Sampling and Learning Mallows and Generalized Mallows Models Under the Cayley Distance

The Mallows and Generalized Mallows models are compact yet powerful and natural ways of representing a probability distribution over the space of permutations. In this paper, we deal with the problems of sampling and learning such distributions when the metric on permutations is the Cayley distance. We propose new methods for both operations, and their performance is shown through several experiments. An application in the field of biology is given to motivate the interest of this model.

[1]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[2]  C. L. Mallows.  NON-NULL RANKING MODELS. I , 1957 .

[3]  R. Duncan Luce,et al.  Individual Choice Behavior , 1959 .

[4]  W. Feller,et al.  An Introduction to Probability Theory and Its Applications, Vol. 1 , 1967 .

[5]  W. Ewens.  The sampling theory of selectively neutral alleles. , 1972, Theoretical population biology.

[6]  R. Plackett.  The Analysis of Permutations , 1975 .

[7]  M. Fligner,et al.  Distance Based Ranking Models , 1986 .

[8]  P. Donnelly,et al.  Partition structures, Polya urns, the Ewens sampling formula, and the ages of alleles. , 1986, Theoretical population biology.

[9]  M. Fligner,et al.  Multistage Ranking Models , 1988 .

[10]  P. Diaconis.  Group representations in probability and statistics , 1988 .

[11]  Joseph S. Verducci,et al.  Probability models on rankings. , 1991 .

[12]  P. Diaconis,et al.  Trailing the Dovetail Shuffle to its Lair , 1992 .

[13]  Martin Schader,et al.  Analyzing and Modeling Data and Knowledge , 1992 .

[14]  Joseph S. Verducci,et al.  Probability Models and Statistical Analyses for Ranking Data , 1992 .

[15]  L. Thurstone.  A law of comparative judgment. , 1994 .

[16]  J. Marden.  Analyzing and Modeling Rank Data , 1996 .

[17]  Pierre Hansen,et al.  Variable Neighborhood Search , 2018, Handbook of Heuristics.

[18]  Herbert S. Wilf.  East Side, West Side . . . - an introduction to combinatorial families-with Maple programming , 1999 .

[19]  P. Diaconis,et al.  Analysis of systematic scan Metropolis algorithms using Iwahori-Hecke algebra techniques , 2004, math/0401318.

[20]  P. Diaconis,et al.  A Bayesian peek into feller volume I , 2002 .

[21]  P. Damien,et al.  Conjugacy class prior distributions on metric‐based ranking models , 2002 .

[22]  P. Pevzner,et al.  Genome-scale evolution: reconstructing gene orders in the ancestral species. , 2002, Genome research.

[23]  John D. Lafferty,et al.  Cranking: Combining Rankings Using Conditional Probability Models on Permutations , 2002, ICML.

[24]  R. Arratia,et al.  Logarithmic Combinatorial Structures: A Probabilistic Approach , 2003 .

[25]  Thomas Brendan Murphy,et al.  Mixtures of distance-based models for ranking data , 2003, Comput. Stat. Data Anal..

[26]  Tayuan Huang,et al.  Metrics on Permutations, a Survey , 2004 .

[27]  Angela D'Elia,et al.  A mixture model for preferences data analysis , 2005, Comput. Stat. Data Anal..

[28]  D. Critchlow.  Ulam's Metric , 2006 .

[29]  Leonidas J. Guibas,et al.  Efficient Inference for Distributions on Permutations , 2007, NIPS.

[30]  Yi Mao,et al.  Non-parametric Modeling of Partially Ranked Data , 2007, NIPS.

[31]  V. Y. Popov,et al.  Multiple genome rearrangement by swaps and by element duplications , 2007, Theor. Comput. Sci..

[32]  Marina Meila,et al.  Estimation and clustering with infinite rankings , 2008, UAI.

[33]  Persi Diaconis,et al.  The Markov chain Monte Carlo revolution , 2008 .

[34]  S. Starr.  THERMODYNAMIC LIMIT FOR THE MALLOWS MODEL ON Sn , 2009, 0904.0696.

[35]  E. Hüllermeier,et al.  A Simple Instance-Based Approach to Multilabel Classification Using the Mallows Model , 2009 .

[36]  Marina Meila,et al.  Tractable Search for Learning Exponential Models of Rankings , 2009, AISTATS.

[37]  Eyke Hüllermeier,et al.  A New Instance-Based Label Ranking Approach Using the Mallows Model , 2009, ISNN.

[38]  Thomas L. Griffiths,et al.  The nested chinese restaurant process and bayesian nonparametric inference of topic hierarchies , 2007, JACM.

[39]  S. Evans,et al.  Trickle-down processes and their boundaries , 2010, 1010.0453.

[40]  Marina Meila,et al.  Dirichlet Process Mixtures of Generalized Mallows Models , 2010, UAI.

[41]  Christian Komusiewicz,et al.  Average parameterization and partial kernelization for computing medians , 2011, J. Comput. Syst. Sci..

[42]  Craig Boutilier,et al.  Learning Mallows Models with Pairwise Preferences , 2011, ICML.

[43]  Alexander Mendiburu,et al.  Introducing the Mallows Model on Estimation of Distribution Algorithms , 2011, ICONIP.

[44]  Peter McCullagh,et al.  Random Permutations and Partition Models , 2011, International Encyclopedia of Statistical Science.

[45]  P. Rinker.  A Mallows model for Coxeter groups and buildings , 2011 .

[46]  Valentin Féray.  Asymptotic behavior of some statistics in Ewens random permutations , 2012, 1201.2157.

[47]  David J. Kriegman,et al.  Locally Uniform Comparison Image Descriptor , 2012, NIPS.

[48]  Alexander Gnedin,et al.  The two-sided infinite extension of the Mallows model for random permutations , 2011, Adv. Appl. Math..

[49]  Marina Meila,et al.  Consensus Ranking with Signed Permutations , 2013, AISTATS.

[50]  Ariel D. Procaccia,et al.  When do noisy votes reveal the truth? , 2013, EC '13.