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Variational Bayesian inference is an important machine-learning tool thatfinds application from statistics to robotics. The goal is to find anapproximate probability density function (PDF) from a chosen family that is insome sense 'closest' to the full Bayesian posterior. Closeness is typicallydefined through the selection of an appropriate loss functional such as theKullback-Leibler (KL) divergence. In this paper, we explore a new formulationof variational inference by exploiting the fact that (most) PDFs are members ofa Bayesian Hilbert space under careful definitions of vector addition, scalarmultiplication and an inner product. We show that, under the right conditions,variational inference based on KL divergence can amount to iterativeprojection, in the Euclidean sense, of the Bayesian posterior onto a subspacecorresponding to the selected approximation family. We work through the detailsof this general framework for the specific case of the Gaussian approximationfamily and show the equivalence to another Gaussian variational inferenceapproach. We furthermore discuss the implications for systems that exhibitsparsity, which is handled naturally in Bayesian space, and give an example ofa high-dimensional robotic state estimation problem that can be handled as aresult. We provide some preliminary examples of how the approach could beapplied to non-Gaussian inference and discuss the limitations of the approachin detail to encourage follow-on work along these lines.

Variational Inference as Iterative Projection in a Bayesian Hilbert Space with Application to Robotic State Estimation