Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models

Haas, Bénédicte; Miermont, Grégory; Pitman, Jim; Winkel, Matthias

doi:10.1214/07-aop377

Cited by 68 publications

(205 citation statements)

References 31 publications

(117 reference statements)

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“…A more general version of the previous statement (i.e., not restricted to binary trees) is shown in Haas et al (2008) by identifying the splitting rules with the transitions of a general fragmentation process (Bertoin 2006). This also allows the authors to study scaling limits of these random tree shapes.…”

Section: Sampling Consistencymentioning

confidence: 94%

Probabilistic models for the (sub)tree(s) of life

Lambert¹

2017

Braz. J. Probab. Stat.

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The goal of these lectures is to review some mathematical aspects of random tree models used in evolutionary biology to model species trees.We start with stochastic models of tree shapes (finite trees without edge lengths), culminating in the β-family of Aldous' branching models.We next introduce real trees (trees as metric spaces) and show how to study them through their contour, provided they are properly measured and ordered.We then focus on the reduced tree, or coalescent tree, which is the tree spanned by species alive at the same fixed time. We show how reduced trees, like any compact ultrametric space, can be represented in a simple way via the so-called comb metric. Beautiful examples of random combs include the Kingman coalescent and coalescent point processes.We end up displaying some recent biological applications of coalescent point processes to the inference of species diversification, to conservation biology and to epidemiology.

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Section: Sampling Consistencymentioning

confidence: 94%

Probabilistic models for the (sub)tree(s) of life

Lambert¹

2017

Braz. J. Probab. Stat.

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“…, are given by (11). This shows that the processΠ, which is specified by the requirement that its restriction to [n] coincides withΠ [n] , is a homogeneous multitype fragmentation with splitting rates (µ i ) i∈{1,...,k} .…”

Section: Sketch Of the Proofmentioning

confidence: 97%

“…As it has been mentioned above, it may be convenient to think of a type as a geometrical shape (see Figure 1 above for an example), but the type can also be used, for instance, to distinguish between active and inactive fragments in a frozen fragmentation (see [10] for a closely related notion in the setting of coalescents). Last but not least, it was observed recently by Haas et al [11] that homogeneous fragmentions bear close connexions with certain continuum random trees, a class of random fractal spaces which has been introduced by Aldous. It is likely that more generally, multitype fragmentations can be used to construct some multifractal continuum random trees, following the analysis developed in [11].…”

Section: Figure 1: Example Of a Fragmentation Of A Square Into Squarementioning

confidence: 99%

“…Last but not least, it was observed recently by Haas et al [11] that homogeneous fragmentions bear close connexions with certain continuum random trees, a class of random fractal spaces which has been introduced by Aldous. It is likely that more generally, multitype fragmentations can be used to construct some multifractal continuum random trees, following the analysis developed in [11].…”

Section: Figure 1: Example Of a Fragmentation Of A Square Into Squarementioning

confidence: 99%

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Homogenenous Multitype Fragmentations

Bertoin

2008

In and Out of Equilibrium 2

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Summary.A homogeneous mass-fragmentation, as it has been defined in [6], describes the evolution of the collection of masses of fragments of an object which breaks down into pieces as time passes. Here, we show that this model can be enriched by considering also the types of the fragments, where a type may represent, for instance, a geometrical shape, and can take finitely many values. In this setting, the dynamics of a randomly tagged fragment play a crucial role in the analysis of the fragmentation. They are determined by a Markov additive process whose distribution depends explicitly on the characteristics of the fragmentation. As applications, we make explicit the connexion with multitype branching random walks, and obtain multitype analogs of the pathwise central limit theorem and large deviation estimates for the empirical distribution of fragments.

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“…Let P 0 = pδ 1 +pδ 0 so that the random mean M (P (σ,1−σ) ) coincides in distribution with a limiting random variable arising in phylogenetic trees as stated in Theorem 20 of [32]. An expression for its probability density function can be deduced as follows…”

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confidence: 99%

Distributional properties of means of random probability measures

Lijoi¹,

Prünster²

2009

Statist. Surv.

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Abstract:The present paper provides a review of the results concerning distributional properties of means of random probability measures. Our interest in this topic has originated from inferential problems in Bayesian Nonparametrics. Nonetheless, it is worth noting that these random quantities play an important role in seemingly unrelated areas of research. In fact, there is a wealth of contributions both in the statistics and in the probability literature that we try to summarize in a unified framework. Particular attention is devoted to means of the Dirichlet process given the relevance of the Dirichlet process in Bayesian Nonparametrics. We then present a number of recent contributions concerning means of more general random probability measures and highlight connections with the moment problem, combinatorics, special functions, excursions of stochastic processes and statistical physics.

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Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models

Cited by 68 publications

References 31 publications

Probabilistic models for the (sub)tree(s) of life

Probabilistic models for the (sub)tree(s) of life

Homogenenous Multitype Fragmentations

Distributional properties of means of random probability measures

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