Mori–Zwanzig Formulation of Bayesian Filtering

Abstract

Most approximate Bayesian filters project each posterior onto a tractable family, and a projection is judged by the information it discards at each step. However, this information bounds the cost of a projection only on average: on a single stream, a projection that discards little now can cost arbitrarily more later. To account for when the cost of a projection arrives, we apply the Mori–Zwanzig formalism of statistical physics to the filter’s own belief. Under the Mori projection defined by the Fisher metric, assumed-density filtering is the Markovian closure of the formalism, and every other projected filter adds a defect to the Markov term. The error of a projected filter is then a propagated sum of its defects and of the returns of what its projections discard. For Gaussian references and linear-Gaussian propagation, we derive the memory in closed form. From its decay rates and a square-integrability condition, we draw a principle for what a filter should resolve: a higher order for smooth non-Gaussianity, and an extra component for changepoints. We further test the principle on real-world data such as DNA copy-number profiles of neuroblastoma tumours. On these profiles, an extra component resolves most changes better than a higher order. We thereby provide an analysis tool for the error of approximate Bayesian filters.

CitationT.-Y. Tsui, K. Kording, J. Gu, L. Liu. (2026). "Mori–Zwanzig Formulation of Bayesian Filtering."