Matt Yglesias makes an interesting point. The "typical" American is white, in that more than half of all Americans are white. The "typical" American is Christian, in that more than half of all Americans are Christian. But does this mean that the "typical" American is a white Christian, in that more than half of all Americans are white Christians? Not necessarily; I don't have the numbers.
Moreover, the "typical" white Christian votes Republican. Thus typical people vote Republican, so the Republicans should have won last night. But they didn't.
The point is that most people are "typical" in some ways, but few people are "typical" in all ways. And a party that is based around just people who are "typical" in all ways (note that I'm not saying this describes the Republican party) is doomed to fail, because most people are unusual along some dimension. I don't think this deserves the name of "paradox", but it's just something worth keeping in mind about How Statistics Work.
05 November 2008
04 November 2008
We win!
I heard that the featured articles at Wikipedia were the articles on McCain and Obama, and for the first time ever they had two featured articles at once.
So I went over to check.
But Wikipedia runs on GMT... and so it's Wednesday there. The featured article is Group (mathematics).
The "We" in the subject line is "mathematicians", of course.
So I went over to check.
But Wikipedia runs on GMT... and so it's Wednesday there. The featured article is Group (mathematics).
The "We" in the subject line is "mathematicians", of course.
Vote!
Scott Aaronson points out that the probability of your vote changing the results of the election scales like N-1/2, where N is the number of people. But the amount of change your vote creates, if it tips the election, scales like N. So the expected amount of change you will cause, by voting, scales like N1/2. That's a big number, so you should vote. If you live in a big country, you should vote more, although that's irrelevant; if any other country is voting today, the US media has ensured that I don't know that.
Of course, N1/2 seems a bit high, and it comes from modeling people as flips of a fair coin; Aaronson points out that under a more realistic prior (due to Andrew Gelman), the expected probability that your vote flips the election is N-1, so the expected amount of change your vote causes doesn't depend on the size of the country.
I won't "officially endorse" anybody. But one of the candidates in the present election trained as a lawyer, taught in a law school for a while, and likes to compare himself to a certain Senator from Illinois. That senator, in order to equip himself to understand the law better, studied Euclid. Being a mathematician, I think this is pretty cool. Who I'm voting for is left as an exercise for the reader.
Of course, N1/2 seems a bit high, and it comes from modeling people as flips of a fair coin; Aaronson points out that under a more realistic prior (due to Andrew Gelman), the expected probability that your vote flips the election is N-1, so the expected amount of change your vote causes doesn't depend on the size of the country.
I won't "officially endorse" anybody. But one of the candidates in the present election trained as a lawyer, taught in a law school for a while, and likes to compare himself to a certain Senator from Illinois. That senator, in order to equip himself to understand the law better, studied Euclid. Being a mathematician, I think this is pretty cool. Who I'm voting for is left as an exercise for the reader.
03 November 2008
AMS: Math in the Media
The AMS puts together a roundup of math in the (mainstream) media.
I think I may be the last person to know this, but in case I'm not, here it is.
I think I may be the last person to know this, but in case I'm not, here it is.
JSTOR question
I just googled JSTOR in order to find it. (Of course, it's at jstor.org.)
Anyway, the Google result I get reads (with links omitted):
Does Google know I'm interested in math, or does it say this to everybody?
As it turns out, I was in fact looking for an article from the Bulletin of the American Mathematical Society, so telling me there were math journals in there worked out well for them.
Anyway, the Google result I get reads (with links omitted):
JSTOR: Home
Scans of print journals, with 10 major math journals (requires subscription).
www.jstor.org/ - Similar pages - Note this
Does Google know I'm interested in math, or does it say this to everybody?
As it turns out, I was in fact looking for an article from the Bulletin of the American Mathematical Society, so telling me there were math journals in there worked out well for them.
A thought on expectation
Political campaigns should not campaign in such a way as to maximize their expected number of votes. Rather, they should campaign in such a way as to maximize their probability of winning.
Early in a campaign, these are pretty much the same thing, because one doesn't know how things are going to play out; late in a campaign they diverge. The candidate that's behind in the polls should, perhaps, start doing things that are, in expectation, Bad Ideas. If there were something that McCain could do that had a 1% chance of swinging 10% of the vote to him in the next 24 hours, and a 99% chance of scaring all the voters away, he should do it. (For example, let's say McCain turns out to secretly be an extraterrestrial; we probably don't want to be ruled by extraterrestrials, but who knows, we might change our mind? Of course, I'm being deliberately silly here.)
This is somewhat analogous to what happens in sports; strategies change late in a game. In the early innings of a baseball game you play, basically, in such a way as to maximize the difference between the expectation of your number of runs and your opponents' number of runs. In the late innings, when you have a better idea of how many runs you need, you change your strategies. See, for example, the bottom of the ninth inning of Game 3 of the World Series; tie game, bases loaded, nobody out. The Rays bring in one of their outfielders to create a fifth infielder; the idea is essentially that they want to maximize the probability of the Phillies scoring zero runs, and if the ball gets into the outfield a run is scoring anyway. (As it turns out, the Phillies won that game -- on a hit in the infield.) Of course, nobody saw that because it happened at two in the morning. Things are different when you're playing for one run.
Basically, what's happening as I write this is that Obama is running out the clock. (Yes, baseball doesn't have a clock. But Obama's a basketball player, so I think he'd like this metaphor.)
What politics and sports have in common, of course, is that there's a huge difference between second place and first place. If you're a company of some sort, does it matter if you sell 1% more or 1% less than your competitor? Not really, although it might have meaning for your pride. But if you're a politician, 1% of the vote makes a big difference.
Early in a campaign, these are pretty much the same thing, because one doesn't know how things are going to play out; late in a campaign they diverge. The candidate that's behind in the polls should, perhaps, start doing things that are, in expectation, Bad Ideas. If there were something that McCain could do that had a 1% chance of swinging 10% of the vote to him in the next 24 hours, and a 99% chance of scaring all the voters away, he should do it. (For example, let's say McCain turns out to secretly be an extraterrestrial; we probably don't want to be ruled by extraterrestrials, but who knows, we might change our mind? Of course, I'm being deliberately silly here.)
This is somewhat analogous to what happens in sports; strategies change late in a game. In the early innings of a baseball game you play, basically, in such a way as to maximize the difference between the expectation of your number of runs and your opponents' number of runs. In the late innings, when you have a better idea of how many runs you need, you change your strategies. See, for example, the bottom of the ninth inning of Game 3 of the World Series; tie game, bases loaded, nobody out. The Rays bring in one of their outfielders to create a fifth infielder; the idea is essentially that they want to maximize the probability of the Phillies scoring zero runs, and if the ball gets into the outfield a run is scoring anyway. (As it turns out, the Phillies won that game -- on a hit in the infield.) Of course, nobody saw that because it happened at two in the morning. Things are different when you're playing for one run.
Basically, what's happening as I write this is that Obama is running out the clock. (Yes, baseball doesn't have a clock. But Obama's a basketball player, so I think he'd like this metaphor.)
What politics and sports have in common, of course, is that there's a huge difference between second place and first place. If you're a company of some sort, does it matter if you sell 1% more or 1% less than your competitor? Not really, although it might have meaning for your pride. But if you're a politician, 1% of the vote makes a big difference.
01 November 2008
Conditional probability is subtle, part 2
Nate Silver returns to the idea of conditional probability: he's saying Pennsylvania is "in play" in this election, not because the polling in Pennsylvania is close, but because conditioned on the election being close, Pennsylvania is close. In general, I've heard quite a few people argue that the candidates should focus on the states that would be close if the election were roughly tied, not the ones that actually are close, because the details of which state a candidate deliberately tries to sway things in only matter in a close election anyway.
Unfortunately, subtleties like this seem to be lost on some of Silver's commenters.
In case you're wondering, my last post titled "Conditional probability is subtle" had nothing to do with politics.
Unfortunately, subtleties like this seem to be lost on some of Silver's commenters.
In case you're wondering, my last post titled "Conditional probability is subtle" had nothing to do with politics.
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