Abstract: Probabilistic models describe complex data generating processes and have been applied to a broad range of fields, such as epidemiology, pharmacology, and astrophysics. Inference for probabilistic models poses significant computational challenges, particularly as models grow in complexity and datasets increase in size. Modern hardware, with its parallelization capabilities, offers new opportunities to accelerate statistical inference. However, many traditional methods are not inherently designed for parallel computation. Markov chain Monte Carlo (MCMC), for instance, typically relies on a few long-running chains. I propose an alternative approach: running hundreds or thousands of shorter chains in parallel. To support this paradigm, I introduce the nested “R-hat,” a novel convergence diagnostic tailored for the many-short-chains regime, paving the way for faster and more automated MCMC.

Next I examine variational inference (VI). VI already leverages the parallelization capacities of modern hardware, however it lacks the theoretical guarantees of MCMC and other statistical methods. I present two key theoretical results: (1) a positive result demonstrating that VI can effectively learn symmetries even under misspecified approximations, and (2) a negative result revealing that factorized (or mean-field) approximations lead to an impossibility theorem, preventing the simultaneous estimation of multiple measures of uncertainty . These findings provide practical guidance for selecting VI’s objective function and approximation family, offering a path toward robust and scalable inference.

https://charlesm93.github.io/