Advances in MCMC Methods with Applications to Particle Filtering, DSMC, and Bayesian Networks

Markov Chain Monte Carlo (MCMC) methods are a class of algorithms for sampling from a desired probability distribution. While there exist many algorithms that attempt to be somewhat universal, these algorithms can struggle for tractability in specific applications. The work in this dissertation is focused on improving MCMC methods in three application areas: Particle Filtering, Direct Simulation Monte Carlo, and Bayesian Networks. In particle filtering, the dimension of the target distribution grows as more data is obtained. As such, sequential sampling methods are necessary in order to have an efficient algorithm. In this thesis, we develop a windowed rejection sampling procedure to get more accurate algorithms while still preserving the necessary sequential structure. Direct Simulation Monte Carlo is a Monte Carlo algorithm for simulating rarefied gas flows. In this dissertation, we review the derivation of the Kac master equation model for 1-dimensional flows. From this, we show how the Poisson process can be exploited to construct a more accurate algorithm for simulating the Kac model. We then develop an epsilon-perfect proof of concept algorithm for the limiting velocity distribution as time goes to infinity. Bayesian Networks (BNs) are graphical models used to represent high dimensional probability distributions. There has been a great deal of interest in learning the structure of a BN from observed data. Here, we do so by walking through the space of graphs by modeling the appearance and disappearance of edges as a birth and death process. We give empirical evidence that this novel jump process approach exhibits better mixing properties than the commonly used Metropolis-Hastings algorithm