More math psychology

Did I really let almost a month elapse without posting? Must be because my life is so wild and exciting. Um, yeah.


Here's a question that's been occupying me for a while -- non-mathematicians' perception of probability. What does a man on the street mean when he says a coin has a 50% chance of landing on heads?


The most likely interpretation is frequentist: if you flip the coin a whole lot of times, you'll see heads about half the time, on average. (How many times is a lot? What does on average mean here? Read my thesis -- at least the section on the Law of Large Numbers.)

But what about one-time events? What does it mean that there's a 30% chance of rain tomorrow? (Tomorrow will only happen once, so any talk of averages is meaningless.) Heck, what does the meteorologist mean by that probability?

Cosma helped me resolve the latter quite satisfactorily in private correspondence (so satisfactorily, in fact, that I feel dumb ever having asked the question). But I turn to the readers:

1. can you make rigorous mathematical sense out of the meteorologist's 30% chance of rain prediction?

2. can you ask your non-mathematician friends what that prediction means to them?

Is There Anything Good About Men?

I know I'm behind on posting about the things I promised I'd post about; rest assured that I'm behind on work stuff as well.

So in the meantime, read this fascinating piece by Roy F. Baumeister. There's too much incisive analysis in there to give a brief sound bite; you'll just have to read the whole thing. Discussion welcome in the comments.

I like to tell myself that if people didn't send me such pointers I'd be 50% more productive...

Psychology of Mathematical Reasoning

Recently, I've found myself needing to explain what it is that mathematicians do. Sometimes I say, "we add really big numbers". You'd think people would laugh (or at least give an incredulous look) -- but how many times have you heard a layman casually comment how "math people" deal with "numbers"?

Actually, number theory is a nice vehicle for giving layfolk a taste of what math is about. Everybody knows about naturals and primes (and if they don't, and you're a mathematician who's been put on the spot, it's something you can explain in under a minute). So I tell people, look: there are obviously infinitely many naturals (for any number there's always a bigger one) and there are also infinitely many primes -- but this latter fact is less obvious and requires proof.

Here is where I've run into unexpected troubles. People have no problem accepting the infinitude of the naturals, but what they have trouble appreciating is that it's not obvious that the primes are infinite. "C'mon -- there are infinitely many numbers, so of course there are infinitely many primes!" I've heard this response from more than one person. "Now wait a minute" I protest. The primes are a subset of the naturals, so a priori, they have every right to be a smaller subset. "OK, gimme an example of a finite subset of the naturals". I'm happy to provide the example {1,2,3,4,5}. "Yeah, but you've constructed it as a finite set, so it doesn't count" is the sort of reply I get.

What seems to be happening is that to an untrained intuition, any subset of the naturals defined by a property without explicit bounds appears to be obviously infinite. Has anyone else encountered this phenomenon? Alexandre? Can my mathematician readers try this out on some non-math friends (no need to obtain signed consent forms) and let me know what you find?

Finally, does anyone have a simple example of a "nontrivially finite" subset of the naturals? That is, a set defined by a (simple!) property P that makes no reference to explicit bounds, yet P is provably finite?

Extreme-point fallacy

First, some administrative notes. Thanks to the readers for the feedback. I got requests for research advice and explanations of techniques from my thesis; I promise to post on both topics shortly.

Do take a look at Daniel's problem in this comment thread. It combines the features of being theoretically interesting and challenging while also being of great practical importance. If anyone has thoughts or literature pointers, please do share them with me or Daniel!

And here's a little teaser to see who's awake. Let D be a subset of R^n of the nicest possible kind: a finitely generated compact convex polytope. Thus, D is nothing more than the convex hull of finitely many points. Let f and g be two convex functions mapping D to R. A mathematician (let's call him Hermann) would like to prove that
f(x) <= g(x)
for all x in D. Now Hermann reasons as follows. He knows (from reading Papadimitriou and Steiglitz, for example) that an affine function achieves its extreme values on the extreme points of a convex domain. From this he concludes that a convex function achieves its maximal values on the extreme points of a convex domain (can you also make this deduction? Hint: the epigraph of a convex function lies above its derivative.). "Aha!" he exclaims. "All I need to do is check that f(x) <= g(x) on the extreme points of D. But there are finitely many of these -- they're just the corners of the polytope!"

You'll agree that verifying f(x)<=g(x) on the finitely many corners of D is, in general, a much simpler task than doing this for all of D. Is there something wrong with Hermann's reasoning, however?

[This is in no way meant to imply that any mathematician named Hermann ever claimed anything of this sort!]

Blog update & open thread

After months of cluelessness, I finally figured out how to make recent comments appear in that tab you now see on the right; many thanks to the anonymous commenter! No more digging through old posts to see who's left a comment (I've occasionally discovered comments on several week old posts.) Any idea how to make that "recent comments" list longer than 5 (blogger doesn't seem to give me that option)?

Following Daniel's suggestion, I'll make this an open thread. Post your problems (open or solved), tell some jokes, start flamewars -- it's all good! [I've never had to remove a comment or "moderate" in general, but don't push me...]

A CMU undergrad (CS) asks for some generic research advice and I promise to devote a post to this shortly.

Trip report + misc.

Back in Israel; it's good to be home. Now that the travels are over, I can look back and say "it wasn't that bad" -- but that's not what I would've told you when I was within a hair of getting arrested for refusing to give up my toothpaste at an airport security screening.

My submission got a "distinguished contribution" award at MLG (in lieu of "best paper" since these are extended abstracts). The analysis and probability workshop at Texas A&M was very intense, reminding me yet again how little of my so-called field I know.

"Meaty" content has been sparse here but I have a good excuse. I'm trying to switch gears from a problem-solving to a paper-writing mode. This requires less creativity and more discipline, but is absolutely indispensable -- both for one's career and for keeping oneself honest.

There's no shortage of high-quality, educational and entertaining material on the web (check out the links on the right), thus I feel no pressure to "deliver" to any readership. Speaking of which, if you're a regular reader of this blog, how about leaving a comment? Just so I know the blog has regular (or any) readers. [Not that a lack of readers has ever discouraged me from writing.] Also, feel free to use the comments to make suggestions regarding what you'd be interested in seeing here. Well enough chit-chat. Back to work.

Blogging from the road

I'm writing from Florence, Italy, where I'm attending the Mining and Learning with Graphs workshop. The paper I'm presenting is the Universal Kernel one. On the off-chance you're reading this and attending the workshop, say hi!

Then it's off to Texas for Concentration Week, namely: "Probability Inequalities with Applications to High Dimensional Phenomena". My talk: Obtaining measure concentration from Markov contraction. Slides will be online soon. [I'm intentionally blurring the distinction between Leonid and Aryeh (they're really the same name); hopefully, this won't cause confusion. Rule of thumb: if we're speaking English, use Leo. If we're speaking Hebrew, Aryeh. If we're speaking Russian, you know what to call me.]