[Paper Review] Jointly Robust Prior for Gaussian Stochastic Process in Emulation, Calibration and Variable Selection

Mengyang Gu|arXiv (Cornell University)|Apr 25, 2018

Probabilistic and Robust Engineering DesignDecision Sciences24 references4 citations

TL;DR

This paper introduces the jointly robust (JR) prior for Gaussian stochastic processes (GaSP) in uncertainty quantification, enabling robust emulation, calibration, and variable selection. The JR prior ensures stable posterior mode estimation, identifies inert inputs automatically, and outperforms the reference prior in predictive accuracy and credible interval coverage while being computationally more efficient.

ABSTRACT

Gaussian stochastic process (GaSP) has been widely used in two fundamental problems in uncertainty quantification, namely the emulation and calibration of mathematical models. Some objective priors, such as the reference prior, are studied in the context of emulating (approximating) computationally expensive mathematical models. In this work, we introduce a new class of priors, called the jointly robust prior, for both the emulation and calibration. This prior is designed to maintain various advantages from the reference prior. In emulation, the jointly robust prior has an appropriate tail decay rate as the reference prior, and is computationally simpler than the reference prior in parameter estimation. Moreover, the marginal posterior mode estimation with the jointly robust prior can separate the influential and inert inputs in mathematical models, while the reference prior does not have this property. We establish the posterior propriety for a large class of priors in calibration, including the reference prior and jointly robust prior in general scenarios, but the jointly robust prior is preferred because the calibrated mathematical model typically predicts the reality well. The jointly robust prior is used as the default prior in two new R packages, called "RobustGaSP" and "RobustCalibration", available on CRAN for emulation and calibration, respectively.

Motivation & Objective

• To address instability in GaSP parameter estimation during emulation and calibration, especially with small sample sizes.
• To develop a prior that maintains robustness in posterior mode estimation while enabling automatic identification of inert inputs.
• To improve predictive accuracy and credible interval coverage in calibration by avoiding overconfidence in uncertainty quantification.
• To provide a computationally efficient alternative to the reference prior with equivalent or better performance.
• To establish posterior propriety for a broad class of priors in calibration, including the JR prior.

Proposed method

• Proposes the jointly robust (JR) prior as a new class of objective priors for GaSP in emulation and calibration.
• Uses a product correlation structure with individual range parameters γₗ and applies a specific parameterization to ensure robustness.
• Employs marginal posterior mode estimation under the JR prior to simultaneously achieve stable estimation and variable selection.
• Derives closed-form derivatives for efficient optimization, enhancing computational speed over the reference prior.
• Applies the JR prior to both emulator and discrepancy function modeling in calibration, improving parameter identifiability.
• Validates the method using simulation examples and real-world case studies, comparing with reference prior and MLE.

Experimental results

Research questions

• RQ1 Can a new objective prior be designed to ensure robust posterior mode estimation in GaSP-based emulation and calibration?
• RQ2 Does the JR prior enable automatic identification of inert inputs without additional computational cost?
• RQ3 How does the JR prior compare to the reference prior in predictive accuracy and uncertainty calibration?
• RQ4 Can the JR prior improve parameter identifiability in calibration under small sample sizes?
• RQ5 What is the computational efficiency of the JR prior relative to the reference prior and MLE?

Key findings

• The JR prior achieves comparable or better predictive performance than the reference prior, with NRMSE of 0.21 in the calibrated model and discrepancy function, compared to 0.28 for the reference prior.
• The JR prior produces 95% posterior credible intervals that cover approximately 98% of held-out data points, indicating well-calibrated uncertainty quantification.
• The reference prior exhibits overconfidence, with only 88% coverage of the 95% credible intervals, due to inflated posterior variance in mean parameters.
• The JR prior is computationally faster than the reference prior due to closed-form derivatives, enabling efficient optimization.
• The marginal posterior mode under the JR prior successfully identifies inert inputs, a capability not consistently provided by the reference prior.
• The JR prior ensures posterior propriety in calibration under general scenarios, supporting valid Bayesian inference.