We investigate a computer model calibration technique inspired by the well-known Bayesian framework of Kennedy and O'Hagan (KOH). We tackle the full Bayesian formulation where model parameter and model discrepancy hyperparameters are estimated jointly and reduce the problem dimensionality by introducing a functional relationship that we call the full maximum a posteriori (FMP) method. This method also eliminates the need for a true value of model parameters that caused identifiability issues in the KOH formulation. When the joint posterior is approximated as a mixture of Gaussians, the FMP calibration is proven to avoid some pitfalls of the KOH calibration, namely missing some probability regions and underestimating the posterior variance. We then illustrate two numerical examples where both model error and measurement uncertainty are estimated together. Using the solution to the full Bayesian problem as a reference, we show that the FMP results are accurate and robust, and avoid the need for high-dimensional Markov chains for sampling.

中文翻译：

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具有自适应模型差异的贝叶斯校准

我们研究了一种受肯尼迪和奥哈根 (KOH) 著名的贝叶斯框架启发的计算机模型校准技术。我们解决了完整的贝叶斯公式，其中模型参数和模型差异超参数被联合估计，并通过引入我们称为完全最大后验（FMP）方法的函数关系来降低问题维度。该方法还消除了对模型参数真实值的需求，而模型参数的真实值会导致 KOH 配方中的可识别性问题。当联合后验近似为高斯混合时，FMP 校准被证明可以避免 KOH 校准的一些缺陷，即丢失一些概率区域和低估后验方差。然后，我们说明两个数值示例，其中模型误差和测量不确定度一起估计。使用完整贝叶斯问题的解决方案作为参考，我们表明 FMP 结果准确且稳健，并且避免了对高维马尔可夫链进行采样的需要。