Bayesian inference is a popular approach to calibrating uncertainties, but it can underpredict such uncertainties when model misspecification is present, impacting its reliability to inform decision making. Recently, the statistics and machine learning communities have developed prediction-oriented inference approaches that provide better calibrated uncertainties and adapt to the level of misspecification present. However, these approaches have yet to be demonstrated in the context of complex scientific applications where phenomena of interest are governed by physics-based models. Such settings often involve single realizations of high-dimensional spatio-temporal data and nonlinear, computationally expensive parameter-to-observable maps. This work investigates variational prediction-oriented inference in problems exhibiting these relevant features; namely, we consider a polynomial model and a contaminant transport problem governed by advection-diffusion equations. The prediction-oriented loss is formulated as the log-predictive probability of the calibration data. We study the effects of increasing misspecification and noise, and we assess approximations of the predictive density using Monte Carlo sampling and component-wise kernel density estimation. A novel aspect of this work is applying prediction-oriented inference to the calibration of model-form uncertainty (MFU) representations, which are embedded physics-based modifications to the governing equations that aim to reduce (but rarely eliminate) model misspecification. The computational results demonstrate that prediction-oriented frameworks can provide better uncertainty characterizations in comparison to standard inference while also being amenable to the calibration of MFU representations.

翻译：贝叶斯推断是校准不确定性的常用方法，但当存在模型误设时，该方法可能低估不确定性，从而影响其为决策提供信息的可靠性。近年来，统计学与机器学习领域提出了面向预测的推断方法，这些方法能够提供校准更优的不确定性估计，并能自适应于存在的误设程度。然而，这些方法尚未在复杂科学应用场景中得到验证，此类场景中感兴趣的现象通常受基于物理的模型所支配。这类问题往往涉及高维时空数据的单次实现，以及非线性、计算代价高昂的参数-观测映射关系。本研究探讨了在具有上述相关特征的问题中应用变分预测导向推断的方法；具体而言，我们考察了一个多项式模型和一个由对流-扩散方程控制的污染物输运问题。预测导向损失函数被构建为校准数据的对数预测概率。我们研究了误设程度和噪声增加的影响，并评估了使用蒙特卡洛采样和分量核密度估计对预测密度的近似效果。本工作的一个创新点在于将预测导向推断应用于模型形式不确定性表征的校准，这些表征是通过对控制方程进行基于物理的修正来实现的，旨在减少（但很少能完全消除）模型误设。计算结果表明，与标准推断方法相比，预测导向框架能够提供更优的不确定性表征，同时也适用于模型形式不确定性表征的校准。