Bayesian analysis of log-linear models with an application

The purpose of this thesis is to study how Lindley's asymptotic and approximate Bayesian methodology can be used in analyzing categorical data using log-linear models. Special attention is paid to the analysis of multinomial categorical data from a Bayesian point of view. The study is mainly based on two papers by Lindley (1964) and Spiegelhalter and Smith (1982). The thesis begins with a thorough discussion of the Bayes factor. Different versions of the Bayes factor and how a Bayes factor can be used to select one of two competing models are presented in detail. In addition, the Savage-Dickey density ratio formula for calculating Bayes factors is discussed. Next Lindley's approximation theorem for the joint posterior distribution of the parameter vector of a multinomial distribution with an improper Dirichlet prior is stated. After that, a more precise and more general version of Lindley's theorem is given with a detailed proof. A discussion of how Spiegelhalter and Smith (1982) use the Savage-Dickey density ratio to calculate Bayes factors for multinomial data under an improper Dirichlet prior is also presented. The last part of the thesis contains an illustration of how Bayesian log-linear models can be used to analyze multinomial data. Lindley's approximation theorem, Spiegelhalter and Smith's method for calculating Bayes factors and an improper Dirichlet prior for the multinomial parameters were utilized for this purpose. The data used for this illustration were collected by the Nutritional Division of the Ministry of Health in Sri Lanka.