A tractable transition function for the Wright-Fisher diffusion with selection via gamma duality

Abstract

The transition function of the Wright-Fisher diffusion with selection is a fundamental object for non-neutral evolutionary modelling and likelihood-based inference, but, unlike in the neutral case, existing representations are impractical or inefficient. Spectral approaches rely on truncation levels that grow with selection strength and the number of alleles, thus leading to rapidly increasing computational costs, while existing duality-based expansions lead to multi-type dual processes whose transition probabilities involve intractable normalising constants. In this paper we obtain a new transition-function representation for the multi-allelic Wright-Fisher diffusion with parent-independent mutation and genic selection that makes transition-law evaluation and simulation practically feasible in regimes where existing methods are costly or unavailable. The key tool is a new gamma duality, obtained by augmenting classical polynomial duality by an exponential tilting and yields a multivariate birth-death dual with fully explicit infinitesimal rates and an entrance-boundary construction. This is used to express the Wright-Fisher transition law as an explicit mixture of standard Dirichlet components, with mixing weights obtained from the dual started from its entrance boundary. We characterise the dual stationary distribution and its small-time behaviour, and develop practical and efficient sampling procedures for both transition and stationary laws, including a fast approximate stationary sampler with explicit error control. Numerical experiments illustrate substantial runtime gains over specialised methods in regimes where they apply, while our approach remains applicable for arbitrary numbers of alleles and selection coefficients.