Occasional reflections on Life, the World, and Mathematics

Posts tagged ‘probability’

The Silver Standard: Stochastics pedagogy

I have written a number of times in support of Nate Silver and his 538 project: Here in general, and here in advance of the 2016 presidential elections. Here I want to make a comment about his salutary contribution to the public understanding of probability.

His first important contribution was to force determinism-minded journalists (and, one hopes, some of their readers) to grapple with the very notion of what a probabilistic prediction means. In the vernacular, “random” seems to mean only a fair coin flip. His background in sports analysis was helpful in this, because a lot of people spend a lot of time thinking about sports, and they are comfortable thinking about the outcomes of sporting contests as random, where the race is not always to the swift nor the battle to the strong, but that’s the way to bet. People understand intuitively that the “best team” will not win every match, and winning 3/4 of a large number of contests is evidence of overwhelming superiority. Analogies from sports and gaming have helped to support intuition, and have definitely improved the quality of discussion over the past decade, at least in the corners of the internet where I hang out.*

Frequently Silver is cited directly for obvious insights like that an 85% chance of winning (like his website’s current predicted probability of the Democrat’s winning the House of Representatives) is like the chance of rolling 1 through 5 on a six-sided die, which is to say, not something you should take for granted. But he has also made a great effort to convey more subtle insights into the nature of probabilistic prediction. I particularly appreciated this article by Silver, from a few weeks ago.

As you see reports about Republicans or Democrats giving up on campaigning in certain races for the House, you should ask yourself whether they’re about to replicate Clinton’s mistake. The chance the decisive race in the House will come somewhere you’re not expecting is higher than you might think…

It greatly helps Democrats that they also have a long tail of 19 “lean R” seats and 48 “likely R” seats where they also have opportunities to make gains. (Conversely, there aren’t that many “lean D” or “likely D” seats that Democrats need to defend.) These races are long shots individually for Democrats — a “likely R” designation means that the Democratic candidate has only between a 5 percent and 25 percent chance of winning in that district, for instance. But they’re not so unlikely collectively: In fact, it’s all but inevitable that a few of those lottery tickets will come through. On an average election night, according to our simulations, Democrats will win about six of the 19 “lean R” seats, about seven of the 48 “likely R” seats — and, for good measure, about one of the 135 “solid R” seats. (That is, it’s likely that there will be at least one total and complete surprise on election night — a race that was on nobody’s radar, including ours.)

This is a more subtle version of the problem that all probabilities get rounded to 0, 1, or 1/2. Conventional political prognosticators evaluate districts as “safe” or “likely” or “toss-up”. The likely or safe districts get written off as certain — which is reasonable from the point of view of individual decision-making — but cumulatively a large number of districts with a 10% chance of being won by the Democrat are simply different from districts with a 0% chance. It’s a good bet that the Republican will win each one, but if you have 50 of them it’s a near certainty that the Democrats will win at least 1, and a strong likelihood they will win 8 or more.

The analogy to lottery tickets isn’t perfect, though. The probabilities here don’t represent randomness so much as uncertainty. After 5 of these “safe” districts go the wrong way, you’re almost certainly going to be able to go back and investigate, and discover that there was a reason why it was misclassified. If you’d known the truth, you wouldn’t have called it safe it all. This enhances the illusion that no one loses a safe seat — only, toss-ups can be mis-identified as safe.

* On the other hand, Dinesh D’Souza has proved himself the very model of a modern right-wing intellectual with this tweet:

 

Feeling good about my chances on this coin flip…

People in the know are starting to think a disastrous “no deal” Brexit is now not at all unlikely. According to UK trade secretary Liam Fox

I have never thought it was much more than 50-50, certainly not much more than 60-40.

The Latvian foreign minister is only slightly more optimistic:

The chances of the UK securing a Brexit deal before it leaves the European Union in March are only 50:50, Latvia’s foreign minister has said ahead of talks with Jeremy Hunt.

Edgars Rinkevics said there was a “very considerable risk” that, with time rapidly running out, Britain could crash out of the bloc without a withdrawal agreement.

But not to worry. Rinkevics went on to say that

having said 50:50, I would say I am remaining optimistic.

I suppose, technically, he is more optimistic than Hunt. Why so gloomy, Jeremy, with your exaggerated estimate of 60% chance of disaster? I think it’s more like 50 percent. That’s a glass half full if ever I saw one…

Of course, an “optimist” is usually thought to be someone who thinks the chances of disaster are significantly less than a coin flip. (more…)

Natural frequencies and individual propensities

I’ve just been reading Gerd Gigerenzer’s book Reckoning with Risk, about risk communication, mainly a plaidoyer for the use of “natural frequencies” in place of probabilities: Statements in the form “In how many cases out of 100 similar cases of X would you expect Y to happen”. He cites one study forensic psychiatry experts who were presented with a case study, and asked to estimate the likelihood of the individual being violent in the next six months. Half the subjects were asked “What is the probability that this person will commit a violent act in the next six months?” The other half were asked “How many out of 100 women like this patient would commit a violent act in the next six months?” Looking at these questions, it was obvious to me that the latter question would elicit lower estimates. Which is indeed what happened: The average response to the first question was about 0.3; the average response to the second was about 20.

What surprised me was that Gigerenzer seemed perplexed by this consistent difference in one direction (though, obviously, not by the fact that the experts were confused by the probability statement). He suggested that those answering the first question were thinking about the same patient being released multiple times, which didn’t make much sense to me.

What I think is that the experts were thinking of the individual probability as a hidden fact, not a statistical statement. Asked to estimate this unknown probability it seems natural that they would be cautious: thinking it’s somewhere between 10 and 30 percent they would not want to underestimate this individual’s probability, and so would conservatively state the upper end. This is perfectly consistent with them thinking that, averaged over 100 cases they could confidently state that about 20 would commit a violent act.

Horse thieves and inverse probabilities

Reading Ron Chernow’s magisterial new biography of Ulysses Grant, I came across this very correct statistical inverse reasoning from the celebrated journalist Horace Greeley (whose role in the high school history curriculum has been reduced to the phrase, “Go West, young man” — that he denied having invented):

All Democrats are not horse thieves, but all horse thieves are Democrats.

This seems like an ironic bon mot, but after he became the Democratic candidate for president against Grant in 1872 he tried to use a milder version unironically as a defence of his new party colleagues:

I never said all Democrats were saloon keepers. What I said was all saloon keepers are Democrats.

Presumably he meant to add that if we knew the base rate of saloonkeeping (or horse thievery) in the population at large, we could calculate from the Democratic vote share the exact fraction of Democrats (and of Republicans) who are saloonkeepers (or horse thieves).

Small samples

New York Republican Representative Lee Zeldin was asked by reporter Tara Golshan how he felt about the fact that polls seem to show that a large majority of Americans — and even of Republican voters — oppose the Republican plan to reduce corporate tax rates. His response:

What I have come in contact with would reflect different numbers. So it would be interesting to see an accurate poll of 100 million Americans. But sometimes the polls get done of 1,000 [people].

Yes, that does seem suspicious, only asking 1,000 people… The 100 million people he has come in contact with are probably more typical.

The view from one-in-a-billion

Nautilus just published my new article, a nontechnical introduction to large deviations: The theory of how unlikely rare events are, and which way a rare event chooses to occur, given that it occurs.

Likely probabilities

The BBC comments on Trump’s latest outrage:

The day after a video tape emerged in which he suggested he could have any woman he wants because he’s a star and so could just “grab them by the pussy”, Mr Trump is in a whole ocean of hot political water.

Enough, quite possibly, to sink any chance he had of winning the White House.

What does it mean to have “quite possibly” no chance? How does it differ from having quite likely a small chance? Or definitely a modest chance?

I suppose this could be describing a combination of an unknown state of facts that determine the probabilities for a random outcome. Imagine a bag of red and blue marbles, from which the election will be determined by picking a random one. (Red for Trump.) She is saying, “Quite possibly there are no red marbles left in the bag.” I’m don’t think that’s a good description of the situation, though. To the extent that Trump has a better chance of winning than you might think by looking at the polls or the near universal opprobrium he is exposed to, it doesn’t seem to me it’s on the basis of facts that are already determined but unknown.

Einstein and the Quantum

I just saw an ad (in Blackwell’s Books) for a book titled Einstein and the Quantum, with a text that began

Einstein himself famously rejected quantum mechanics with his God does not play dice theory…

Putting aside the fact that “God does not play dice with the universe” is a quip, not a theory, I’m fascinated by this extreme statement of a calumny on Einstein that I knew as standard when I first learned about quantum mechanics from popular science in the 1970s, that the old man, despite his revolutionary past (and he was only in his late 40s) simply lacked the intellectual flexibility to keep up, rejected the new science, and was proved wrong by the march of progress.

In fact, that famous remark (from a 1926 letter to Max Born) acknowledged up front that the emerging probabilistic view of quantum mechanics was proving very useful. He simply rejected the willingness to deny a micro-level interpretation. (And the so-called Copenhagen “Interpretation” of quantum mechanics is really an anti-interpretation, a programmatic refusal to interpret. For more comments on the pedagogical function, see here.) The fact that this approach went from strength to strength as a calculating tool does not mean that its interpretive framework, the one that said that probabilities are the fundamental objects and there is no use going deeper, has been proved, any more than the success of Maxwell’s equations proved the existence of molecular vortices in the luminiferous aether. In particular, proponents of the Copenhagen Interpretation have tended to ignore the fact that they are helping themselves to a supposedly primitive concept, probability, that is actually complex, strange, and sorely in need of physical foundations.

Certainly one powerful strain of modern work on the foundations of physics — in particular, the Everett interpretation of quantum mechanics (cf. David Wallace’s The Emergent Multiverse) also rejects the notion that there is some randomness at the core of quantum mechanics, and takes as a point of departure the entanglement theory first proposed in the Einstein-Podolsky-Rosen thought experiment.

* Einstein wrote, “Die Theorie liefert viel, aber dem Geheimnis des Alten bringt sie uns kaum näher. Jedenfalls bin ich überzeugt, daß der nicht würfelt.“ Literally: “The theory gives us much, but it hardly brings us nearer to the Ancient One’s secret. In any case, I am convinced that he does not throw dice.”

On the downgrade

Further reflections on non-transitive folk probability

Continuing my thoughts about zero-one probability from here, I come to the recent decision of Standard & Poor’s to lower their rating of US treasury debt. There are plenty of reasons to doubt their judgement,  both because they’ve been absurdly wrong in the past (subprime mortgage backed securities were AAA, but treasury bills are risky?), because they can’t read budget estimates or can’t do basic arithmetic, because they are trying to project political trends, which they surely know even less about than about arithmetic, or because the people who work there are generally known to be pretty dim. But from a probabilist’s point of view what’s strange is the timing. Whatever you may think of the recent deal to avoid the US defaulting on its debt, it did avoid defaulting on its debt. Surely the likelihood of a default went down after the deal was passed. So why is the credit rating lower this week than it was last week? Now, this is all perfectly consistent with the view that S&P is not actually making a prediction of future default probability, but simply seeking the best opportunity to promote its wares. Certainly, the way they operate is not the like someone trying to give what will be perceived as neutral advice; they act more like central bankers, timing their announcements to try to move markets and (above all) seem relevant. They’re reminiscent of the folktale of the rooster who threatens to withhold his crowing, which inevitably will forestall the sun rise. The other animals plead with him to relent, but it’s a threat that only works as long as the rooster is modest enough to recognise that he can’t hold out forever. In the case of the US treasury bonds, S&P held out, and still the sun rose.

But there is something about their approach that seems to make sense to intelligent people, and not purely idiosyncratic. I’m reminded of Tversky’s famous conjunction fallacy, with studies seeming to show that people’s everyday probability intuitions don’t necessary satisfy the apparently inevitable law of conjunction: The probability of A or B must be bigger than the probability of A and the probability of B. Here we see intuitions of probability that don’t seem to satisfy the law of total expectation: If  are possible future states of the world, and is the probability of event A conditional on  happening, then the probability of event A now must be some kind of average of these conditional probabilities.

downgrade

Credit and Credibility

Are banks crazy or a cartel?

bommel3

When a government (let us say, in Athens) could possibly renege on promises made to banks who loaned them money or bought their bonds, which that government is unable to fulfill without draconian cuts to public services, all right-thinking people attack the feckless politicians and threaten a collapse of confidence and the world economy. This is a DEFAULT! Other governments and the IMF might jump in to pour money into the state coffers, on the condition that they flow out the other end into the investors’ pockets.

But when a government (could be in Athens, or in London, or for that matter Madison, Wisconsin) has made promises of pensions to government employees, but has failed to fund them adequately, it is short-sighted and greedy for these civil servants to insist on these promises being honoured.

Why is a default such a terrible thing? Because, they say, if the country defaults on its debts it will be shut out of the credit markets. Hmmm. Let’s suppose it is true. Why? Suppose you own a bank, and your thinking of lending to one of two countries, let’s call one of them Piigsia and the other Sameria. Both are heavily indebted. Piigsia introduces crushing austerity measures, while Sameria repudiates its sovereign debt. Which of those countries would you rather loan your bank’s money to? The one that’s shown a great willingness to pay off its debts but is financially crushed, or the one who may be more likely to try to weasel out of its debts, but is eminently capable of paying. Solvency is not merely (or even primarily) a state of mind. I mean, what good is it to have the current government express a willingness to pay off its debts, knowing that it’s likely to be punished by voters for these “good” intentions? Maybe they just don’t want to be serial defaulters, so having avoided defaulting this time will encourage them to default on the next batch of loans.

As for Sameria, it sucks for the other banks that have lost their money, but why should I give up a chance to make a good profit for the sake of punishing Sameria for hosing my competitors? In fact, in a competitive market, why shouldn’t I be happy that my competitors have made a loss, and just try to get better conditions for my loan?

(more…)

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