@article{discovery1576406,
          volume = {34},
            note = {Original content from this work may be used under the terms of the Creative  Commons Attribution 3.0 licence (http://creativecommons.org/licenses/by/3.0). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.},
       publisher = {IOP PUBLISHING LTD},
            year = {2018},
           title = {Variational Gaussian approximation for Poisson data},
         journal = {Inverse Problems},
          number = {2},
           month = {February},
        keywords = {variational Gaussian approximation, Poisson data, hierarchical modeling, Kullback-Leibler divergence, alternating direction maximization},
             url = {https://doi.org/10.1088/1361-6420/aaa0ab},
            issn = {1361-6420},
        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.},
          author = {Arridge, SR and Ito, K and Jin, B and Zhang, C}
}