Markov Chain Monte Carlo using couplings toward scalable statistical inference

This project develops a statistical toolbox for the approximation of probability distributions that commonly arise in data analysis. The problem of approximating probabilities arise in many tasks of data science: in Bayesian statistics and its many variants, in classical hypothesis testing with p-values, in likelihood-based methods when the model involves latent variables, in models with intractable likelihoods, in the construction of knockoffs for principled variable selection, for example. State-of-the-art methods for such approximations include Markov chain Monte Carlo (MCMC), where a Markov chain is generated in such a way that it converges to the probability of interest as the length of the sequence goes to infinity. This stands at odds with modern developments in computing hardware, which provide an increasing number of parallel processors, but where each process has a stagnating clock speed. Methods that are amenable to parallel computing must emerge to help scientists in all fields to make the most of their data. The project builds upon a framework called Unbiased Markov chain Monte Carlo (UMCMC), in which accuracy improves arbitrarily with the number of parallel runs. Each run involves the generation of coupled Markov chains for a random time horizon.Part 1 develops UMCMC to realize its potential as a comprehensive basis for probabilistic computation on modern hardware. The project includes theoretical analyses of cost and measures of efficiency, and methodological innovations towards adaptive, efficient, robust and convenient computation.Part 2 contributes to the applicability of UMCMC, by conceptualizing the design of coupled Markov transitions, and considering a number of challenging settings: distributions supported on submanifolds and their application in economics, distributions on graphs and their applications in the fight against malaria, and Bayesian nonparametric models for cell type deconvolution from transcriptomics data.