Bayesian Modelling and Inference for Statistical Models with Intractable Likelihoods

Intractable probability models play a significant role in various fields of statistical analysis, such as the social sciences, ecology, epidemiology, and genetics, among others. These are models in which the probability distribution cannot be written in closed-form, an aspect that challenges usual inferential methods. This thesis focused on the Bayesian analysis of models where intractability comes from a non-analytical normalisation constant. This class of problems is known as being doubly-intractable, referring to the unavailability of the likelihood model in addition to the evidence. This thesis aims to develop novel statistical approaches for three distinct modelling challenges pertaining to multivariate data in the form of proportions, dependent counts, and rankings. Each of the methods offers distinct contributions to their respective fields, yet are connected by their shared characteristic of being doubly-intractable. Hence, one goal of the thesis is to elegantly handle Bayesian inference via tools including Markov Chain Monte Carlo in the absence of closed-form expressions for the likelihood of such models. The contributions of this thesis are the following. First, a model-based clustering approach for proportions is developed motivated by a real data problem on the relative abundance of coral-building species at locations in Australia’s Great Barrier Reef. In a hierarchical formulation, compositions are assumed random observation from a finite Dirichlet Mixture model, and cluster allocations follow a spatial process prior. Spatial modelling is achieved with a Potts distribution, which is an intractable statistical model. The proposed model acknowledges the location of proportions, offering a probabilistic approach for clustering spatially distributed compositions. The second thesis outcome is the introduction of a multivariate generalisation of the Conway-Maxwell (COM)-Poisson distribution. The COM-Poisson is a distribution for integer-valued data that extends the equidispersed Poisson distribution to over and under-dispersion. By developing their d−variate version, we extend its scope to the multivariate context. The proposed MultCOMP distribution admits a flexible covariance structure, and depends on a number of intractable terms. Finally, the thesis contributes to ranking data modelling in the context of rank aggregation. Rankings are an expression of opinion regarding n objects, where an assessor arranges items from best to worse. In rank aggregation, we are concerned with summarising the preferences of multiple assessors, received in the form of rankings. A distance-based model is developed for situations when relative preferences are not well-defined for all ranked alternatives. In this case, we allow for the estimation of a consensus which accommodates the possibility of non-decisiveness, or ties. The resulting model is an intractable one which offers an extension of the Mallows (Mallows, 1957) distribution, and a robust representation of preferences in terms of ordered groups.