Tag Archives: 25-sigma

The worst discrepancy in science?

In New Eyes on the Universe I repeat the oft-quoted claim that the cosmological constant is the source of the biggest discrepancy between theory and observation in all of science. The discrepancy arises because cosmologists measure the cosmological constant to be 10-29 g/cm3. (At least, this is the value assuming that it’s a cosmological constant that’s causing the accelerating expansion of the Universe. It’s possible that dark energy has some explanation other than a positive cosmological constant, but that seems increasingly unlikely: we seem to live in a Universe with a non- zero cosmological constant, and this constant is blowing the Universe apart.) Physicists, using our best theories of Nature – quantum field theories – calculate the value of the cosmological constant to be about 1091 g/cm3. That means theory differs from observation by a factor of 10120.

The discrepancy between theory and observation here is so bad that it’s a joke. Such large numbers simply don’t happen often in physics. (For comparison, the number of particles in the observable universe is about 1080. This is inconceivably smaller than 10120.) It tells us that physics needs some new ideas – and physicists have indeed been looking around for decades for those ideas. For an increasing number of physicists it means we need to embrace anthropic thinking; others have not given up on the idea that a unified understanding of gravity and quantum theory will explain the value of the cosmological constant; or perhaps some as-yet undreamed of theory will solve the problem.

Whatever, it’s difficult to imagine how a discrepancy in science could possibly be bigger than 10120. But I recently came across an error that dwarfs even this. (Whether you think my example comes from ‘real’ science depends on your opinion of economics. It’s said that astrology was invented to make economists look good, but let’s give them the benefit of the doubt.)

In his entertaining and informative book about the 2007/2008 meltdown of the global financial system, Whoops: Why Everyone Owes Everyone and No One Can Pay, John Lanchester presents a quote made by David Viniar, Chief Financial Officer of Goldman Sachs. Viniar was trying to explain why, in August 2007, the premier Goldman Sachs hedge fund was worth about one quarter less than it had been worth at the start of the year. Viniar said:  “We were seeing things that were 25 standard deviation moves, several days in a row.”

Viniar’s statement is so wrong that it’s difficult to put into words. But I’ll have a go.

What is the probability of observing one of these 25-sigma events that Viniar was seeing so frequently? Well, economists often base their models on a Gaussian distribution of events, and in that case standard deviations are easy to calculate. (Actually, economists also use a ‘fat-tailed’ distribution. If Viniar based his remarks on a fat-tailed distribution then that would have the effect of softening the numbers I give below, but it wouldn’t in any way affect the conclusions. In the same way, there are ways of presenting the ‘cosmological constant argument’ that reduces the discrepancy between theory and observation to a mere 1030 – but that of course doesn’t solve the underlying problem.) Assuming that there are 250 trading days in a year, then on the basis of a Gaussian distribution one would expect to see:

  • a 2-sigma event on one trading day out of about 44
  • a 3-sigma event on one day out of about 741 days
  • a 4-sigma event about once every 31560 days
  • a 5-sigma event about once every 3483046 days
  • a 6-sigma event about once every 1009976678 days

You no doubt can see where this pattern is heading. The probabilities get very small very quickly. I’d never really thought about the probability of a 25-sigma event, since for all practical purposes it’s zero. But it’s possible to do the sums, and it turns out that someone like Viniar should have expected to see a “25 standard deviation move” on one trading day out of 3.1 x 10136. (For the purposes of comparison, about 5 x 1012 days have elapsed since the Big Bang.) And Viniar, the lucky chap, was seeing such an incredibly, unbelievably, humongously improbable event happening on a daily basis!

The discrepancy between observation and the predictions of basic economic models completely dwarfs the discrepancy between observation and physicists’ prediction of the value of the cosmological constant. It’s not even close. The difference seems to be that physicists know their models are wrong, and are trying hard to fix them; economists at hedge funds are still investing other people’s money using demonstrably incorrect models. David Viniar remains CFO of Goldman Sachs.