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

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 (i) a jump Gauss-Markov system under a certain factored Markov process approximation, and (ii) in a Gauss-Markov model with log-polynomial likelihoods under a Gauss--Markov constraint on the variational approximation. It is demonstrated that the value functional recursions are tractable in both cases. The resulting estimators are examined in simulation studies and are found to be of adequate quality in comparison to sensible baselines.