COMBINING PUSH-FORWARD MEASURES AND BAYES' RULE TO CONSTRUCT CONSISTENT SOLUTIONS TO STOCHASTIC INVERSE PROBLEMS
COMBINING PUSH-FORWARD MEASURES AND BAYES' RULE TO CONSTRUCT CONSISTENT SOLUTIONS TO STOCHASTIC INVERSE PROBLEMS
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DOI：
10.1137/16m1087229
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发表时间：
2018-01-01
影响因子：
3.1
通讯作者：
Wildey, T.
中科院分区：
文献类型：
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作者：
Butler, T.;Jakeman, J.;Wildey, T.
我们提出并提出了一种由感兴趣的随机观测数据推断确定性模型参数的反问题的数值求解方法。作为概率度量给出的解决方案是使用针对可测量地图的贝叶斯更新方法来导出的，该方法找到后验概率度量，该后验概率度量在通过确定性模型传播时产生与数据上观察到的概率度量完全匹配的前推度量。我们找到这样的后验度量的方法，我们称之为一致贝叶斯推理或基于前推的推理，很简单，只需要计算由先验概率度量和确定性模型组合而成的前推概率度量。我们证明了观测相容后验的存在唯一性，并给出了稳定性分析和误差分析。我们还讨论了一致性贝叶斯推理、经典/统计贝叶斯推理和最近发展的一种测度论推理方法之间的关系。最后，给出了分析和数值结果，突出了一致性贝叶斯方法的某些性质，以及该方法与上述两种推理方案的区别。
We formulate, and present a numerical method for solving, an inverse problem for inferring parameters of a deterministic model from stochastic observational data on quantities of interest. The solution, given as a probability measure, is derived using a Bayesian updating approach for measurable maps that finds a posterior probability measure that when propagated through the deterministic model produces a push-forward measure that exactly matches the observed probability measure on the data. Our approach for finding such posterior measures, which we call consistent Bayesian inference or push-forward based inference, is simple and only requires the computation of the push-forward probability measure induced by the combination of a prior probability measure and the deterministic model. We establish existence and uniqueness of observation-consistent posteriors and present both stability and error analyses. We also discuss the relationships between consistent Bayesian inference, classical/statistical Bayesian inference, and a recently developed measure-theoretic approach for inference. Finally, analytical and numerical results are presented to highlight certain properties of the consistent Bayesian approach and the differences between this approach and the two aforementioned alternatives for inference.