Variational Gaussian approximation for Poisson data
Variational Gaussian approximation for Poisson data
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DOI：
10.1088/1361-6420/aaa0ab
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发表时间：
2018-02-01
期刊：
影响因子：
2.1
通讯作者：
Zhang, Chen
中科院分区：
文献类型：
--
作者：
Arridge, Simon R.;Ito, Kazufumi;Zhang, Chen
泊松模型经常被用来描述计数数据，但在贝叶斯上下文中，它会导致分析上难以处理的后验概率分布。在这项工作中，我们分析了变分高斯近似的后验分布所产生的泊松模型与高斯先验。这是通过寻求最佳高斯分布来实现的，该最佳高斯分布使从后验分布到近似的Kullback-Leibler发散最小化，或者等效地使模型证据的下限最大化。我们得到了一个明确的表达式的下限，并证明了最佳高斯逼近的存在性和唯一性。下界泛函可以被看作是经典吉洪诺夫正则化的一个变体，它也惩罚协方差。在此基础上，提出了一种有效的交替方向最大化算法，并分析了其收敛性。我们讨论了通过前向算子的低秩结构和协方差的稀疏性来降低计算复杂度的策略。进一步，作为下界的一个应用，我们讨论了先验分布中超参数选择的层次贝叶斯模型，并提出了确定超参数的单调收敛算法.我们提出了广泛的数值实验来说明高斯近似和算法。
The Poisson model is frequently employed to describe count data, but in a Bayesian context it leads to an analytically intractable posterior probability distribution. In this work, we analyze a variational Gaussian approximation to the posterior distribution arising from the Poisson model with a Gaussian prior. This is achieved by seeking an optimal Gaussian distribution minimizing the Kullback-Leibler divergence from the posterior distribution to the approximation, or equivalently maximizing the lower bound for the model evidence. We derive an explicit expression for the lower bound, and show the existence and uniqueness of the optimal Gaussian approximation. The lower bound functional can be viewed as a variant of classical Tikhonov regularization that penalizes also the covariance. Then we develop an efficient alternating direction maximization algorithm for solving the optimization problem, and analyze its convergence. We discuss strategies for reducing the computational complexity via low rank structure of the forward operator and the sparsity of the covariance. Further, as an application of the lower bound, we discuss hierarchical Bayesian modeling for selecting the hyperparameter in the prior distribution, and propose a monotonically convergent algorithm for determining the hyperparameter. We present extensive numerical experiments to illustrate the Gaussian approximation and the algorithms.