Bayesian Methods for Censored Categorical Data

Abstract

Abstract Bayesian methods are given for finite-category sampling when some of the observations suffer missing category distinctions. Dickey's (1983) generalization of the Dirichlet family of prior distributions is found to be closed under such censored sampling. The posterior moments and predictive probabilities are proportional to ratios of B. C. Carlson's multiple hypergeometric functions. Closed-form expressions are developed for the case of nested reported sets, when Bayesian estimates can be computed easily from relative frequencies. Effective computational methods are also given in the general case. An example involving surveys of death-penalty attitudes is used throughout to illustrate the theory. A simple special case of categorical missing data is a two-way contingency table with cross-classified count data xij (i = 1, …, r; j = 1, …, c), together with supplementary trials counted only in the margin distinguishing the rows, yi (i = 1, …, r). There could also be further supplementary trials reported only by counts distinguishing the columns, Zj (j = 1, …, c). Under assumptions that the censoring process itself is "noninformative" regarding the category probabilities θ ij (e.g., the report for each possible outcome might be nonrandom and prespecified), the Bayesian inference regarding the θ ij 's would be based on the likelihood function Such a likelihood is ordinarily considered intractable and unsuited for Bayesian conjugate prior inference. We develop a Bayesian conjugate theory, however, by recognizing the complete integrals of such functions as Carlson functions and the posterior distributions resulting from Dirichlet prior distributions as known generalized Dirichlet distributions. The corresponding posterior density functions are similar in form to the likelihood, and these constitute a family of distributions closed under sampling and tractable in various senses, including the convenient computability of moments and modes.