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Statistical inference for continuous-time Markov chains is simplified using efficient algorithms for large, sparse rate matrices. This approach enables faster, more accurate analysis of complex systems like population genetics and epidemic modeling.

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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.