Proximal Approximate Inference in State-Space Models
Hany Abdulsamad, Ángel F. García-Fernández, Simo Särkkä
TL;DR
This work reframes Bayesian smoothing in nonlinear, non-Gaussian state-space models as a dynamic, entropic proximal optimization problem. By enforcing KL-based trust regions and employing forward, reverse, or hybrid Gauss–Markov factorizations, it yields efficient forward–backward recursive smoothing algorithms that operate in log-space and scale linearly with horizon. Themethod leverages two practical posterior approximations—Generalized Statistical Linear Regression and Fourier–Hermite moment matching—to obtain tractable quadratic potentials and closed-form updates for Gaussian marginals. Together, these elements provide a principled, scalable framework that unifies classical smoothing with modern variational inference, enabling accurate inference in complex temporal models with strong computational properties.
Abstract
We present a class of algorithms for state estimation in nonlinear, non-Gaussian state-space models. Our approach is based on a variational Lagrangian formulation that casts Bayesian inference as a sequence of entropic trust-region updates subject to dynamic constraints. This framework gives rise to a family of forward-backward algorithms, whose structure is determined by the chosen factorization of the variational posterior. By focusing on Gauss--Markov approximations, we derive recursive schemes with favorable computational complexity. For general nonlinear, non-Gaussian models we close the recursions using generalized statistical linear regression and Fourier--Hermite moment matching.