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Variational inference as iterative projection in a Bayesian Hilbert space with application to robotic state estimation

Published online by Cambridge University Press:  24 October 2022

Timothy D. Barfoot Open the ORCID record for Timothy D. Barfoot [Opens in a new window]  and
Gabriele M. T. D’Eleuterio
Timothy D. Barfoot*
Affiliation:
University of Toronto Institute for Aerospace Studies, 4925 Dufferin Street, Ontario, M3H 5T6, Canada
Gabriele M. T. D’Eleuterio
Affiliation:
University of Toronto Institute for Aerospace Studies, 4925 Dufferin Street, Ontario, M3H 5T6, Canada
*
*Corresponding author. E-mail: tim.barfoot@utoronto.ca
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Abstract

Variational Bayesian inference is an important machine learning tool that finds application from statistics to robotics. The goal is to find an approximate probability density function (PDF) from a chosen family that is in some sense “closest” to the full Bayesian posterior. Closeness is typically defined through the selection of an appropriate loss functional such as the Kullback-Leibler (KL) divergence. In this paper, we explore a new formulation of variational inference by exploiting the fact that (most) PDFs are members of a Bayesian Hilbert space under careful definitions of vector addition, scalar multiplication, and an inner product. We show that, under the right conditions, variational inference based on KL divergence can amount to iterative projection, in the Euclidean sense, of the Bayesian posterior onto a subspace corresponding to the selected approximation family. We work through the details of this general framework for the specific case of the Gaussian approximation family and show the equivalence to another Gaussian variational inference approach. We furthermore discuss the implications for systems that exhibit sparsity, which is handled naturally in Bayesian space, and give an example of a high-dimensional robotic state estimation problem that can be handled as a result. We provide some preliminary examples of how the approach could be applied to non-Gaussian inference and discuss the limitations of the approach in detail to encourage follow-on work along these lines.

Keywords

Aitchison geometryBayesian Hilbert spacesvariational inferenceBayesian inferencecompositional datastochastic algebra
Type
Research Article
Information
Robotica , Volume 41 , Issue 2 , February 2023 , pp. 632 - 667
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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