Scalable computation of predictive probabilities in probit models with Gaussian process priors

TL;DR

This paper introduces scalable, closed-form methods for computing predictive probabilities in probit Gaussian process models, enabling efficient analysis of high-dimensional binary data.

Contribution

It derives closed-form expressions for predictive probabilities and develops scalable Monte Carlo and variational methods for high-dimensional settings.

Findings

Methods scale to high dimensions where previous solutions are impractical

Closed-form expressions improve accuracy and computational efficiency

Empirical studies demonstrate practical applicability

Abstract

Predictive models for binary data are fundamental in various fields, and the growing complexity of modern applications has motivated several flexible specifications for modeling the relationship between the observed predictors and the binary responses. A widely-implemented solution is to express the probability parameter via a probit mapping of a Gaussian process indexed by predictors. However, unlike for continuous settings, there is a lack of closed-form results for predictive distributions in binary models with Gaussian process priors. Markov chain Monte Carlo methods and approximation strategies provide common solutions to this problem, but state-of-the-art algorithms are either computationally intractable or inaccurate in moderate-to-high dimensions. In this article, we aim to cover this gap by deriving closed-form expressions for the predictive probabilities in probit Gaussian processes that rely either on cumulative distribution functions of multivariate Gaussians or on functionals of multivariate truncated normals. To evaluate these quantities we develop novel scalable solutions based on tile-low-rank Monte Carlo methods for computing multivariate Gaussian probabilities, and on mean-field variational approximations of multivariate truncated normals. Closed-form expressions for the marginal likelihood and for the posterior distribution of the Gaussian process are also discussed. As shown in simulated and real-world empirical studies, the proposed methods scale to dimensions where state-of-the-art solutions are impractical.