Monte Carlo Methods for Discrete Problems

Abstract

This thesis applies Monte Carlo methods to discrete estimation problems. Applications of Markov Chain Monte Carlo (MCMC) to such problems are difficult, as the state-space is a set of discrete points. Markov kernels are hard to design on such spaces. Applications of sequential Monte Carlo are difficult, because most descriptions of such problems as a sequence of random variables tend to be artificial, and therefore unhelpful. This thesis is divided into five chapters, presented in the order in which they were written. In the first chapter, we describe the application of conditional Monte Carlo to a simple random graph model. The aim is to estimate the probability that the random graph model is connected. This conditional Monte Carlo method is then extended to the continuous analog of the random graph model.In the second chapter, sequential Monte Carlo is applied to the same random graph model. With some creativity, it is possible to describe the problem using a sequence of random variables. This sequence has an independence property that can be exploited for more efficient estimation. It is also useful for importance sampling. This section shows that classical sequential Monte Carlo ideas can, with some difficulty, be applied to discrete estimation problems.In the third chapter, we describe the incorporation of without-replacement sampling into sequential Monte Carlo algorithms. We take a sampling-design approach, and demonstrate that recent work in the field of sequential Monte Carlo can be viewed as an application of multi-stage sampling and the Horvitz-Thompson estimator. This approach results in a simple interpretation of the optimality result of Fearnhead and Clifford (2003), and a novel variance estimator based on multi-stage cluster sampling.In the fourth chapter, we describe the application of the without-replacement sampling algorithms from the third chapter to the counting of binary contingency tables. The importance sampling density of Y. Chen, Diaconis, Holmes, and Liu (2005) has generally excellent performance for this problem. Bez´akov´a, Sinclair, Stefankoviˇc, and Vigoda (2012) showed ˇ that there is a class of pathological problems for which the importance sampling density performs arbitrarily badly. We show that the addition of without-replacement sampling to the importance sampling density results in significantly improved performance for these pathological problems.In the fifth chapter, we describe the application of MCMC methods to the problem of counting the number of decomposable graphs with a given number of vertices and edges. Counting these collections of graphs is important in applying the size-based prior in Bayesian graphical modeling. We describe a faster mixing version of the MCMC algorithm of Armstrong, Carter, Wong, and Kohn (2009), based on the vertex incremental algorithm of Berry, Heggernes, and Villanger (2006).