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Peter Harremoës

Publications and source records attributed to Peter Harremoës.

29 records · Page 2Linked to original sources
arXiv2014

Mutual information of Contingency Tables and Related Inequalities

For testing independence it is very popular to use either the $χ^{2}$-statistic or $G^{2}$-statistics (mutual information). Asymptotically both are $χ^{2}$-distributed so an obvious question is which of the two statistics that has a distribution that is closest to the $χ^{2}$-distribution. Surprisingly the distribution of mutual information is much better approximated by a $χ^{2}$-distribution than the $χ^{2}$-statistic. For technical reasons we shall focus on the simplest case with one degree of freedom. We introduce the signed log-likelihood and demonstrate that its distribution function can be related to the distribution function of a standard Gaussian by inequalities. For the hypergeometric distribution we formulate a general conjecture about how close the signed log-likelihood is to a standard Gaussian, and this conjecture gives much more accurate estimates of the tail probabilities of this type of distribution than previously published results. The conjecture has been proved numerically in all cases relevant for testing independence and further evidence of its validity is given.

math.ST↗
arXiv2013

Extendable MDL

In this paper we show that combination of the minimum description length principle and a exchange-ability condition leads directly to the use of Jeffreys prior. This approach works in most cases even when Jeffreys prior cannot be normalized. Kraft's inequality links codes and distributions but a closer look at this inequality demonstrates that this link only makes sense when sequences are considered as prefixes of potential longer sequences. For technical reasons only results for exponential families are stated. Results on when Jeffreys prior can be normalized after conditioning on a initializing string are given. An exotic case where no initial string allow Jeffreys prior to be normalized is given and some way of handling such exotic cases are discussed.

cs.IT↗
arXiv2012

Information Divergence is more chi squared distributed than the chi squared statistics

For testing goodness of fit it is very popular to use either the chi square statistic or G statistics (information divergence). Asymptotically both are chi square distributed so an obvious question is which of the two statistics that has a distribution that is closest to the chi square distribution. Surprisingly, when there is only one degree of freedom it seems like the distribution of information divergence is much better approximated by a chi square distribution than the chi square statistic. For random variables we introduce a new transformation that transform several important distributions into new random variables that are almost Gaussian. For the binomial distributions and the Poisson distributions we formulate a general conjecture about how close their transform are to the Gaussian. The conjecture is proved for Poisson distributions.

math.ST↗
arXiv2011

Lower bounds on Information Divergence

In this paper we establish lower bounds on information divergence from a distribution to certain important classes of distributions as Gaussian, exponential, Gamma, Poisson, geometric, and binomial. These lower bounds are tight and for several convergence theorems where a rate of convergence can be computed, this rate is determined by the lower bounds proved in this paper. General techniques for getting lower bounds in terms of moments are developed.

cs.IT↗
arXiv2011

Is Zero a Natural Number?

It is argued that zero should be considered as a cardinal number but not an ordinal number. One should make a clear distinction between order types that are labels for well-ordered sets and ordinal numbers that are labels for the elements in these sets.

math.HO↗
arXiv2010

Joint Range of f-divergences

We provide a general method for evaluation of the joint range of f-divergences for two different functions f. Via topological arguments we prove that the joint range for general distributions equals the convex hull of the joint range achieved by the distributions on a two-element set. The joint range technique provides important inequalities between different f-divergences with various applications in information theory and statistics.

cs.IT↗
arXiv2010

Rényi Divergence and Majorization

Rényi divergence is related to Rényi entropy much like information divergence (also called Kullback-Leibler divergence or relative entropy) is related to Shannon's entropy, and comes up in many settings. It was introduced by Rényi as a measure of information that satisfies almost the same axioms as information divergence. We review the most important properties of Rényi divergence, including its relation to some other distances. We show how Rényi divergence appears when the theory of majorization is generalized from the finite to the continuous setting. Finally, Rényi divergence plays a role in analyzing the number of binary questions required to guess the values of a sequence of random variables.

cs.IT↗
arXiv2010

On Bahadur Efficiency of Power Divergence Statistics

It is proved that the information divergence statistic is infinitely more Bahadur efficient than the power divergence statistics of the orders $α>1$ as long as the sequence of alternatives is contiguous with respect to the sequence of null-hypotheses and the the number of observations per bin increases to infinity is not very slow. This improves the former result in Harremoës and Vajda (2008) where the the sequence of null-hypotheses was assumed to be uniform and the restrictions on on the numbers of observations per bin were sharper. Moreover, this paper evaluates also the Bahadur efficiency of the power divergence statistics of the remaining positive orders $0< α\leq 1.$ The statistics of these orders are mutually Bahadur-comparable and all of them are more Bahadur efficient than the statistics of the orders $α> 1.$ A detailed discussion of the technical definitions and conditions is given, some unclear points are resolved, and the results are illustrated by examples.

math.ST↗
arXiv2009

Joint Range of Rényi Entropies

The exact range of the joined values of several Rényi entropies is determined. The method is based on topology with special emphasis on the orientation of the objects studied. Like in the case when only two orders of Rényi entropies are studied one can parametrize upper and lower bounds but an explicit formula for a tight upper or lower bound cannot be given.

cs.IT↗
arXiv2009

Properties of Classical and Quantum Jensen-Shannon Divergence

Jensen-Shannon divergence (JD) is a symmetrized and smoothed version of the most important divergence measure of information theory, Kullback divergence. As opposed to Kullback divergence it determines in a very direct way a metric; indeed, it is the square of a metric. We consider a family of divergence measures (JD_alpha for alpha>0), the Jensen divergences of order alpha, which generalize JD as JD_1=JD. Using a result of Schoenberg, we prove that JD_alpha is the square of a metric for alpha lies in the interval (0,2], and that the resulting metric space of probability distributions can be isometrically embedded in a real Hilbert space. Quantum Jensen-Shannon divergence (QJD) is a symmetrized and smoothed version of quantum relative entropy and can be extended to a family of quantum Jensen divergences of order alpha (QJD_alpha). We strengthen results by Lamberti et al. by proving that for qubits and pure states, QJD_alpha^1/2 is a metric space which can be isometrically embedded in a real Hilbert space when alpha lies in the interval (0,2]. In analogy with Burbea and Rao's generalization of JD, we also define general QJD by associating a Jensen-type quantity to any weighted family of states. Appropriate interpretations of quantities introduced are discussed and bounds are derived in terms of the total variation and trace distance.

quant-ph↗