Learning and Updating of Uncertainty in Dirichlet Models

摘要

In this paper we analyze the problem of learning and updating of uncertainty in Dirichlet models, where updating refers to determining the conditional distribution of a single variable when some evidence is known. We first obtain the most general family of prior-posterior distributions which is conjugate to a Dirichlet likelihood and we identify those hyperparameters that are influenced by data values. Next, we describe some methods to assess the prior hyperparameters and we give a numerical method to estimate the Dirichlet parameters in a Bayesian context, based on the posterior mode. We also give formulas for updating uncertainty by determining the conditional probabilities of single variables when the values of other variables are known. A time series approach is presented for dealing with the cases in which samples are not identically distributed, that is, the Dirichlet parameters change from sample to sample. This typically occurs when the population is observed at different times. Finally, two examples are given that illustrate the learning and updating processes and the time series approach.