Mutual Information (MI) is a fundamental measure of dependence between random variables, but its practical application is limited because it is difficult to calculate in many circumstances. Variational methods offer one approach by introducing an approximate distribution to create various bounds on MI, which in turn is an easier optimization problem to solve. In practice, the variational distribution chosen is often a Gaussian, which is convenient but lacks flexibility in modeling complicated distributions. In this paper, we introduce new classes of variational estimators based on Normalizing Flows that extend the previous Gaussian-based variational estimators. Our new estimators maintain many of the same theoretical guarantees while simultaneously enhancing the expressivity of the variational distribution. We experimentally verify that our new methods are effective on large MI problems where discriminative-based estimators, such as MINE and InfoNCE, are fundamentally limited. Furthermore, we compare against a diverse set of benchmarking tests to show that the flow-based estimators often perform as well, if not better, than the discriminative-based counterparts. Finally, we demonstrate how these estimators can be effectively utilized in the Bayesian Optimal Experimental Design setting for online sequential decision making.

Computing mutual information (MI) of random variables lacks a closed-form in nontrivial models. Variational MI approximations are widely used as flexible estimators for this purpose, but computing them typically requires solving a costly nonconvex optimization. We prove that a widely used class of variational MI estimators can be solved via moment matching operations in place of the numerical optimization methods that are typically required. We show that the same moment matching solution yields variational estimates for so-called “implicit” models that lack a closed form likelihood function. Furthermore, we demonstrate that this moment matching solution has multiple orders of magnitude computational speed up compared to the standard optimization based solutions. We show that theoretical results are supported by numerical evaluation in fully parameterized Gaussian mixture models and a generalized linear model with implicit likelihood due to nuisance variables. We also demonstrate on the implicit simulation-based likelihood SIR epidemiology model, where we avoid costly likelihood free inference and observe many orders of magnitude speedup.

Gaussian mixture models (GMMs) are fundamental to machine learning due to their flexibility as approximating densities. However, uncertainty quantification of GMMs remains a challenge as differential entropy lacks a closed form. This paper explores polynomial approximations, specifically Taylor and Legendre, to the GMM entropy from a theoretical and practical perspective. We provide new analysis of a widely used approach due to Huber et al.(2008) and show that the series diverges under simple conditions. Motivated by this divergence we provide a novel Taylor series that is provably convergent to the true entropy of any GMM. We demonstrate a method for selecting a center such that the series converges from below, providing a lower bound on GMM entropy. Furthermore, we demonstrate that orthogonal polynomial series result in more accurate polynomial approximations. Experimental validation supports our theoretical results while showing that our method is comparable in computation to Huber et al. We also show that in application, the use of these polynomial approximations, such as in Nonparametric Variational Inference by Gershamn et al. (2012), rely on the convergence of the methods in computing accurate approximations. This work contributes useful analysis to existing methods while introducing a novel approximation supported by firm theoretical guarantees.