Dario Spanò

Diffusion processes and coalescent trees

We dedicate this paper to Sir John Kingman on his 70th Birthday. In modern mathematical population genetics the ancestral history of a population of genes back in time is described by John Kingman’s coalescent tree. Classical and modern approachesmodel gene frequencies by diffusion processes. This paper, which is partly a review, discusses how coalescent processes are dual to diffusion processes in an analytic and probabilistic sense. Bochner (1954) and Gasper (1972) were interested in characterizations of processes with Beta stationary distributions and Jacobi polynomial eigenfunctions. We discuss the connection with Wright-Fisher diffusions and the characterization of these processes. Subordinated Wright-Fisher diffusions are of this type. An Inverse Gaussian subordinator is interesting and important in subordinated Wright-Fisher diffusions and is related to the Jacobi Poisson Kernel in orthogonal polynomial theory. A related time-subordinated forest of non-mutant edges in the Kingman coalescent is novel.

Orthogonal polynomial kernels and canonical correlations for Dirichlet measures

We consider a multivariate version of the so-called Lancaster problem of characterizing canonical correlation coefficients of symmetric bivariate distributions with identical marginals and orthogonal polynomial expansions. The marginal distributions examined in this paper are the Dirichlet and the Dirichlet multinomial distribution, respectively, on the continuous and the N-discrete d-dimensional simplex. Their infinite-dimensional limit distributions, respectively, the Poisson–Dirichlet distribution and Ewens’s sampling formula, are considered as well. We study, in particular, the possibility of mapping canonical correlations on the d-dimensional continuous simplex (i) to canonical correlation sequences on the d+1-dimensional simplex and/or (ii) to canonical correlations on the discrete simplex, and vice versa. Driven by this motivation, the first half of the paper is devoted to providing a full characterization and probabilistic interpretation of n-orthogonal polynomial kernels (i.e., sums of products of orthogonal polynomials of the same degree n) with respect to the mentioned marginal distributions. We establish several identities and some integral representations which are multivariate extensions of important results known for the case d=2 since the 1970s. These results, along with a common interpretation of the mentioned kernels in terms of dependent Polya urns, are shown to be key features leading to several non-trivial solutions to Lancaster’s problem, many of which can be extended naturally to the limit as d→∞.

Multivariate Jacobi and Laguerre polynomials, infinite-dimensional extensions, and their probabilistic connections with multivariate Hahn and Meixner polynomials

Multivariate versions of classical orthogonal polynomials such as Jacobi, Hahn, Laguerre, Meixner are reviewed and their connection explored by adopting a probabilistic approach. Hahn and Meixner polynomials are interpreted as posterior mixtures of Jacobi and Laguerre polynomials, respectively. By using known properties of Gamma point processes and related transformations, a new infinite-dimensional version of Jacobi polynomials is constructed with respect to the size-biased version of the Poisson-Dirichlet weight measure and to the law of the Gamma point process from which it is derived.

A vector of Dirichlet processes

Random probability vectors are of great interest especially in view of their application to statistical inference. Indeed, they can be used for determining the de Finetti mixing measure in the representation of the law of a partially exchangeable array of random elements taking values in a separable and complete metric space. In this paper we describe a construction of a vector of Dirichlet processes based on the normalization of completely random measures that are jointly infinitely divisible. After deducing the form of the Laplace exponent of the vector of the gamma completely random measures, we study some of their distributional properties. Our attention particularly focuses on the dependence structure and the specific partition probability function induced by the proposed vector.

Exact simulation of the Wright–Fisher diffusion

The Wright-Fisher family of diffusion processes is a widely used class of evolutionary models. However, simulation is difficult because there is no known closed-form formula for its transition function. In this article we demonstrate that it is in fact possible to simulate exactly from a broad class of Wright-Fisher diffusion processes and their bridges. For those diffusions corresponding to reversible, neutral evolution, our key idea is to exploit an eigenfunction expansion of the transition function; this approach even applies to its infinite-dimensional analogue, the Fleming-Viot process. We then develop an exact rejection algorithm for processes with more general drift functions, including those modelling natural selection, using ideas from retrospective simulation. Our approach also yields methods for exact simulation of the moment dual of the Wright-Fisher diffusion, the ancestral process of an infinite-leaf Kingman coalescent tree. We believe our new perspective on diffusion simulation holds promise for other models admitting a transition eigenfunction expansion.

Wright-Fisher construction of the two-parameter Poisson-Dirichlet diffusion

The two-parameter Poisson–Dirichlet diffusion, introduced in 2009 by Petrov, extends the infinitely-many-neutral-alleles diffusion model, related to Kingman’s one-parameter Poisson–Dirichlet distribution and to certain Fleming–Viot processes. The additional parameter has been shown to regulate the clustering structure of the population, but is yet to be fully understood in the way it governs the reproductive process. Here we shed some light on these dynamics by formulating a K-allele Wright–Fisher model for a population of size N, involving a uniform mutation pattern and a specific state-dependent migration mechanism. Suitably scaled, this process converges in distribution to a K-dimensional diffusion process as N → ∞. Moreover, the descending order statistics of the K-dimensional diffusion converge in distribution to the two-parameter Poisson–Dirichlet diffusion as K → ∞. The choice of the migration mechanism depends on a delicate balance between reinforcement and redistributive effects. The proof of convergence to the infinite-dimensional diffusion is nontrivial because the generators do not converge on a core. Our strategy for overcoming this complication is to prove a priori that in the limit there is no “loss of mass”, i.e., that, for each limit point of the sequence of finite-dimensional diffusions (after a reordering of components by size), allele frequencies sum to one.

Conjugacy properties of time-evolving Dirichlet and gamma random measures

We extend classic characterisations of posterior distributions under Dirichlet process and gamma random measures priors to a dynamic framework. We consider the problem of learning, from indirect observations, two families of time-dependent processes of interest in Bayesian nonparametrics: the first is a dependent Dirichlet process driven by a Fleming–Viot model, and the data are random samples from the process state at discrete times; the second is a collection of dependent gamma random measures driven by a Dawson–Watanabe model, and the data are collected according to a Poisson point process with intensity given by the process state at discrete times. Both driving processes are diffusions taking values in the space of discrete measures whose support varies with time, and are stationary and reversible with respect to Dirichlet and gamma priors respectively. A common methodology is developed to obtain in closed form the time-marginal posteriors given past and present data. These are shown to belong to classes of finite mixtures of Dirichlet processes and gamma random measures for the two models respectively, yielding conjugacy of these classes to the type of data we consider. We provide explicit results on the parameters of the mixture components and on the mixing weights, which are time-varying and drive the mixtures towards the respective priors in absence of further data. Explicit algorithms are provided to recursively compute the parameters of the mixtures. Our results are based on the projective properties of the signals and on certain duality properties of their projections.

Inference and rare event simulation for stopped Markov processes via reverse-time sequential Monte Carlo

We present a sequential Monte Carlo algorithm for Markov chain trajectories with proposals constructed in reverse time, which is advantageous when paths are conditioned to end in a rare set. The reverse time proposal distribution is constructed by approximating the ratio of Green’s functions in Nagasawa’s formula. Conditioning arguments can be used to interpret these ratios as low-dimensional conditional sampling distributions of some coordinates of the process given the others. Hence, the difficulty in designing SMC proposals in high dimension is greatly reduced. Empirically, our method outperforms an adaptive multilevel splitting algorithm in three examples: estimating an overflow probability in a queueing model, the probability that a diffusion follows a narrowing corridor, and the initial location of an infection in an epidemic model on a network.

Wright-Fisher diffusion bridges

The trajectory of the frequency of an allele which begins at x at time 0 and is known to have frequency z at time T can be modelled by the bridge process of the Wright-Fisher diffusion. Bridges when x=z=0 are particularly interesting because they model the trajectory of the frequency of an allele which appears at a time, then is lost by random drift or mutation after a time T. The coalescent genealogy back in time of a population in a neutral Wright-Fisher diffusion process is well understood. In this paper we obtain a new interpretation of the coalescent genealogy of the population in a bridge from a time t∈(0,T). In a bridge with allele frequencies of 0 at times 0 and T the coalescence structure is that the population coalesces in two directions from t to 0 and t to T such that there is just one lineage of the allele under consideration at times 0 and T. The genealogy in Wright-Fisher diffusion bridges with selection is more complex than in the neutral model, but still with the property of the population branching and coalescing in two directions from time t∈(0,T). The density of the frequency of an allele at time t is expressed in a way that shows coalescence in the two directions. A new algorithm for exact simulation of a neutral Wright-Fisher bridge is derived. This follows from knowing the density of the frequency in a bridge and exact simulation from the Wright-Fisher diffusion. The genealogy of the neutral Wright-Fisher bridge is also modelled by branching Pólya urns, extending a representation in a Wright-Fisher diffusion. This is a new very interesting representation that relates Wright-Fisher bridges to classical urn models in a Bayesian setting.

Bayesian non-parametric inference for Λ-coalescents : posterior consistency and a parametric method

We investigate Bayesian non-parametric inference of the