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.
Highlights
Bayesian statistical methods provide a formal mechanism for taking into account prior knowledge
As to simplify the separate tasks that an assessor must perform, the elicitation process for a multivariate prior has been decomposed into a number of processes for eliciting univariate beta distributions
Two novel methods have been proposed for reconciling the beta marginal distributions into a multivariate prior distribution
Summary
Bayesian statistical methods provide a formal mechanism for taking into account prior knowledge. The elicitation method of Kadane et al (1980) has been designed to encode a positive-definite covariance matrix of a multivariate t-distribution as a conjugate prior for the hyperparameters of a normal multiple linear regression model. Their method can be useful in a variety of multivariate elicitation problems that require the assessment of positive-definite matrices (Garthwaite et al 2005, 2013). No other elicitation method for Gaussian copulas seems to encode a positive-definite correlation matrix using quartile assessments To fill this gap, the second proposed method in this paper elicits a Gaussian copula function with beta marginal distributions as a prior distribution for multinomial models.
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