Standard Bayesian inference schemes are infeasible for inverse problems with computationally expensive forward models. A common solution is to replace the model with a cheaper surrogate. To avoid overconfident conclusions, it is essential to acknowledge the surrogate approximation by propagating its uncertainty. At present, a variety of distinct uncertainty propagation methods have been suggested, with little understanding of how they vary. To fill this gap, we propose a mixture distribution termed the expected posterior (EP) as a general baseline for uncertainty-aware posterior approximation, justified by decision theoretic and modular Bayesian inference arguments. We then investigate the expected unnormalized posterior (EUP), a popular heuristic alternative, analyzing when it may deviate from the EP baseline. Our results show that this heuristic can break down when the surrogate uncertainty is highly non-uniform over the design space, as can be the case when the log-likelihood is emulated by a Gaussian process. Finally, we present the random kernel preconditioned Crank-Nicolson (RKpCN) algorithm, an approximate Markov chain Monte Carlo scheme that provides practical EP approximation in the challenging setting involving infinite-dimensional Gaussian process surrogates.

翻译：标准的贝叶斯推断方法对于前向模型计算成本高昂的反问题是不可行的。常见的解决方案是用更经济的代理模型替代原始模型。为避免得出过于自信的结论，必须通过传播代理模型的不确定性来体现其近似误差。目前已有多种不同的不确定性传播方法被提出，但对其差异性的理解尚不充分。为填补这一空白，我们提出一种称为期望后验的混合分布，作为不确定性感知后验近似的一般基准，其合理性可通过决策理论和模块化贝叶斯推断论证得到支撑。随后我们研究了期望未归一化后验这一常用的启发式替代方法，分析了其可能偏离期望后验基准的情况。研究结果表明，当代理模型不确定性在设计空间内高度非均匀时（例如使用高斯过程模拟对数似然函数的情形），该启发式方法可能失效。最后，我们提出了随机核预条件Crank-Nicolson算法，这是一种近似的马尔可夫链蒙特卡罗方案，能够在涉及无限维高斯过程代理模型的复杂场景中实现实用的期望后验近似。