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- W173566686 abstract "Independence Models for Integer Points of Polytopes by Austin Warren Shapiro Chair: Alexander I. Barvinok The integer points of a high-dimensional polytope P are generally difficult to count or sample uniformly. We consider a class of low-complexity random models for these points which arise from an entropy maximization problem. From these models, by way of “anti-concentration” results for sums of independent random variables, we derive general, efficiently computable upper bounds on the number of integer points of P . We make a detailed study of contingency tables with bounded entries, which are the integer points of a transportation polytope truncated by a cuboid. We provide efficiently computable estimates for the logarithm of the number of m × n tables with specified row and column sums r1, . . . , rm, c1, . . . , cn and bounds on the entries. These estimates are asymptotic as m,n→∞ simultaneously, given that no ri (resp., cj) is allowed to exceed a fixed multiple of the average row sum (resp., column sum). As an application, we consider a random, uniformly selected table with entries ≤ κ having a given sum. Responding to questions raised by Diaconis and Efron in the context of statistical significance testing, we show that the occurrence of row sums r1, . . . , rm is positively correlated with the occurrence of column sums c1, . . . , cn when κ ≥ 2 and r1, . . . , rm, c1, . . . , cn are sufficiently extreme. We give evidence that the opposite is true for near-average values of r1, . . . , rm, c1, . . . , cn." @default.
- W173566686 created "2016-06-24" @default.
- W173566686 creator A5050469262 @default.
- W173566686 date "2011-01-01" @default.
- W173566686 modified "2023-09-26" @default.
- W173566686 title "Independence Models for Integer Points of Polytopes." @default.
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