epistomology


26
Apr 13

Manifesto addition: “N is always finite”

Added one more point to my manifesto:

“N” is always finite. Probability theory gives us many powerful limit theorems. These tell us what to expect, mathematically, as the number of trials goes to infinity. That fine, but we don’t live our lives at the limit. In the real world, asymptotics are much more important, as are non-parametric inference and tools like bootstrap to maximize the information extracted from small sample sizes. Even in those real life situations when N could increase without limit, the growth of the sample size itself can change the problem. Toss a die enough times and the edges will start to wear off, changing the frequency that each side comes up.

Had to add it after reading the phrase “if we let N go to infinity” one to many times.


18
Apr 13

Sudden clarity about the null hypothesis

Can’t take credit for this realization (I studied at an “Orthodox” shop), and the “Clarance” wishes to be anonymous, so send all karma points to your favorite (virtual) charity.


27
Mar 13

Minding the reality gap

Officially, unemployment in the US is declining. It’s fallen from a high of 9.1% a couple years ago, to 7.8% in recent months. This would be good news, if the official unemployment rate measured unemployment, in the everyday sense of the word. It doesn’t. The technical definition of “U3″ unemployment, the most commonly reported figure, excludes people who’ve given up looking for work, those who’ve retired early due to market conditions, and workers so part time they clock in just one hour per week.

Most critically, unemployment excludes the 14 million American on disability benefits, a number which has quadrupled over the last 30 years. If you include just this one segment of the population in the official numbers, the unemployment rate would double. On Saturday, This American Life devoted their entire hour to an exploration of this statistic. Russ Robert’s, who’s podcast I’ve recommend in the past, discussed the same topic last year. Despite the magnitude of the program and the scale of the change, these are the only outlets I know of to report on the disability number, and on the implications it has for how we interpret the decline in U3 unemployment.

Targeting the number, not the reality
Statistics, in the sense of numerical estimates, are measures which attempt to condense the complex world of millions of people into a single data point. Honest statistics come with margins of error (the most honest indicate, at least qualitatively, a margin of error for their margin of error). But even the best statistical measures are merely symptoms of some underlying reality; they reflect some aspect of the reality as accurately as possible. The danger with repeated presentations of any statistic (as in the quarterly, monthly, and even hourly reporting of GDP, unemployment, and Dow Jones averages), is that we start to focus on this number by itself, regardless of the reality it was created to represent. It’s as if the patient has a high fever and all anyone talks about is what the thermometer says. Eventually the focus becomes, “How do we get the thermometer reading down?” All manner of effort goes into reducing the reading, irrespective of the short, and certainly long-term, health of the patient. When politicians speak about targeting unemployment figures, this is what they mean, quite literally. Their goal is to bring down the rate that gets reported by the Bureau of Labor Statistics, the number discussed on television and in every mainstream source of media.

Politicians focus on high profile metrics, and not the underlying realities, because the bigger and more complicated the system, the easier it is to tweak the method of measurement or its numeric output, relative to the difficulty of fixing the system itself. Instead of creating conditions which allow for growth in employment (which would likely require a reduction in politicians’ legislative and financial powers), the US has quietly moved a huge segment of its population off welfare, which counts against unemployment, and into disability and prisons — the incarcerated also don’t count in U3, whether they are slaving away behind bars or not.

How metrics go bad
Over time, all social metrics diverge from the reality they were created to reflect. Sometimes this is the result of a natural drift in the underlying conditions; the metric no longer captures the same information it had in the past, or no longer represents the broad segment of society it once did. For example, the number of physical letters delivered by the postal service no longer tracks the level of communication between citizens.

Statistics and the reality they were designed to represent are also forced apart through deliberate manipulation. Official unemployment figures are just one example of an aggressively targeted/manipulated metric. Another widely abused figure is the official inflation rate, or core Consumer Price Index. This measure excludes food and energy prices, for the stated reason that they are highly volatile. Of course, these commodities represent a significant fraction of nearly everyone’s budget, and their prices can be a leading indicator of inflation. The CPI also uses a complex formula to calculate “hedonics,” which mark down reported prices based on how much better the new version of a product is compared to the old one (do a search for “let them eat iPads”).

I don’t see it as a coincidence that unemployment and inflation figures are among the most widely reported and the most actively manipulated. In fact, I take the following to be an empirical trend so strong I’m willing to call it a law: the greater the visibility of a metric, the more money and careers riding on it, the higher the likelihood it will be “targeted.” In this light, the great scandal related to manipulation of LIBOR, a number which serves as pivot point for trillions of dollars in contracts, is that the figure was assumed to be accurate to begin with.

Often the very credibility of the metric, built up over time by its integrity and ability to reflect an essential feature of the underlying reality, is cashed in by those who manipulate it. Such was the case with the credit ratings agencies: after a long run of prudent assessments, they relaxed their standards for evaluating mortgage bundles, cashing in on the windfall profits generated by the housing bubble.

Why we don’t see the gaps
It might seem like the disconnect between a statistic and reality would cause a dissonance that, once large enough to be clearly visible, would lead to reformulation of the statistic, bringing it back in line with the underlying fundamentals. Clearly there are natural pressures in that direction. For example, people laid off at the beginning of a recession are unlikely to believe that the recovery has begun until they themselves go back to work. Their skepticism of the unemployment figure erodes its credibility. Unfortunately, two powerful forces work against the re-alignment of metric and reality: the first related to momentum and our blindness to small changes, the second having to do with the effects of reflexivity and willful ignorance.

In terms of inertia, humans have a built-in tendency to believe that what has been will continue to be. More sharply, the longer a trend has continued, the longer we presume it will continue — if it hasn’t happened yet, how could it happen now? Laplace’s rule of succession is our best tool for estimating probabilities under the assumption of a constant generating process, one that spits out a stream of conditionally independent (exchangeable) data points. But the rule of succession fails utterly, at times spectacularly, when the underlying conditions change. And underlying conditions always change!

These changes, when they come slowly, pass under our radar. Humans are great at noticing large differences from one day to the next, but poor at detecting slow changes over long periods of time. Ever walked by an old store with an awning or sign that’s filthy and falling apart? You wonder how the store owner could fail to notice the problem, but there was never any one moment when it passed from shiny and new to old and decrepit. If you think you’d never be as blind as that shop keeper, look down at your keyboard right now. As with our environment, if the gap between statistic and reality changes slowly, over time, we may not see the changes. Meanwhile, historical use of the statistic lends weight to it’s credibility, reducing the chance that we’d notice or question the change — it has to be right, it’s what we’ve always used!

The perceived stability of slowly changing systems encourages participants to depend on or exploit it. This, in turn, can create long term instabilities as minor fluctuations trigger extreme reactions on the part of participants. Throughout the late 20th century and the first years of the 21st, a large number of investors participated in the “Carry Trade,” a scheme which depended on the long term stability of the Yen, and of the differential between borrowing rates in Japan and interest rates abroad. When conditions changed in 2008, investors “unwound” these trades at full speed, spiking volatility and encouraging even more traders to exit their positions as fast as they could.

These feedback loops are an example of reflexivity, the tendency in some complex systems for perception (everyone will panic and sell) to affect reality (everyone panics and sells). Reflexivity can turn statistical pronouncements into self-fulfilling prophecies, at least for a time. The belief that inflation is low, if widespread, can suppress inflation in and of itself! If I believe that the cash in my wallet and the deposits in my bank account will still be worth essentially the same amount tomorrow or in a year, then I’m less likely to rush out to exchange my currency for hard goods. Conversely, once it’s clear that my Bank of Zimbabwe Bearer Cheques have a steeply declining half-life of purchasing power, then I’m going to trade these paper notes for tangible goods as quickly as possible, nominal price be damned!

Don’t look down

If perception can shape reality, then does the gap between reality and statistic matter? Clearly, the people who benefit most from the status quo do their best to avoid looking down, lest they encourage others to do the same. More generally, though, can we keep going forward so long as we don’t look down, like Wile E. Coyote chasing the road runner off a cliff?

The clear empirical answer to that questions is: “Yes, at least for a while.” The key is that no one knows how long this while can last, nor is it clear what happens when the reckoning comes. Despite what ignorant commentators might have said ex post facto, by 2006 there was wide understanding that housing prices were becoming un-sustainably inflated. In 2008, US prices crashed back down to earth. North of the border, in Canada, the seemingly equally inflated housing market stumbled, shrugged, then continued along at more level, but still gravity-defying trajectory.

The high cost of maintaining the facade
Even as the pressures to close the gap grow along with its size, the larger the divergence between official numbers and reality, the greater the pressures to keep up the facade. If the fictional single entity we call “the economy” appears to be doing better, politicians get re-elected and consumers spend more money. When the music finally stops, so too will the gravy-train for a number of vested interests. So the day of reckoning just keeps getting worse and worse as more and more resources go into maintaining the illusion, into reassuring the public that nothing’s wrong, into extending, pretending, and even, if need be, shooting the messenger.

It’s not just politicians and corporations who become invested in hiding and ignoring the gap. We believe official statistics because we want to believe them, and we act as if we believe them because we believe that others believe them. We buy houses or stocks at inflated prices on the hope that someone else will buy them from us at an even more inflated price.

My (strong) belief is that most economic and political Black Swans are the result of mass delusion, based on our faith in the quality and meaning of prominently reported, endlessly repeated, officially sanctioned statistics. The illustration at the beginning of this post comes from a comic I authored about a character who makes his living off just this gap between official data and the reality on the ground, a gap that always closes, sooner or later, making some rich and toppling others.


11
Dec 12

“We didn’t even bother to get the $7 coffee”

A couple weeks ago I highlighted the recommendation that researchers test their models (and the processes which generated them!) against random noise. This is an important “reality check” of their methods, to see how susceptible they are to detecting something in nothing. In the video above, Jimmy Kimmel gives a nice illustration of how this idea could be extended to a taste test, or any survey where participants are asked to differentiate between samples. Kimmel’s experiment also gives a nice illustration of how humans can be primed to find what we expect to find, even if it’s not there.


31
Oct 12

Recommendation of the week

“[I]f you have performed any statistical analysis that is more complex than calculating the mean and the standard deviation, you should perform the same analysis on noise to make sure that whatever effect you observe is indeed a unique feature of your data and not an artefact of the analysis.”

Found this one over at Stefan’s sieste blog. I couldn’t agree more, especially now that computers and big data sets entice us to make ever more complex models. Oh, and that’s not a bad thing! As I’ve argued, we’ll need to give up on simple, easy to interpret models in order to get more predictive power.

I’d go even more meta than Stefan and argue that you should re-test your entire model-creating process on noise (perhaps he meant this with his quote). If you started with a data set, then ran a stepwise variable selection algorithm, then added in a new non-linear term to get a better fit, do the same on noise, trying to get the best fit. Are you able to get a statistically significant result? Better still, run the same procedure on different types of noise, not just Gaussian White (I know, sounds like something you’d load into a syringe. Normality, the gateway drug?).