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<https://discovery.ucl.ac.uk/id/eprint/1576406> <http://purl.org/ontology/bibo/abstract> "The Poisson model is frequently employed to describe count data, but in a Bayesian context it leads to&#13; an analytically intractable posterior probability distribution. In this work, we analyze a variational Gaussian&#13; approximation to the posterior distribution arising from the Poisson model with a Gaussian prior. This is&#13; achieved by seeking an optimal Gaussian distribution minimizing the Kullback-Leibler divergence from&#13; the posterior distribution to the approximation, or&#13; equivalently maximizing the lower bound for the model evidence. We derive an explicit expression for&#13; the lower bound, and show the existence and uniqueness of the optimal Gaussian approximation. The lower&#13; bound functional can be viewed as a variant of classical Tikhonov regularization that penalizes also the&#13; covariance. Then we develop an efficient alternating direction maximization algorithm for solving&#13; the optimization problem, and analyze its convergence. We discuss strategies for reducing the computational&#13; complexity via low rank structure of the forward operator and the sparsity of the covariance. Further, as an&#13; application of the lower bound, we discuss hierarchical Bayesian modeling for selecting the&#13; hyperparameter in the prior distribution, and propose a monotonically convergent algorithm for determining&#13; the hyperparameter. We present extensive numerical experiments to illustrate the Gaussian approximation and the algorithms."^^<http://www.w3.org/2001/XMLSchema#string> .
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