Filtering coupled Wright-Fisher diffusions.

Area of Science:

• Population Genetics
• Mathematical Biology
• Stochastic Processes

Background:

• Coupled Wright-Fisher diffusions model allele frequency evolution across multiple loci.
• These models are complex, with weakly coupled dynamics influenced by interaction coefficients.
• Filtering these diffusions within a hidden Markov model framework is crucial for parameter inference.

Purpose of the Study:

• To derive filtering and smoothing distributions for coupled Wright-Fisher diffusions with parent-independent mutation.
• To adapt duality methods for analyzing these unobserved diffusion states.
• To enable parameter inference in complex genetic models.

Main Methods:

• Utilized recently introduced duality methods to derive filtering and smoothing distributions.
• Modeled population sampling via multinomial distributions at discrete time points.
• Developed algorithms based on mixtures of tilted Dirichlet kernels and sequential updating of weights.

Main Results:

• Derived filtering and smoothing distributions as countable mixtures of tilted products of Dirichlet kernels.
• Described the structure of mixing weights and their sequential update mechanisms.
• Provided pseudo-code for algorithm implementation and discussed handling of intractable quantities.

Conclusions:

• The developed filtering and smoothing distributions are key for parameter inference in coupled Wright-Fisher models.
• The method offers a way to analyze complex genetic drift and selection scenarios.
• The study provides a practical algorithmic framework, illustrated with synthetic data.