On Eliciting Some Prior Distributions for Multinomial Models

In Bayesian analysis of multinomial models, an important assessment task is to elicit an informative joint prior distribution for multinomial probabilities. We start by introducing a method to elicit a univariate beta distribution for the probability of each category using probability quartiles. Three different multivariate priors are introduced using the elicited betas. As a tractable conjugate prior, we elicit the hyperparameters of the Dirichlet distribution from those of the univariate betas through some forms of reconciliation using least-squares techniques. However, the Dirichlet distribution is not flexible enough to represent prior information. So, the proposed method is also designed to elicit a generalized Dirichlet distribution, known as Connor-Mosimann distribution, which is also a conjugate prior that has a larger number of parameters and hence a more flexible dependence structure. Moreover, we use the beta marginal distributions to construct a Gaussian copula as a multivariate normal distribution function that binds these marginals to form their joint multivariate distribution. The proposed method elicits a positive-definite correlation matrix of this Gaussian copula. All proposed methods are designed to be used with the aid of interactive graphical software written in Java.