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Variational bayesian approximation. A rigorous approach

Bahraini, A ; Sharif University of Technology | 2022

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  1. Type of Document: Article
  2. Publisher: Publishing House of the Romanian Academy , 2022
  3. Abstract:
  4. We apply the theory of optimal transport to study mathematical properties of mean field variational Bayesian approximation. It turns out that if K +C > 0 where C is the convexity coefficient of −log p and K is a lower bound for the Ricci curvature of the underlying parameter space, then the corresponding system of equations of variational Bayesian approximation admits a unique solution. The uniqueness property in presence of symmetry leads to preservation of mode. As an explicit application we correct Bayesian Gaussian Mixture model in such a way that it turns into a convex model while its (unique) maximum likelihood solution coincides asymptotically with the true solution. Using convexity it is possible to prove asymptotic accuracy of the mode obtained by mean field variational Bayesian approximation. This seems to be the first rigorous proof for this fundamental fact which was expected based on several experimental computations. © 2022, Publishing House of the Romanian Academy. All rights reserved
  5. Keywords:
  6. Gaussian mixture model ; Geodesic convexity ; MFVBA ; pptimal transport ; Ricci curvature
  7. Source: Proceedings of the Romanian Academy Series A - Mathematics Physics Technical Sciences Information Science ; Volume 23, Issue 2 , 2022 , Pages 107-112 ; 14549069 (ISSN)
  8. URL: https://academiaromana.ro/sectii2002/proceedings/doc2022-2/02-Bahraini.pdf