Filtering coupled Wright-Fisher diffusions

TL;DR

This paper develops a filtering method for coupled Wright-Fisher diffusions, enabling inference of allele frequencies over time in multi-locus genetic models using duality techniques and mixture distributions.

Contribution

It introduces a novel filtering approach for coupled Wright-Fisher diffusions, deriving explicit mixture representations and algorithms for sequential inference and parameter estimation.

Findings

Derived countable mixture representations of filtering distributions.

Provided algorithms for sequential updating of mixture weights.

Illustrated the method with synthetic data simulations.

Abstract

Coupled Wright-Fisher diffusions have been recently introduced to model the temporal evolution of finitely-many allele frequencies at several loci. These are vectors of multidimensional diffusions whose dynamics are weakly coupled among loci through interaction coefficients, which make the reproductive rates for each allele depend on its frequencies at several loci. Here we consider the problem of filtering a coupled Wright-Fisher diffusion with parent-independent mutation, when this is seen as an unobserved signal in a hidden Markov model. We assume individuals are sampled multinomially at discrete times from the underlying population, whose type configuration at the loci is described by the diffusion states, and adapt recently introduced duality methods to derive the filtering and smoothing distributions. These respectively provide the conditional distribution of the diffusion states given past data, and that conditional on the entire dataset, and are key to be able to perform parameter inference on models of this type. We show that for this model these distributions are countable mixtures of tilted products of Dirichlet kernels, and describe their mixing weights and how these can be updated sequentially. The evaluation of the weights involves the transition probabilities of the dual process, which are not available in closed form. We lay out pseudo codes for the implementation of the algorithms, discuss how to handle the unavailable quantities, and briefly illustrate the procedure with synthetic data.