An Introduction to Bayesian Inference via Variational Approximations

Markov chain Monte Carlo (MCMC) methods have facilitated an explosion of interest in Bayesian methods. MCMC is an incredibly useful and important tool but can face difficulties when used to estimate complex posteriors or models applied to large data sets. In this paper, we show how a recently developed tool in computer science for fitting Bayesian models, variational approximations, can be used to facilitate the application of Bayesian models to political science data. Variational approximations are often much faster than MCMC for fully Bayesian inference and in some instances facilitate the estimation of models that would be otherwise impossible to estimate. As a deterministic posterior approximation method, variational approximations are guaranteed to converge and convergence is easily assessed. But variational approximations do have some limitations, which we detail below. Therefore, variational approximations are best suited to problems when fully Bayesian inference would otherwise be impossible. Through a series of examples, we demonstrate how variational approximations are useful for a variety of political science research. This includes models to describe legislative voting blocs and statistical models for political texts. The code that implements the models in this paper is available in the supplementary material.