Advances in Sequential Monte Carlo Methods and Site-Dependent DNA Evolution Models

Abstract

This thesis collects results regarding finite sample error bounds for sequential Monte Carlo (SMC) methods and models of DNA evolution under site dependence. We firstprovide novel finite sample error bounds for SMC estimators in settings where the target probability distribution is multimodal. Next, we turn to the problem of computing marginal sequence likelihoods under models of DNA evolution that incorporate site dependence. In particular, we propose a randomized approximation scheme for approximating marginal sequence likelihoods using importance sampling ideas and derive explicit upper and lower bounds on the sample size required to obtain an accurate estimate. In addition, we extend the importance sampling approach to an SMC setting, provide finite sample error bounds for the corresponding estimator, and establish explicit conditions under which SMC approximates marginal sequence likelihoods efficiently in non-trivial settings. Finally, we develop a stochastic model of B cell affinity maturation to explore how site dependence in DNA evolution may be leveraged to aid in the induction of HIV broadly neutralizing antibodies using sequential vaccine design strategies.