Filtering the Wright-Fisher diffusion

TL;DR

This paper develops an exact filtering method for a Wright-Fisher diffusion process observed indirectly through noisy measurements, expressing the filtering distributions as finite mixtures of parametric distributions.

Contribution

It introduces a computable filtering framework for Wright-Fisher diffusions with specific observation models, enabling exact calculations of filtering, prediction, and smoothing distributions.

Findings

Filtering distributions are finite mixtures of parametric distributions.

The method allows exact computation of filtering, prediction, and smoothing.

The number of mixture components varies over iterations.

Abstract

We consider a Wright-Fisher diffusion (x(t)) whose current state cannot be observed directly. Instead, at times t1 < t2 < . . ., the observations y(ti) are such that, given the process (x(t)), the random variables (y(ti)) are independent and the conditional distribution of y(ti) only depends on x(ti). When this conditional distribution has a specific form, we prove that the model ((x(ti), y(ti)), i 1) is a computable filter in the sense that all distributions involved in filtering, prediction and smoothing are exactly computable. These distributions are expressed as finite mixtures of parametric distributions. Thus, the number of statistics to compute at each iteration is finite, but this number may vary along iterations.