in terms of survival and hazard gradient constraints are given. The information analysis of contingency tables is briefly described. An example illustrates applications of Dirichlet prior for inferences about information measures of contingency tables and Kendall's tau for ordinal data. Results that compare solicitation of the mean and mode vectors for the Dirichlet model are given and are illustrated through an example from a study of business executives' environmental uncertainty in the aftermath of 9/11 terrorist attacks. The probability integral transformation is proposed for facilitating implementation of the maximum entropy Dirichlet process prior procedure for inference about the fit of continuous distributions to the data.

Estimation of Group Priorities and Value of Information

“…Information measures of Dirichlet distribution is given in Ebrahimi et al . () who extended results by Lindley () on binomial sampling and beta density.…”

“…Ebrahimi et al . () discusses information properties of the Dirichlet family and emphasizes the applicability of entropy methods when used as a prior distribution with multinomial data and thus is different than what is considered here. To the best of our knowledge, this paper is the first to develop information measures for predictive distributions on the basis of a Dirichlet likelihood.…”

Mixtures of Gaussian Processes for regression under multiple prior distributions

“…= log Γ( α0 ) − A proof can be found in [26]. The derivatives of the (log-) gamma and digamma functions can be calculated via the polygamma function and are implemented in most modern libraries for automatic differentiation.…”