q-赫尔米特多项式混沌近似在基于q-高斯先验的贝叶斯反演中的应用
Q-Hermite polynomials chaos approximation of likelihood function based on q-Gaussian prior in Bayesian inversion
AI总结：
本文基于q-高斯先验分布，利用q-赫尔米特多项式混沌近似方法加速后验分布采样，并研究近似似然函数与截断先验分布下后验分布的KL散度收敛性，同时获得总变差和Hellinger度量下的收敛性。
摘要：
在实际应用中，先验分布的构造和后验分布采样的加速通常是工程中贝叶斯反演算法的两个关键问题。本文引入q-高斯分布作为反问题的先验分布，并讨论基于谱似然近似的加速算法。我们主要关注在使用近似似然函数和截断先验分布时，后验分布在KL散度意义下的收敛性。此外，还获得了总变差和Hellinger度量下的收敛性。最后给出了两个数值例子。
英文摘要：
In real applications, the construction of prior and acceleration of sampling for posterior are usually two key points of Bayesian inversion algorithm for engineers. In this paper, q-analogy of Gaussian distribution, q-Gaussian distribution, is introduced as the prior of inverse problems. And an acceleration algorithm based on spectral likelihood approximation is discussed. We mainly focus on the convergence of the posterior distribution in the sense of Kullback-Leibler divergence when approximated likelihood function and truncated prior distribution are used. Moreover, the convergence in the sense of total variation and Hellinger metric is obtained. In the end two numerical examples are displayed.
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