On Eliciting Logistic Normal Priors for Multinomial Models

Multinomial models arise when there is a set of complementary and mutually exclusive categories and each observation falls into one of these categories. Such models are used in many scientific and industrial applications. For example, they are frequently applied to the compositions of rocks in geology, to patterns of consumer selection preferences in microeconomics, and to voting behavior in political science. Other examples arise in medicine, psychology and biology. Here we are concerned with the simplest case, where each observation has the same probability of falling into any specified category and observations are independent of each other. Then observations follow a multinomial distribution with, say, probability pi that an observation falls in the ith category. We suppose that an expert’s opinions about the pi are to be quantified as a prior distribution for use in a Bayesian analysis. To quantify the expert’s knowledge, the approach we adopt is to specify a distribution to represent her opinion and then ask her to make assessments that determine appropriate values for the parameters of that distribution. It might seem natural to use the Dirichlet distribution to represent an expert’s opinion, as this is the conjugate prior distribution for the parameters of a multinomial distribution. Elicitation methods that follow this approach have been proposed. O’Hagan et al. (2006) discussed two main available methods for Dirichlet elicitation, the method of Dickey et al. (1983) and that of Chaloner and Duncan (1987). However, while the standard Dirichlet distribution offers tractability and mathematical simplicity, it has been criticized as too inflexible to represent a broad range of prior information about the parameters of multinomial models [e.g. Aitchison (1986), O’Hagan and Forster (2004)]. Its main drawback is that it has a limited number of parameters – if there are k categories then the corresponding Dirichlet distribution has only k parameters. Typically they will be too few to represent an expert’s opinions about the means, variances and covariances of the pi. In particular, the dependence structure between Dirichlet variates cannot be determined independently of its mean values; Dirichlet variates are always negatively correlated, which may not represent prior belief. Motivated by these deficiencies, several authors have constructed new families of distributions for proportions that allow more general types of dependence structures [e.g. Leonard (1975), Tian et al. (2010)]. Some of these new distributions are direct generalizations of the standard Dirichlet distribution [e.g. Connor and Mosimann (1969)]. Generalized, nested or mixed forms of the Dirichlet distribution have been introduced and proposed as suitable prior distributions. For more details on possible prior distributions for multinomial models see, for example, O’Hagan and Forster (2004). Here we model expert opinion by a logistic normal distribution [Aitchison (1986)]. This has a large number of parameters and gives a prior distribution with a much more flexible dependence structure. Eliciting parameters of multivariate distributions is not, in general, an easy task, especially if variates are not independent [O’Hagan et al. (2006)]. In the case of multinomial models, a particularly difficulty is to elicit assessments that satisfy all the constraints of mathematical coherence. Some of these constraints are obvious; the probabilities of each category must be non-negative and sum to one, for example. Others are less obvious. For example, if there are only two categories, the lower quartile for one category and the upper quartile of the other category must add to one. As