(327a) Hierarchical Bayesian Optimization of Gray-Box Models

In chemical engineering, computer models based on physicochemical knowledge are frequently used to predict behavior in areas where data is unavailable. However, if the underlying model structure is misspecified, the predictions may differ significantly from reality. To address this issue, statisticians Kennedy and O'Hagan developed a Bayesian framework for model calibration that utilizes an additive semi-parametric (i.e., gray-box) structure^[1]. In this approach, the unknown model is the sum of a misspecified computer model and a nonparametric bias term that corrects for systematic discrepancy between the data and the model. Furthermore, the nonparametric bias term follows a Gaussian process (GP) prior, and its posterior is inferred jointly with the unknown parameters of the computer model. While technically well-defined in Bayesian inference, joint estimation generally results in an unidentifiable model, thereby fragmenting meaningful uncertainty quantification^[2,3]. Although identifiable frameworks have been developed^[3,4], this issue is often overlooked by practitioners. Additionally, there is a gap in using identifiable gray-box models for applications such as adaptive design of experiments.

Recently, Bayesian optimization (BO) has emerged as an adaptive sampling strategy for optimizing black-box functions^[5]. In BO, the unknown objective function is approximated using a GP surrogate model, which enables the construction of an inexpensive acquisition function. The acquisition function samples the design space with the fitted GP by balancing the tradeoff between exploration (sampling where uncertainty is high) and exploitation (sampling where the objective mean is high). The expected improvement acquisition function^[6] is commonly used due to its ease of implementation^[7]. In this work, we demonstrate how to perform BO of identifiable gray-box models to enable adaptive design of experiments under uncertainty. Using the identifiable model calibration framework introduced by Wong et al.^[3], we first estimate the misspecified model parameters using frequentist methods to construct the mean of the GP. We then use a hierarchical adaptation of the expected improvement acquisition function^[8] to recommend the most informative experiments, while incorporating prior beliefs on the bias term. Numerical experiments show that this approach improves upon BO with ordinary kriging models^[9] for a reaction engineering case study.

[1] Kennedy MC, O'Hagan A. Bayesian calibration of computer models. Journal of the Royal Statistical Society Series B: Statistical Methodology. 2001;63(3):425-64.

[2] Brynjarsdóttir J, OʼHagan A. Learning About Physical Parameters: The Importance of Model Discrepancy. Inverse Problems. 2014;30(11):114007.

[3] Wong RKW, Storlie CB, Lee TCM. A Frequentist Approach to Computer Model Calibration. Journal of the Royal Statistical Society Series B: Statistical Methodology. 2017;79(2):635-48.

[4] Tuo R, Jeff Wu CF. A Theoretical Framework for Calibration in Computer Models: Parameterization, Estimation and Convergence Properties. SIAM/ASA Journal on Uncertainty Quantification. 2016;4(1):767-95.

[5] Greenhill S, Rana S, Gupta S, Vellanki P, Venkatesh S. Bayesian Optimization for Adaptive Experimental Design: A Review. IEEE Access. 2020;8:13937-48.

[6] Jones DR, Schonlau M, Welch WJ. Efficient Global Optimization of Expensive Black-Box Functions. Journal of Global Optimization. 1998;13(4):455–92.

[7] Wang K, Dowling AW. Bayesian Optimization for Chemical Products and Functional Materials. Current Opinion in Chemical Engineering. 2022;36:100728.

[8] Chen Z, Mak S, Wu CFJ. A Hierarchical Expected Improvement Method for Bayesian Optimization. arXiv:191107285. 2022.

[9] Olea RA. Geostatistics for Engineers and Earth Scientists. Springer Science & Business Media; 2012.