使用传输图估计贝叶斯反演中的系统误差
Estimating systematic errors in Bayesian inversion using transport maps
AI总结：
本文提出结合贝叶斯推断、模型误差校正与传输采样方案的统一框架，利用两种损失训练传输图以估计贝叶斯反演的系统误差，经数值示例验证了方法的有效性。
摘要：
在间接测量中，需通过求解逆问题确定待求参数，通常采用贝叶斯框架。测量过程的精确数值模拟往往计算量巨大，因此必须依赖近似模型，这些替代模型（surrogates）会引入额外的模型误差，进而可能扭曲所得的参数分布。此外，即便有替代模型带来的额外速度提升，通过马尔可夫链蒙特卡洛（Markov chain Monte Carlo）等传统方法确定后验分布仍可能成本高昂，尤其是针对复杂的后验形状时。本文提出了一个统一框架，将贝叶斯推断、模型误差校正与基于传输的采样方案相结合，以解决上述问题。为训练该传输方案，研究了两种不同的损失函数：一种等价于与传输问题相关的Kullback-Leibler散度，另一种基于该损失的上界，通常称为证据下界（evidence lower bound）。研究表明，基于后者训练传输方案会极大改变优化空间，可能在近似目标后验时引入不期望的偏差。将所提方法的计算成本与已有的成熟方法进行了比较，并通过数值示例验证了理论结果。
英文摘要：
In indirect measurements, the sought parameters have to be determined by solving an inverse problem, typically in a Bayesian framework. Often, the accurate numerical simulation of the measuring process is computationally demanding, making it necessary to rely on approximate models. These surrogates, however, introduce an additional model error and thus may distort the resulting parameter distribution. Moreover, even with the additional speed granted by the surrogate, posterior determination through conventional means such as Markov chain Monte Carlo might be cost intensive, specifically for complicated posterior shapes. In this paper, we propose a unified framework that combines Bayesian inference, model error correction and a transport-based sampling scheme to address these issues. To train the transport scheme, we investigate two different losses: one equivalent to the Kullback-Leibler divergence associated to the transport problem and one based on an upper bound of this loss, generally known as the evidence lower bound. We demonstrate that training the transport based on the latter changes the optimisation landscape drastically, potentially introducing an undesired bias in approximating the target posterior. We compare the computational cost of our approach with established methods and underline the theoretical results with numerical examples.
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