Population Monte Carlo methods with applications in Bayesian statistics

The Metropolis-Hastings (MH) algorithm of Hastings (1970) is a Markov chain Monte Carlo method that has proven to be a flexible and effective tool to simulate from probability distributions. However, Metropolis-Hastings can generate Markov chains with high degrees of autocorrelation or that have difficult escaping local modes. Population Monte Carlo methods are one class of methods that have been proposed to improve shortcomings in the original Metropolis-Hastings algorithm. These approaches combine information from a collection of previously simulated values in order to improve the way new values are proposed or accepted into the chain. This work discusses and evaluates existing population Monte Carlo methods and presents a generalization of the Metropolis-Hastings algorithm called the sample Metropolis-Hastings (SMH) algorithm. At each iteration, SMH proposes a new value based on information stored in a sample of simulated particles and replaces a relatively poor member of the sample when the proposed value is accepted using SMHs novel acceptance probability. SMH can be implemented without much fine-tuning or prior knowledge of the target distribution and produces an i.i.d. sample from the target distribution upon convergence. Examples discussed in this work show that SMH has the potential to mix well and explore regions of the state space that a conventional Metropolis-Hastings algorithm would have trouble reaching, even when the target distribution is multimodal. This work discusses properties of the sample Metropolis-Hastings algorithm and other population Monte Carlo methods and demonstrates their use in sampling from complex target distributions. The method is applied to examples with multimodal distributions and scenarios with multimodal or constrained likelihoods