Infinite Dimensional空间上概率测度的Kullback-Leibler逼近
Kullback-Leibler Approximation for Probability Measures on Infinite Dimensional Spaces
摘要：
在各种应用中，重要的是从无限维空间上的概率测度$μ$中提取信息。例子包括贝叶斯方法的逆问题和可能的条件）连续时间马尔可夫过程。然后，可能感兴趣的是从一个简单的度量类中找到近似于$μ$的度量$ν$。这个问题的研究的情况下，Kullback-Leibler散度是用来衡量质量的近似。采用变分法的观点，并详细研究了从高斯测度集中选择$ν$的特殊情况。基本的存在性和唯一性定理的建立，以及最小序列的性质。此外，参数化类的高斯通过平均值和逆协方差的介绍，需要正则化的解释，并正则化的最小化进行了详细研究。从这项工作中产生的变分法框架提供了适当的基础计算算法。
英文摘要：
In a variety of applications it is important to extract information from a probability measure $μ$ on an infinite dimensional space. Examples include the Bayesian approach to inverse problems and possibly conditioned) continuous time Markov processes. It may then be of interest to find a measure $ν$, from within a simple class of measures, which approximates $μ$. This problem is studied in the case where the Kullback-Leibler divergence is employed to measure the quality of the approximation. A calculus of variations viewpoint is adopted and the particular case where $ν$ is chosen from the set of Gaussian measures is studied in detail. Basic existence and uniqueness theorems are established, together with properties of minimising sequences. Furthermore, parameterisation of the class of Gaussians through the mean and inverse covariance is introduced, the need for regularisation is explained, and a regularised minimisation is studied in detail. The calculus of variations framework resulting from this work provides the appropriate underpinning for computational algorithms.
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