Objective Bayesian Analysis of a Cokriging Model for Hierarchical Multifidelity Codes

TL;DR

This paper introduces new closed-form formulas for predictive means and variances in autoregressive cokriging models, and develops objective Bayesian priors that ensure proper posteriors, improving computational efficiency and reliability in multifidelity modeling.

Contribution

It derives closed-form predictive formulas and proposes objective Bayesian priors that guarantee proper posteriors for autoregressive cokriging models.

Findings

Closed-form formulas depend only on correlation parameters.

Common priors like constant and inverse correlation are improper.

Objective priors such as reference and Jeffreys yield proper posteriors.

Abstract

Autoregressive cokriging models have been widely used to emulate multiple computer models with different levels of fidelity. The dependence structures are modeled via Gaussian processes at each level of fidelity, where covariance structures are often parameterized up to a few parameters. The predictive distributions typically require intensive Monte Carlo approximations in previous works. This article derives new closed-form formulas to compute the means and variances of predictive distributions in autoregressive cokriging models that only depend on correlation parameters. For parameter estimation, we consider objective Bayesian analysis of such autoregressive cokriging models. We show that common choices of prior distributions, such as the constant prior and inverse correlation prior, typically lead to improper posteriors. We also develop several objective priors such as the independent reference prior and the independent Jeffreys prior that are shown to yield proper posterior distributions. This development is illustrated with a borehole function in an eight-dimensional input space and applied to an engineering application in a six-dimensional input space.