Abstract

Variational inference (vi) is widely used as an efficient alternative to Markov chain Monte Carlo. It posits a family of approximating distributions $q$ and finds the closest member to the exact posterior $p$ . Closeness is usually measured via a divergence $D(q||p)$ from $q$ to $p$ . While successful, this approach also has problems. Notably, it typically leads to underestimation of the posterior variance. In this paper we propose chivi, a black-box variational inference algorithm that minimizes $D_{\chi}(p||q)$ , the $\chi$ -divergence from $p$ to $q$ . chivi minimizes an upper bound of the model evidence, which we term the $\chi$ upper bound (cubo). Minimizing the cubo leads to improved posterior uncertainty, and it can also be used with the classical vi lower bound (elbo) to provide a sandwich estimate of the model evidence. We study chivi on three models: probit regression, Gaussian process classification, and a Cox process model of basketball plays. When compared to expectation propagation and classical vi, chivi produces better error rates and more accurate estimates of posterior variance.