Gaussian limiting distributions for the number of components in combinatorial structures

Flajolet, Philippe; Soria, Michèle

doi:10.1016/0097-3165(90)90056-3

Cited by 105 publications

(106 citation statements)

References 16 publications

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“…10 the tree function T x = , W , 1 , x , used in the enumeration of trees and graphs on sets of labeled vertices [Wright, 1977, Janson et al, 1993 and in computing the distribution of cycles in random mappings [Flajolet and Soria, 1990]. Connections to dynamical stability via the W function and to sparse graph enumeration via the T function are very intriguing and may lead to arguments as to whether the entropic prior is optimal for learning concise sparse models.…”

Section: Graph Theorymentioning

confidence: 99%

Structure Learning in Conditional Probability Models via an Entropic Prior and Parameter Extinction

Brand

1999

Neural Computation

126

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We introduce an entropic prior for multinomial parameter estimation problems and solve for its maximum a posteriori (MAP) estimator. The prior is a bias for maximally structured and minimally ambiguous models. In conditional probability models with hidden state, iterative MAP estimation drives weakly supported parameters toward extinction, effectively turning them off. Thus structure discovery is folded into parameter estimation. We then establish criteria for simplifying a probabilistic model's graphical structure by trimming parameters and states, with a guarantee that any such deletion will increase the posterior probability of the model. Trimming accelerates learning by sparsifying the model. All operations monotonically and maximally increase the posterior probability, yielding structure-learning algorithms only slightly slower than parameter estimation via expectation-maximization (EM), and orders of magnitude faster than search-based structure induction. When applied to hidden Markov model (HMM) training, the resulting models show superior generalization to held-out test data. In many cases the resulting models are so sparse and concise that they are interpretable, with hidden states that strongly correlate with meaningful categories. Neural ComputationThis work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to copy in whole or in part without payment of fee is granted for nonprofit educational and research purposes provided that all such whole or partial copies include the following: a notice that such copying is by permission of Mitsubishi Electric Research Laboratories, Inc.; an acknowledgment of the authors and individual contributions to the work; and all applicable portions of the copyright notice. Copying, reproduction, or republishing for any other purpose shall require a license with payment of fee to Mitsubishi Electric Research Laboratories, Inc. All rights reserved. AbstractWe introduce an entropic prior for multinomial parameter estimation problems and solve for its maximum a posteriori (MAP) estimator. The prior is a bias for maximally structured and minimally ambiguous models. In conditional probability models with hidden state, iterative MAP estimation drives weakly supported parameters toward extinction, effectively turning them off. Thus structure discovery is folded into parameter estimation. We then establish criteria for simplifying a probabilistic model's graphical structure by trimming parameters and states, with a guarantee that any such deletion will increase the posterior probability of the model. Trimming accelerates learning by sparsifying the model. All operations monotonically and maximally increase the posterior probability, yielding structure-learning algorithms only slightly slower than parameter estimation via expectation-maximization (EM), and orders of magnitude faster than search-based structure induction. When applied to hidden Markov model (HMM) training, the resulting models show superior generalization to held-ou...

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Section: Graph Theorymentioning

confidence: 99%

Structure Learning in Conditional Probability Models via an Entropic Prior and Parameter Extinction

Brand

1999

Neural Computation

126

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“…The reason is simply that the corresponding exponential generating function, according to (2.8) and well-known techniques for enumerating labeled configurations, is exp(y ln(l/(l -T(z)))), and this is just (0.5). Stepanov [13] showed that these coefficients are asymptotically normal with mean and variance | In n + 0( 1 ) ; Flajolet and Soria [4] extended this to a general result on the number of components in random labeled structures.…”

Section: Tree Polynomialsmentioning

confidence: 99%

“…Some sort of smoothness condition is necessary for the validity of (4.9); we cannot conclude that xn/n approaches a limit if we know only that J2m>l £"m/w3/2 exists. For example, we might have f 0, if « is not a power of 2 ; (4)(5)(6)(7)(8)(9)(10)(11)(12)(13) C" = \3k/k2, if«-/.…”

Section: Asymptotic Lemmasmentioning

confidence: 99%

A Recurrence Related to Trees

Pittel¹

1989

Proceedings of the American Mathematical Society

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“…Among all possible multisets of total weight n, we select one at random, and let Y } be the number of objects of weighty included; the joint distribution of these counts is given by (1"1). See Flajolet and Soria [14] and Arratia and Tavare [1] for probabilistic treatments of multisets in general. WithiV 9 (i) given by (1)(2), where q is any positive integer, the total number of possible multisets of weight n is q n , and (1"3) is valid.…”

Section: D\ndmentioning

confidence: 99%

On random polynomials over finite fields

Arratia

Barbour²,

Tavaré

1993

Math. Proc. Camb. Phil. Soc.

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We consider random monic polynomials of degree n over a finite field of q elements, chosen with all q n possibilities equally likely, factored into monic irreducible factors. More generally, relaxing the restriction that q be a prime power, we consider that multiset construction in which the total number of possibilities of weight n is q n . We establish various approximations for the joint distribution of factors, by giving upper bounds on the total variation distance to simpler discrete distributions. For example, the counts for particular factors are approximately independent and geometrically distributed, and the counts for all factors of sizes 1,2, ...,b, where b = O(n/logn), are approximated by independent negative binomial random variables. As another example, the joint distribution of the large factors is close to the joint distribution of the large cycles in a random permutation. We show how these discrete approximations imply a Brownian motion functional central limit theorem and a Poisson-Dirichlet limit theorem, together with appropriate error estimates. We also give Poisson approximations, with error bounds, for the distribution of the total number of factors.

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Gaussian limiting distributions for the number of components in combinatorial structures

Cited by 105 publications

References 16 publications

Structure Learning in Conditional Probability Models via an Entropic Prior and Parameter Extinction

Structure Learning in Conditional Probability Models via an Entropic Prior and Parameter Extinction

A Recurrence Related to Trees

On random polynomials over finite fields

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