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Stochastic Filtering of Reaction Networks Partially Observed in Time Snapshots

Muruhan Rathinam, Mingkai Yu

TL;DR

The paper tackles state inference for stochastic reaction networks with exact partial observations at time snapshots. It introduces a Targeting Algorithm that enforces observations by conditioning reaction counts through a Girsanov-change-of-measure framework using inhomogeneous Poisson proposals, and it rigorously justifies the method while comparing it to naive filtering. The approach includes an end-point step to satisfy the final observation and a Poisson-bridge interpolation step for interpolation between snapshots, with extensions to multiple observations and two-stage variants. Empirical results across several canonical reaction networks show that targeting improves effective sample size and accuracy, especially for larger copy numbers and rarer observations, while offering practical trade-offs with computational cost and resampling strategies.

Abstract

Stochastic reaction network models arise in intracellular chemical reactions, epidemiological models and other population process models, and are a class of continuous time Markov chains which have the nonnegative integer lattice as state space. We consider the problem of estimating the conditional probability distribution of a stochastic reaction network given exact partial state observations in time snapshots. We propose a particle filtering method called the targeting method. Our approach takes into account that the reaction counts in between two observation snapshots satisfy linear constraints and also uses inhomogeneous Poisson processes as proposals for the reaction counts to facilitate exact interpolation. We provide rigorous analysis as well as numerical examples to illustrate our method and compare it with other alternatives.

Stochastic Filtering of Reaction Networks Partially Observed in Time Snapshots

TL;DR

The paper tackles state inference for stochastic reaction networks with exact partial observations at time snapshots. It introduces a Targeting Algorithm that enforces observations by conditioning reaction counts through a Girsanov-change-of-measure framework using inhomogeneous Poisson proposals, and it rigorously justifies the method while comparing it to naive filtering. The approach includes an end-point step to satisfy the final observation and a Poisson-bridge interpolation step for interpolation between snapshots, with extensions to multiple observations and two-stage variants. Empirical results across several canonical reaction networks show that targeting improves effective sample size and accuracy, especially for larger copy numbers and rarer observations, while offering practical trade-offs with computational cost and resampling strategies.

Abstract

Stochastic reaction network models arise in intracellular chemical reactions, epidemiological models and other population process models, and are a class of continuous time Markov chains which have the nonnegative integer lattice as state space. We consider the problem of estimating the conditional probability distribution of a stochastic reaction network given exact partial state observations in time snapshots. We propose a particle filtering method called the targeting method. Our approach takes into account that the reaction counts in between two observation snapshots satisfy linear constraints and also uses inhomogeneous Poisson processes as proposals for the reaction counts to facilitate exact interpolation. We provide rigorous analysis as well as numerical examples to illustrate our method and compare it with other alternatives.
Paper Structure (24 sections, 8 theorems, 103 equations, 2 figures, 10 tables, 3 algorithms)

This paper contains 24 sections, 8 theorems, 103 equations, 2 figures, 10 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Reaction Networks and The Filtering Problem
  3. The Filtering Algorithm
  4. Mathematical Setup
  5. The Naive Algorithm
  6. Effective Sample Size
  7. Conditional Propensity Function
  8. The Targeting Problem and the Targeting Algorithm
  9. First step of the Targeting Algorithm
  10. Second step of the Targeting Method
  11. Multiple Observation Snapshots
  12. Examples
  13. Example 1: Pure-death model
  14. Example 2: Reversible Reaction $S_1 \longleftrightarrow S_2$
  15. Example 3: $S_1 \longleftrightarrow S_2, \quad S_1+S_2 \longleftrightarrow S_3$
  16. ...and 9 more sections

Key Result

Lemma 1

For $i \in {\mathbb N}$, $z_0=(x_0,y_0) \in {\mathbb Z}_+^{n}$ and $k' \in {\mathbb Z}_+^{m_1}$

Figures (2)

  • Figure 1: Example 1. Pure death example. Estimated conditional distribution compared with the actual conditional distribution. A common observation. $x_0 = 1000$, $c=2$, $T=0.5$, $t=0.2$. The estimate came with $95\%$ confident interval, which was estimated based on $N_r = 100$ trials, while each trial used filter sample size $N_s = 1000$.
  • Figure 2: Convergence of various simulation methods as sample size increase. The Total Variation Error (TVE) against $\frac{1}{N_s}$ in log scale. We used Example 3 a common observation to illustrate. We took $Z(0)=(20, 20, 20)$, $c=(0.5, 1, 0.1, 1)$, $t = T = 1$, $\Delta t = 0.1$, $y_T = 24$. We used $N_r = 100$ trials, filter sample sizes were $N_s = 1000, 4000, 16000, 64000$.

Theorems & Definitions (12)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Theorem 1
  • ...and 2 more