Variational Inference for Uncertainty Quantification: an Analysis of Trade-offs

TL;DR

This paper analyzes how variational inference's choice of divergence affects the accuracy of uncertainty estimates, especially under non-factorized target distributions, revealing inherent trade-offs and limitations.

Contribution

It provides a theoretical analysis of the trade-offs in variational inference when approximating non-factorized distributions using factorized families, focusing on uncertainty estimation.

Findings

Factorized VI can accurately estimate only one of three uncertainty measures.

The choice of divergence influences which uncertainty measure is better estimated.

Empirical results support the theoretical ordering of divergences based on uncertainty estimation.

Abstract

Given an intractable distribution $p$, the problem of variational inference (VI) is to find the best approximation from some more tractable family $Q$. Commonly, one chooses $Q$ to be a family of factorized distributions (i.e., the mean-field assumption), even though $p$ itself does not factorize. We show that this mismatch can lead to an impossibility theorem: if $p$ does not factorize and furthermore has a non-diagonal covariance matrix, then any factorized approximation $q\!\in\!Q$ can correctly estimate at most one of the following three measures of uncertainty: (i) the marginal variances, (ii) the marginal precisions, or (iii) the generalized variance (which for elliptical distributions is closely related to the entropy). In practice, the best variational approximation in $Q$ is found by minimizing some divergence $D(q,p)$ between distributions, and so we ask: how does the choice of divergence determine which measure of uncertainty, if any, is correctly estimated by VI? We consider the classic Kullback-Leibler divergences, the more general $\alpha$-divergences, and a score-based divergence which compares $\nabla \log p$ and $\nabla \log q$. We thoroughly analyze the case where $p$ is a Gaussian and $q$ is a (factorized) Gaussian. In this setting, we show that all the considered divergences can be ordered based on the estimates of uncertainty they yield as objective functions for VI. Finally, we empirically evaluate the validity of this ordering when the target distribution $p$ is not Gaussian.