Beyond Marginal Uncertainty: How Accurately can Bayesian Regression Models Estimate Posterior Predictive Correlations?

TL;DR

This paper evaluates how accurately Bayesian regression models can estimate predictive correlations, proposing efficient metrics and a benchmark task to improve uncertainty estimation in deep learning.

Contribution

It introduces two new metrics, meta-correlations and cross-normalized likelihoods, for directly assessing predictive correlation estimates in Bayesian models.

Findings

Bayesian models vary in correlation estimation accuracy

TAL outperforms traditional active learning in utilizing uncertainty

Proposed metrics align well with TAL performance

Abstract

While uncertainty estimation is a well-studied topic in deep learning, most such work focuses on marginal uncertainty estimates, i.e. the predictive mean and variance at individual input locations. But it is often more useful to estimate predictive correlations between the function values at different input locations. In this paper, we consider the problem of benchmarking how accurately Bayesian models can estimate predictive correlations. We first consider a downstream task which depends on posterior predictive correlations: transductive active learning (TAL). We find that TAL makes better use of models' uncertainty estimates than ordinary active learning, and recommend this as a benchmark for evaluating Bayesian models. Since TAL is too expensive and indirect to guide development of algorithms, we introduce two metrics which more directly evaluate the predictive correlations and which can be computed efficiently: meta-correlations (i.e. the correlations between the models correlation estimates and the true values), and cross-normalized likelihoods (XLL). We validate these metrics by demonstrating their consistency with TAL performance and obtain insights about the relative performance of current Bayesian neural net and Gaussian process models.