Part 1: Starting from Zero
Part 2: Meet Eustace
Part 3: Bernoulli Trials
Part 4: Why Randomness is Not All Equally Random

Have you ever been told what to do with your life? A particular college, major, grad school, career? Isn’t it annoying, that someone would presume to plot out your life for you, as if you had no say in the matter? Probability theory has a term for this (the plotting, not the annoyance): strong solution.

A strong solution is any specified trajectory for a random process. In our coin flipping game it would be the realized sequence of heads and tails. Of course Eustace can’t know such a path in advance. The best he can do is to construct a distribution of possible outcomes. This distribution is a weak solution, which is defined, not by its path, which is unknown, but only by the moments of a probability distribution. If Eustace knows a random process is stationary, he has confidence that the moments of the process will converge to the same values every time. The coin flipping game, for instance, is stationary: its long term average winnings, given a fair coin, will always converge to zero. Looking into an uncertain future, Eustace is always limited to a weak solution: it is specified by the expectations, or moments, of the underlying random process. The actual path remains a mystery.

So far we haven’t given poor Eustace much help. A weak “solution” is fine for mathematics; but being a mere cloud of possibilities, it is, from Eustace’s point of view, no solution at all. (A Eustace entranced by the weak solution is what we commonly call a perfectionist.) Sooner rather than later, he must risk a strong solution. He must chart a course: he must act.

Well then, on what basis? Probability theory has a term for this too. The accumulated information on which Eustace can base his course is called a filtration. A filtration represents all information available to Eustace when he chooses a course of action. Technically, it is an algebra (sigma algebra) defined on an increasing set of information (Borel set). The more of the available filtration Eustace uses, the better he does in the casino.

In the coin flipping game, Eustace’s filtration includes, most obviously, the record of the previous flips. Of course in this case the filtration doesn’t help him predict the next flip, but it does help him predict his overall wins and losses. If Eustace wins the first flip (t=1), he knows that after the next flip (t=2), he can’t be negative. This is more information than he had when he started (t=0). If the coin is fair, Eustace has an equal likelihood of winning or losing \$1. Therefore, the expected value of his wealth at any point is simply what he has won up to that point. The past reveals what Eustace has won. The future of this stationary distribution is defined by unchanging moments. In a fair game, Eustace can expect to make no money no matter how lucky he feels.

His filtration also includes, less obviously, the constraints of the game itself. Recall that if he wins \$100 he moves to a better game and if he loses \$100 he’s out in the street. To succeed he must eliminate paths that violate known constraints; a path to riches, for instance, that requires the casino to offer an unlimited line of credit is more likely a path to the poorhouse.

We can summarize all of this with two simple equations:

E(wealth@t | F@t-1) = wealth@t-1 (first moment)
variance(wealth@t|F@t-1) = 1 (second moment)

The expected wealth at any time t is simply the wealth Eustace has accumulated up until time t-1. E is expected value. t is commonly interpreted as a time index. More generally, it is an index that corresponds to the size of the filtration, F. F accumulates the set of distinguishable events in the realized history of a random process. In our coin game, the outcome of each flip adds information to Eustace’s filtration.

We have also assumed that when Eustace’s wealth reaches zero he must stop playing. Game over. There is always a termination point, though it need not always be zero; maybe Eustace needs to save a few bucks for the bus ride home. Let’s give this point a name; call it wealthc (critical). Introducing this term into our original equation for expected wealth, we now have:

max E(wealth@t – wealthc | F@t-1)

His thermodynamic environment works the same way. In the casino, Eustace can’t blindly apply any particular strong solution — an a priori fixed recipe for a particular sequence of hits and stands at the blackjack table. Each card dealt in each hand will, or should, influence his subsequent actions in accordance with the content of his filtration. The best strategy is always the one with max E(wealth@t|F@t-1) at each turn. In this case, F@t-1 represents the history of dealt cards.

As Eustace graduates to higher levels of the casino, the games become more complex. Eustace needs some way of accommodating histories: inflexibility is a certain path to ruin. Card-counters differ from suckers at blackjack only by employing a more comprehensive model that adapts to the available filtration. They act on more information — the history of the cards dealt, the number of decks in the chute, the number of cards that are played before a reshuffle. By utilizing all the information in their filtration, card counters can apply the optimal strong solution every step of the way.

In the alpha casino, Eustace encounters myriad random processes. His ability to mediate the effects of these interactions is a direct consequence of the configuration of his alpha model. The best he can hope to do is accommodate as much of the filtration into this model as he can to generate the best possible response. Suboptimal responses will result in smaller gains or greater losses of alpha. We will take up the policy implications, as one of my readers put it, of all this in Part 6.

Disclaimer: Although I use the language of agency — to know, to act, to look into the future — nothing in this discussion is intended to impute agency, or consciousness, or even life, to Eustace. One could speak of any inanimate object with a feedback mechanism — a thermostat, a coffeemaker — in exactly the same way. Unfortunately English does not permit discussing these matters in any other terms. Which is why I sometimes want to run shrieking back to my equations. You may feel otherwise.

Watch a drop of rain trace down a window-pane. Being acquainted with gravity, you might expect it to take a perfectly straight path, but it doesn’t. It zigs and zags, so its position at the bottom of the pane is almost never a plumb drop from where it began.

Or graph a series of Bernoulli trials. Provided the probability of winning is between 0 and 1, the path, again, will veer back and forth unpredictably.

You are observing Gaussian randomness in action. All Gaussian processes are Lipschitz continuous, meaning, approximately, that you can draw them without lifting your pencil from the paper.

The most famous and widely studied of all Gaussian processes is Brownian motion, discovered by the biologist Robert Brown in 1827, which has had a profound impact on almost every branch of science, both physical and social. Its first important applications were made shortly after the turn of the last century by Louis Bachelier and Albert Einstein.

Bachelier wanted to model financial markets; Einstein, the movement of a particle suspended in liquid. Einstein was looking for a way to measure Avogadro’s number, and the experiments he suggested proved to be consistent with his predictions. Avogadro’s number turned out be very large indeed — a teaspoon of water contains about 2x10E23 molecules.

Bachelier hoped that Brownian motion would lead to a model for security prices that would provide a sound basis for option pricing and hedging. This was finally realized, sixty years later, by Fischer Black, Myron Scholes and Robert Merton. It was Bachelier’s idea that led to the discovery of non-anticipating strategies for tackling uncertainty. Black et al showed that if a random process is Gaussian, it is possible to construct a non-anticipating strategy to eliminate randomness.

Theory and practice were reconciled when Norbert Wiener directed his attention to the mathematics of Brownian motion. Among Wiener’s many contributions is the first proof that Brownian motion exists as a rigorously defined mathematical object, rather than merely as a physical phenomenon for which one might pose a variety of models. Today Wiener process and Brownian motion are considered synonyms.

Back to Eustace.

Eustace plays in an uncertain world, his fortunes dictated by random processes. For any Gaussian process, it is possible to tame randomness without anticipating the future. Think of the quadrillions of Eustaces floating about, all encountering continuous changes in pH, salinity and temperature. Some will end up in conformations that mediate the disruptive effects of these Gaussian fluctuations. Such conformations will have lower overall volatility, less positive entropy, and, consequently, higher alpha.

Unfortunately for Eustace, all randomness is not Gaussian. Many random processes have a Poisson component as well. Unlike continuous Gaussian processes, disruptive Poisson processes exhibit completely unpredictable jump discontinuities. You cannot draw them without picking up your pencil.

Against Poisson events a non-anticipating strategy, based on continuous adjustment, is impossible. Accordingly they make trouble for all Eustaces, even human beings. Natural Poisson events, like tornadoes and earthquakes, cost thousands of lives. Financial Poisson events cost billions of dollars. The notorious hedge fund Long-Term Capital Management collapsed because of a Poisson event in August 1998, when the Russian government announced that it intended to default on its sovereign debt. Bonds that were trading around 40 sank within minutes to single digits. LTCM’s board members, ironically, included Robert Merton and Myron Scholes, the masters of financial Gaussian randomness. Yet even they were defeated by Poisson.

All hope is not lost, however, since any Poisson event worth its salt affects its surroundings by generating disturbances before it occurs. Eustaces configured to take these hints will have a selective advantage. Consider a moderately complex Eustace — a wildebeest, say. For wildebeests, lions are very nasty Poisson events; there are no half-lions or quarter-lions. But lions give off a musky stink and sometimes rustle in the grass before they pounce, and wildebeests that take flight on these signals tend to do better in the alpha casino than wildebeests that don’t.

Even the simplest organisms develop anti-Poisson strategies. For example, pH levels and salinity are mediated by buffers while capabilities like chemotaxis are a response to Poisson dynamics.

A successful Eustace must mediate two different aspects of randomness: Gaussian and Poisson. Gaussian randomness continuously generates events while Poisson randomness intermittently generates events. On the one hand, Gaussian strategies can be adjusted constantly; on the other, a response to a Poisson event must be based on thresholds for signals. Neither of these configurations is fixed. Eustace is a collection of coupled processes. So in addition to external events, some processes may be coupled to other internal processes that lead to configuration changes.

We will call this choreographed ensemble of coupled processes an alpha model. Within the context of our model, we can see a path to the tools of information theory where the numerator for alpha represents stored information and new information, and the denominator represents error and noise. The nature of this path will be the subject of the next installment.