SIAM/ASA Journal on Uncertainty Quantification, Volume 8, Issue 4, Page 1383-1413, January 2020.
In many applications, Bayesian inverse problems can give rise to probability distributions which contain complexities due to the Hessian varying greatly across parameter space. This complexity often manifests itself as lower-dimensional manifolds on which the likelihood function is invariant, or varies very little. This can be due to trying to infer unobservable parameters, or due to sloppiness in the model which is being used to describe the data. In such a situation, standard sampling methods for characterizing the posterior distribution, which do not incorporate information about this structure, will be highly inefficient. In this paper, we seek to develop an approach to tackle this problem when using adaptive importance sampling methods by employing optimal transport maps to simplify posterior distributions which are concentrated on lower-dimensional manifolds. This approach is applicable to a whole range of problems for which Monte Carlo Markov chain methods mix slowly. We demonstrate the approach by considering inverse problems arising from partially observed stochastic reaction networks. In particular, we consider systems which exhibit multiscale behavior, but for which only the slow variables in the system are observable. We demonstrate that certain multiscale approximations lead to more consistent approximations of the posterior than others. The use of optimal transport maps stabilizes the ensemble transform adaptive importance sampling method and allows for efficient sampling with smaller ensemble sizes. This approach allows us to take advantage of the large increases of efficiency when using adaptive importance sampling methods for previously intractable Bayesian inverse problems with complex posterior structure.

中文翻译：

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运输地图加速的自适应重要性采样及其在多尺度随机反应网络引起的逆问题中的应用

SIAM / ASA不确定性量化杂志，第8卷，第4期，第1383-1413页，2020年1月。
在许多应用中，由于Hessian在整个参数空间中的变化很大，贝叶斯逆问题会引起包含复杂性的概率分布。这种复杂性通常表现为似然函数不变或变化很小的低维流形。这可能是由于试图推断无法观察到的参数，或者是由于用于描述数据的模型过于草率。在这种情况下，用于表征后验分布的标准采样方法（其中未包含有关此结构的信息）将非常无效。在本文中，当使用自适应重要性抽样方法时，我们寻求开发一种方法来解决此问题，方法是采用最佳运输图来简化集中在低维流形上的后验分布。这种方法适用于蒙特卡洛·马尔科夫链方法混合缓慢的所有问题。我们通过考虑由部分观察到的随机反应网络引起的逆问题来证明该方法。特别地，我们考虑表现出多尺度行为的系统，但是对于这些系统，只有系统中的慢变量才是可观察到的。我们证明，某些多尺度逼近比其他逼近导致更一致的后验逼近。最佳运输图的使用稳定了整体变换自适应重要性采样方法，并允许以较小的整体大小进行有效采样。当使用自适应重要性采样方法来解决先前具有复杂后结构的棘手贝叶斯逆问题时，这种方法使我们能够利用效率的大幅提高。