Uncertainty quantification under recommended IMQ weight hyperparameters for RCGP

Determine whether the recommended hyperparameter choices β = σ/√2 and c(x) = Q_N(1−ε), used in the inverse multi-quadric weight function w_IMQ(x, y) = β (1 + ((y − m(x))^2)/(c(x)^2))^(−1/2) for robust and conjugate Gaussian process regression (RCGP) with centering at the prior mean m(x), yield sensible uncertainty quantification for the posterior and predictive distributions.

Background

Robust and conjugate Gaussian processes (RCGPs) achieve robustness via a generalised Bayes loss with an inverse multi-quadric (IMQ) weight function. Prior work recommended centering the IMQ at the prior mean m(x), fixing β = σ/√2, and choosing the shrinking parameter c based on the assumed outlier proportion ε through c(x) = Q_N(1 − ε). These choices strongly affect predictive variance and thus uncertainty quantification.

The paper explicitly notes that it is not clear whether these recommended settings provide sensible uncertainty quantification and that the prior work did not paper this question. While the present paper proposes an adaptive, sequential alternative in the spatio-temporal setting, the cited uncertainty quantification question for the original RCGP hyperparameter recommendations remains explicitly identified as unresolved in the literature.