This course covers the theoretical and applied foundations of Bayesian statistical analysis with an emphasis on computational tools for Bayesian hierarchical models. We will discuss model checking, model assessment, and model comparison. The course will cover Bayesian stochastic simulation (Markov chain Monte Carlo) in depth with an orientation towards deriving important properties of the Gibbs sampler and the Metropolis Hastings algorithm. Extensions and hybrids will be discussed. We will then use Markov chain Monte Carlo tools to fit linear and nonlinear specifications with multiple levels, longitudinal features, and non-normal distributional assumptions. Lectures will include theoretical discussions of modeling and estimation as well as practical guidance for fitting complex multilevel models with R and JAGS software. Applications will be drawn from political science, sociology, epidemiology, economic policy, and public administration.