Nonparametric Bayesian Calibration of Computer Models

TL;DR

This work develops a nonparametric Bayesian framework for calibrating computer models by treating parameter distributions under an empirical stochastic inverse problem (eSIP). It proves the existence of a unique posterior with an explicit density formula via disintegration of measures, establishes almost-everywhere continuity, and shows a maximum-entropy property for the uniform prior. A practical estimator based on reweighting random samples, plus an alternative accept-reject method, is analyzed and shown to be asymptotically unbiased and strongly consistent, with convergence governed by the accuracy of the data-driven density estimation. The approach is demonstrated through examples including an exponential-decay SIP and a falling-ball experiment, illustrating contour-induced posterior structure and practical considerations when data are limited. These results offer a robust, principled pathway for nonparametric calibration of complex computer models with quantified uncertainty and forecasting capabilities.

Abstract

Calibration of computer models is a key step in making inferences, predictions, and decisions for complex science and engineering systems. We formulate and analyze a nonparametric Bayesian methodology for computer model calibration. This paper presents a number of key results including; establishment of a unique nonparametric Bayesian posterior corresponding to a chosen prior with an explicit formula for the corresponding conditional density; a maximum entropy property of the posterior corresponding to the uniform prior; the almost everywhere continuity of the density of the nonparametric posterior; and a comprehensive convergence and asymptotic analysis of an estimator based on a form of importance sampling. We illustrate the problem and results using several examples, including a simple experiment.

Figures (15)

• Figure 1: We show $32$ generalized contours for the map $Q$ for the exponential decay model in Example \ref{['EX:expdecay']} corresponding to $T=.5$ (left) and $T=2$ (center). Each contour curve corresponds to a different value of $q$. On the right, we plot the arclength of the generalized contours $Q^{-1}(q)$ against $q$ for $T=2$.
• Figure 2: Left: empirical density for $P^{\mathrm{d}}_\Lambda$ for Example \ref{['EX:expdecay']} computed using $200,000$ points. Center: empirical density for $P_{\mathcal{D}}$ at $T=.5$. Right: empirical density for $P_{\mathcal{D}}$ at $T=2$.
• Figure 3: Left and center: Heatmaps of estimators of the posterior for the exponential decay model using the uniform prior for $T=.5$ (left) and $T=2$ (center). Right: Overlays of the two heatmaps. Event A is relatively low probability for both posteriors, Event B is relatively high probability for the posterior for time $.5$ but relatively low probability for the posterior for time $2$, while Event C is the reverse. Event D is relatively high probability for both posteriors.
• Figure 4: The posterior for the exponential decay model using the uniform prior for $T=2$ together with the sample points (red) from the data generating distribution used to create the simulated output data.
• Figure 5: Left: we illustrate disintegration when the conditional probabilities are written in terms of densities. The probability on $\Lambda$ is represented by a density shaded in green and the probability on $\mathcal{D}$ is represented by a density shaded in blue. The disintegrated density on $Q^{-1}(q)$ is indicated with a darker shade of green. Right: we illustrate the estimation by random sampling. The observed data is indicated by dark red points on $\mathcal{D}$ and the associated empirical estimator is shown in blue relative to the partition $\{I_i\}$. We show the samples $\{\lambda_i\}$ in $\Lambda$ as points and $Q^{-1}(I_i)$ is shaded in gold. Figure is adapted from kimspaper.

Theorems & Definitions (43)

• Definition 1.1: empirical Stochastic Inverse Problem (eSIP)
• Definition 1.2: Stochastic Inverse Problem (SIP)
• Example 1
• Remark 1.3
• Remark 1.4
• Remark 2.2
• Theorem 2.3
• Example 2
• Theorem 2.4
• Example 3