In Bayesian statistics we are interested in the posterior distribution of parameters. In simple cases we can derive analytical expressions for the posterior. However in most situations, the posterior expectations cannot be calculated analytically due to the complexity of the integrals. This thesis develops some new methodologies for applied problems which deal with multidimensional parameters, complex model structures and complex likelihood functions. The first project is concerned with the simulation of one-day cricket matches. Given that only a finite number of outcomes can occur on each ball, a discrete generator on a finite set is developed where the outcome probabilities are estimated from historical data. The probabilities depend on the batsman, the bowler, the number of wickets lost, the number of balls bowled and the innings. The proposed simulator appears to do a reasonable job at producing realistic results. The simulator allows investigators to address complex questions involving one-day cricket matches. The second project investigates the suitability of Dirichlet process priors in the Bayesian analysis of network data. Dirichlet process priors allow the researcher to weaken prior assumptions by going from a parametric to a semiparametric framework. This is important in the analysis of network data where complex nodal relationships rarely allow a researcher the confidence in assigning parametric priors. The Dirichlet process also provides a clustering mechanism which is often suitable for network data where groups of individuals in a network can be thought of as arising from the same cohort. The approach is highlighted on two network models and implemented using WinBUGS. The third project develops a Bayesian latent variable model to analyze ordinal survey data. The data are viewed as multivariate responses arising from a class of continuous latent variables with known cut-points. Each respondent is characterized by two parameters that have a Dirichlet process as their joint prior distribution. The proposed mechanism adjusts for classes of personality traits. As the resulting posterior distribution is complex and high-dimensional, posterior expectations are approximated by MCMC methods. The methodology is tested through simulation studies and illustrated using student feedback data from course evaluations at Simon Fraser University.
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Thesis advisor: Swartz, Tim
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