Tensor-train approximation of the chemical master equation and its application for parameter inference

Abstract

In this work, we perform Bayesian inference tasks for the chemical master equation in the tensor-train format. The tensor-train approximation has been proven to be very efficient in representing high dimensional data arising from the explicit representation of the chemical master equation solution. An additional advantage of representing the probability mass function in the tensor train format is that parametric dependency can be easily incorporated by introducing a tensor product basis expansion in the parameter space. Time is treated as an additional dimension of the tensor and a linear system is derived to solve the chemical master equation in time. We exemplify the tensor-train method by performing inference tasks such as smoothing and parameter inference using the tensor-train framework. A very high compression ratio is observed for storing the probability mass function of the solution. Since all linear algebra operations are performed in the tensor-train format, a significant reduction of the computational time is observed as well.

Tensor-train approximation of the chemical master equation and its application for parameter inference

“…Numerical experiments have been performed to showcase the advantages of the proposed framework in terms of accuracy and computational efficiency Ion et al (2021). Among them, we present here only he SEIQR model.…”

“…A probabilistic description of the distribution over the parameter space (called posterior) can be obtained using Bayes rule. As presented in Ion et al (2021), updating the posterior implies solving the CME and constructing the likelihood (conditional probability of observing the data given the underlying state of the system). Both of the steps are efficiently performed using the TT-format without being affected by the curse of dimensionality, since both the observation model and the CME operator can be computed directly in the TT-format.…”

“…Unfortunately, the computational cost grows exponentially with the number of species, due to the fact that the system states must be labeled explicitly to cast the CME into a ODE. A framework for performing Bayesian inference tasks for the parameter-dependent CME is suggested, by exploiting the so called tensor-train (TT) decomposition to approximate the joint distribution over the CME states and parameters (see Ion et al (2021)). For that purpose, we construct an explicit representation of the evolution operator in the TT format and show that it can be constructed without ever assembling the corresponding matrix.…”

Neural-network solutions to stochastic reaction networks

“…Given such an property, one can further conduct an estimation on choosing the proper N to control the truncation error [30], and then N can be adjusted accordingly for the neural network without losing the flexibility on learning the joint distribution. Besides, compared with the method based on the tensor network [33,34], the advantages of the VAN mainly include its generality. The neural-network ansatz is more flexible to represent complex probability distributions.…”

“…Another class of methods truncate the CME into a state space covering the majority of the probability distribution, including the finite state projection method [26,27], the sliding window method [28], and the ACME method [29,30] based on the finite-buffer technique [31]. Further advances employ the Krylov subspace approximation [32] and the tensor-train representations [33,34]. However, computational cost of these methods is still prohibitive to reach high accuracy when both the number (type) and counts of species become large [35].…”