On Solutions to Multivariate Maximum α-Entropy Problems

Costa, Jose A.; Hero, Alfred O.; Vignat, Christophe

doi:10.1007/978-3-540-45063-4_14

Cited by 60 publications

(64 citation statements)

References 15 publications

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“…Note the symmetry between (39)- (40) and (18)- (19) respectively that can be explained by remarking the symmetry between the Student-t and Student-r variables as evoked in [37].…”

Section: The General Student-r Casementioning

confidence: 99%

“…where A m is a scalar inverse Gamma random variable 6 invΓ m 2 , 2 with shape parameter m/2 and scale parameter 1/2, independent of the zero-mean Gaussian vector G n with identity covariance matrix (see also [31,37,39]). In the particular Cauchy case m = 1, one recovers the fact that A 1 is a Lévy variable [40].…”

Section: Convergence In Distributionmentioning

confidence: 99%

“…This representation was given in [37] for m integer, but it holds for non integer values of m as well. Now, variable C m = G n 2 + B m,n is Gamma distributed Γ m+2 2 , 2 and then, with the same technique as in the Student-t case, m+2…”

Section: Convergence In Distributionmentioning

confidence: 99%

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On classes of non-Gaussian asymptotic minimizers in entropic uncertainty principles

Zozor

Vignat

2007

Physica A: Statistical Mechanics and its Applications

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In this paper we revisit the Bialynicki-Birula & Mycielski uncertainty principle [1] and its cases of equality. This Shannon entropic version of the well-known Heisenberg uncertainty principle can be used when dealing with variables that admit no variance. In this paper, we extend this uncertainty principle to Rényi entropies. We recall that in both Shannon and Rényi cases, and for a given dimension n, the only case of equality occurs for Gaussian random vectors. We show that as n grows, however, the bound is also asymptotically attained in the cases of n-dimensional Student-t and Student-r distributions. A complete analytical study is performed in a special case of a Student-t distribution. We also show numerically that this effect exists for the particular case of a n-dimensional Cauchy variable, whatever the Rényi entropy considered, extending the results of Abe [2] and illustrating the analytical asymptotic study of the student-t case. In the Student-r case, we show numerically that the same behavior occurs for uniformly distributed vectors. These particular cases and other ones investigated in this paper are interesting since they show that this asymptotic behavior cannot be considered as a "Gaussianization" of the vector when the dimension increases.

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“…Note the symmetry between (39)- (40) and (18)- (19) respectively that can be explained by remarking the symmetry between the Student-t and Student-r variables as evoked in [37].…”

Section: The General Student-r Casementioning

confidence: 99%

Section: Convergence In Distributionmentioning

confidence: 99%

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On classes of non-Gaussian asymptotic minimizers in entropic uncertainty principles

Zozor

Vignat

2007

Physica A: Statistical Mechanics and its Applications

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“…is also the maximum Rényi entropy distribution under the constraints E{XX } = and n n + 2 < α < 1 (see Costa et al (2003)). It can be seen, from the nonsymmetric Bregman divergence measure, that…”

Section: The Nonlinear Fokker-planck Equation and Entropymentioning

confidence: 99%

Student processes

Heyde

Leonenko

2005

Adv. Appl. Probab.

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“…Rényi entropy maximizers under moment constraints are distributions with a power-law decay (when α < 1). See Costa et al [22] or Johnson and Vignat [20]. Many statistical physicists have studied this principle in the hope that it may "explain" the emergence of power-laws in many naturally occurring physical and socio-economic systems, beginning with Tsallis [23].…”

Section: Introductionmentioning

confidence: 99%

Minimization Problems Based on Relative <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>-Entropy I: Forward Projection

Kumar

Sundaresan

2015

IEEE Trans. Inform. Theory

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Minimization problems with respect to a one-parameter family of generalized relative entropies are studied. These relative entropies, which we term relative α-entropies (denoted Iα), arise as redundancies under mismatched compression when cumulants of compressed lengths are considered instead of expected compressed lengths. These parametric relative entropies are a generalization of the usual relative entropy (Kullback-Leibler divergence). Just like relative entropy, these relative α-entropies behave like squared Euclidean distance and satisfy the Pythagorean property. Minimizers of these relative α-entropies on closed and convex sets are shown to exist. Such minimizations generalize the maximum Rényi or Tsallis entropy principle. The minimizing probability distribution (termed forward Iα-projection) for a linear family is shown to obey a power-law. Other results in connection with statistical inference, namely subspace transitivity and iterated projections, are also established. In a companion paper, a related minimization problem of interest in robust statistics that leads to a reverse Iα-projection is studied.

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On Solutions to Multivariate Maximum α-Entropy Problems

Cited by 60 publications

References 15 publications

On classes of non-Gaussian asymptotic minimizers in entropic uncertainty principles

On classes of non-Gaussian asymptotic minimizers in entropic uncertainty principles

Student processes

Minimization Problems Based on Relative <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>-Entropy I: Forward Projection

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