Multilevel models (also called hierarchical linear models, random effects regressions, and mixed effects models) are extremely useful in handling hierarchical datasets in statistical applications. Bayesian analysis is a widely accepted formulation for estimating unknown parameters from data. Bayesian methods have found great success in statistical practice. Modern Bayesian inference generally entails computing simulation draws of the parameters from the posterior distribution. However, we often have computational problems such as covergence, computation time, and storage space. The main purpose of this thesis is to propose some computational strategies to handle these problems and discuss using such strategies to simulate from the posterior distribution of the parameters in hierarchical models with respect to specific examples. Two basic computational strategies are discussed. First, we discuss transformed and parameter-expanded Gibbs samplers far multilevel linear and generalized linear models which are used to speed convergence. Second, we propose sampling for Bayesian computation with large datasets. We illustrate with practical examples from our work in statistical application.
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