Month: November 2012

for generating this report, and NPR should be ashamed of themselves for running it on their flagship news program Morning Edition. For that matter, so should John Grotzinger, the NASA scientist interviewed in the segment.

Grotzinger says they recently put a soil sample in SAM, and the analysis shows something remarkable. “This data is gonna be one for the history books. It’s looking really good,” he says.

Grotzinger can see the pained look on my face as I wait, hoping he’ll tell me what the heck he’s found, but he’s not providing any more information.

So why doesn’t Grotzinger want to share his exciting news? The main reason is caution. Grotzinger and his team were almost stung once before. When SAM analyzed an air sample, it looked like there was methane in it, and at least here on Earth, some methane comes from living organisms.

Either they’ve got something amazing, in which case it’ll still be amazing once they’ve released it, or they don’t, in which case this is just a bit of hype that does a bit more to erode the credibility of scientists everywhere. There’s no scenario in which this report has any positive effect.

 

[I just noticed this sitting in the “Drafts” section of my blog. (You can tell it’s old, because I mention something that I’m “going to be teaching in the fall.”) I don’t know why I didn’t post it at the time. Did I notice something wrong with the physics, and put it aside until I fixed it? I don’t think so — it looks right to me.

This is a subject lots of people have written about, and what I say here is just a summary of some standard stuff, but it’s an incredibly cool subject, so if you don’t already know all this, check it out.]

 

Einstein’s big problem with quantum physics was that it involved “spooky action at a distance.”  His most famous quote on quantum physics is about randomness — “God does not play dice” — but in fact he seems to have been much more bothered by nonlocality than randomness.

The sort of nonlocality that bothered Einstein, and that bothers many other people, is kind of hard to explain.  In particular, one key ingredient is a thing called Bell’s Inequality, which is somewhat technical.  There are some very good explanations out there, especially a couple by N.D. Mermin, e.g., one in the American Journal of Physics (paywall) and one that originally appeared in Physics Today.  But recently I came across a nice way of thinking about it in an unexpected place: a puzzle blog.  Even though this way of formulating the result seems to be well known in certain circles (and is apparently close to what Bell wrote in his original paper), I’d never encountered it before.

If you want to understand this stuff, you can read any or all of the above.  But I’m going to try to summarize the main idea here too, mostly because I’m going to be teaching this topic in the fall, and I can use the practice.  So here goes.

Let me start with a quote from the puzzle blog:

Suppose three friends A, B and C take a test with 100 yes-no questions. If you compare the answers given by A and B, 98 of the 100 are the same. Likewise, if you compare the answers given by B and C, again 98 of the 100 are the same. What is the minimum number of questions that A and C have answered in the same way?

The answer, of course, is 96. Now let’s connect this with quantum physics.

In quantum physics,  particles like electrons have a property called “spin.”  The main thing you need to know about electron spin is that when you measure it you always get one of two values, which are usually called “spin-up” and “spin-down.”

Spin-up and spin-down have to be specified relative to some chosen axis: you can pick any direction you want, and measure the spin of an electron along an axis oriented in that direction, and you’ll get either spin-up or spin-down.  (If the axis is horizontal, then the terminology is kind of stupid: it’d make much more sense to call it “spin-left” and “spin-right”.  But we still say “up” and “down” even in this case.)

It’s relatively easy (or so they tell me) to produce pairs of particles that are “entangled,” meaning that their spins are related to each other.  In particular, it’s possible to produce a pair of particles that have the following properties:

1. There’s no way to predict in advance what the result will be of any spin measurement on either of the particles.
2. If you measure the spin of one particle, the spin of the other particle, measured about the same axis, is guaranteed to be the same.

(Actually, it’s technically easier to create pairs where the spins are guaranteed to be opposite, but it’s possible to flip one around after the fact, and anyway it’s easier to explain this way.)

So far, there’s nothing “spooky” going on. In fact, these electrons are just like my socks, which  — take my word for it — are not at all spooky.  You don’t know what color my socks are, but knowing what a fastidious dresser I am, you can bet that they match.  So there’s no way to predict the result of one sock-color measurement, but once you’ve measured one you can predict the other with certainty.

(If you actually tried to do this experiment, you’d have a problem: I’m not wearing socks at the moment.  The sock analogy comes from an essay by John Bell, by the way, although I think in his case he imagined someone whose socks always failed to match.)

Einstein believed that the electrons in this system really were just like my socks: my socks have a definite color, even if you don’t know it, and the electrons (according to Einstein) have a definite spin-value even before we’ve measured it.  If we repeat this experiment many times, each pair of electrons will leave the source with a plan in mind about whether to say “up” or “down” when the spin is measured.  That plan may be chosen randomly, but the two particles have the same plan, so of course they give the same answer.  The name for this point of view is “local realism,” by the way.

Bell’s inequality says that local realism is impossible.  To see why, we have to bring in the fact that the spins can be measured about different axes — that is, that we can rotate our measuring apparatus before the electron hits it.  Rule 2 above says what happens when spins of both particles are measured with respect to the same axis: the results agree 100% of the time.  It doesn’t matter what axis we choose, as long as it’s the same for both measurements.  But what if we rotate one measurement apparatus relative to the other?  There’s another rule for that:

3. If the two measurement axes are rotated with respect to each other by an angle x, then the results of the measurements will agree cos²(x/2) of the time.

In particular, say one axis is tilted 16.3 degrees away from the other.  Then that number works out to 0.98.  That is, the results will agree 98% of the time.

Now suppose we have a machine that can mass-produce pairs of entangled particles.  We can imagine doing an experiment in which we measure the spin of either particle about a vertical axis.   (It doesn’t matter which particle: we know they’ll come out the same.)  The series of outcomes of those measurements (up, down, down, up, up, down, down, down, …) is like student B’s test.

Now suppose that instead we measure the first particle’s spin about an axis that’s tilted 16.3 degrees to the left.  The resulting sequence is like student A’s test: it’ll agree with the first list 98% of the time.

Finally, suppose that instead we measure the second particle’s spin about an axis that’s tilted 16.3 degrees to the right.  The results here are (you won’t be surprised to hear) like student C’s test.  Once again, there’s a 98% match with the first list.

Here’s the thing: there’s nothing stopping us from measuring both the  second and third lists simultaneously (particle 1 with an axis tilted to the left, and particle 2 with an axis tilted to the right).  If we do, we find that the results match only 92% of the time.  (That’s rule 3, but with x=32.6 degrees, which is how far apart these two axes are.)  That should be impossible: if both sets of tilted-axis results are 98% correlated with the (hypothetical) vertical-axis results, then they must be at least 96% correlated with each other.

In summary, there is no way to consistently assign values to all three sets of possible measurements (vertical axis, tilted to the left, tilted to the right) that agrees with the probabilities found in quantum mechanics.  In effect, that means that the particle doesn’t “decide” whether it’s going to tell you it’s spin-up or spin-down until you decide which axis you’re going to measure.  And yet the pairs of particles manage to decide the same way, even if they’re very far apart.

By the way, Bell’s inequality was originally a theoretical result.  Rules 1-3 were known to be the predictions made by quantum mechanics, but they hadn’t been tested experimentally at the time.  So originally all you could conclude was that either quantum physics was wrong or local realism was wrong.  But later, the experiment was done, and the quantum physics predictions were confirmed to be correct.  So local realism is wrong, and electrons are not like my socks.

A couple of quick pre-election notes:

 

1. If I could outlaw one type of political journalism, it’d be stories of the form “This election will be decided by voters of Type X” (Rural voters, working-class voters, etc.) In past elections, we were told that the election would be decided by soccer moms, NASCAR dads, security moms, and a host of others.

The problem with these statements is not that they’re false — it’s that they’re true but vacuous. What does it mean to say that the election will be decided by young, unmarried women? It means that, if young, unmarried women tend to vote more than expected for a candidate, then that candidate will win. But in a close election, that’s equally true for all groups.

The problem is that “if more people vote for your guy than the other guy, then your guy will win” doesn’t make for a compelling campaign narrative, so the poor beleaguered horserace journalist has to come up with some novel-sounding theory about some particular group that’s going to “decide” the election. Either that or they could find something useful and important to write about.

 

2. One bit of good news from the campaign: Bayesian reasoning has gone mainstream. Look at all the attention Nate Silver‘s been getting.

It won’t surprise anyone who knows me to hear that I love Silver’s approach to political analysis. I’ll take a clear model built on data over 100 pundits’ gut feelings.

A number of Republican-leaning types hate Silver, because his model consistently says Obama’s more likely to win. It’s true that Silver is openly pro-Obama, and it’s true that people have a tendency to (consciously or unconsciously) skew things in the direction they prefer. But there are several reasons one shouldn’t make too much of this argument:

• Other quantitative, statistics-based models give predictions similar to Silver’s or even more pro-Obama, as do the betting markets. See, for example, the Princeton Election Consortium.
• Silver’s reputation as a prediction whiz is on the line. His motivation to be accurate is probably stronger than his motivation to boost his guy.
• Silver laid out his methodology explicitly and publicly quite early on, and as far as I know there’s no evidence he’s changed it.

Of course, Silver’s model might be wrong nonetheless. Some people have said silly things like “We’ll know next week how good his model was,” but of course we won’t. As Ezra Klein put it,

If Mitt Romney wins on election day, it doesn’t mean Silver’s model was wrong. After all, the model has been fluctuating between giving Romney a 25 percent and 40 percent chance of winning the election. That’s a pretty good chance! If you told me I had a 35 percent chance of winning a million dollars tomorrow, I’d be excited. And if I won the money, I wouldn’t turn around and tell you your information was wrong. I’d still have no evidence I’d ever had anything more than a 35 percent chance.

One of the sillier critiques of Silver came from Josh Gerstein of Politico:

Isn’t the basic problem with the Nate Silver prediction in question, and the critique, that it puts a percentage on a one-off event?

Of course we use probabilities to describe one-off events all the time. Does Gerstein listen to weather forecasts?

Then there’s the New York Times’s public editor, Margaret Sullivan, who thinks Silver violated journalistic ethics when he offered to make a bet against Joe Scarborough. Scarborough insists that anyone who thinks the race is anything other than a tossup is an idiot, so Silver offered him an even-money bet on the race (with the loser donating money to charity, so neither stands to gain personally).

Sullivan’s point of view on this strikes me as extremely silly. Offering to bet is a standard rhetorical trick when having arguments about probabilities. If you really believe in your probabilistic statement, you should be willing to use it as the basis of a bet.

I don’t think I’ve ever used vulgar language in this blog before, but here goes: by far the best way to put this point is Alex Tabarrok’s line : A bet is a tax on bullshit. If all pundits who opine about the race had to put bets on their predictions, we’d be a lot better off.