Filtered Markovian Projection: Dimensionality Reduction in Filtering for Stochastic Reaction Networks
Chiheb Ben Hammouda, Maksim Chupin, Sophia Münker, Raúl Tempone
TL;DR
The paper addresses the curse of dimensionality in filtering partially observed SRNs by introducing Filtered Markovian Projection (FMP), a principled framework that reduces the effective state dimension while preserving the targeted marginal conditional distributions. It presents two MP-based filters: the Unconditional MP (UMP), which is simple but can be inconsistent for filtering, and the Consistent Conditional MP (CMP), which enforces consistency with the observed trajectory via a new FMP theorem. CMP combines a particle filter to estimate projected propensities with a reduced-dimension FFSP for the marginal filtering equations, achieving significant computational gains (up to orders of magnitude) while maintaining accuracy in the marginal distributions. Theoretical results establish consistency and an $O(M^{-1/2})$ convergence rate with respect to the number of particles, and numerical examples (e.g., bistable gene networks and linear cascades) demonstrate substantial speedups and improved tail probability estimation relative to FFSP and standard PF. The work paves the way for scalable, consistent filtering in high-dimensional SRNs and related stochastic systems.
Abstract
Stochastic reaction networks (SRNs) model stochastic effects for various applications, including intracellular chemical or biological processes and epidemiology. A typical challenge in practical problems modeled by SRNs is that only a few state variables can be dynamically observed. Given the measurement trajectories, one can estimate the conditional probability distribution of unobserved (hidden) state variables by solving a stochastic filtering problem. In this setting, the conditional distribution evolves over time according to an extensive or potentially infinite-dimensional system of coupled ordinary differential equations with jumps, known as the filtering equation. The current numerical filtering techniques, such as the filtered finite state projection (D'Ambrosio et al., 2022), are hindered by the curse of dimensionality, significantly affecting their computational performance. To address these limitations, we propose to use a dimensionality reduction technique based on the Markovian projection (MP), initially introduced for forward problems (Ben Hammouda et al., 2024). In this work, we explore how to adapt the existing MP approach to the filtering problem and introduce a novel version of the MP, the Filtered MP, that guarantees the consistency of the resulting estimator. The novel consistent MP filter employs a reduced-variance particle filter for estimating the jump intensities of the projected model and solves the filtering equations in a low-dimensional space. The analysis and empirical results highlight the superior computational efficiency of projection methods compared to the existing filtered finite state projection in the large dimensional setting.