Fig. 1: At the heart of stochastic modeling and analysis of biological systems is the mathematical problem of solving chemical master equations (CMEs).

A Inference problem associated with single-cell measurement data. To solve this inference problem, researchers need to solve CMEs to obtain single-cell distributions for many candidate models and then select the model based on the computed distributions. B Stochastic filtering based on single-cell time-course data. The aim is to compute the conditional probability distribution of unobserved species within a cell using the cell’s time-course data. The problem is often solved in a recursive manner involving prediction steps and correction steps. The prediction steps need to solve CMEs to obtain the predicted probability distributions. C Our divide-and-conquer idea for solving large-scale CMEs. Our method utilizes Rao-Blackwellization and stochastic filtering (for conditional independence) to divide the original CME into several manageable sub-problems for low dimensional subsystems.