Recent Advances in the Elicitation of Uncertainty Distributions from Experts for Multinomial Probabilities

In this chapter, we consider the problem of the elicitation and specification of an uncertainty distribution based on expert judgements, which may be a subjective prior distribution in a Bayesian analysis, for a set of probabilities which are constrained to sum to one. A typical context for this is as a prior distribution for the probabilities in a multinomial model. The Dirichlet distribution has long been advocated as a natural way to represent the uncertainty distribution over the probabilities in this context. The relatively small number of parameters allows for specification based on relatively few elicited quantities but at the expense of a very restrictive structure. We detail recent advances in elicitation for the Dirichlet distribution and recently proposed alternative approaches, which offer greater flexibility at the expense of added complexity. In order of increasing flexibility, they are the generalised Dirichlet distribution, multivariate copulas and vines. An extension of multinomial models containing covariates is discussed.