Fragmentation process, pruning poset for rooted forests, and Möbius inversion

Abstract

We consider a discrete-time Markov chain, called fragmentation process, that describes a specific way of successively removing objects from a linear arrangement. The process arises in population genetics and describes the ancestry of the genetic material of individuals in a population experiencing recombination. We aim at the law of the process over time. To this end, we investigate sets of realisations of this process that agree with respect to a specific order of events and represent each such set by a rooted (binary) tree. The probability of each tree is, in turn, obtained by Möbius inversion on a suitable poset of all rooted forests that can be obtained from the tree by edge deletion; we call this poset the pruning poset. Dependencies within the fragments make it difficult to obtain explicit expressions for the probabilities of the trees. We therefore construct an auxiliary process for every given tree, which is i.i.d. over time, and which allows to give a pathwise construction of realisations that match the tree.

A probabilistic analysis of a continuous-time evolution in recombination

“…It can be observed that the structure of the formulae of P(F t (T I )) is reminiscent of an inclusion-exclusion. This is to be expected, as it has been studied and proven [6] to be the case for single-crossover. It remains an open problem to give the same interpretation in the general partition framework we have developed.…”

“…Our objective is to obtain a formula for the law of the fragmentation process, that is, to be able to compute P(X t = δ) for any δ ∈ Y * (G ρ ). In [6] a formula is obtained on the single-crossover case, that is when G ρ only contains partitions of the type {{1, .., .m}, {m + 1, ..., n} : 1 < m < n}. In this case the main idea is to code the embedded jump chain of the fragmentation process by taking advantage of the fact that once the sites split they become independent.…”

“…As said, the equation (1) was studied in a general framework in [5], and its solution was given by a recursion formula using tools from combinatorics and differential equation. Also this solution was studied via a Markov fragmentation process in the particular case of single-crossover in [6] and [4] by using techniques from probability and graph theory. It remained open if the general case can be also studied using this kind of tools to obtain an explicit solution instead of a recursive one.…”