Design-marginal calibration of Gaussian process predictive distributions: Bayesian and conformal approaches
This work addresses calibration issues in Gaussian process predictions for interpolation problems, which is incremental as it builds on existing conformal and Bayesian approaches.
The paper tackles the calibration of Gaussian process predictive distributions in interpolation settings by introducing two methods, cps-gp and bcr-gp, which achieve finite-sample marginal calibration and control dispersion and tail behavior, as demonstrated through numerical experiments on benchmark functions with metrics like coverage and scaled continuous ranked probability score.
We study the calibration of Gaussian process (GP) predictive distributions in the interpolation setting from a design-marginal perspective. Conditioning on the data and averaging over a design measure μ, we formalize μ-coverage for central intervals and μ-probabilistic calibration through randomized probability integral transforms. We introduce two methods. cps-gp adapts conformal predictive systems to GP interpolation using standardized leave-one-out residuals, yielding stepwise predictive distributions with finite-sample marginal calibration. bcr-gp retains the GP posterior mean and replaces the Gaussian residual by a generalized normal model fitted to cross-validated standardized residuals. A Bayesian selection rule-based either on a posterior upper quantile of the variance for conservative prediction or on a cross-posterior Kolmogorov-Smirnov criterion for probabilistic calibration-controls dispersion and tail behavior while producing smooth predictive distributions suitable for sequential design. Numerical experiments on benchmark functions compare cps-gp, bcr-gp, Jackknife+ for GPs, and the full conformal Gaussian process, using calibration metrics (coverage, Kolmogorov-Smirnov, integral absolute error) and accuracy or sharpness through the scaled continuous ranked probability score.