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Reversible Markov structures on divisible set partitions

Part of: Markov processes Combinatorial probability Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  30 March 2016

Harry Crane  and
Peter McCullagh
Harry Crane*
Affiliation:
Rutgers University
Peter McCullagh*
Affiliation:
University of Chicago
*
∗ Postal address: Department of Statistics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA. Email address: [email protected]
∗∗ Postal address: Department of Statistics, University of Chicago, Eckhart Hall, 5734 S. University Avenue, Chicago, IL 60637, USA.
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Abstract

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We study k-divisible partition structures, which are families of random set partitions whose block sizes are divisible by an integer k = 1, 2, …. In this setting, exchangeability corresponds to the usual invariance under relabeling by arbitrary permutations; however, for k > 1, the ordinary deletion maps on partitions no longer preserve divisibility, and so a random deletion procedure is needed to obtain a partition structure. We describe explicit Chinese restaurant-type seating rules for generating families of exchangeable k-divisible partitions that are consistent under random deletion. We further introduce the notion of Markovian partition structures, which are ensembles of exchangeable Markov chains on k-divisible partitions that are consistent under a random process of Markovian deletion. The Markov chains we study are reversible and refine the class of Markov chains introduced in Crane (2011).

Keywords

Markovian partition structureexchangeable partition structureEwens-Pitman partitionChinese restaurant processdivisible partitiongroup-divisible association scheme

MSC classification

Primary: 60C05: Combinatorial probability 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60B99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 3 , September 2015 , pp. 622 - 635
Copyright
Copyright © 2015 by the Applied Probability Trust 

References

Bailey, R. A. (2004). Association Schemes: Designed Experiments, Algebra and Combinatorics (Camb. Studies Adv. Math. 84). Cambridge University Press.CrossRefGoogle Scholar
Bertoin, J. (2008). Two-parameter Poisson–Dirichlet measures and reversible exchangeable fragmentation–coalescence processes. Combin. Prob. Comput. 17 329-337.CrossRefGoogle Scholar
Crane, H. (2011). A consistent Markov partition process generated from the paintbox process. J. Appl. Prob. 48 778-791.Google Scholar
Crane, H. (2013). Permanental partition models and Markovian Gibbs structures. J. Statist. Phys. 153 698-726.CrossRefGoogle Scholar
Crane, H. (2013). Some algebraic identities for the α-permanent. Linear Algebra Appl. 439 3445-3459.Google Scholar
Crane, H. (2014). The cut-and-paste process. Ann. Prob. 42 1952-1979.CrossRefGoogle Scholar
Crane, H. (2015). Clustering from categorical data sequences. J. Amer. Statist. Assoc. 110 810-823.Google Scholar
Donnelly, P. and Grimmett, G. (1993). On the asymptotic distribution of large prime factors. J. London Math. Soc. 47 395-404.CrossRefGoogle Scholar
Efron, B. and Thisted, R. (1976). Estimating the number of unseen species: how many words did Shakespeare know? Biometrika 63 435-447.Google Scholar
Ewens, W. J. (1972). The sampling theory of selectively neutral alleles. Theoret. Pop. Biol. 3 87-112, 240, 376.CrossRefGoogle ScholarPubMed
Fisher, R. A., Corbet, A. S. and Williams, C. B. (1943). The relation between the number of species and the number of individuals in a random sample of an animal population. J. Animal Ecol. 12 42-58.Google Scholar
Gnedin, A., Haulk, C. and Pitman, J. (2010). Characterizations of exchangeable partitions and random discrete distributions by deletion properties. In Probability and Mathematical Genetics (London Math. Soc. Lecture Note Ser. 378), Cambridge University Press, pp. 264–298.CrossRefGoogle Scholar
Kingman, J. F. C. (1978). Random partitions in population genetics. Proc. R. Soc. London A 361 1-20.Google Scholar
Kingman, J. F. C. (1978). The representation of partition structures. J. London Math. Soc. (2) 18 374-380.Google Scholar
McCullagh, P. and Yang, J. (2008). How many clusters? Bayesian Anal. 3 101-120.CrossRefGoogle Scholar
Pitman, J. (2006). Combinatorial Stochastic Processes (Lecture Notes Math. 1875), Springer, Berlin.Google Scholar
Sloane, N. (2010). The on-line encyclopedia of integer sequences. Available at http://www.oeis.org/ Google Scholar
Thisted, R. and Efron, B. (1987). Did Shakespeare write a newly-discovered poem? Biometrika 74 445-455.Google Scholar
Wilf, H. S. (2006). Generatingfunctionology, 3rd edn. A K Peters, Wellesley, MA.Google Scholar