Actually, the example of Bayesian reasoning in the post below doesn’t really give the flavor of Tenenbaum’s work. Here’s something a little closer. Abe and Becky each roll a die 12 times. Abe’s rolls come up

1,2,3,4,5,6,1,2,3,4,5,6.

Becky gets

2,3,3,1,6,2,5,2,4,4,1,6.

Both of these results occurs with probability (1/6)^12, or about one in two billion. So why is it that Abe finds his result startling, while Becky doesn’t, when the two outcomes are equally unlikely?

Extra credit: what does this have to do with arguments about intelligent design?

(If you like the extra credit question, you might want to read my colleague Elliot Sober’s papers on the topic, or even buy his book!)

Our friend Josh Tenenbaum, a psychologist at M.I.T., is in town to plenarize this Thursday at 3pm at the First Annual UW Cognitive Science Conference. He hasn’t posted a title, but he’ll be talking about his research on Bayesian models of cognition. What makes a model “Bayesian” is a close attention to a priori probabilities, usually called priors.

For instance: suppose you’re an infant, trying to figure out how language works. You notice that when you wake up, your father points out the window and says “Hello, sun!” Pretty soon you figure out that the bright light he’s pointing at is called “sun.” But the evidence presented to you is just as consistent with the theory that, when you’ve just woken up, the word “sun” refers to the bright light out the window — but before bedtime it means “the stuffed monkey on the dresser.” How, infant, do you pick the former theory over the latter? Because you have some set of priors which tells you that words are very unlikely to refer to different things at different times of day. You don’t learn this principle from experience — you start with it, which is what makes it a “prior.” According to Chomsky-style linguistics, you are born with lots of priors about language — you know, for instance, that there’s a fixed order in which the subject, object, and verb of a sentence are supposed to come. If you didn’t have all these priors, you wouldn’t have a chance of learning to talk; there’s an infinitude of theories of language, all consistent with the evidence you encounter. Your priors are what allows you to narrow that down to the one that’s vastly more likely than the sometimes-it’s-a-monkey alternatives.

I don’t think Josh is going to talk about language acquisition, but he is going to talk about the ways that lots of interesting cognitive processes can be described in Bayesian terms, and how to check empirically that these Bayesian descriptions are accurate. Recommended to anyone who likes mathy thinking about thinking.

I spent last Wednesday morning working in the profoundly pleasant Prairie Cafe in Middleton Hills. This is the kind of unassuming place that you’d assume would make really first-rate breakfast and soups and maybe a heavily besprouted chicken-salad sandwich, but where you might hesitate to order a hot lunch. In fact, the corned beef hash, while homemade, was just so-so, while the reuben was really first-rate. The cold black-bean and corn salad that came alongside in lieu of coleslaw was even better, a crisp contrast to the thoroughly correct hot goopiness of the reuben.

Middleton Hills, it turns out, is a Duany Plater-Zyberk development in the “New Urbanist” style. Which means mixed retail and housing, walkability, density, stores fronting directly on sidewalks, cheap houses and expensive ones on the same block, and so on. Basically, if you take every feature of America’s soul-killing suburbs that people like to complain about, invert them, and build housing developments based on the result, you get something like New Urbanism.

As for me, I grew up in one of America’s soul-killing suburbs, and I like them! One of the nicest features of the Near West Side of Madison is that you can get on your bike and be in an authentically urban landscape in 15 minutes; or, after a 15-minute drive in the other direction, you can pull up in the oversized parking lot outside the even more oversized grocery store and load your station wagon until it groans.

Anyway, Middleton Hills. My first impression is that it’s charming; the houses all share a mild kind of Prairie style, but no two on the block look exactly alike. The main drag, Frank Lloyd Wright Boulevard, winds around a big and agreeably wild pond; lots of cattails, lots of birds, grass not too kempt. The street names do a good job of congratulating you for your participation in sustainable development — John Muir Drive, Aldo Leopold Way, and, best of all, Diversity Road.

My second impression is that it’s completely empty. You can see that the streets are laid out to encourage pedestrianism and unplanned human interaction, as in Princeton, a favorite town of Duany Plater-Zyberk’s, and mine. But at three in the afternoon, the only people I saw were a trickle of kids coming home from school, and a birdwatcher. The birdwatcher and I watched a sandhill crane for a few minutes. Then I sat down to continue revising a long-overdue paper with Michel and Venkatesh about sums of three squares. (Among other things, the paper features a careful explanation of the group structure — more properly, torsor structure — on the set of representations of a squarefree integer n as the sum of three squares. More on this when the paper’s finished.)

What makes Princeton’s streets lively and new-urban, of course, is that it has a big and interesting downtown, whose shops and restaurants serve not just Princetonians but residents of the surrounding towns. Middleton Hills has a grocery store, the Prairie Cafe, a pizza place, and a Starbucks — not enough to draw foot traffic away from Madison, or, for that matter, downtown Middleton. If this post pulls in a throng of reuben-lovers, I guess I’ll have done my bit for the New Urbanism.

I finished reading the red book, so I’m now a complete authority on my classmates in the Harvard-Radcliffe Class of 1993.

• If you are a woman who went to Harvard and you’re not presently working outside the house, you call yourself a “stay-at-home mom” — but if you’re a woman married to a male alum and in the same situation, your husband might call you a “homemaker,” or even, in one case, a “housewife.”
• Triathlon is startlingly popular among my classmates. Alternative explanation: each and every person who completed a triathlon mentioned this fact in their entry. Actually, both might apply. Travel to non-western countries enjoys a similar status.
• Funniest sentence (intentional division) “I’m still in Washington, DC, eager to prove that a Jewish man with a law degree can make it in this town.”
• Funniest sentence (unintentional division) “Needless to say I am quite astonished that fifteen years have passed since we first set foot on campus.”
• Most of the really angry people don’t write in, so kudos to the person who started her entry “If you really gave a shit about what I have been doing with my life, I’d have heard from you by now.”
• Harvard does thousands of great things to its students and a few bad things, one of which is to promote the idea that the people eating Chickwiches on either side of us are fated to be the rulers of the world we’ll live in as adults. Not true, it turns out. On first glance, I think the members of our class most notable to the world at large are the executive producer of the Daily Show and the minority whip of the Florida State Senate.
• I sometimes buy old Harvard reunion books at used bookstores; these make very absorbing reading, if you dont mind the fact that you’re almost certainly going through a dead person’s memorabilia. The books from 35th and 50th reunions muse a lot about the meaning of it all — this is totally absent from the present book, apart from a few classmates who switched religions, or obtained one for the first time.
• Finally, if you’re reading this and you went to Harvard, please do click the “Chickwiches” link. It’s horrifying and disgusting in all the right ways.

Which are actual Britax carseats and which did I make up?

• Diplomat
• Monarch
• Overlord
• Emperor
• Royal
• Regent
• Viceroy
• Tyrant
• Viscount
• Chancellor
• Sultan
• Husky

It means “after whatever fashion.” I didn’t learn this in school — my high-school Latin stopped somewhere around “Petrus Cuniculus in arbo cum eius tres fratribus, Flopsus, Mopsus, Cottontalusque habuit.” I saw it on a slide in a history-of-mathematics lecture and had to go find out what it meant. Emmanuel just pointed out to me that, while “quomodocumque” hasn’t yet made it as an English loan-word, the OED does have the related “quomodocunquize,” meaning “to make money however one can.” The most recent cite is 1652, but in these near-recessionary times, quomodocunquizing — or even all-out quomodocunquism — may be poised for a comeback.

Dear fashionable women of Madison,

I can’t help noticing that lately you’ve been wearing a lot of big, blocky off-white trench coats with giant buttons and double-wide belts. I’m sorry to have to be the one to tell you that you look like a cross between my dad in 1979 and a subway flasher. Please reconsider.

Sincerely yours,

Quomodocumque

Here’s a funny question. Let f in C[x] be a squarefree polynomial of degree at least 6. Let S be the set of complex numbers t such that the Jacobian of the hyperelliptic curve

is not simple. Is S always finite? Even more, is there a bound on |S| which doesn’t depend on f, or depends only on the degree of f?

This question comes from the introduction to “Non-simple abelian varieties in a family: geometric and analytic approaches” , a new paper by me, Christian Elsholtz, Chris Hall, and Emmanuel Kowalski. In its original form this was a four-author, six-page paper — fortunately we’ve now added enough material to make the ratio a bit more respectable!

The paper isn’t about complex algebraic geometry at all — it explains how to get bounds on S when f has rational coefficients and t ranges over rational numbers, which is quite a different story. The point of the paper is partly to prove some theorems and partly to make a metamathematical point — that problems of this kind can be approached via either arithmetic geometry or analytic number theory, and that the two approaches have complementary strengths and weaknesses. Bounds from arithmetic geometry are stronger but less uniform; bounds from analytic number theory are weaker but have better uniformity.

Here’s my favorite example of this phenomenon. Let X be a smooth plane curve over Q of degree d at least 4. Then by Faltings’ Theorem we know that X has only finitely many rational points.

On the other hand, a beautiful theorem of Heath-Brown tells us that the number of rational points on X with coordinates of height at most B is at most C B^(2/d), for some constant C depending only on d. At first, this seems to give much less than Faltings. After all, as B gets larger and larger, the upper bound given by Heath-Brown gets arbitrarily large — whereas we know by Faltings that there are only finitely many points on the whole curve, no matter how large we allow the coordinates to be.

But note that the constant in Heath-Brown’s result doesn’t depend on the curve X. It is what’s called a uniform bound. Faltings’ theorem, by contrast, gives an upper bound on the number of points which depends very badly on the choice of X. Depending on what you’re trying to accomplish, you might be willing to sacrifice uniformity to get finiteness — or the reverse. But it’s best to have both options at hand.

Is it possible to have uniformity and finiteness simultaneously? Conjecturally, yes. Caporaso, Harris, and Mazur showed that, conditional on Lang’s conjecture, there is a constant B(g) such that every genus-g curve X/Q has at most B(g) rational points. The Caporaso-Harris-Mazur paper came out when I was in graduate school, and the idea of such a uniform bound was considered so wacky that CHM was thought of as evidence against Lang’s conjecture. Joe Harris used to wander around the department, buttonholing graduate students and encouraging us to cook up examples of genus-g curves with arbitrarily many points, thus disproving Lang. We all tried, and we all failed — as did many more experienced people. And nowadays, the idea that there might be a uniform bound for the number of rational points on a genus-g curve is considered fairly reputable, even among people who have their doubts about Lang’s conjecture. As far as I know, the world record for the number of rational points on a genus-2 curve is 588, due to Kulesz. Can you beat it?

The other day we took CJ to a house with a friendly, but barking, golden retriever. CJ was a little intimidated, and at one point really scared when the dog took a run at him.

Later we had this conversation:

Me: CJ, were you a little scared of the dog?

CJ: I was scared but Daddy was not scared.

Me: Maybe when you’re a little older you won’t be scared.

CJ: When I am a little older and Daddy is a baby we will go in the house with the dog and I will not be scared but Daddy will be scared!

Update:  Toby asks in comments whether it’s common for children to have this misperception.  Fortunately, the indispensable I Used To Believe, the online repository of weird childhood beliefs, is here to help.  We quickly find:

• “I used to belive as i got older my parents would get younger and i would have to take care of them as babies until there body couldn’t age as fast then they would die”
• “I used to believe that when I got to a certain age, my older brother would ‘turn small’ and I would be able to be the older sibling. I believed that this cycle would last forever.”
• “i used to believe that my brother and I were going to switch places with my parents - that as we got bigger, they would get smaller. I was under 5, because my other brother had not been born yet - and it seemed to make sense.”
• “When I was about 3 or 4 years old, I believed that, just as little kids were in the process of growing bigger and older, adults were in the process of growing younger and littler. So I thought that my parents would one day be little children again.”

And that’s just from the first 5 pages out of 27 on the topic of “growing older.”  Much more common, apparently, is the belief that you switch genders as you age - CJ has expressed some thoughts along these lines, too, now that I think about it.

It’s snowing here.

That is all.