We propose a new Bayesian model calibration formalism as an alternative to the Kennedy O'Hagan (KOH) framework which we term integrated bias with full uncertainty (IBFU). In KOH, calibration parameters are modeled as fixed, but unknown distributions with relatively weak prior constraints, and their posteriors are inferred jointly with an additive discrepancy Gaussian Process (GP). This formulation often provides limited regularization and leads to confounding pathologies when applied to inexact models with sparse, noisy measurements. By contrast, we represent each calibration parameter as the sum of a fixed best estimate value and a parameter correction represented by an independent GP over the input space, equipped with strong shrinkage priors. Any residual discrepancy that cannot be addressed via parameter correction is captured by an additive discrepancy GP operating on the simulator, similar to KOH. We then impose orthogonality constraints to mitigate confounding between the simulator and modeled additive discrepancy and colinearity between model parameters. Imposing strong complexity shrinkage via conservative hyperpriors forces the mean parameter correction to remain flat across the domain, resulting in predictions that essentially converge with the KOH formulation. However, upon relaxing complexity shrinkage, should the data provide evidence that the effective calibration parameter varies across the domain, the mean parameter correction is allowed to become a function of the domain in a controlled, structured manner. In this sense, our approach is more universal: it effectively nests KOH as a special case while extending it to input dependent calibration, and it is more tightly constrained, because it anchors the true values around the best estimates and the shrinkage prior actively regularizes the calibration parameters.

翻译：我们提出一种新的贝叶斯模型校准形式主义，作为Kennedy-O'Hagan（KOH）框架的替代方案，称为全不确定性集成偏差（IBFU）。在KOH中，校准参数被建模为固定但分布未知的先验，且先验约束相对较弱，其后验分布与加性不一致高斯过程（GP）联合推断。该公式在面对稀疏、含噪测量的非精确模型时，往往仅提供有限的正则化，并导致混淆病态问题。相比之下，我们将每个校准参数表示为一个固定最佳估计值与一个参数校正项之和，其中参数校正项由输入空间上的独立GP表示并配备强收缩先验。任何无法通过参数校正解决的残差不一致性，则通过作用于模拟器的加性不一致GP捕获（类似于KOH）。我们进一步施加正交性约束，以缓解模拟器与建模加性不一致之间的混淆以及模型参数间的共线性。通过保守超先验施加强复杂性收缩，迫使均值参数校正项在整个定义域内保持平坦，使得预测结果本质上收敛于KOH公式。然而，若数据提供证据表明有效校准参数随定义域变化，则在放松复杂性收缩后，均值参数校正项被允许以受控的结构化方式成为定义域的函数。从这个意义上说，我们的方法更具通用性：它实质上将KOH作为特例嵌套其中，同时将其扩展至输入依赖型校准，且约束更严格——因为它将真实值锚定在最佳估计值周围，且收缩先验主动对校准参数进行正则化。