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.
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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.
| Type: | Article |
|---|---|
| Title: | Variational Gaussian approximation for Poisson data |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1088/1361-6420/aaa0ab |
| Publisher version: | https://doi.org/10.1088/1361-6420/aaa0ab |
| Language: | English |
| Additional information: | 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. |
| Keywords: | variational Gaussian approximation, Poisson data, hierarchical modeling, Kullback–Leibler divergence, alternating direction maximization |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Computer Science |
| URI: | https://discovery.ucl.ac.uk/id/eprint/1576406 |
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