# Can we trust Bayesian uncertainty quantification from Gaussian process priors with squared exponential covariance kernel?

## TL;DR

This paper critically examines the reliability of Bayesian credible sets derived from Gaussian process priors with squared exponential kernels, revealing overconfidence issues and proposing modifications for improved uncertainty quantification.

## Contribution

It demonstrates that standard hyper-parameter estimation leads to overconfident credible sets and proposes logarithmic adjustments to achieve reliable, adaptive uncertainty quantification.

## Key findings

- Credible sets are overconfident with maximum marginal likelihood hyper-parameter estimation.
- Logarithmic inflation of credible sets improves coverage accuracy.
- Results extend beyond Gaussian white noise to regression and classification models.

## Abstract

We investigate the frequentist coverage properties of credible sets resulting in from Gaussian process priors with squared exponential covariance kernel. First we show that by selecting the scaling hyper-parameter using the maximum marginal likelihood estimator in the (slightly modified) squared exponential covariance kernel the corresponding credible sets will provide overconfident, misleading uncertainty statements for a large, representative subclass of the functional parameters in context of the Gaussian white noise model. Then we show that by either blowing up the credible sets with a logarithmic factor or modifying the maximum marginal likelihood estimator with a logarithmic term one can get reliable uncertainty statement and adaptive size of the credible sets under some additional restriction. Finally we demonstrate on a numerical study that the derived negative and positive results extend beyond the Gaussian white noise model to the nonparametric regression and classification models for small sample sizes as well.