SEQUENTIAL MONTE CARLO METHODS FOR PROBLEMS ON FINITE STATE-SPACES

Abstract

In recent years, sequential Monte Carlo (SMC) methods are amongst the most widely used computational techniques. In this thesis, we make efforts on the development and applications of SMC methods for problems on finite state-spaces. Firstly, we consider SMC methods for approximating the likelihood of the network model. We prove that the relative variance of SMC estimates grow polynomially in the size of networks. Then we develop particle Markov chain Monte Carlo (PMCMC) algorithms to perform Bayesian inference. Secondly, we propose an adaptive SMC algorithm to calculate the permanent of binary {0,1} matrices. We establish its convergence and show that one needs a computational cost O(n^4log^4(n)) to achieve an arbitrarily small relative variance. Thirdly, we present SMC algorithms to approximate a-permanents of positive a and matrices with non-negative entries. We also provide statistical estimations of densities of boson point processes and MCMC methods to fit the associated model to data.