December 10, 2011

Notes on stable distributions

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After attending a recent talk at TTI about dimension reduction by Moses Charikar in which he mentioned the special role stable distributions play, I made a note to freshen up my own scattershot knowledge of facts about stable distributions. Of course, things got too busy and the the note ended up on my sub-list of to-do items that get infinitely postponed. However, I’ve been saved by a recent post to ArXiV by Svante Janson, who does all sorts of interesting work on these cool objects called graphons (the limits of infinite graph processes) :

Stable Distributions
Svante Janson

We give some explicit calculations for stable distributions and convergence to them, mainly based on less explicit results in Feller (1971). The main purpose is to provide ourselves with easy reference to explicit formulas. (There are no new results.)

All (or at least most) of the facts I wanted in one place! Hooray!

He starts with infinitely divisible distributions (e.g. Gaussian, Poisson, Gamma) and then talks about \alpha-stable distributions and the uniqueness of the corresponding measures for \alpha \in (0,2] (the case \alpha = 2 gives the Gaussian. I’m still reading it (bits at a time), but it’s great to have little surveys like this — broadens the mind, builds character, &c.

 

November 14, 2011

The basketball strike and some confusing lingo

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Via Deadspin I saw this AP article on the latest twist in the NBA labor dispute and this tweet from columnist Adrian Wojnarowski : “The chances of losing the entire 2011-12 season has suddenly become the likelihood.” Assuming we correct to “likelihood,” what does this mean from a statistical standpoint? Is this frequentist analysis of a Bayesian procedure? Help me out folks…

 

November 2, 2011

HGR maximal correlation and the ratio of mutual informations

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From one of the presentation of Zhao and Chia at Allerton this year, I was made aware of a paper by Elza Erkip and Tom Cover on “The efficiency of investment information” that uses one of my favorite quantities, the Hirschfeld–Gebelein–Rényi maximal correlation; I first discovered it in this gem of a paper by Witsenhausen.

The Hirschfeld–Gebelein–Rényi maximal correlation \rho_m(X,Y) between two random variables X and Y is

\sup_{f \in \mathcal{F}_X, g \in \mathcal{G}_Y} \mathbb{E}[ f(X) g(Y) ]

where \mathcal{F}_X is all real-valued functions such that \mathbb{E}[ f(X) ] = 0 and \mathbb{E}[ f(X)^2 ] = 1 and \mathcal{G}_Y is all real valued functions such that \mathbb{E}[ g(Y) ] = 0 and \mathbb{E}[ g(Y)^2 ] = 1. It’s a cool measure of dependence that covers discrete and continuous variables, since they all get passed through these “normalizing” f and g functions.

The fact in the Erkip-Cover paper is this one:

sup_{ P(z|y) : Z \to Y \to X } \frac{I(Z ; X)}{I(Z ; Y)} = \rho_m(X,Y)^2.

That is, the square of the HGR maximal correlation is the best (or worst, depending on your perspective) ratio of the two sides in the Data Processing Inequality:

I(Z ; Y) \ge I(Z ; X).

It’s a bit surprising to me that this fact is not as well known. Perhaps it’s because the “data processing” is happening at the front end here (by choosing P(z|y)) and not the actual data processing Y \to X which is given to you.

 

November 1, 2011

Readings

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Tiassa, by Steven Brust. As Cosma puts it, mind candy, and only worth reading if you’ve read the other 10 books in the series. Quite enjoyable, however.

Kraken, by China Miéville. A rollicking adventure involving a giant squid, horrific monsters and gruesome deaths, a dark underbelly of London, the end of the world, and… a ghost piggie. Among other things. I enjoyed it.

Hindoo Holiday, by J.R. Ackerley. A travelogue of a gay Englishman who becomes an attaché to a gay Raja in a princely state in the early 20th century. Often full of colonial condescension (though in a light tone) about things Indian. Most of us are tragically sad of buffoonish. The homosexuality is not overt but explicit enough that the book was censored when published. Still, it’s an interesting historical read, just because it is so weird.

The Lost Promise of Civil Rights, by Risa Goluboff. A really fascinating book about the history of civil rights litigation in the US from Lochner to Brown. The term “civil rights” was in a state of flux during that era, transitioning from a labor-based understanding to discrimination-based standing. The main players were the Justice Department’s Civil Rights Service and the NAACP. By choosing which cases to pursue and which arguments to advance, they explored different visions of what civil rights could mean and why they were rights in the first place. In particular, the NAACP did not take on many labor cases because they were actively pursuing a litigation agenda that culminated in Brown. The decision in Brown and subsequent decisions shaped our modern understanding of civil rights as grounded in stopping state-sanctioned discrimination. However, the “lost promise” in the title shows what was lost in this strategy — the state-sponsored parts of Jim Crow were taken down, but the social institutions that entrench inequality were left.

The Devil in the White City, by Erik Larson. I had to read this since I just moved to Chicago and I work right near Jackson Park. This was a very engaging read (Larson just has that “style”) but a bit creepy in that “watched too many episodes of Dexter” way. I enjoyed it a little less than Thunderstruck, but I had more professional attachment to that one.

 

October 30, 2011

Banff blog

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I figured I would blog about this week’s workshop at Banff in a more timely fashion. Due to the scheduling of flights out of Calgary, I will have to miss the last day of talks. The topics of people’s presentations varied rather widely, and many were not about the sort of Good-Turing estimator setup. Sometimes it was a bit hard to see how to see how the problems or approaches were related (not that they had to be directly), but given that the crowd had widely varying backgrounds, presenters had a hard time because the audience had to check in a new set of notation or approach for every talk. The advantage is that there were lots of questions — the disadvantage is that people insisted on “finishing” their presentations. By mid-week my brain was over-full, and a Wednesday afternoon hike up Sulphur Mountain was the perfect solution.

The view from Sulpur Mountain

The view from Sulpur Mountain

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October 23, 2011

Banfffffffffffffff

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I’ve just arrived in chilly but beautiful Banff for a workshop on Information theory and statistics for large alphabets. I’m looking forward to it, although I will have to miss the last day due to the timing of flights out of Calgary that get me to Chicago before midnight. My itineraries there and back seem especially perverse : ORD-SEA-YYC and YYC-SFO-ORD. However, thanks to the new gig I have a new laptop with a functional battery so I am doing a bit more busy-work and less New Yorker reading in the plane. I might try to write a bit more about the topics in the workshop — although the topic seems focused, there are a wide range of approaches and angles to take on the problem of estimating probabilities/prevalences in situations where you may not get to see each outcome once. Certainly I hope I can get the journal version of a paper from last year’s Allerton squared away.

 

October 23, 2011

Bayesianism in philosophy

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In an effort to get myself more philosophically informed with regards to probability and statistics, I’ve been reading about various notions and their discontents, such as symmetry, or Bayesianism, or p-values. I was delighted to find this recent pair of papers (part I,part II) by fellow Berkeley-ite and occasional puzzle-partner Kenny Easwaran (now a prof at USC) on Bayesianism in Philosophy Compass. In the first paper he goes through basic tenets of Bayesian approaches to probability in terms of subjective belief, and their philosophical justification via rational actions or “Dutch book” arguments and representation theorems. What’s also interesting from a scientific view (somewhat off-topic from the article) is the angle being advanced (some might say “pushed”) by some cognitive scientists that people are actually doing some kind of Bayesian conditionalization in certain tasks (here’s a plug for my buddy Pradeep‘s work). The second article talks about the difficulties in developing a consistent and quantitative “confirmation theory” in Bayesianism. In different fields there are different questions how how to do this, and as Kenny points out, the anti-Bayesians in different fields are different — the null-position is not necessarily frequentism.

They’re a relatively quick read, and I think provide some different perspectives for those of us who usually see these concepts in our little fiefdoms.

 

October 19, 2011

Quote of the day : the ethical “success” of probability

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I am reading Ian Hacking’s The Taming of Chance, which I picked up from The Seminary Coop upon arriving here. They just had it on the shelf! The book, as he puts it, is a way to understand why probability has been “an incredible success story” in the realms of metaphysics, epistemology, logic, and ethics. By success he means probabilistic ideas have radically changed these areas. On the last point:

Ethics is in part the study of what to do. Probability cannot dictate values, but it now lies at the basis of all reasonable choice made by officials. No public decision, no risk analysis, no environmental impact, no military strategy can be conducted without decision theory couched in terms of probabilities. By covering opinion with a veneer of objectivity, we replace judgement by computation.

 

October 14, 2011

Allerton 2011

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Despite Yury‘s attempts to get me to “stop blogging,” here is my much-delayed recap of Allerton. Because of moving to TTI-Chicago right after Allerton and then almost immediately shuttling back to UCSD for the iDASH Privacy Workshop, things have been a bit delayed. I could only attend for two days but I wanted to highlight a few interesting talks that I saw. More Allerton blogging was done by Michael Mitzenmacher (part 1, part 2) and a bit by Maxim Raginsky about his talk on causal calculus (since he blogged about it I don’t have to, ha!). The conference has gotten too big and the rooms are too small to hold the audience, so it probably is time to move the thing. We have similar issues at ITA and the 2012 ITA Workshop is moving off campus next year (you heard it here first, folks!)

But here are some interesting talks I saw:

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October 2, 2011

Linkage

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I will post more about Allerton soon (I’m still on the road), but I wanted to clear out some old links before doing that. I’m starting my new gig at TTIC this week, and the last few weeks have been a whirlwind of travel and internetlessness, so blogging has been curtailed.

And a (not-so-recent) tour around the ArXiV — I haven’t had a chance to read these yet, but maybe once I am settled…

 

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