Development and Analysis of Markov Chains on Manifolds - Karimian Sichani, Elnaz (Université d'Ottawa | University of Ottawa, 2024)
Random sampling from a probability distribution is a fundamental computational task that is widely applied in various disciplines. It has applications in statistics, machine learning, probability, and other areas that involve stochastic modeling. MCMC is a way to (approximately) sample. Given a target distribution, the MCMC method typically consists of two phases. First, construct a Markov chain whose stationary distribution is either the target distribution or a close approximation of it. Next, simulate the chain for a sufficient number of steps to ensure that it has mixed and produced an approximate sample from the target distribution. However, for an MCMC algorithm to be efficient, the Markov chain must quickly reach its steady state. This is a major concern in computational science. Therefore, understanding the time it takes for the Markov chain to reach its equilibrium distribution, known as the ``mixing time,'' is essential. This thesis explores the most commonly employed techniques for bounding the mixing time of Markov chains, such as the spectral and profile methods, geometric bounds, comparison methods, coupling, and decomposition techniques. We present some new results on convergence bounds of continuous-state Markov chains and prove analogous relationships between different notions of distance from stationarity in discrete to continuous state spaces. We then investigate the sharpness of the spectral profile bound on the mixing time of continuous-state Markov chains and find that it is precise up to a factor of $\log \log$ of the initial density. This finding has applications in the comparison of Markov chains. Finally, in this thesis, we discuss the application of simulation models and model fitting in synthetic data generation (SDG). We propose a generative model for producing synthetic longitudinal data that integrates efficient simulation techniques, including MCMC, sequential trees, and copula, to sample a latent factor matrix within the framework of the state-of-the-art generalized canonical polyadic (GCP) tensor decomposition.
Keywords: Markov chains; Mixing time; Simulation