Exact simulation of coupled Wright–Fisher diffusions

Abstract

In this paper an exact rejection algorithm for simulating paths of the coupled Wright–Fisher diffusion is introduced. The coupled Wright–Fisher diffusion is a family of multivariate Wright–Fisher diffusions that have drifts depending on each other through a coupling term and that find applications in the study of networks of interacting genes. The proposed rejection algorithm uses independent neutral Wright–Fisher diffusions as candidate proposals, which are only needed at a finite number of points. Once a candidate is accepted, the remainder of the path can be recovered by sampling from neutral multivariate Wright–Fisher bridges, for which an exact sampling strategy is also provided. Finally, the algorithm’s complexity is derived and its performance demonstrated in a simulation study.

Unbiased likelihood-based estimation of Wright-Fisher diffusion processes

“…The rejection scheme is based on Lemma 1 above and uses candidate paths distributed according to WF , , , for which an exact simulation procedure exists, see Algorithms 4 and 5 in [8] and Algorithms 5 and 6 in [9] for a multivariate generalization. Note that, if assumptions 3. where we recognize the right hand side as…”

“…and = + , rendering an exact computation of E [•] impossible. However, there exists an exact sampling strategy for drawing samples from , see Algorithm 2 in [8] and Algorithm 3 in [9], and, given , = 1, . .…”

“…Following the recent results on exact simulation for coupled Wright-Fisher diffusions [9], one can write an equivalent to Lemma 1 above, and build an unbiased estimator for each corresponding likelihood contribution as in (5). Given = 1, .…”

“…Our estimating approach is based on the Simultaneous Acceptance Method (SAM) presented in [7]. In short, our method provides an unbiased estimator of the likelihood of model ( 1), based on Monte Carlo samples drawn from the recently developed exact simulation algorithms for Wright-Fisher diffusions, see [8], [9]. Uniform convergence results of our Monte Carlo estimator to the true likelihood function ensure convergence of the maximizers and are used to show consistency of our maximum likelihood estimator (MLE).…”

Statistical inference for stochastic differential equations

“…With EA3, exact simulation of a very large class of univariate diffusions is possible, with minimal assumptions on , in addition to the ones needed for existence of a solution. The work paved by EA1, EA2, and EA3 has led to a collection of novel algorithms for exact simulation of jump diffusions (Broadie & Kaya, 2006; Casella & Roberts, 2011; Gonçalves & Roberts, 2014; Pollock et al, 2016), the Wright–Fisher diffusion (Garca‐Pareja et al, 2019; Jenkins & Spano, 2017; Zhao et al, 2014), and others.…”