The Coin Game, II

Good answers to the last question! I think I perhaps put my thumb on the scale too much by naming a variable p.

Let me try another version in the form of a dialogue.

ME: Hey in that other room somebody flipped a fair coin. What would you say is the probability that it fell heads?

YOU: I would say it is 1/2.

ME: Now I’m going to give you some more information about the coin. A confederate of mine made a prediction about whether the coin would fall head or tails and he was correct. Now what would you say is the probability that it fell heads?

YOU: Now I have no idea, because I have no information about the propensity of your confederate to predict heads.

This is in fact what Joyce thinks you should say. White disagrees. But I think they both agree that it feels weird to say this, whether or not it’s correct.

Why would it not feel weird? I think Qiaochu’s comment in the previous thread gives a clue. He writes:

Re: the update, no, I don’t think that’s strange. You gave me some weird information and I conditioned on it. Conditioning on things changes my subjective probabilities, and conditioning on weird things changes my subjective probabilities in weird ways.

In other words, he takes it for granted that what you are supposed to do is condition on new information. Which is obviously what you should do in any context where you’re dealing with mathematical probability satisfying the usual axioms. Are we in such a context here? I certainly don’t mean “you have no information about Coin 2” to mean “Coin 2 falls heads with probability p where p is drawn from the uniform distribution (or Jeffreys, or any other specified distribution, thanks Ben W.) on [0,1]” — if I meant that, there could be no controversy!

I think as mathematicians we are very used to thinking that probability as we know it is what we mean when we talk about uncertainty. Or, to the extent we think we’re talking about something other than probability, we are wrong to think so. Lots of philosophers take this view. I’m not sure it’s wrong. But I’m also not sure it’s right. And whether it’s wrong or right, I think it’s kind of weird.

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The coin game

Here is a puzzling example due to Roger White.

There are two coins.  Coin 1 you know is fair.  Coin 2 you know nothing about; it falls heads with some probability p, but you have no information about what p is.

Both coins are flipped by an experimenter in another room, who tells you that the two coins agreed (i.e. both were heads or both tails.)

What do you now know about Pr(Coin 1 landed heads) and Pr(Coin 2 landed heads)?

(Note:  as is usual in analytic philosophy, whether or not this is puzzling is itself somewhat controversial, but I think it’s puzzling!)

Update: Lots of people seem to not find this at all puzzling, so let me add this. If your answer is “I know nothing about the probability that coin 1 landed heads, it’s some unknown quantity p that agrees with the unknown parameter governing coin 2,” you should ask yourself: is it strange that someone flipped a fair coin in another room and you don’t know what the probability is that it landed heads?”

Relevant readings: section 3.1 of the Stanford Encyclopedia of Philosophy article on imprecise probabilities and Joyce’s paper on imprecise credences, pp.13-14.

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Vampire post gets Brooksed

A while ago I read a great paper by the philosopher L. A. Paul and wrote this post about it, asking:  is the experience of becoming a vampire analogous in important ways to the experience of becoming a parent?  When deciding whether to become a vampire, is it relevant what human you thinks about being a vampire, or only what future vampire you would think about being a vampire?

Paul liked the example and was kind enough to include (her much deeper and more fully worked-out version of) it in her book, Transformative Experience.

And now David Brooks, the official public philosopher de nous jours, has devoted a whole column to Paul’s book!  And he leads with the vampires!

Let’s say you had the chance to become a vampire. With one magical bite you would gain immortality, superhuman strength and a life of glamorous intensity. Your friends who have undergone the transformation say the experience is incredible. They drink animal blood, not human blood, and say everything about their new existence provides them with fun, companionship and meaning.

Would you do it? Would you consent to receive the life-altering bite, even knowing that once changed you could never go back?

The difficulty of the choice is that you’d have to use your human self and preferences to try to guess whether you’d enjoy having a vampire self and preferences. Becoming a vampire is transformational. You would literally become a different self. How can you possibly know what it would feel like to be this different version of you or whether you would like it?

Brooks punts on the actually difficult questions raised by Paul’s book, counseling you to cast aside contemplation of your various selves’ preferences and do as objective moral standards demand.  But Paul makes it clear (p.19) that “in the circumstances I am considering… there are no moral or religious rules that determine just which act you should choose.”

Note well, buried in the last paragraph:

When we’re shopping for something, we act as autonomous creatures who are looking for the product that will produce the most pleasure or utility. But choosing to have a child or selecting a spouse, faith or life course is not like that.

Choosing children, spouses, and vocations are discussed elsewhere in the piece, but choosing a religion is not.  And yet there it is in the summation.  The column is yet more evidence for my claim that David Brooks will shortly announce — let’s say within a year — that he’s converting to Christianity.  Controversial predictions!  And vampires!  All part of the Quomodocumque brand.

 

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John Kasich’s blue collar roots

From today’s NYT profile of John Kasich:

As the people of Ohio already know — and Republican voters elsewhere are just beginning to find out — Gov. John R. Kasich grew up in working-class McKees Rocks, Pa., the son of a postal worker and the grandson of a coal miner. His grandfather was so poor, Mr. Kasich recently told voters in New Hampshire, that he would bring home scraps of his lunch to share with his children.

“They would even be able to taste the coal mine in that lunch,” Mr. Kasich said. “Some of you can relate to that.”

As a congressman and as governor, Mr. Kasich has made hardscrabble stories of life in McKees Rocks a cornerstone of his political biography.

Another kind of politician would have as cornerstone of his political biography: “My grandfather was a coal miner and was miserably poor, but my father was able to get a stable, well-paying job with the federal government, which is a big part of the reason I was able to get out of McKees Rocks and go to Ohio State, major in political science instead of something practical, and become a state senator when I was 26.”

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Pila on a “modular Fermat equation”

I like this paper by Pila that just went up on the arXiv, which shows the way that you can get Diophantine consequences from the rapid progress being made in theorems of Andre-Oort type.  (I also want to blog about Tsimerman + Zhang + Yuan on “average Colmez” and Andre-Oort, maybe later!)

Pila shows that if N and M are sufficiently large primes, you can’t have elliptic curves E_1/Q and E_2/Q such that E_1 has an N-isogenous curve E_1 -> E’_1, E_2 has an M-isogenous curve E_2 -> E’_2, and j(E’_1) + j(E’_2) = 1.  (It seems to me the proof uses little about this particular algebraic relation and would work just as well for any f(j(E’_1),j(E’_2)) whose vanishing didn’t cut out a modular curve in X(1) x X(1).)  (This is “Fermat-like” in that it asserts finiteness of rational points on a natural countable family of high-genus curves; a more precise analogy is explained in the paper.)

How this works, loosely:  suppose you have such an (E_1, E_2).  A theorem of Kühne guarantees that E_1 and E_2 are not both CM (I didn’t know this!) It follows (WLOG assume N > M) that the N-isogenies of E_1 are defined over a field of degree at least N^a for some small a (Pila uses more precise bounds coming from a recent paper of Najman.)  So the Galois conjugates of (E’_1, E’_2) give you a whole bunch of algebraic points (E”_1, E”_2) with j(E”_1) + j(E”_2) = 1.

So what?  Rational curves have lots of low-height algebraic points.  But here’s the thing.  These isogenous choices of (E’_1, E’_2) aren’t just any algebraic points on X(1) x X(1); they represent pairs of elliptic curves drawn from a {\em fixed pair of isogeny classes}.  Let H be the hyperbolic plane as usual, and write (z,w) for a point on H x H corresponding to (E’_1, E’_2).  Then the other choices (E”_1, E”_2) correspond to points (gz,hw) with g,h in GL(Q).  GL(Q), not GL(R)!  That’s what we get from working in a fixed isogeny class.  And these points satisfy

j(gz) + j(hw) = 1.

To sum up:  you have a whole bunch of rational points (g,h) on GL_2 x GL_2.  These points are pretty low height (for this Pila gestures at a paper of his with Habegger.)  And they lie on the surface j(gz) + j(hw) = 1.  But this surface is a totally non-algebraic thing, because remember, j is a transcendental function on H!  So (Pila’s version of) the Ax-Lindemann theorem generates a contradiction; a transcendental curve can’t have too many low-height rational points.

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More is less: Ted Olson on Citizens United

I saw Ted Olson and David Boies talk about the Citizens United decision at the Aspen Ideas Festival a couple of months ago.  Olson likes the decision, and he was passionate and funny in its defense.  “The more speech we have, the better,” he said.  And who can disagree?  The antidote to bad speech is good speech, marketplace of ideas, etc.

It wasn’t until I was on my way home, esprit de l’airplane, that it occurred to me to think about the followup case, Arizona Free Enterprise Club vs. Bennettdecided a year after Citizens United with the same five justices in the majority.  In that case, the Court found unconstitutional an Arizona law that provided government funds to publicly funded candidates allowing them to match any spending by a self-funded candidate exceeding a specified cap.  Here the Court managed to reason that adding more speech, funded by the state, added up to less speech.  They argued that a wealthy candidate whose every ad was matched by an equally well-funded opposition ad would refrain from campaigning at all — the self-funded candidates so inconfident in the strength of their ideas, apparently, as to prefer silence to both camps getting equal time.

It’s pretty starkly different from Olson’s let-a-hundred-flowers-bloom philosophy.  The Court called the Arizona law a “burden” on free speech, though of course it in no way prevented self-funded candidates from spending and speaking.  Unless you take the view that free speech responded to is effectively cancelled or suppressed, precisely the opposite of Olson’s attitude.  I wonder what he thinks about this decision?  Is the right to free speech a right to be heard, or a right to drown out?

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Natural accumulation

Don’t even ask me how I fell down this rabbit hole in the middle of August but I was trying to understand the legal requirements in Wisconsin and other states concerning shoveling snow off the city sidewalk in front of your house.  It turns out there’s no state law requiring this (though there are city ordinances in Madison and Milwaukee to this effect.)

More:  there’s a 1956 Wisconsin Supreme Court case, Walley v. Patake, which holds that a property owner isn’t liable if they fail to shovel the sidewalk abutting their property, and someone falls there and is injured, as long as the snow and ice is “natural accumulation” — that is, it’s a different story if there’s a huge heap of ice on the sidewalk because you piled it there when you shoveled your driveway.  In Hagerty v. Village of Bruce (1978) the Wisco Supremes clarified that even when the landowner is violating a city law by not shoveling, they still don’t take on liability.  The theory here is that the liability for injury on a public walkway belongs to the city, and the city can’t delegate it; the point of the shoveling law is to require landowners to act so as to make injuries less likely, but that’s all; the city is still liable.

In Ohio (Brinkman v. Ross, 1993) you are not even liable when someone slips on the ice on your own property, as long as it’s natural accumulation.  I wonder to what extent this is the case in other states?  I wonder if there’s a law professor somewhere in America who’s an expert on icy sidewalk liability?

 

 

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Devil math!

The Chinese edition of How Not To Be Wrongpublished by CITAC and translated by Xiaorui Hu, comes out in a couple of weeks.

ChineseCover

The Chinese title is

魔鬼数学

or

“Mo gui shu xue”

which means “Devil mathematics”!  Are they saying I’m evil?  Apparently not.  My Chinese informants tell me that in this context “Mo gui” should be read as “magical/powerful and to some extent to be feared” but not necessarily evil.

One thing I learned from researching this is that the Mogwai from Gremlins are just transliterated “Mo gui”!  So don’t let my book get wet, and definitely don’t read it after midnight.

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How to succeed in business without really dying

The New York Times reports that people with a longer workweek have more strokes.

People who work 55 hours or more per week have a 33 percent greater risk of stroke and a 13 percent greater risk of coronary heart disease than those working standard hours, researchers reported on Wednesday in the Lancet.

The new analysis includes data on more than 600,000 individuals in Europe, the United States and Australia, and is the largest study thus far of the relationship between working hours and cardiovascular health.

If for some reason you’re looking to write a contrarian “opposition to universal healthcare from the left” editorial, start right here!  When health insurance is tied to employment, as in the US model, businesses have some incentive to avoid workplace environments that leave their employees broken husks likely to require expensive long-term late-life care.  Once you break that link, businesses are free to work people until they stroke out, with the cost externalized to the health care system.

(Of course, an actual left take on this would no doubt involve heavier regulation on businesses to mitigate unhealthy workplace practices, expanding on things like OSHA, child labor laws, etc., but let’s not let that get in the way of a contrarian spin!)

“Losers often grow up to be writers”

My first cousin once removed Marilyn Sachs, on writing:

One final word of encouragement to those of you who are cowardly, cry babies, and liars, as I was. These are extremely promising qualities for future writers. If you are a coward, you will probably spend more time at the library than you would ordinarily, and if you tell lies, it just shows that you have an imagination even if others don’t always appreciate it. Cry babies tend to be sensitive, which is also a plus for writers. When I grew up, I found that I had become a great expert on bullies, and my books are full of them.

So, don’t feel you have to be smart, beautiful, brave and popular to become a writer. Or even to be a good speller. Losers often grow up to be writers, which means we have the final word.

Her books are mostly for kids.  Have you read them, parents?  Some of the classics:  Laura’s Luck (1965), my favorite alienated-kids-at-summer-camp book.  The Fat Girl (1984), a truly creepy YA novel about brutal psychosexual guerilla war in high school.  The Bear’s House (1972).  I remember almost nothing about this but just hearing the title makes me choke up so I know it was really sad.

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