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

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