# The Information Structuralist

## ISIT 2011: favorite talks

Obligatory disclaimer: YMMV, “favorite” does not mean “best,” etc. etc.

• Emmanuel Abbe and Andrew Barron, “Polar coding schemes for the AWGN channel” (pdf)
• The problem of constructing polar codes for channels with continuous input and output alphabets can be reduced, in a certain sense, to the problem of constructing finitely supported approximations to capacity-achieving distributions. This work analyzes several such approximations for the AWGN channel. In particular, one approximation uses quantiles and approaches capacity at a rate that decays exponentially with support size. The proof of this fact uses a neat trick of upper-bounding the Kullback-Leibler divergence by the chi-square distance and then exploiting the law of large numbers.

• Tom Cover, “On the St. Petersburg paradox”
• A fitting topic, since this year’s ISIT took place in St. Petersburg! Tom has presented a reformulation of the problem underlying this (in)famous paradox in terms of finding the best allocation of initial capital so as to optimize various notions of relative wealth. This reformulation obviates the need for various extra assumptions, such as diminishing marginal returns (i.e., concave utilities), and thus provides a means of resolving the paradox from first principles.

• Paul Cuff, Tom Cover, Gowtham Kumar, Lei Zhao, “A lattice of gambles”
• There is a well-known correspondence between martingales and “fair” gambling systems. Paul and co-authors explore another correspondence, between fair gambles and Lorenz curves used in econometric modeling, to study certain stochastic orderings and transformations of martingales. There are nice links to the theory of majorization and, through that, to Blackwell’s framework for comparing statistical experiments in terms of their expected risks.

• Ioanna Ioannou, Charalambos Charalambous, Sergey Loyka, “Outage probability under channel distribution uncertainty” (pdf; longer version: arxiv:1102.1103)
• The outage probability of a general channel with stochastic fading is the probability that the conditional input-output mutual information given the fading state falls below the given rate. In this paper, it is assumed that the state distribution is not known exactly, but there is an upper bound on its divergence from some fixed “nominal” distribution (this model of statistical uncertainty has been used previously in the context of robust control). The variational representation of the divergence (as a Legendre-Fenchel transform of the moment-generating function) then allows for a clean asymptotic analysis of the outage probability.

• Mohammad Naghshvar, Tara Javidi, “Performance bounds for active sequential hypothesis testing”
• Mohammad and Tara show how dynamic programming techniques can be used to develop tight converse bounds for sequential hypothesis testing problems with feedback, in which it is possible to adaptively control the quality of the observation channel. This viewpoint is a lot cleaner and more conceptually straightforward than “classical” proofs based on martingales (à la Burnashev). This new technique is used to analyze asymptotically optimal strategies for sequential $M$-ary hypothesis testing, variable-length coding with feedback, and noisy dynamic search.

• Chris Quinn, Negar Kiyavash, Todd Coleman, “Equivalence between minimal generative model graphs and directed information graphs” (pdf)
• For networks of interacting discrete-time stochastic processes possessing a certain conditional independence structure (motivating example: discrete-time approximations of smooth dynamical systems), Chris, Negar and Todd show the equivalence between two types of graphical models for these networks: (1) generative models that are minimal in a certain “combinatorial” sense and (2) information-theoretic graphs, in which the edges are drawn based on directed information.

• Ofer Shayevitz, “On Rényi measures and hypothesis testing” (long version: arxiv:1012.4401)
• Ofer obtained a new variational characterization of Rényi entropy and divergence that considerably simplifies their analysis, in many cases completely replacing delicate arguments based on Taylor expansions with purely information-theoretic proofs. He also develops a new operational characterization of these information measures in terms of distributed composite hypothesis testing.