贝叶斯推理与反演中的变分先验替换
Variational Prior Replacement in Bayesian Inference and Inversion
AI总结：
针对贝叶斯推理中先验信息变更需重算、成本高昂的问题，提出变分先验替换（VPR）方法，可在不重算原推理的情况下替换后验中的先验信息，经二维地震全波形反演验证，精度与独立重算相当且计算效率大幅提升。
摘要：
许多科学研究需要利用记录到的数据估计一组模型参数的取值。在贝叶斯推理中，观测数据与先验知识的信息相结合，通过计算后验概率分布函数以概率方式更新模型参数，先验信息通常由先验概率分布来描述。在科研项目开展过程中，常会出现需要更改先验信息的情况。然而，求解单个贝叶斯推理问题的计算成本通常很高，因为这一般需要抽取大量模型样本，且必须模拟每个样本为真实情况时本应记录到的数据集。因此，每次先验信息变化时都重新计算贝叶斯推理解的成本会极高。\n我们提出了一种数学公式，可利用变分方法更改嵌入在解中的先验信息，无需重新计算原始贝叶斯推理。该方法从已求得的后验分布中移除原有先验信息，替换为新的先验信息，因此我们将该方法命名为变分先验替换（variational prior replacement，VPR）。我们通过一个二维地震全波形反演算例验证了VPR的效果：VPR得到的后验解与采用不同先验分布求解独立推理问题得到的结果相近。前者在笔记本电脑上数分钟即可完成，而后者需要使用高性能计算资源运算数天。我们通过对比三种不同类型先验信息（均匀先验、平滑先验和地质先验分布）得到的后验解，证明了该方法的价值。
英文摘要：
Many scientific investigations require that the values of a set of model parameters are estimated using recorded data. In Bayesian inference, information from both observed data and prior knowledge is combined to update model parameters probabilistically by calculating the posterior probability distribution function. Prior information is often described by a prior probability distribution. Situations arise in which we wish to change prior information during the course of a scientific project. However, estimating the solution to any single Bayesian inference problem is often computationally costly, as it typically requires many model samples to be drawn, and the data set that would have been recorded if each sample was true must be simulated. Recalculating the Bayesian inference solution every time prior information changes can therefore be extremely expensive. We develop a mathematical formulation that allows the prior information that is embedded within a solution, to be changed using variational methods, without recalculating the original Bayesian inference. In this method, existing prior information is removed from a previously obtained posterior distribution and is replaced by new prior information. We therefore call the methodology variational prior replacement (VPR). We demonstrate VPR using a 2D seismic full waveform inversion example, in which VPR provides similar posterior solutions to those obtained by solving independent inference problems using different prior distributions. The former can be completed within minutes on a laptop computer, whereas the latter requires days of computations using high-performance computing resources. We demonstrate the value of the method by comparing the posterior solutions obtained using three different types of prior information: uniform, smoothing and geological prior distributions.
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