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A geometric variational approach to Bayesian inference

Saha, Abhijoy; Bharath, Karthik; Kurtek, Sebastian

Authors

Abhijoy Saha

Sebastian Kurtek



Abstract

We propose a novel Riemannian geometric framework for variational inference in Bayesian models based on the nonparametric Fisher–Rao metric on the manifold of probability density functions. Under the square-root density representation, the manifold can be identified with the positive orthant of the unit hypersphere S ∞ in L 2 , and the Fisher–Rao metric reduces to the standard L 2 metric. Exploiting such a Riemannian structure, we formulate the task of approximating the posterior distribution as a variational problem on the hypersphere based on the α-divergence. This provides a tighter lower bound on the marginal distribution when compared to, and a corresponding upper bound unavailable with, approaches based on the Kullback–Leibler divergence. We propose a novel gradient-based algorithm for the variational problem based on Fréchet derivative operators motivated by the geometry of S ∞ , and examine its properties. Through simulations and real data applications, we demonstrate the utility of the proposed geometric framework and algorithm on several Bayesian models.

Citation

Saha, A., Bharath, K., & Kurtek, S. (2020). A geometric variational approach to Bayesian inference. Journal of the American Statistical Association, 115(530), 822-835. https://doi.org/10.1080/01621459.2019.1585253

Journal Article Type Article
Acceptance Date Jan 30, 2019
Online Publication Date Mar 26, 2019
Publication Date 2020
Deposit Date Mar 8, 2019
Publicly Available Date Mar 27, 2020
Journal Journal of the American Statistical Association
Print ISSN 0162-1459
Electronic ISSN 1537-274X
Publisher Taylor & Francis Open
Peer Reviewed Peer Reviewed
Volume 115
Issue 530
Pages 822-835
DOI https://doi.org/10.1080/01621459.2019.1585253
Keywords Infinite-dimensional Riemannian optimization; Gradient ascent algorithm; Square- root density; Bayesian density estimation; Bayesian logistic regression
Public URL https://nottingham-repository.worktribe.com/output/1620463
Publisher URL https://www.tandfonline.com/doi/full/10.1080/01621459.2019.1585253
Additional Information This is an Accepted Manuscript of an article published by Taylor & Francis in Journal of the American Statistical Association on 26.03.2019 , available online: http://www.tandfonline....0/01621459.2019.1585253

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