Marginally-calibrated deep distributional regression

TL;DR

This paper introduces a scalable deep distributional regression method that produces marginally calibrated predictive distributions, improving uncertainty quantification in neural network regression especially for likelihood-free inference applications.

Contribution

It proposes a novel approach combining a Gaussian prior on final layer weights with a non-parametric marginal distribution to achieve marginal calibration in deep neural network regression.

Findings

Improved uncertainty quantification in ecological time series applications.

Scalable method suitable for likelihood-free inference.

Avoids manual specification of summary statistics.

Abstract

Deep neural network (DNN) regression models are widely used in applications requiring state-of-the-art predictive accuracy. However, until recently there has been little work on accurate uncertainty quantification for predictions from such models. We add to this literature by outlining an approach to constructing predictive distributions that are `marginally calibrated'. This is where the long run average of the predictive distributions of the response variable matches the observed empirical margin. Our approach considers a DNN regression with a conditionally Gaussian prior for the final layer weights, from which an implicit copula process on the feature space is extracted. This copula process is combined with a non-parametrically estimated marginal distribution for the response. The end result is a scalable distributional DNN regression method with marginally calibrated predictions, and our work complements existing methods for probability calibration. The approach is first illustrated using two applications of dense layer feed-forward neural networks. However, our main motivating applications are in likelihood-free inference, where distributional deep regression is used to estimate marginal posterior distributions. In two complex ecological time series examples we employ the implicit copulas of convolutional networks, and show that marginal calibration results in improved uncertainty quantification. Our approach also avoids the need for manual specification of summary statistics, a requirement that is burdensome for users and typical of competing likelihood-free inference methods.