eprintid: 1576406 rev_number: 32 eprint_status: archive userid: 608 dir: disk0/01/57/64/06 datestamp: 2017-10-01 03:46:49 lastmod: 2021-12-05 00:44:27 status_changed: 2018-01-30 17:19:17 type: article metadata_visibility: show creators_name: Arridge, SR creators_name: Ito, K creators_name: Jin, B creators_name: Zhang, C title: Variational Gaussian approximation for Poisson data ispublished: pub divisions: UCL divisions: B04 divisions: C05 divisions: F48 keywords: variational Gaussian approximation, Poisson data, hierarchical modeling, Kullback–Leibler divergence, alternating direction maximization 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. abstract: 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. date: 2018-02 date_type: published publisher: IOP PUBLISHING LTD official_url: https://doi.org/10.1088/1361-6420/aaa0ab oa_status: green full_text_type: pub language: eng primo: open primo_central: open_green article_type_text: Article verified: verified_manual elements_id: 1515690 doi: 10.1088/1361-6420/aaa0ab lyricists_name: Arridge, Simon lyricists_name: Jin, Bangti lyricists_name: Zhang, Chen lyricists_id: SRARR14 lyricists_id: BJINX59 lyricists_id: CZHAB51 actors_name: Flynn, Bernadette actors_id: BFFLY94 actors_role: owner full_text_status: public publication: Inverse Problems volume: 34 number: 2 article_number: 025005 pages: 29 issn: 1361-6420 citation: Arridge, SR; Ito, K; Jin, B; Zhang, C; (2018) Variational Gaussian approximation for Poisson data. Inverse Problems , 34 (2) , Article 025005. 10.1088/1361-6420/aaa0ab <https://doi.org/10.1088/1361-6420%2Faaa0ab>. Green open access document_url: https://discovery.ucl.ac.uk/id/eprint/1576406/1/Arridge_Variational_Gaussian_approximation.pdf