This paper proposes an effective treatment of hyperparameters in the Bayesian inference of a scalar field from indirect observations. Obtaining the joint posterior distribution of the field and its hyperparameters is challenging. The infinite dimensionality of the field requires a finite parametrization that usually involves hyperparameters to reflect the limited prior knowledge. In the present work, we consider a Karhunen-Lo{\`e}ve (KL) decomposition for the random field and hyperparameters to account for the lack of prior knowledge of its autocovariance function. The hyperparameters must be inferred. To efficiently sample jointly the KL coordinates of the field and the autocovariance hyperparameters, we introduce a change of measure to reformulate the joint posterior distribution into a hierarchical Bayesian form. The likelihood depends only on the field's coordinates in a fixed KL basis, with a prior conditioned on the hyperparameters. We exploit this structure to derive an efficient Markov Chain Monte Carlo (MCMC) sampling scheme based on an adapted Metropolis-Hasting algorithm. We rely on surrogate models (Polynomial Chaos expansions) of the forward model predictions to further accelerate the MCMC sampling. A first application to a transient diffusion problem shows that our method is consistent with other approaches based on a change of coordinates (Sraj et al., 2016). A second application to a seismic traveltime tomography highlights the importance of inferring the hyperparameters.

翻译：本文提出了一种有效处理超参数的方法，用于根据间接观测对标量场进行贝叶斯推断。获取场及其超参数的联合后验分布具有挑战性。场的无限维性要求采用有限参数化方法，通常需要引入超参数来反映有限的先验知识。在本研究中，我们采用Karhunen-Loève (KL)分解对随机场和超参数进行建模，以解决其自协方差函数先验知识缺失的问题。超参数必须通过推断确定。为了高效联合采样场的KL坐标和自协方差超参数，我们提出一种测度变换方法，将联合后验分布重构为分层贝叶斯形式。似然函数仅依赖于固定KL基下的场坐标，其先验分布以超参数为条件。我们利用这一结构，基于改进的Metropolis-Hasting算法推导出高效的马尔可夫链蒙特卡洛（MCMC）采样方案，并采用前向模型预测的代理模型（多项式混沌展开）进一步加速MCMC采样。首次应用于瞬态扩散问题的实例表明，该方法与其他基于坐标变换的方法（Sraj等，2016）结果一致。第二次应用于地震走时层析成像的实例则凸显了推断超参数的重要性。