Advanced Monte Carlo Methods for Complex Statistical Models

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Bayesian statistics has formed a unified framework for many statistical models, focusing on computing posterior distributions. The posterior computation is typically carried out by running Markov chain Monte Carlo (MCMC) algorithms. However, it is challenging to design efficient MCMC proposals for sophisticated models that require computing high-dimensional posteriors with multiple modes. Sequential Monte Carlo (SMC) methods are an alternative class of sampling algorithms that have become very popular in recent years. The objective of my research program is to develop novel SMC methodologies and combine them with MCMC for inferences of realistically complex statistical models that are traditionally approached by poorly performed MCMC methods. I will explore original applications with these methods, specifically focusing on two application areas: phylogenetics and evaluation of diagnostic tests. In phylogenetics, I propose a general evolutionary model that can incorporate various types of mutation events to study how species evolve through time and how closely species are related to each other. I also propose a novel SMC method that can efficiently sample phylogenetic trees and the mutation events. My proposed research can improve the efficiency of large-scale genetic data analysis. In the other application area - evaluation of diagnostic tests, I propose a general multivariate probit latent class model with mixed effects and a combination of SMC and MCMC for implementation in order to improve disease diagnosis and to evaluate new diagnostic tests when no gold standard diagnostic tests are available. In summary, my research program can develop efficient computing methods for inferences of sophisticated statistical modeling, which replace many heuristic or conventional methods and solve significant problems in phylogenetics and in evaluating medical diagnostic tests. My proposed methodologies can also be applied to other scientific fields such as computational biology.