Efficient statistical inference for stochastic reaction processes

We address the problem of estimating unknown model parameters and state variables in stochastic reaction processes when only sparse and noisy measurements are available. Using an asymptotic system size expansion for the backward equation we derive an efficient approximation for this problem. We demonstrate the validity of our approach on model systems and generalize our method to the case when some state variables are not observed.

💡 Research Summary

The paper tackles the challenging problem of jointly estimating unknown kinetic parameters and hidden state trajectories in stochastic reaction processes (SRPs) when only sparse, noisy observations are available. Traditional Bayesian approaches require solving both the forward and backward master equations, which quickly become intractable for realistic systems due to the curse of dimensionality and the need for extensive Monte‑Carlo sampling. To overcome these limitations, the authors develop a novel approximation based on a system‑size expansion (also known as the van Kampen Ω‑expansion) applied to the backward equation.

In the large‑system‑size limit (characterized by a scaling parameter N), the conditional probability ψ(x,t) that the future observations will be generated from state x at time t can be approximated by a Gaussian distribution whose mean μ(t) and covariance Σ(t) evolve according to deterministic differential equations. The mean follows the macroscopic rate equations (the law of mass action), while the covariance satisfies a linear matrix differential equation involving the Jacobian of the mean dynamics and a diffusion matrix that captures intrinsic reaction noise. This pair of equations is essentially a continuous‑time Kalman‑type filter for the backward problem.

When an observation y_k arrives at time t_k, the Gaussian ψ is updated using Bayes’ rule. The observation model is assumed to be additive Gaussian noise on a possibly nonlinear measurement function h(x). Linearizing h around the current mean yields a standard Kalman update: the mean and covariance are corrected by the Kalman gain computed from the measurement Jacobian H and the combined process‑measurement covariance. Between observations, μ and Σ are propagated forward in time by integrating the deterministic backward equations.

Parameter inference is performed within an Expectation‑Maximization (EM) framework. In the E‑step, the current parameter set θ is used to compute the Gaussian backward messages, which provide sufficient statistics E