## ISIT 2013: two plenaries on concentration of measure

Of the five plenary talks at this year’s ISIT, two were about concentration of measure: Katalin Marton’s Shannon lecture on “Distance-divergence inequalities” and Gabor Lugosi’s talk on “Concentration inequalities and the entropy method” the next morning. Since the topic of measure concentration is dear to my heart, I thought I would write down a few unifying themes.

## ITA 2012: some evanescent reflections

Getting settled in at my new academic home, with all the attendant administrivia, has sucked up almost two months of missed blogging opportunities. So, here goes.

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## Updates! Get your updates here!

Just a couple of short items, while I catch my breath.

1. First of all, starting January 1, 2012 I will find myself amidst the lovely cornfields of Central Illinois, where I will be an assistant professor in the Department of Electrical and Computer Engineering at UIUC. This will be a homecoming of sorts, since I have spent three years there as a Beckman Fellow. My new home will be in the Coordinated Science Laboratory, where I will continue doing (and blogging about) the same things I do (and blog about).

2. Speaking of Central Illinois, last week I was at the Allerton Conference, where I had tried my best to preach Uncle Judea‘s gospel to ~~anyone willing to listen~~ information theorists and their fellow travelers. The paper, entitled “Directed information and Pearl’s causal calculus,” is now up on arxiv, and here is the abstract:

Probabilistic graphical models are a fundamental tool in statistics, machine learning, signal processing, and control. When such a model is defined on a directed acyclic graph (DAG), one can assign a partial ordering to the events occurring in the corresponding stochastic system. Based on the work of Judea Pearl and others, these DAG-based “causal factorizations” of joint probability measures have been used for characterization and inference of functional dependencies (causal links). This mostly expository paper focuses on several connections between Pearl’s formalism (and in particular his notion of “intervention”) and information-theoretic notions of causality and feedback (such as causal conditioning, directed stochastic kernels, and directed information). As an application, we show how conditional directed information can be used to develop an information-theoretic version of Pearl’s “back-door” criterion for identifiability of causal effects from passive observations. This suggests that the back-door criterion can be thought of as a causal analog of statistical sufficiency.

If you had seen my posts on stochastic kernels, directed information, and causal interventions, you will, more or less, know what to expect.

Incidentally, due to my forthcoming move to UIUC, this will be my last Allerton paper!

## 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)
- Tom Cover, “On the St. Petersburg paradox”
- Paul Cuff, Tom Cover, Gowtham Kumar, Lei Zhao, “A lattice of gambles”
- Ioanna Ioannou, Charalambos Charalambous, Sergey Loyka, “Outage probability under channel distribution uncertainty” (pdf; longer version: arxiv:1102.1103)
- Mohammad Naghshvar, Tara Javidi, “Performance bounds for active sequential hypothesis testing”
- Chris Quinn, Negar Kiyavash, Todd Coleman, “Equivalence between minimal generative model graphs and directed information graphs” (pdf)
- Ofer Shayevitz, “On Rényi measures and hypothesis testing” (long version: arxiv:1012.4401)

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.

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.

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.

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 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 -ary hypothesis testing, variable-length coding with feedback, and noisy dynamic search.

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 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.

## ISIT 2011: plenaries, the Shannon lecture

Better late than never, right? Besides, Anand all but made sure that I would blog about it eventually.

I have attended this year’s Shannon lecture and all the plenaries except for Wojtek Szpankowski‘s talk on the information theory of algorithms and combinatorics (I bravely fought the jetlag and lost), but you can watch it here. Here are, shall we say, some impressionistic sketches based on the notes I was taking.

## Autumn travels

I have been on the road for the past few days. First I went to Washington DC to visit University of Maryland at College Park and to present my work on empirical processes and typical sequences at their Information and Coding Theory Seminar. A scientist’s dream — two hours in front of a blackboard, no slides!

And now I find myself amid the luscious cornfields of Central Illinois. That’s right, until Friday I’m in Urbana-Champaign for the annual Allerton conference. This year, Todd Coleman (UIUC), Giacomo Como (MIT), and I have co-organized a session on Information Divergence and Stochastic Dynamical Systems, which promises to be quite interesting — it will feature invited talks on Bayesian inference and evolutionary dynamics, reinforcement learning, optimal experimentation, opinion dynamics in social networks, signaling in decentralized control, and optimization of observation channels in control problems. If you happen to be attending Allerton this year, come on by!

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