Learning stochastic filtering

We quantify the performance of approximations to stochastic filtering by the Kullback-Leibler divergence to the optimal Bayesian filter. Using a two-state Markov process that drives a Brownian measurement process as prototypical test case, we compare two stochastic filtering approximations: a static low-pass filter as baseline, and machine learning of Voltera expansions using nonlinear Vector Auto Regression (nVAR). We highlight the crucial role of the chosen performance metric, and present two solutions to the specific challenge of predicting a likelihood bounded between $0$ and $1$.