2007, Birkner, Matthias, Blath, Jochen

One of the main problems in mathematical genetics is the inference of evolutionary parameters of a population (such as the mutation rate) based on the observed genetic types in a finite DNA sample. If the population model under consideration is in the domain of attraction of a classical Fleming-Viot process, then the standard means to describe the corresponding genealogy is Kingman's coalescent. For this process, powerful inference methods are well-established. An important feature of this class of models is, roughly speaking, that the number of offspring of each individual is small when compared to the total population size. Recently, more general population models have been studied, in particular in the domain of attraction of so-called generalised Lambda Fleming-Viot processes, as well as their (dual) genealogies, given by the so-called Lambda-coalescents. Moreover, Eldon & Wakeley (2006) have provided evidence that such more general coalescents, which allow m ultiple collisions, might actually be more adequate to describe real populations with extreme reproductive behaviour, in particular many marine species. In this paper, we extend methods of Ethier & Griffiths (1987) and Griffiths & Tavaré (1994) to obtain a likelihood based inference method for general Lambda-coalescents. In particular, we obtain a method to compute (approximate) likelihood surfaces for the observed type probabilities of a given sample. We argue that within the (vast) family of Lambda-coalescents, the parametrisable sub-family of Beta$(2-alpha,alpha)$-coalescents, where $alpha in (1,2]$, are of particular biological relevance. We apply our method in this case to simulated and real data (taken from Árnason (2004)). We conclude that for populations with extreme reproductive behaviour, the Kingman-coalescent as standard model might have to be replaced by more general coalescents, in particular by Beta$(2-alpha,alpha)$-coalescents.

2007, Birkner, Matthias, Blath, Jochen

We show how Donnelly and Kurtz' (modified) lookdown construction for measure-valued processes can be used to analyse the longterm- and scaling properties of spatially stable generalised $Lambda$-Fleming Viot processes, exhibiting a rare ``natural'' example of a scaling family converging in f.d.d. sense, but not in any of Skorohod's topologies on path space. This completes results of Fleischmann and Wachtel (2004) about the spatial Neveu process and complements results of Dawson and Hochberg (1982) about the classical Fleming Viot process. The lookdown construction provides an elegant machinery and clear intuition to describe the path properties of the family in terms of a ``flicker effect'', clarifying ``what can go wrong.''

2007, Birkner, Matthias, Blath, Jochen

One of the central problems in mathematical genetics is the inference of evolutionary parameters of a population (such as the mutation rate) based on the observed genetic types in a finite DNA sample. If the population model under consideration is in the domain of attraction of the classical Fleming-Viot process, such as the Wright-Fisher- or the Moran model, then the standard means to describe its genealogy is Kingman's coalescent. For this coalescent process, powerful inference methods are well-established. An important feature of the above class of models is, roughly speaking, that the number of offspring of each individual is small when compared to the total population size, and hence all ancestral collisions are binary only. Recently, more general population models have been studied, in particular in the domain of attraction of so-called generalised $Lambda$-Fleming-Viot processes, as well as their (dual) genealogies, given by the so-called $Lambda$-coalescents, which allow multiple collisions. Moreover, Eldon and Wakeley (2006) provide evidence that such more general coalescents might actually be more adequate to describe real populations with extreme reproductive behaviour, in particular many marine species. In this paper, we extend methods of Ethier and Griffiths (1987) and Griffiths and Tavaré (1994, 1995) to obtain a likelihood based inference method for general $Lambda$-coalescents. In particular, we obtain a method to compute (approximate) likelihood surfaces for the observed type probabilities of a given sample. We argue that within the (vast) family of $Lambda$-coalescents, the parametrisable sub-family of Beta$(2-alpha, alpha)$-coalescents, where $alpha in (1,2]$, are of particular relevance. We illustrate our method using simulated datasets, thus obtaining maximum-likelihood estimators of mutation and demographic parameters.

2008, Birkner, Matthias, Blath, Jochen, Möhle, Martin, Steinrücken, Matthias, Tams, Johanna

Let $Lambda$ be a finite measure on the unit interval. A $Lambda$-Fleming-Viot process is a probability measure valued Markov process which is dual to a coalescent with multiple collisions ($Lambda$-coalescent) in analogy to the duality known for the classical Fleming Viot process and Kingman's coalescent, where $Lambda$ is the Dirac measure in $0$. We explicitly construct a dual process of the coalescent with simultaneous multiple collisions ($Xi$-coalescent) with mutation, the $Xi$-Fleming-Viot process with mutation, and provide a representation based on the empirical measure of an exchangeable particle system along the lines of Donnelly and Kurtz (1999). We establish pathwise convergence of the approximating systems to the limiting $Xi$-Fleming-Viot process with mutation. An alternative construction of the semigroup based on the Hille-Yosida theorem is provided and various types of duality of the processes are discussed. In the last part of the paper a populations is considered which undergoes recurrent bottlenecks. In this scenario, non-trivial $Xi$-Fleming-Viot processes naturally arise as limiting models.

2007, Birkner, Matthias, Blath, Jochen

We review recent progress in the understanding of the interplay between population models, measure-valued diffusions, general coalescent processes and inference methods for evolutionary parameters in population genetics. Along the way, we will discuss the powerful and intuitive (modified) lookdown construction of Donnelly and Kurtz, Pitman's and Sagitov's $Lambda$-coalescents as well as recursions and Monte Carlo schemes for likelihood-based inference of evolutionary parameters based on observed genetic types.