Advances and Applications in Perfect Sampling
Perfect sampling algorithms are Markov Chain Monte Carlo (MCMC) methods without statistical error. The latter are used when one needs to get samples from certain (non-standard) distributions. This can be accomplished by creating a Markov chain that has the desired distribution as its stationary distribution, and by running sample paths "for a long time", i.e. until the chain is believed to be in equilibrium. The question "how long is long enough?" is generally hard to answer and the assessment of convergence is a major concern when applying MCMC schemes. This issue completely vanishes with the use of perfect sampling algorithms which - if applicable - enable exact simulation from the stationary distribution of a Markov chain.
In this thesis, we give an introduction to the general idea of MCMC methods and perfect sampling. We develop advances in this area and highlight applications of these advances to two relevant problems.
As advances, we discuss and devise several variants of the well-known Metropolis-Hastings algorithm which address accuracy, applicability, and computational cost of this method. We also describe and extend the idea of slice coupling, a technique which enables one to couple continuous sample paths of Markov chains for use in perfect sampling algorithms.
As a first application, we consider Bayesian variable selection. The problem of variable selection arises when one wants to model the relationship between a variable of interest and a subset of explanatory variables. In the Bayesian approach one needs to sample from the posterior distribution of the model and this simulation is usually carried out using regular MCMC methods. We significantly expand the use of perfect sampling algorithms within this problem using ideas developed in this thesis.
As a second application, we depict the use of these methods for the interacting fermion problem. We employ perfect sampling for computing self energy through Feynman diagrams using Monte Carlo integration techniques.
We conclude by stating two open questions for future research.
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defense talk: slides