arXiv subjects
Publications and source records attributed to Peter Harremoës.
The Radon-Nikodym theorem plays a significant role in the definition of Shannon entropy, f-divergences, and other basic quantities in information theory. The existence of Radon Nikodym derivates appear in many text books in measure theory but in text books on probability or information theory it is often omitted because the proof is often considered to be too difficult.
Shannon based his information theory on the notion of probability measures as it we developed by Kolmogorov. In this paper we study some fundamental problems in information theory based on expectation measures. In the theory of expectation measures it is natural to study data sets where no randomness is present and it is also natural to study information theory for point processes as well as sampling where the sample size is not fixed. Expectation measures in combination with Kraft's Inequality can be used to clarify in which cases probability measures can be used to quantify randomness.
Information projections have found important applications in probability theory, statistics, and related areas. In the field of hypothesis testing in particular, the reverse information projection (RIPr) has recently been shown to lead to growth-rate optimal (GRO) e-statistics for testing simple alternatives against composite null hypotheses. However, the RIPr as well as the GRO criterion are undefined whenever the infimum information divergence between the null and alternative is infinite. We show that in such scenarios, under some assumptions, there still exists a measure in the null that is closest to the alternative in a specific sense. Whenever the information divergence is finite, this measure coincides with the usual RIPr. It therefore gives a natural extension of the RIPr to certain cases where the latter was previously not defined. This extended notion of the RIPr is shown to lead to optimal e-statistics in a sense that is a novel, but natural, extension of the GRO criterion. We also give conditions under which the (extension of the) RIPr is a strict sub-probability measure, as well as conditions under which an approximation of the RIPr leads to approximate e-statistics. For this case we provide tight relations between the corresponding approximation rates.
Rate distortion theory was developed for optimizing lossy compression of data, but it also has a lot of applications in statistics. In this paper we will see how rate distortion theory can be used to analyze a complicated data set involving orientations of early Islamic mosques. The analysis involves testing, identification of outliers, choice of compression rate, calculation of optimal reconstruction points, and assigning "descriptive confidence regions" to the reconstruction points. In this paper the focus will be on the methods, so the integrity of the data set and the interpretation of the results will not be discussed.
Information theory is built on probability measures and by definition a probability measure has total mass 1. Probability measures are used to model uncertainty, and one may ask how important it is that the total mass is one. We claim that the main reason to normalize measures is that probability measures are related to codes via Kraft's inequality. Using a minimum description length approach to statistics we will demonstrate with that measures that are not normalized require a new interpretation that we will call the Poisson interpretation. With the Poisson interpretation many problems can be simplified. The focus will shift from from probabilities to mean values. We give examples of improvements of test procedures, improved inequalities, simplified algorithms, new projection results, and improvements in our description of quantum systems.
The hypergeometric distributions have many important applications, but they have not had sufficient attention in information theory. Hypergeometric distributions can be approximated by binomial distributions or Poisson distributions. In this paper we present upper and lower bounds on information divergence. These bounds are important for statistical testing and a better understanding of the notion of exchange-ability.
The principle called information causality has been used to deduce Tsirelson's bound. In this paper we derive information causality from monotonicity of divergence and relate it to more basic principles related to measurements on thermodynamic systems. This principle is more fundamental in the sense that it can be formulated for both unipartite systems and multipartite systems while information causality is only defined for multipartite systems. Thermodynamic sufficiency is a strong condition that put severe restrictions to shape of the state space to an extend that we conjecture that under very weak regularity conditions it can be used to deduce the complex Hilbert space formalism of quantum theory. Since the notion of sufficiency is relevant for all convex optimization problems there are many examples where it does not apply.
Statistical inference may follow a frequentist approach or it may follow a Bayesian approach or it may use the minimum description length principle (MDL). Our goal is to identify situations in which these different approaches to statistical inference coincide. It is proved that for exponential families MDL and Bayesian inference coincide if and only if the renormalized saddle point approximation for the conjugated exponential family is exact. For 1-dimensional exponential families the only families with exact renormalized saddle point approximations are the Gaussian location family, the Gamma family and the inverse Gaussian family. They are conjugated families of the Gaussian location family, the Gamma family and the Poisson-exponential family. The first two families are self-conjugated implying that only for the two first families the Bayesian approach is consistent with the frequentist approach. In higher dimensions there are more examples.
Recently it has been demonstrated that the Shannon entropy or the von Neuman entropy are the only entropy functions that generate a local Bregman divergences as long as the state space has rank 3 or higher. In this paper we will study the properties of Bregman divergences for convex bodies of rank 2. The two most important convex bodies of rank 2 can be identified with the bit and the qubit. We demonstrate that if a convex body of rank 2 has a Bregman divergence that satisfies sufficiency then the convex body is spectral and if the Bregman divergence is monotone then the convex body has the shape of a ball. A ball can be represented as the state space of a spin factor, which is the most simple type of Jordan algebra. We also study the existence of recovery maps for Bregman divergences on spin factors. In general the convex bodies of rank 2 appear as faces of state spaces of higher rank. Therefore our results give strong restrictions on which convex bodies could be the state space of a physical system with a well-behaved entropy function.
For convex optimization problems Bregman divergences appear as regret functions. Such regret functions can be defined on any convex set but if a sufficiency condition is added the regret function must be proportional to information divergence and the convex set must be spectral. Spectral set are sets where different orthogonal decompositions of a state into pure states have unique mixing coefficients. Only on such spectral sets it is possible to define well behaved information theoretic quantities like entropy and divergence. It is only possible to perform measurements in a reversible way if the state space is spectral. The most important spectral sets can be represented as positive elements of Jordan algebras with trace 1. This means that Jordan algebras provide a natural framework for studying quantum information. We compare information theory on Hilbert spaces with information theory in more general Jordan algebras, and conclude that much of the formalism is unchanged but also identify some important differences.
The notion of Bregman divergence and sufficiency will be defined on general convex state spaces. It is demonstrated that only spectral sets can have a Bregman divergence that satisfies a sufficiency condition. Positive elements with trace 1 in a Jordan algebra are examples of spectral sets, and the most important example is the set of density matrices with complex entries. It is conjectured that information theoretic considerations lead directly to the notion of Jordan algebra under some regularity conditions.
In this paper we present various new inequalities for tail proabilities for distributions that are elements of the most improtant exponential families. These families include the Poisson distributions, the Gamma distributions, the binomial distributions, the negative binomial distributions and the inverse Gaussian distributions. All these exponential families have simple variance functions and the variance functions play an important role in the exposition. All the inequalities presented in this paper are formulated in terms of the signed log-likelihood. The inequalities are of a qualitative nature in that they can be formulated either in terms of stochastic domination or in terms of an intersection property that states that a certain discrete distribution is very close to a certain continuous distribution.
It is well-known that there are a number of relations between theoretical finance theory and information theory. Some of these relations are exact and some are approximate. In this paper we will explore some of these relations and determine under which conditions the relations are exact. It turns out that portfolio theory always leads to Bregman divergences. The Bregman divergence is only proportional to information divergence in situations that are essentially equal to the type of gambling studied by Kelly. This can be related an abstract sufficiency condition.
In this paper we establish lower bounds on information divergence of a distribution on the integers from a Poisson distribution. These lower bounds are tight and in the cases where a rate of convergence in the Law of Thin Numbers can be computed the rate is determined by the lower bounds proved in this paper. General techniques for getting lower bounds in terms of moments are developed. The results about lower bound in the Law of Thin Numbers are used to derive similar results for the Central Limit Theorem.
Logarithmic score and information divergence appear in both information theory, statistics, statistical mechanics, and portfolio theory. We demonstrate that all these topics involve some kind of optimization that leads directly to the use of Bregman divergences. If a sufficiency condition is also fulfilled the Bregman divergence must be proportional to information divergence. The sufficiency condition has quite different consequences in the different areas of application, and often it is not fulfilled. Therefore the sufficiency condition can be used to explain when results from one area can be transferred directly from one area to another and when one will experience differences.
We study the existence or absence of non-Shannon inequalities for variables that are related by functional dependencies. Although the power-set on four variables is the smallest Boolean lattice with non-Shannon inequalities there exist lattices with many more variables without non-Shannon inequalities. We search for conditions that ensures that no non-Shannon inequalities exist. It is demonstrated that 3-dimensional distributive lattices cannot have non-Shannon inequalities and planar modular lattices cannot have non-Shannon inequalities. The existence of non-Shannon inequalities is related to the question of whether a lattice is isomorphic to a lattice of subgroups of a group.
Rényi divergence is related to Rényi entropy much like Kullback-Leibler divergence 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 Kullback-Leibler divergence, and depends on a parameter that is called its order. In particular, the Rényi divergence of order 1 equals the Kullback-Leibler divergence. We review and extend the most important properties of Rényi divergence and Kullback-Leibler divergence, including convexity, continuity, limits of $σ$-algebras and the relation of the special order 0 to the Gaussian dichotomy and contiguity. We also show how to generalize the Pythagorean inequality to orders different from 1, and we extend the known equivalence between channel capacity and minimax redundancy to continuous channel inputs (for all orders) and present several other minimax results.
Pinsker's widely used inequality upper-bounds the total variation distance $||P-Q||_1$ in terms of the Kullback-Leibler divergence $D(P||Q)$. Although in general a bound in the reverse direction is impossible, in many applications the quantity of interest is actually $D^*(P,\eps)$ --- defined, for an arbitrary fixed $P$, as the infimum of $D(P||Q)$ over all distributions $Q$ that are $\eps$-far away from $P$ in total variation. We show that $D^*(P,\eps)\le C\eps^2 + O(\eps^3)$, where $C=C(P)=1/2$ for "balanced" distributions, thereby providing a kind of reverse Pinsker inequality. An application to large deviations is given, and some of the structural results may be of independent interest. Keywords: Pinsker inequality, Sanov's theorem, large deviations