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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 approaches model gene frequencies by diffusion processes. This paper, which is partly a review, discusses how coalescent… (More)

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… (More)

We consider a random partition Π of N = {1, 2, . . .} such that, for each n, its restriction Πn to [n] = {1, . . . , n} is given by an exchangeable Gibbs partition with parameters α, V for α ∈ (−∞, 1] and V = (Vn,k) defined recursively by setting V1,1 = 1 and Vn,k = (n− αk)Vn+1,k + Vn+1,k+1 k ≤ n = 1, 2, . . . (Gnedin and Pitman 2006). By ranking the blocks… (More)

The Wright-Fisher family of diffusion processes is a class of evolutionary models widely used in population genetics, with applications also in finance and Bayesian statistics. Simulation and inference from these diffusions is therefore of widespread interest. However, simulating a Wright-Fisher diffusion is difficult because there is no known closed-form… (More)

- Jochen Blath, Adrián González Casanova, Noemi Kurt, Dario Spanò
- J. Applied Probability
- 2013

We present a new model for seed banks, where direct ancestors of individuals may have lived in the near as well as the very far past. The classical Wright-Fisher model, as well as a seed bank model with bounded age distribution considered by Kaj, Krone and Lascoux (2001) are special cases of our model. We discern three parameter regimes of the seed bank age… (More)

- Jere Koskela, Paul Jenkins, Dario Spanò
- J. Applied Probability
- 2015

Full likelihood inference under Kingman’s coalescent is a computationally challenging problem to which importance sampling (IS) and the product of approximate conditionals (PAC) method have been applied successfully. Both methods can be expressed in terms of families of intractable conditional sampling distributions (CSDs), and rely on principled… (More)

We investigate Bayesian non-parametric inference for the Λ-measure of Λ-coalescent processes parametrised by probability measures on the unit interval and provide an implementable, provably consistent MCMC inference algorithm. We give verifiable criteria on the prior for posterior consistency when observations form a time series, and prove that any… (More)

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… (More)

We develop particle Gibbs samplers for static-parameter estimation in discretely observed piecewise deterministic process (PDPs). PDPs are stochastic processes that jump randomly at a countable number of stopping times but otherwise evolve deterministically in continuous time. A sequential Monte Carlo (SMC) sampler for filtering in PDPs has recently been… (More)