Tag Archives: probability

Subjective probabilities: point/counterpoint

  • Adam Elga:  “Subjective Probabilities Should Be Sharp” — at least for rational agents, who are vulnerable to a kind of Dutch Book attack if they insist that there are observable hypotheses whose probability can not be specified as a real number.
  • Cosma Shalizi:  “On the certainty of the Bayesian Fortune-Teller” — People shouldn’t call themselves Bayesians unless they’re committed to the view that all observable hypotheses have sharp probabilities — even if they present their views in some hierarchical way “the probability that the probability is p is f(p)” you can obtain whatever expected value you want by integrating over the distribution.  On the other hand, if you reject this view, you are not really a Bayesian and you are probably vulnerable to Dutch Book as in Elga, but Shalizi is at ease with both of these outcomes.
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Aggregating degrees of belief: a puzzle

There are two events X and Y whose probability you’d like to estimate.  So you ask a hundred trusted, reasonable people what they think.  Half of them say that the probability of X and the probability of Y are both 90%, and the probability of both X and Y occurring is 81%.  The other half say that P(X) = 10%, P(Y) = 10%, and P(X and Y) = 1%.

What is your best estimate of P(X), P(Y), and P(X and Y)?

If you said “50%, 50%, 41%,” does it bother you that you deem these events not to be independent, even though every single person you polled said the opposite?  If not, what did you say?

(The subtext of this post is:  is the “Independence of Irrelevant Alternatives” axiom in Arrow’s theorem a good idea?  Feel free to discuss that too.)

 

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Close election? Flip for it

I have an op-ed in today’s Washington Post advocating the use of randomization devices to determine the winner of close elections.

Some will balk at the idea of choosing our leaders by chance. But that’s actually the coin flip’s most important benefit! Close elections are already determined by chance. Bad weather in the big city, a busted voting machine in an outlying town, a poorly designed ballot leading elderly Jews to vote for Pat Buchanan — any of these chance events can make the difference when the electorate is stuck at 50-50.

A note for the many people who either e-mailed me or posted comments to say that I was a nutty leftist who would never have written this if the more liberal candidate in the Wisconsin supreme court election were ahead: in fact, I pitched this when Kloppenburg appeared to be leading by 200 votes.  The correction of the Waukesha numbers, which made the election much less close, was thus inconvenient for both Kloppenburg and me.  But I just rejiggered the piece to place a greater emphasis on very close votes from the past (Franken-Coleman, Bush-Gore.)  Not sure why I neglected to include the equally tight Gregoire-Rossi WA-GOV race.  Or how I failed to notice that Charles Seife, whose book I quote in the piece, also wrote essentially the same editorial in the New York Times two years ago.

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Condorcet on French mathematics as underdog

Quite striking and strange for a modern mathematician to read the following, from Condorcet’s 1787  lectures to the Lycee:

It is to French mathematicians that we owe the theory of probability calculus.  This is perhaps worth saying.  Other nations, and often even Frenchmen themselves, have reproached us for lacking the gift of invention, granting us only the ability to perfect other people’s discoveries…

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Condorcet was an interesting dude

I knew about him only in relation with the voting paradox.  But he also wrote The Future Progress of the Human Mind (1795), a utopian tract featuring surprsingly modern stuff like this:

No one has ever believed that the human mind could exhaust all the facts of nature, all the refinements of measuring and analyzing these facts, the inter relationship of objects, and all the possible combinations of ideas….

But because, as the number of facts known increases, man learns to classify them, to reduce them to more general terms; because the instruments and the methods of observation and exact measurement are at the same time reaching a new precision; . . . the truths whose discovery has cost the most effort, which at first could be grasped only by men capable of profound thought, are soon carried further and proved by methods that are no longer beyond the reach of ordinary intelligence. If the methods that lead to new combinations are exhausted, if their application to problems not yet solved requires labors that exceed the time or the capacity of scholars, soon more general methods, simpler means, come to open a new avenue for genius….

The organic perfectibility or degeneration of races in plants and animals may be regarded as one of the general laws of nature.

This law extends to the human species; and certainly no one will doubt that progress in medical conservation [of life], in the use of healthier food and housing, a way of living that would develop strength through exercise without impairing it by excess, and finally the destruction of the two most active causes of degradation-misery and too great wealth-will prolong the extent of life and assure people more constant health as well as a more robust constitution. We feel that the progress of preventive medicine as a preservative, made more effective by the progress of reason and social order, will eventually banish communicable or contagious illnesses and those diseases in general that originate in climate, food, and the nature of work. It would not be difficult to prove that this hope should extend to almost all other diseases, whose more remote causes will eventually be recognized. Would it be absurd now to suppose that the improvement of the human race should be regarded as capable of unlimited progress? That a time will come when death would result only from extraordinary accidents or the more and more gradual wearing out of vitality, and that, finally, the duration of the average interval between birth and wearing out has itself no specific limit whatsoever? No doubt man will not become immortal, but cannot the span constantly increase between the moment he begins to live and the time when naturally, without illness or accident, he finds life a burden?

Read a longer excerpt here.

The voting paradoxes are found in Condorcet’s 1785 treatise Essay on the Application of Analysis to the Probability of Majority Decisions. But the main body of the book isn’t about voting paradoxes; it’s an attempt to provide mathematical backing for democratic theory.  Condorcet argued that the probability of the majority holding the wrong position was much smaller than the chance that the minority would be in the wrong.  So democracy is justified not only on principle, but because it is more likely to yield true beliefs on the part of the government.  I learned this, and other interesting facts, from Trevor Pateman’s article “Majoritarianism,” which presents Condorcet as a kind of quantitative version of Rousseau.  

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The Google puzzle and the perils of averaging ratios

The following brain-teaser has been going around, identified as a question from a Google interview (though there’s some controversy about whether Google actually uses questions like this.)

There’s a certain country where everybody wants to have a son. Therefore each couple keeps having children until they have a boy; then they stop. What fraction of the population is female?

Steve Landsburg posted a version of this question on his blog.  “The answer they expect,” he writes, “is simple, definitive, and wrong… Are you smarter than the folks at Google?  What’s the answer?”

Things quickly went blooey.  Google’s purported answer — fiercely argued for by lots of Landsburg’s readers — is 1/2.  Landsburg said the right answer was less.  A huge comment thread and many follow-up posts ensued.  Lubos Motl took time out from his busy schedule of yelling at mathematicians about string theory to yell at Landsburg about probability theory.  Landsburg offered to bet Motl, or anybody else, $15,000 that a computer simulation would demonstrate the correctness of his answer.

What’s going on here?  How could a simple probability question have stirred up such a ruckus?

Here’s Landsburg’s explanation of the question:

What fraction of the population should we expect to be female? That is, in a large number of similar countries, what would be the average proportion of females?

If G is the number of girls, and B the number of boys, Landsburg is asking for the expected value E(G/(G+B)).  And let’s get one thing straight:  Landsburg is absolutely right about this expected value.  For any finite number of families, it is strictly less than 1/2.  (See the related Math Overflow thread for a good explanation.)  Landsburg has very patiently knocked down the many wrong arguments to the contrary in his comments section.  Anybody who bets against him, on his terms, is going to lose.

Nonetheless, I’m about to explain why Landsburg is wrong.

You see, Google’s version of the question doesn’t specify anything about expectation.  They might just as well have meant:  “What is the proportion of the expected number of females in the expected population?”  Which is to say, “What is E(G)/E(G) + E(B)”?  And the answer to that question is 1/2.  Just to emphasize the subtlety involved here:

On average, the number of boys and the number of girls are the same.  Furthermore, the proportion of girls is, on average, less than 1/2.

Weird, right?  E(G)/E(G) + E(B) isn’t what Landsburg was asking for — but, if Google’s answer was 1/2, it’s presumably the question they had in mind.  To accuse them of getting their own question “wrong” is a bit rich.

But let me go all in — I actually think Landsburg’s interpretation of the question is not only different from Google’s, but in some ways inferior!  Because averaging ratios with widely ranging denominators is kind of a weird thing to do.  You can certainly compute the average population density of all the U.S. states — but should you? What meaning or use would the result have?

I had a really pungent example ready to deploy, which illustrates the perils of averaging ratios and explains why Landsburg’s version of the question was a little weird.  Then I went to the Joint Meetings before getting around to writing this post.  And when I got back, I discovered that Landsburg had posted the same example on his own blogin support of his point of view!  Awesome.  Here it is:

There’s a certain country where everybody wants to have a son. Therefore each couple keeps having children until they have a boy; then they stop. In expectation, what is the ratio of boys to girls?

The answer to this question is, of course, infinity; in a finite population there might be no girls, so B/G is infinite with some positive probability, so E(B/G) is infinite as well.

But the correctness of that answer surely tells us this is a terrible question!  Averaging is a terribly cruel thing to do to a bunch of ratios.  One zero denominator and you’ve wiped out your entire dataset.

What if Landsburg had phrased his new question along the lines of Google’s original puzzle?

There’s a certain country where everybody wants to have a son. Therefore each couple keeps having children until they have a boy; then they stop. What is the ratio of boys to girls in this country?

Honest question:  does Landsburg truly think that infinity is the only “right answer” to this question?  Does he think infinity is a good answer?  Would he hire a person who gave that answer?  Would you?

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Reader survey: how seriously do you take expected utility?

Slate reposted an old piece of mine about the lottery, on the occasion of tonight’s big Mega Millions drawing.  This prompted an interesting question on Math Overflow:

I have often seen discussions of what actions to take in the context of rare events in terms of expected value. For example, if a lottery has a 1 in 100 million chance of winning, and delivers a positive expected profit, then one “should” buy that lottery ticket. Or, in a an asteroid has a 1 in 1 billion chance of hitting the Earth and thereby extinguishing all human life, then one “should” take the trouble to destroy that asteroid.

This type of reasoning troubles me.

Typically, the justification for considering expected value is based on the Law of Large Numbers, namely, if one repeatedly experiences events of this type, then with high probability the average profit will be close to the expected profit. Hence expected profit would be a good criterion for decisions about common events. However, for rare events, this type of reasoning is not valid. For example, the number of lottery tickets I will buy in my lifetime is far below the asymptotic regime of the law of large numbers.

Is there any justification for using expected value alone as a criterion in these types of rare events?

This, to me, is a hard question.  Should one always, as the rationality gang at Less Wrong likes to say, “shut up and multiply?” Or does multiplying very small probabilities by very large values inevitably yield confused and arbitrary results?

UpdateCosma Shalizi’s take on lotteries and utilities, winningly skeptical as usual.

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October 2010 linkdump

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Adverb placement and probability

Just a weird and somehow illuminating syntactic trip-up:  per the odds of the moment at 538, it is correct to say both that

England, the United States, and Slovenia will all probably advance

and

England, the United States, and Slovenia will definitely not all advance.

There’s no paradox here, just a reminder to take care about the ambiguity in English-language descriptions of probabilities.

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The Challenger disaster was not caused by Russian roulette

Malcolm Gladwell wrote:

It doesn’t take much imagination to see how risk homeostasis applies to NASA and the space shuttle. In one frequently quoted phrase, Richard Feynman, the Nobel Prize- winning physicist who served on the Challenger commission, said that at NASA decision-making was “a kind of Russian roulette.” When the O-rings began to have problems and nothing happened, the agency began to believe that “the risk is no longer so high for the next flights,” Feynman said, and that “we can lower our standards a little bit because we got away with it last time.” But fixing the O-rings doesn’t mean that this kind of risk-taking stops. There are six whole volumes of shuttle components that are deemed by NASA to be as risky as O-rings. It is entirely possible that better O-rings just give NASA the confidence to play Russian roulette with something else.

If this is really what Feynman said, wasn’t he wrong?  In Russian roulette, you know there’s one bullet in the gun.  The chance of a catastrophe is just one in six the first time you put the gun to your head; but if you survive the first try, you know the round is in one of the remaining five chambers and the chance of death next time you pull the trigger climbs to 20%.  The longer you play, the more likely disaster becomes.

But what if you don’t know how many chambers are loaded?  Suppose you play “Bayes Roulette,” in which the number of bullets is equally likely to be anywhere from 1 to 6.  Then the chance of survival on the first try is (5/6) if there’s 1 bullet in the cylinder, (4/6) if 2 bullets, and so on, for a total of

(1/6)(5/6) + (1/6)(4/6) + … (1/6)(0/6) = 5/12

which is about 41%.  Pretty bad.  But let’s say you pull the trigger once and live.  Now by Bayes’ theorem, the chance that there’s 1 bullet in the cylinder is

Pr(1 bullet in cylinder and I survived the first try) / P(I survived the first try)

or

(5/36)/(5/12) = 1/3.

Similarly, the chance that there are 5 bullets in the cylinder is

(1/36)/(5/12) = 1/15.

And the chance that there were 6 bullets in the cylinder is 0, because if there had been, well, you would be a former Bayesian.

All in all, your chance of surviving the next shot is

(5/15)*(4/5) + (4/15)*(3/5) + (3/15)*(2/5) + (2/15)*(1/5) + (1/15)*0= 8/15.

In other words, once you survive the first try, you’re more likely, not less, to survive the next one; because you’ve increased the odds that the gun is mostly empty.

Or suppose the gun is either fully loaded or empty, but you don’t know which.  The first time you pull the trigger, you have no idea what your odds of death are.  But the second time, you know you’re completely safe.

I think the space shuttle are a lot more like Bayes Roulette than Russian Roulette.  You don’t know how likely an O-ring failure is to cause a crash, just as you don’t know how many bullets are in the gun.  And if the O-rings fail now and then, with no adverse consequences, you are in principle perfectly justified in worrying less about O-rings.  If you shoot yourself four times and no bullet comes out, you ought to be getting more confident the gun is empty.

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