MCMC methods: graph samplers, invariance tests and epidemic models

Abstract

Markov Chain Monte Carlo (MCMC) techniques are used ubiquitously for simulation-based inference. This thesis provides novel contributions to MCMC methods and their application to graph sampling and epidemic modeling. The first topic considered is that of sampling graphs conditional on a set of prescribed statistics, which is a difficult problem arising naturally in many fields: sociology (Holland and Leinhardt, 1981), psychology (Connor and Simberloff, 1979), categorical data analysis (Agresti, 1992) and finance (Squartini et al., 2018, Gandy and Veraart, 2019) being examples. Bespoke MCMC samplers are proposed for this setting. The second major topic addressed is that of modeling the dynamics of infectious diseases, where MCMC is leveraged as the general inference engine. The first part of this thesis addresses important problems such as the uniform sampling of graphs with given degree sequences, and weighted graphs with given strength sequences. These distributions are frequently used for exact tests on social networks and two-way contingency tables. Another application is quantifying the statistical significance of patterns observed in real networks. This is crucial for understanding whether such patterns indicate the presence of interesting network phenomena, or whether they simply result from less interesting processes, such as nodal-heterogeneity. The MCMC samplers developed in the course of this research are complex, and there is great scope for conceptual, analytic, and implementation errors. This motivates a chapter that develops novel tests for detecting errors in MCMC implementations. The tests introduced are unique in being exact, which allows us to keep the false rejection probability arbitrarily low. Rather than develop bespoke samplers, as in the first part of the thesis, the second part leverages a standard MCMC framework Stan (Stan Development Team, 2018) as the workhorse for fitting state-of-the-art epidemic models. We present a general framework for semi-mechanistic Bayesian modeling of infectious diseases using renewal processes. The term semi-mechanistic relates to statistical estimation within some constrained mechanism. This research was motivated by the ongoing SARS-COV-2 pandemic, and variants of the model have been used in specific analyses of Covid-19. We present epidemia, an R package allowing researchers to leverage the epidemic models. A key goal of this work is to demonstrate that MCMC, and in particular, Stan’s No-U-Turn (Hoffman and Gelman, 2014) sampler, can be routinely employed to fit a large-class of epidemic models. A second goal is to make the models accessible to the general research community, through epidemia.