Statistical Inference on Random Structures
Randomness for a statistician must have some structure. In traditional combinatorics the word random means uniform distribution on a set which may be the set of all graphs with n vertices, the set of all permutations of the numbers N = (1,2,..., n), the s
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Horizons of Combinatorics Balatonalmadi pp.37-66.
STATISTICAL INFERENCE ON RANDOM STRUCTURES
VILLa CSISZAR , LIDIA REJTO alld GABOR TUSNADY
INTRODUCTION
Randomness for a statistician must have SOlne structure. In traditional combinatorics the word randoln meallS uniforln distribution on a set which may be the set of all graphs with n vertices , the set of all permutations of the numbers N == (1 , 2,... , n) , the set of all partitions of N , or any other set of simple structure. In practice the statistician meets a subset of the structures and she or he is interested in the question , what was the mechanism which generated the sample. Uniform distribution and independence are shapeless alld they have low complexity for catching the character of sanlpIes produced by real life situations. In [12] Persi Diaconis investigated a sample consistillg of the votes in an election of the American Psychological Association. The sample was illvestigated by others but without achieving a reasollable goodness of fit , because the present collection of distribution of permutations is not large enough. Investigating the sample we found a hidden property leadillg to a new class of distributions of permutations. CIa잃ical
statistics developed around the multidimensional Gaussian distribution. Even in Euclidean space the family of useful distributions is still meager. On otIler saluple spaces tIle collection of distributions is mucll less developed. Graphs appear in applications as structured relatiolls. In many cases rather heavy simplifications are needed for reducing the complexity of the investigated situation to a graph. One source of our interest in graphs is the system of metabolic interactions , which may have some fractal structure: the enzynlatic interactions may be leveled , they lnay be sensitive for situations , their control might be hierarcllic. Challges of tIle concelltratioll
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Csiszar, L. Rejto and G. Tusnady
of different enzymes in a cell follow their dynamical rule what is reflected imperfectly in the graph of enzymatic interactions. In moderll combinatorics the stochastic method is rapidly extending. We shall use the ideas of papers [7] , [8] and [9] written by Christian Borgs , Jennifer Chayes , Laszlo Lovasz , Vera T. Sos , Balazs Szegedy and Katalin Vesztergo l1lbi in defining new classes of random permutations.
SVD of real matrices. Let M be an arbitrary digital picture: a face , a tree , a hill or SOllle other natural object which is not very complicated. Let us suppose that the colours are ordered according to their wave lengths and M is an m times n real lnatrix containing the codes of the colours in the individual pixels. Let Q be a random permutation of the integers 1,... , m and (3 of 1,... ， η. Let R(i ,.1 )==M( α(i) ， (3(.1) )
be the randolnly reordered copy of M. How can we reconstruct M from R? One possible lllethod is the singular value decomposition (SVD) of R which is invariant under random perlllutations. The singular values of matrices M and R are identical. We refer to them as th
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