Direct statistical inference for finite Markov jump processes via the matrix exponential.

Area of Science:

• Computational Statistics
• Mathematical Biology
• Applied Probability

Background:

• Statistical inference for continuous-time Markov chains (CTMCs) is often complex due to computational challenges with large state spaces.
• Existing methods like particle Markov Chain Monte Carlo (MCMC) can be computationally intensive and difficult to implement.
• Rate matrices ($$) in CTMCs, especially those from reaction networks, are often sparse, offering potential for computational efficiency.

Purpose of the Study:

• To develop and demonstrate fast, robust, and accurate algorithms for evaluating the product of a vector and the exponential of a large, sparse rate matrix.
• To facilitate direct statistical inference for CTMCs, bypassing computationally expensive methods.
• To showcase the practical application of these algorithms in population genetics and epidemic modeling.

Main Methods:

• Implementation of novel algorithms based on efficient linear algebra tools that exploit matrix sparsity.
• Focus on the accurate computation of the matrix exponential of sparse rate matrices.
• Demonstration using a model for allele mixing in a population and a Susceptible-Infectious-Removed (SIR) epidemic model.

Main Results:

• The developed algorithms provide fast, robust, and accurate computation of the required matrix-vector products.
• The approach simplifies direct statistical inference for CTMCs with large state spaces.
• Successful application to biological models, including population genetics and epidemiology.

Conclusions:

• Efficient computational tools can significantly simplify statistical inference for large, sparse continuous-time Markov chains.
• This method offers a more accessible alternative to complex inference schemes like particle MCMC.
• The approach is broadly applicable to various scientific domains involving dynamic systems modeled by CTMCs.