Occasional reflections on Life, the World, and Mathematics

Posts tagged ‘probability’

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.


Credit and Credibility

Are banks crazy or a cartel?


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?


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