On choosing and bounding probability metrics

When studying convergence of measures, an important issue is the choice of probability metric. In this review, we provide a summary and some new results concerning bounds among ten important probability metrics/distances that are used by statisticians and probabilists. We focus on these metrics because they are either well-known, commonly used, or admit practical bounding techniques. We summarize these relationships in a handy reference diagram, and also give examples to show how rates of convergence can depend on the metric chosen.Comment: To appear, International Statistical Review. Related work at http://www.math.hmc.edu/~su/papers.htm

On Heavy-Quark Free Energies, Entropies, Polyakov Loop, and AdS/QCD

In this paper we explore some of the features of a heavy quark-antiquark pair at finite temperature using a five-dimensional framework nowadays known as AdS/QCD. We shall show that the resulting behavior is consistent with our qualitative expectations of thermal gauge theory. Some of the results are in good agreement with the lattice data that provides additional evidence for the validity of the proposed model.Comment: 15 pages, 10 figures; v2: comments added, misprints correcte

Vanishing largest Lyapunov exponent and Tsallis entropy

We present a geometric argument that explains why some systems having vanishing largest Lyapunov exponent have underlying dynamics aspects of which can be effectively described by the Tsallis entropy. We rely on a comparison of the generalised additivity of the Tsallis entropy versus the ordinary additivity of the BGS entropy. We translate this comparison in metric terms by using an effective hyperbolic metric on the configuration/phase space for the Tsallis entropy versus the Euclidean one in the case of the BGS entropy. Solving the Jacobi equation for such hyperbolic metrics effectively sets the largest Lyapunov exponent computed with respect to the corresponding Euclidean metric to zero. This conclusion is in agreement with all currently known results about systems that have a simple asymptotic behaviour and are described by the Tsallis entropy.Comment: 15 pages, No figures. LaTex2e. Some overlap with arXiv:1104.4869 Additional references and clarifications in this version. To be published in QScience Connec

Failure of Standard Thermodynamics in Planck Scale Black Hole System

The final stage of the black hole evaporation is a matter of debates in the existing literature. In this paper, we consider this problem within two alternative approaches: noncommutative geometry(NCG) and the generalized uncertainty principle(GUP). We compare the results of two scenarios to find a relation between parameters of these approaches. Our results show some extraordinary thermodynamical behavior for Planck size black hole evaporation. These extraordinary behavior may reflect the need for a fractal nonextensive thermodynamics for Planck size black hole evaporation process.Comment: 26 Pages, 10 Figures, Revised and References adde

Hawking radiation as tunneling from a Vaidya black hole in noncommutative gravity

In the context of a noncommutative model of coordinate coherent states, we present a Schwarzschild-like metric for a Vaidya solution instead of the standard Eddington-Finkelstein metric. This leads to the appearance of an exact $(t - r)$ dependent case of the metric. We analyze the resulting metric in three possible causal structures. In this setup, we find a zero remnant mass in the long-time limit, i.e. an instable black hole remnant. We also study the tunneling process across the quantum horizon of such a Vaidya black hole. The tunneling probability including the time-dependent part is obtained by using the tunneling method proposed by Parikh and Wilczek in terms of the noncommutative parameter $\sigma$. After that, we calculate the entropy associated to this noncommutative black hole solution. However the corrections are fundamentally trifling; one could respect this as a consequence of quantum inspection at the level of semiclassical quantum gravity.Comment: 19 pages, 5 figure

Jensen Shannon divergence as a measure of the degree of entanglement

The notion of distance in Hilbert space is relevant in many scenarios. In particular, distances between quantum states play a central role in quantum information theory. An appropriate measure of distance is the quantum Jensen Shannon divergence (QJSD) between quantum states. Here we study this distance as a geometrical measure of entanglement and apply it to different families of states.Comment: 5 pages, 2 figures, to appear in the special issue of IJQI "Noise, Information and Complexity at Quantum Scale", eds. S. Mancini and F. Marcheson

Sequence alignment, mutual information, and dissimilarity measures for constructing phylogenies

Existing sequence alignment algorithms use heuristic scoring schemes which cannot be used as objective distance metrics. Therefore one relies on measures like the p- or log-det distances, or makes explicit, and often simplistic, assumptions about sequence evolution. Information theory provides an alternative, in the form of mutual information (MI) which is, in principle, an objective and model independent similarity measure. MI can be estimated by concatenating and zipping sequences, yielding thereby the "normalized compression distance". So far this has produced promising results, but with uncontrolled errors. We describe a simple approach to get robust estimates of MI from global pairwise alignments. Using standard alignment algorithms, this gives for animal mitochondrial DNA estimates that are strikingly close to estimates obtained from the alignment free methods mentioned above. Our main result uses algorithmic (Kolmogorov) information theory, but we show that similar results can also be obtained from Shannon theory. Due to the fact that it is not additive, normalized compression distance is not an optimal metric for phylogenetics, but we propose a simple modification that overcomes the issue of additivity. We test several versions of our MI based distance measures on a large number of randomly chosen quartets and demonstrate that they all perform better than traditional measures like the Kimura or log-det (resp. paralinear) distances. Even a simplified version based on single letter Shannon entropies, which can be easily incorporated in existing software packages, gave superior results throughout the entire animal kingdom. But we see the main virtue of our approach in a more general way. For example, it can also help to judge the relative merits of different alignment algorithms, by estimating the significance of specific alignments.Comment: 19 pages + 16 pages of supplementary materia