SMILE

Stochastic Models for the Inference of Life Evolution

Presentation

SMILE is an interdisciplinary research group gathering probabilists, statisticians, bio-informaticians and biologists.
SMILE is affiliated to the Stochastics and Biology group of LPSM (Lab of Probability, Statistics and Modeling) at Sorbonne Université (ex Université Pierre et Marie Curie Paris 06).
SMILE is hosted within the CIRB (Center for Interdisciplinary Research in Biology) at Collège de France.
SMILE is supported by Collège de France and CNRS.
Visit also our homepage at CIRB.

Directions

SMILE is hosted at Collège de France in the Latin Quarter of Paris. To reach us, go to 11 place Marcelin Berthelot (stations Luxembourg or Saint-Michel on RER B).
Our working spaces are rooms 107, 121 and 122 on first floor of building B1 (ask us for the code). Building B1 is facing you upon exiting the traversing hall behind Champollion's statue.

Contact

You can reach us by email (amaury.lambert - at - upmc.fr) or (smile - at - listes.upmc.fr).

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Publication

2018

Exchangeable coalescents, ultrametric spaces, nested interval-partitions: A unifying approach

Kingman's representation theorem (Kingman 1978) states that any exchangeable partition of \$$\mathbb{N}\$$ can be represented as a paintbox based on a random mass-partition. Similarly, any exchangeable composition (i.e.\ ordered partition of \$$\mathbb{N}\$$) can be represented as a paintbox based on an interval-partition (Gnedin 1997. Our first main result is that any exchangeable coalescent process (not necessarily Markovian) can be represented as a paintbox based on a random non-decreasing process valued in interval-partitions, called nested interval-partition, generalizing the notion of comb metric space introduced by Lambert & Uribe Bravo (2017) to represent compact ultrametric spaces. As a special case, we show that any \$$\Lambda\$$-coalescent can be obtained from a paintbox based on a unique random nested interval partition called \$$\Lambda\$$-comb, which is Markovian with explicit semi-group. This nested interval-partition directly relates to the flow of bridges of Bertoin & Le~Gall (2003). We also display a particularly simple description of the so-called evolving coalescent by a comb-valued Markov process. Next, we prove that any measured ultrametric space \$$U\$$, under mild measure-theoretic assumptions on \$$U\$$, is the leaf set of a tree composed of a separable subtree called the backbone, on which are grafted additional subtrees, which act as star-trees from the standpoint of sampling. Displaying this so-called weak isometry requires us to extend the Gromov-weak topology, that was initially designed for separable metric spaces, to non-separable ultrametric spaces. It allows us to show that for any such ultrametric space \$$U\$$, there is a nested interval-partition which is 1) indistinguishable from \$$U\$$ in the Gromov-weak topology; 2) weakly isometric to \$$U\$$ if \$$U\$$ has complete backbone; 3) isometric to \$$U\$$ if \$$U\$$ is complete and separable.

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