Fractal Press

The Mercury News is reporting a new study by the US Geological Survey that predicts “California faces a 99.7 percent chance of big quake by 2037.”

How the notion of “probability” is interpreted in these studies is a mystery. The power-law tails on these processes have a very low exponent and the ability to calibrate the probability in the tails shrinks very fast as one moves down the tail.

D. A. Freedman and P.B. Stark at Berkeley looked at this problem in a paper titled “What is the chance of an earthquake?” This 2001 paper studied an earlier similar forecast.

From the paper, on the probability forecast:

There is no straightforward interpretation of the USGS probability forecast. Many steps involve models that are largely untestable; modeling choices often seem arbitrary. Frequencies are equated with probabilities, fiducial distributions are used, outcomes are assumed to be equally likely, and sub jective probabilities are used in ways that violate Bayes rule.

…and:

Philosophical difficulties aside, the numerical probability values seem rather arbitrary.

They conclude with this:

Probabilities are a distraction. Instead of making forecasts, the USGS could help to improve building codes and to plan the government’s response to the next large earthquake. Bay Area
residents should take reasonable precautions, including bracing and bolting their homes as well as securing water heaters, bookcases, and other heavy objects. They should keep first aid supplies, water, and food on hand. They should largely ignore the USGS probability forecast.

Invest in preparation instead of building models and forecasting rare events. This false precision is dangerous if it informs decision making.

Ever notice that even in pretty large groups a sudden hush can fall as everyone stops talking at the same time? I’ve noticed this in large classrooms filled with boisterous kids to formal dinners with great conversationalists.

When it does occur, the group usually just looks around, smiles, and moves on. The din of chatter fills the air again. The members of the group would find it impossible to find a specific cause for the sudden silence. In fact, every individual in the group who is a part of this process knows that there is no specific cause - they don’t even think about looking for one.

Now consider a blind man at the same table. He can only observe the sounds and the atmosphere but cannot see. He is in no position to rule out a large number of possible causes. While the rest of the non-blind people can see and confirm that nothing extraordinary occurred to cause the sudden silence, he cannot do this and so his mind will race to figure out what the cause was.

He will consider many hypothesis: maybe somebody spilt a drink, maybe somebody did something inappropriate, maybe a large group of waiters showed up to serve them, maybe the lights went out for a fraction of a second and came back on …

He will not consider that there is no cause. He will not consider that the question “Why?” is simply unanswerable.

In the case above, us humans are participants in the process so we have some insight into the lack of causality. In many other endeavors we are simply external observers or lack the self-awareness to consider the lack of causality. Socio-economic processes are highly complex and distributed. It is very possible that these are environments where “Why?” is unanswerable. But we may be like blind men looking for causes where none exist.

In any situation where a large investment is being made based on an answer to the question “Why?,” it may be prudent to ask if that question is answerable to begin with. Historians, economists and scientists who study history looking for non-proximate causes may be looking for an answer that does not exist (or is unknowable, they are epistemologically equivalent).

Zero-Sum games with results being decided far into the future can trick participants into thinking they’ve just entered a win-win transaction where both of them come out ahead. This is similar to a conman “overpaying” for goods with counterfeit money - both the conman and the merchant go home happy. The merchant is in for a rude surprise only after he tries to use that money. The pain he experiences is in direct proportion to the happiness he felt earlier.

Derivatives transaction, stock market transaction with a fixed money-supply, and forex speculation are all zero-sum games. Derivatives are the most interesting because they let traders push the uncertainty out far into the future. Consider two traders who enter a derivatives transaction with an expiration date two years out. If the market is illiquid, they can both happily “mark to model” and book paper profits if they use different models. In fact, since they entered the transaction in the first place, they probably are using different models.

Even “mark to market” transactions are susceptible to model-error, just less so. The market as a whole is an average of many models, any sharp change in this average will expose participants to model error. Even if the merchant was paid in real dollars for the asset, he can still lose out if rampant inflation sets in and the asset’s value doesn’t change but the cash in his hands is steadily becoming worthless. The model that the market used for currencies changed and the change did him in.

As in the counterfeit money transaction, the person who is assuming a tightly-constrained game (ludic fallacy) will emerge as the loser. False premises are usually the root problem (these dollar bills are real, cash will be worth the same tomorrow, this cannot happen in this market, …).

In fact, the presence of win-win players in what is known to be a zero-sum game can be used as a diagnostic tool to prepare for massive reality-checks in the future. Hedge Funds and Investment Banks all tend to trade with each other. And they particularly enjoy derivatives transaction. If most hedge funds end up showing stellar returns for an extended period of time, it shouldn’t come as a big surprise that at least some of them have been borrowing from destiny.

The problem in the world of finance is not just limited to using the Black-Scholes Model in options pricing. The “volatility smile” may or may not make up for the deficiencies in the model, but there are other far more insidious implications of assuming the world of the gaussian is relevant in the real world.

Consider the implications of infinite variance. Any distribution where the integral over the second moment does not converge (Pareto distribution with tail exponent < 2 will do it) will have an infinite or undefined variance. This has implications outside of the variance. It also invalidates the correlation measures, R-squared and other statistics.

A simple way to see this is to consider two random variables X and Y that are driven by stable infinite variance distributions. Let us see whether a correlation measure makes any sense. Remember that variance goes as X^2 and Y^2 and the correlation measures go as X*Y.

We have:

• X^2 ~= Variance of variable X
• Y^2 ~= Variance of variable Y
• (X+Y)^2 ~= Variance of combined stable distributed variables
• X*Y ~= Correlation measure

X*Y = 0.5*[(X+Y)^2 - X^2 - Y^2]

The sum of two random variables under stable distributions with infinite variance will result in a random variable driven by a stable distribution with infinite variance as well. So, all the terms on the right-hand-side of the equation above are infinite (undefined). This means the X*Y measure is undefined as well.

So correlation measures fall apart in domains with infinite variance. Distributions with the tail-exponent < 2 will have infinite variance, but given that estimating the tail exponent is notoriously difficult [1], any over-reliance on correlation measures is highly fragile to estimation errors.

In real-world samples of fractal domains (infinite variance domains), the correlation measure will show very high sensitivity to sample sizes and windows - it will jump all over the place with small changes in sample size.

Nassim Taleb’s arguments against the gaussian go far beyond those involving the volatility smile in options pricing. Once you identify a domain as possibly heavy-tailed be suspicious of any statistical measures: the “Sharpe Ratio,” R-squared, etc. There is little evidence that they work well in these domains. Over-reliance on these in decision-making (such as judging the performance of a hedge fund) can be disastrous.

[1] Levy-Stable Distributions Revisited: Tail Index > 2 Does Not Exclude The Levy Stable Regime (Weron 2001)

Note: I saw this framing of the connection between correlation measures and variance in one of Mandelbrot’s papers, I forgot which one.

I think I am beginning to understand how specialization is rewarded in the economy. Entities that specialize are essentially collecting the premium from selling put options on the dimension in which they are specializing. They are very vulnerable to abrupt state-changes and they are rewarded for this risk.

In case you are unfamiliar with put options: they are simply insurance against the drop in value of some asset or variable. Buying a put option against a stock is like buying insurance in case the stock goes below a certain threshold. This insurance comes at a cost: the premium you pay for the put option. Options sellers pocket the premium upfront but expose themselves to the risk that the option will be triggered if the asset or variable drops below the threshold.

People, organizations, or countries that specialize are selling insurance (put options) against the dimension in which they are specializing. A country that follows Ricardo’s law of comparative advantage and decides to specialize completely in agriculture even with some mineral and mining resources available, will find itself extremely vulnerable to an abrupt change of state (say a crop disease gets entrenched in the country) when agriculture becomes impossible and the lack of investment in mining infrastructure renders it suddenly uncompetitive in the global economy.

In terms of careers, abrupt state-changes historically have included the growth of outsourcing, changes in the economy (IBM in 1990), and new technology (ice sellers and refrigerators).

Specialization comes with increased rewards, but it is a tradeoff with a decrease in robustness against the unknown. In other words, the premium on specialization can come from money borrowed from destiny.

I’m not sure people recognize this as a tradeoff, as hordes of youngsters are encouraged to begin specializing earlier and earlier without a second thought.

Thinking about it in these terms makes it so clear. People who specialize are making an extremely concentrated bet that the area under question will still be relevant many years from now, if not, they are toast. That’s a pretty big risk if you consider it on a long enough time-line.

PhD students are dare-devil risk takers. Who would have expected it?