Bayesian Modeling and Computations

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The broad objective of my research proposal is the development and implementation of complex statistical models with a focus on computation in Bayesian settings. I attempt to work in subject areas corresponding to real problems. In the subject area of social networks, I am investigating the assessment of ``accuracy'' corresponding to interpersonal perceptions. Triadic data are the richest type of interpersonal perception data where all individuals in a study have ratings on every pair of individuals. The assessment of accuracy is a fundamental problem and is one of the oldest issues in social and personal psychology. Are people's perceptions of others valid? The approach that I am considering extends a traditional random effects model where analysis has remained illusive from a classical point of view. Various modeling assumptions and the introduction of prior distributions leads to complex and high dimensional Bayesian models. A second project involves the modeling and analysis of a special type of categorical data. In surveys, categorical data are often misclassified. For example, a subject whose ``true'' classification is the first category may be incorrectly classified in the second category. In such cases, severely biased estimators can occur when the effect of misclassification is ignored. In this project, I will attempt to account for misclassification for data extending from a single multinomial cohort to the case of subject-specific covariates. In addition, the presence of gold standard data will be considered. The complexity of the models brings nonidentifiability issues to the forefront and the need to elicit subjective prior distributions. Statistics in sport is a general application area in which I am involved. One problem that I plan to pursue concerns optimal team selection in Twenty20 cricket. An innovation in this work is that traditional cricket statistics are not used in determining player worth. Instead, run differential with and without a given player in the lineup is the true measure of value. I propose an empirical Bayes procedure to infer batting and bowling characteristics of players. Via simulation, these characteristics can be used to assess the quality of given lineups. Lineups may then be optimized over a huge combinatorial space via fine tuning of the simulated annealing algorithm. This work may benefit teams in terms of roster selection. A methodological project which I am considering concerns the development of importance sampling algorithms. Importance sampling is a fundamental sampling strategy that is important in the approximation of integrals arising in Bayesian statistics. Importance sampling has two specific advantages over popular Markov chain methods: (i) generated variates are independent which simplifies error assessment and (ii) there is no need to diagnose convergence to stationarity. Nevertheless, it is sometimes said that importance sampling does not work well in high dimensions. I do not believe this sentiment to be entirely true. Rather, importance sampling has not been implemented using sufficiently rich families for higher dimensional problems. Typically, the multivariate normal and Student distributions have been used for importance sampling. It is my goal to develop multivariate importance sampling algorithms using alternative multivariate distributions such as skew-symmetric families. I also plan on developing an adaptive component to importance sampling whereby the importance sampler is improved over successive iterations. Obtaining proofs of the convergence of the adaptive algorithm forms part of the project.