多输出信号贝叶斯反演问题中协方差矩阵不确定性分析的自适应后验分布
Adaptive posterior distributions for uncertainty analysis of covariance matrices in Bayesian inversion problems for multioutput signals
AI总结：
本文针对多输出非线性模型参数与噪声协方差矩阵的贝叶斯反演问题，提出自适应目标自适应重要性采样（ATAIS）方法，通过分块推断与样本回收重加权实现完整贝叶斯分析，并辅以ILIS算法验证其优势。
摘要：
本文研究对非线性多输出模型的参数以及不同输出信号的协方差矩阵进行贝叶斯推断的问题。我们提出了一种用于多变量贝叶斯反演问题的自适应重要性采样（AIS）方案，该方案基于两个主要思想：将感兴趣的变量分成两个块，并利用已知的解析优化公式进行推断。我们同时估计多变量非线性模型的未知参数和噪声的协方差矩阵。在所提推断方案的第一部分中，设计了一种称为自适应目标自适应重要性采样（ATAIS）的新型AIS技术，该技术在非线性模型参数上的IS技术与噪声协方差矩阵的频率学派方法之间迭代交替。在所提推断方案的第二部分中，考虑协方差矩阵上的先验密度，并将ATAIS获得的样本云进行回收和重新加权，以获得对模型参数和协方差矩阵的完整贝叶斯研究。ATAIS是本文的主要贡献。此外，还提出了倒置分层重要性采样（ILIS），作为一种可能具有吸引力的算法（但基于概念上更简单的思想）。不同的数值例子展示了所提方法的优势。
英文摘要：
In this paper we address the problem of performing Bayesian inference for the parameters of a nonlinear multi-output model and the covariance matrix of the different output signals. We propose an adaptive importance sampling (AIS) scheme for multivariate Bayesian inversion problems, which is based in two main ideas: the variables of interest are split in two blocks and the inference takes advantage of known analytical optimization formulas. We estimate both the unknown parameters of the multivariate non-linear model and the covariance matrix of the noise. In the first part of the proposed inference scheme, a novel AIS technique called adaptive target adaptive importance sampling (ATAIS) is designed, which alternates iteratively between an IS technique over the parameters of the non-linear model and a frequentist approach for the covariance matrix of the noise. In the second part of the proposed inference scheme, a prior density over the covariance matrix is considered and the cloud of samples obtained by ATAIS are recycled and re-weighted to obtain a complete Bayesian study over the model parameters and covariance matrix. ATAIS is the main contribution of the work. Additionally, the inverted layered importance sampling (ILIS) is presented as a possible compelling algorithm (but based on a conceptually simpler idea). Different numerical examples show the benefits of the proposed approaches
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