Stochastic Models for the Inference of Life Evolution


SMILE is an interdisciplinary research group gathering probabilists, statisticians, bio-informaticians and biologists.
SMILE is affiliated to the Stochastics and Biology group of LPMA (Lab of Probability and Random Models) at UPMC (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.


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.


You can reach us by email (amaury.lambert - at - or (smile - at -

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Time Reversal Dualities for some Random Forests

We consider a random forest \$$\mathcal{F}^*\$$, defined as a sequence of i.i.d. birth-death (BD) trees, each started at time 0 from a single ancestor, stopped at the first tree having survived up to a fixed time \$$T\$$. We denote by \$$\left(\xi^*_t, 0\leq t \leq T \right)\$$ the population size process associated to this forest, and we prove that if the BD trees are supercritical, then the time-reversed process \$$\left(\xi^*_{T-t}, 0 \leq t \leq T\right)\$$, has the same distribution as \$$\left(\widetilde\xi^*_t, 0 \leq t \leq T\right)\$$, the corresponding population size process of an equally defined forest \$$\widetilde{\mathcal{F}}^*\$$, but where the underlying BD trees are subcritical, obtained by swapping birth and death rates or equivalently, conditioning on ultimate extinction. We generalize this result to splitting trees (i.e. life durations of individuals are not necessarily exponential), provided that the i.i.d. lifetimes of the ancestors have a specific explicit distribution, different from that of their descendants. The results are based on an identity between the contour of these random forests truncated up to \$$T\$$ and the duality property of L\'evy processes. This identity allows us to also derive other useful properties such as the distribution of the population size process conditional on the reconstructed tree of individuals alive at \$$T\$$, which has potential applications in epidemiology.



The species problem from the modeler's point of view

How to define and delineate species is a long-standing question sometimes called the species problem. In modern systematics, species should be groups of individuals sharing characteristics inherited from a common ancestor which distinguish them from other such groups. A good species definition should thus satisfy the following three desirable properties: (A) Heterotypy between species, (B) Homotypy within species and (E) Exclusivity, or monophyly, of each species. In practice, systematists seek to discover the very traits for which these properties are satisfied, without the a priori knowledge of the traits which have been responsible for differentiation and speciation nor of the true ancestral relationships between individuals. Here to the contrary, we focus on individual-based models of macro-evolution, where both the differentiation process and the population genealogies are explicitly modeled, and we ask: How and when is it possible, with this significant information, to delineate species in a way satisfying most or all of the three desirable properties (A), (B) and (E)? Surprisingly, despite the popularity of this modeling approach in the last two decades, there has been little progress or agreement on answers to this question. We prove that the three desirable properties are not in general satisfied simultaneously, but that any two of them can. We show mathematically the existence of two natural species partitions: the finest partition satisfying (A) and (E) and the coarsest partition satisfying (B) and (E). For each of them, we propose a simple algorithm to build the associated phylogeny. We stress that these two procedures can readily be used at a higher level, namely to cluster species into monophyletic genera. The ways we propose to phrase the species problem and to solve it should further refine models and our understanding of macro-evolution.

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