SUBTREE PRUNE AND REGRAFT: A REVERSIBLE REAL TREE-VALUED MARKOV PROCESS

By Steven N. Evans, A. Winter. 2005

83 citations recorded by the source corpus.

We use Dirichlet form methods to construct and analyze a reversible Markov process, the stationary distribution of which is the Brownian continuum random tree. This process is inspired by the subtree prune and regraft (SPR) Markov chains that appear in phylogenetic analysis. A key technical ingredient in this work is the use of a novel Gromov–Hausdorff type distance to metrize the space whose elements are compact real trees equipped with a probability measure. Also, the investigation of the Dirichlet form hinges on a new path decomposition of the Brownian excursion.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/29f35ae41ee8c4abc624ac55ed2a27f3d68e86bd

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Citations in the corpus

Growth of Lévy trees (2005, result): “We refer to [7–9] for general results concerning real trees, [31,32] for applications of real trees to group theory and to [10,11,13,14,16] and also [20] for a probabilistic use of real trees.”
A new family of Markov branching trees: the alpha-gamma model (2008, methodology): “Several methods to formalize this have been developed [2, 10, 13].”
Scaling limits of Markov branching trees with applications to Galton–Watson and random unordered trees (2010, methodology): “A simple adaptation of the results of [21] and Section 6 in [29] (in order to take into account the particular role of the distinguished point ρ) shows that:”
The local weak limit of the minimum spanning tree of the complete graph (2013, background): “tle risk of ambiguity. If X = (X;d;) then we write mass(X) = (X). There are several natural distances on compact measured metric spaces that generalize the Gromov{Hausdor distance, see for instance [28,74,52,1]. The presentation we adopt is still dierent from these references, but closest in spirit ”
Hyperconvexity and tight-span theory for diversities (2010, background): “See [37] for a related measure on real trees.”
Random real trees (2006, background): “This makes it very natural to consider random variables or even random processes taking values in the space T ([27] gives a particularly nice example of such a process, which combines root growth and regrafting and converges in distribution to the CRT, see also [28] for further developments along th”
Spinal partitions and invariance under re-rooting of continuum random trees (2007, methodology): “An alternative formalism can be considered, via the set of equivalence classes of compact rooted R-trees endowed with the Gromov–Hausdorff distance, as developed in [16, 17].”
Regenerative tree growth: Binary self-similar continuum random trees and Poisson--Dirichlet compositions (2008, methodology): “More specifically, it is shown in [9] that the distance function”
Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models (2006, methodology): “A natural possibility would be to use Evans’ and Winter’s separable and complete metric structure [18] on the space of ‘weighted R-trees’, although we would have to incorporate the fact that our trees are rooted.”
Tree-valued Fleming–Viot dynamics with mutation and selection (2011, background): “Note that the resulting processes are among the first tree-valued stochastic processes in the literature [but see also Zambotti (2001, 2002, 2003), Evans, Pitman and Winter (2006), Evans and Winter (2006), Evans and Lidman (2007)].”