Eliciting Dirichlet and Gaussian copula prior distributions for multinomial models

Abstract

In this paper, we propose novel methods of quantifying expert opinion about prior distributions for multinomial models. Two different multivariate priors are elicited using median and quartile assessments of the multinomial probabilities. First, we start by eliciting a univariate beta distribution for the probability of each category. Then we elicit the hyperparameters of the Dirichlet distribution, as a tractable conjugate prior, from those of the univariate betas through various forms of reconciliation using least-squares techniques. However, a multivariate copula function will give a more flexible correlation structure between multinomial parameters if it is used as their multivariate prior distribution. So, second, we use beta marginal distributions to construct a Gaussian copula as a multivariate normal distribution function that binds these marginals and expresses the dependence structure between them. The proposed method elicits a positive-definite correlation matrix of this Gaussian copula. The two proposed methods are designed to be used through interactive graphical software written in Java.

Quality-adjusted life year difference in patients with predominant negative symptoms of schizophrenia treated with cariprazine and risperidone

“…Experts were asked to give their opinion about the probabilities of patients with PNS of schizophrenia moving from a certain Mohr-Lenert health state to another when treated with a second-generation antipsychotic drug, like risperidone. Data from the expert interviews were gathered using the Prior Elicitation Graphical Software which is capable of modeling opinions about multinomial probabilities by a Dirichlet prior distribution, implementing methodology for assessing a subjective (personal opinion) distribution for the parameters of a multinomial model [18] using Bayesian mathematics.…”

Using informative prior based on expert opinion in Bayesian estimation of the transition probability matrix in Markov modelling—an example from the cost-effectiveness analysis of the treatment of patients with predominantly negative symptoms of schizophrenia with cariprazine

“…Recently, a freeware application was published by Elfadaly and Garthwaite to aid in eliciting Dirichlet and Gaussian copula prior distributions [8]. The proposed method elicits hyperparameters of the Dirichlet distribution from those of its marginal beta distributions through forms of reconciliation that use least-squares techniques.…”

“…An additional criterion at the selection was to have considerable research experience. The prior probabilities were elicited using Prior Elicitation Graphical Software (PEGS) [8]. The feasibility of using the application, and the process of the elicitation was pilot tested without the involvement of the experts.…”

Prior Elicitation, Assessment and Inference with a Dirichlet Prior

“…As such, it seems better to use an elicitation method that fits well with the geometry of the Dirichlet family. If it is felt that more is known a priori than a Dirichlet prior can express, then it is appropriate to contemplate using some other family of priors, see, for example, Elfadaly and Garthwaite [4,5]. Given the conjugacy property of Dirichlet priors and their common usage, the focus here is on devising elicitation algorithms that work well with this family.…”

“…A notable aspect of their approach is that it also works for the Connor-Mosimann distribution, a generalization of the Dirichlet, and in that case no reconciliation is required. Similarly, Elfadaly and Garthwaite [5] base the elicitation on the three quartiles of the marginal beta distributions of the p i which, while independent of order, still requires reconciliation to ensure that the elicited marginals correspond to a Dirichlet. In addition, the elicitation procedure based on the conditionals is extended to develop an elicitation procedure for a more flexible prior based on a Gaussian copula.…”