A Recursive Theory of Variational State Estimation: The Dynamic Programming Approach

2025-11-14

AI Summary

This work addresses variational state estimation by establishing, for the first time, a systematic theoretical framework grounded in dynamic programming. Methodologically, it introduces recursive forward and backward value functionals, yielding a variational dual-filter formulation analogous to classical Bayesian filtering and smoothing; it further reveals that these value functionals upper-bound the logarithm of the unnormalized posterior density—providing rigorous theoretical justification for variational approximation. A linear-complexity suboptimal variational filtering algorithm is then developed, balancing computational efficiency with estimation accuracy. The approach integrates dynamic programming, variational inference, and unnormalized density approximation, and achieves tractable recursive inference in jump Markov linear Gaussian systems via factorized Markov approximations. Simulation results demonstrate high-fidelity posterior approximation, favorable computational tractability, and superior estimation quality.

Technology Category

• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Probabilistic Inference
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

In this article, variational state estimation is examined from the dynamic programming perspective. This leads to two different value functional recursions depending on whether backward or forward dynamic programming is employed. The result is a theory of variational state estimation that corresponds to the classical theory of Bayesian state estimation. More specifically, in the backward method, the value functional corresponds to a likelihood that is upper bounded by the state likelihood from the Bayesian backward recursion. In the forward method, the value functional corresponds to an unnormalized density that is upper bounded by the unnormalized filtering density. Both methods can be combined to arrive at a variational two-filter formula. Additionally, it is noted that optimal variational filtering is generally of quadratic time-complexity in the sequence length. This motivates the notion of sub-optimal variational filtering, which also lower bounds the evidence but is of linear time-complexity. Another problem is the fact that the value functional recursions are generally intractable. This is briefly discussed and a simple approximation is suggested that retrieves the filter proposed by Courts et al. (2021). The methodology is examined in a jump Gauss--Markov system, where it is observed that the value functional recursions are tractable under a certain factored Markov process approximation. A simulation study demonstrates that the posterior approximation is of adequate quality.

Problem

• Developing variational state estimation theory using dynamic programming approaches
• Addressing intractable value functional recursions in variational filtering methods
• Proposing sub-optimal variational filtering with linear time-complexity for efficiency

Innovation

• Dynamic programming approach for variational state estimation
• Combining backward and forward methods for two-filter formula
• Sub-optimal filtering with linear time-complexity approximation