Monte Carlo Markov chain sampling for Bayesian computation, with applications to constrained parameter spaces

Ming-Hui Chen, Purdue University

Abstract

Recently, Monte Carlo Markov chain sampling methods have become widely used for evaluating multidimensional integrals $\int\sb{R\sp{k}} h({\underline x}) f({\underline x})d{\underline x},$ where f is a density function. If f is a Bayesian posterior density, then the above integral is a posterior expectation. The most common Markov chain samplers include Metropolis-Hastings (Metropolis et al. 1953 and Hastings 1970), Gibbs (Geman and Geman 1984 and Gelfand and Smith 1990), and Hit-and-Run (Belisle, Romeijn and Smith 1990). This thesis contains some contributions to further extend these Monte Carlo Markov chain sampling methods. Variations of the Hit-and-Run (H&R) sampler are considered. The convergence results of Belisle, Romeijn and Smith (1990) are generalized to unbounded regions and to unbounded integrands. Three other variations that are intended to reduce point-estimator variance: conditional expectation in the random direction, sampling in multiple directions, and adaptive external control variates are also proposed. The performance of three Monte Carlo Markov chain samplers--the Gibbs sampler, the H&R sampler, and the Metropolis sampler--is considered. Based on bivariate normal examples, the Gibbs and Hit-and-Run samplers are empirically compared. For zero correlation, the Gibbs sampler provides independent data, resulting in better performance than H&R. As the absolute value of the correlation increases, the H&R performance improves, with H&R being substantially better for correlations above 0.9. An importance weighted marginal density estimation (IWMDE) method is proposed. An IWMDE is obtained by averaging many dependent observations of the ratio of the full joint posterior densities multiplied by a weighting conditional density. Asymptotic properties of the IWMDE and guidelines for choosing a weighting conditional density are investigated. A bivariate normal model and a constrained linear multiple regression model are used to illustrate how to derive the IWMDEs for the marginal posterior densities. A hybrid Markov chain sampling scheme that combines the Gibbs sampler and the Hit-and-Run sampler is developed. This hybrid algorithm is well-suited to Bayesian computation for constrained parameter spaces and has been utilized in two applications: (i) a constrained linear multiple regression problem and (ii) prediction for a multinomial distribution with constrained parameter space.

Degree

Ph.D.

Advisors

Schmeiser, Purdue University.

Subject Area

Statistics

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