Bellman filtering and smoothing for state-space models

TL;DR

This paper introduces a Bellman-based framework for filtering, smoothing, and parameter estimation in general state-space models that may be nonlinear, non-Gaussian, and even degenerate. By approximating the Bellman value function with a multivariate quadratic form, the authors obtain a forward recursion with per-step complexity $O(m^3)$, enabling scalable filtering in high-dimensional settings and recovering the Kalman filter as a special case. A smoothing extension yields RTS-like results under linear-Gaussian dynamics, while parameter estimation is performed via a fast, approximate maximum likelihood approach that leverages the filter outputs. Simulation and two real-world applications demonstrate that the Bellman filter achieves competitive accuracy with substantial computational savings and robust performance in high-dimensional and degenerate settings, with smoothing delivering notable improvements in state estimation. The work suggests a versatile, efficient alternative to simulation-based filters, with practical impact for large-scale time-series analysis in engineering, economics, and climate science.

Abstract

This paper presents a new filter for state-space models based on Bellman's dynamic-programming principle, allowing for nonlinearity, non-Gaussianity and degeneracy in the observation and/or state-transition equations. The resulting Bellman filter is a direct generalisation of the (iterated and extended) Kalman filter, enabling scalability to higher dimensions while remaining computationally inexpensive. It can also be extended to enable smoothing. Under suitable conditions, the Bellman-filtered states are stable over time and contractive towards a region around the true state at every time step. Static (hyper)parameters are estimated by maximising a filter-implied pseudo log-likelihood decomposition. In univariate simulation studies, the Bellman filter performs on par with state-of-the-art simulation-based techniques at a fraction of the computational cost. In two empirical applications, involving up to 150 spatial dimensions or highly degenerate/nonlinear state dynamics, the Bellman filter outperforms competing methods in both accuracy and speed.

Figures (3)

• Figure 1: Illustration of mean squared error (MSE) contraction due to inequality \ref{['inequality simplified']}
• Figure 2: MAE of filtered states and filtering times (in seconds per sample)
• Figure 3: Expanding-window parameter estimation results for model \ref{['katzfuss1']}--\ref{['katzfuss3']}

Theorems & Definitions (12)

• Definition 1: Value function
• Proposition 1: Filtering using Bellman's equation
• Remark 1
• Corollary 1: Kalman filter as a special case
• Theorem 1: Contractivity of the mean squared error
• Proposition 2: Uniformly bounded MSE
• Theorem 2: Stability of the time-invariant Bellman filter.
• Proposition 3: Bellman's forward and backward recursions.
• Proposition 4: Bellman smoother with linear Gaussian state equation
• Corollary 2