Generalized Markov branching trees. (English) Zbl 1425.60032
Summary: Motivated by the gene tree/species tree problem from statistical phylogenetics, we extend the class of Markov branching trees to a parametric family of distributions on fragmentation trees that satisfies a generalized Markov branching property. The main theorems establish important statistical properties of this model, specifically necessary and sufficient conditions under which a family of trees can be constructed consistently as sample size grows. We also consider the question of attaching random edge lengths to these trees.
MSC:
| 60G09 | Exchangeability for stochastic processes |
| 60G05 | Foundations of stochastic processes |
| 60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |
Keywords:
Markov branching tree; exchangeable random partition; exchangeable fragmentation; beta-splitting model; coalescent process; hidden Markov modelReferences:
| [1] | Abraham, R.,Delmas, J.-F. and He, H. (2012).Pruning Galton‒Watson trees and tree-valued Markov processes.Ann. Inst. H. Poincaré Prob. Statist.48,688-705. · Zbl 1256.60028 |
| [2] | Aldous, D. (1996).Probability distributions on cladograms.In Random Discrete Structures(IMA Vol. Math. Appl. 76),Springer,New York,pp.1-18. · Zbl 0841.92015 |
| [3] | Aldous, D. (1998).Tree-valued Markov chains and Poisson Galton‒Watson distributions.In Microsurveys in Discrete Probability(DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 41),American Mathematical Society,Providence, RI,pp.1-20. · Zbl 0913.60067 |
| [4] | Aldous, D. and Pitman, J. (1998).Tree-valued Markov chains derived from Galton‒Watson processes.Ann. Inst. H. Poincaré Statist.34,637-686. · Zbl 0917.60082 |
| [5] | Aldous, D.,Krikun, M. and Popovic, L. (2008).Stochastic models for phylogenetic trees on higher-order taxa.J. Math. Biol.56,525-557. · Zbl 1141.92027 |
| [6] | Berestycki, N. and Pitman, J. (2007).Gibbs distributions for random partitions generated by a fragmentation process.J. Statist. Phys.127,381-418. · Zbl 1126.82013 |
| [7] | Bertoin, J. (1996).Lévy Processes.Cambridge University Press. · Zbl 0861.60003 |
| [8] | Bertoin, J. (2006).Random Fragmentation and Coagulation Processes(Camb. Stud. Adv. Math. 102).Cambridge University Press. · Zbl 1107.60002 |
| [9] | Burke, C. J. and Rosenblatt, M. (1958).A Markovian function of a Markov chain.Ann. Math. Statist.29,1112-1122. · Zbl 0100.34402 |
| [10] | Crane, H. (2013).Consistent Markov branching trees with discrete edge lengths.Electron. Commun. Prob.18,14pp. · Zbl 1306.60126 |
| [11] | Crane, H. (2013).Some algebraic identities for the α-permanent.Linear Algebra Appl.439,3445-3459. · Zbl 1283.15026 |
| [12] | Crane, H. (2014).The cut-and-paste process.Ann. Prob.42,1952-1979. · Zbl 1317.60034 |
| [13] | Crane, H. (2015).Clustering from categorical data sequences.J. Amer. Statist. Assoc.110,810-823. · Zbl 1373.62304 |
| [14] | Crane, H. (2015).Generalized Ewens‒Pitman model for Bayesian clustering.Biometrika102,231-238. · Zbl 1345.62089 |
| [15] | Crane, H. (2016).The ubiquitous Ewens sampling formula.Statist. Sci.31,1-19.(Rejoinder: 31,37-39.) · Zbl 1442.60010 |
| [16] | Evans, S. N. (2008).Probability and Real Trees(Lecture Notes Math. 1920).Springer,Berlin. |
| [17] | Evans, S. N. and Winter, A. (2006).Subtree prune and regraft: a reversible real tree-valued Markov process.Ann. Prob.34,918-961. · Zbl 1101.60054 |
| [18] | Ewens, W. J. (1972).The sampling theory of selectively neutral alleles.Theoret. Pop. Biol.3,87-112. · Zbl 0245.92009 |
| [19] | Felsenstein, J. (2004).Inferring Phylogenies.Sinauer,Sunderland. |
| [20] | Haas, B. and Miermont, G. (2012).Scaling limits of Markov branching trees with applications to Galton‒Watson and random unordered trees.Ann. Prob.40,2589-2666. · Zbl 1259.60033 |
| [21] | Haas, B.,Miermont, G.,Pitman, J. and Winkel, M. (2008).Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models.Ann. Prob.36,1790-1837. · Zbl 1155.92033 |
| [22] | Hudson, R. (1983).Properties of a neutral allele model with intragenic recombination.Theoret. Pop. Biol.23,183-201. · Zbl 0505.62090 |
| [23] | Kingman, J. F. C. (1982).The coalescent.Stoch. Process. Appl.13,235-248. · Zbl 0491.60076 |
| [24] | McCullagh, P.,Pitman, J. and Winkel, M. (2008).Gibbs fragmentation trees.Bernoulli14,988-1002. · Zbl 1158.60373 |
| [25] | McVean, G. and Cardin, N. (2005).Approximating the coalescent with recombination.Phil. Trans. R. Soc. London360,1387-1393. |
| [26] | Pitman, J. (1995).Exchangeable and partially exchangeable random partitions.Prob. Theory Relat. Fields102,145-158. · Zbl 0821.60047 |
| [27] | Pitman, J. (2006).Combinatorial Stochastic Processes(Lecture Notes Math. 1875).Springer,Berlin. · Zbl 1103.60004 |
| [28] | Tajima, F. (1983).Evolutionary relationship of DNA sequences in finite populations.Genetics105,437-460. |
| [29] | Wang, Y.et al. (2014).A new method for modeling coalescent processes with recombination.BMC Bioinformatics15,12pp. |
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