Tuesday, September 23, 2008

An Altogether Appropriate Admonishment

Some Notes on Determinism vs Probability

There is an irresistible force within each human that compels them to find reason in the randomness around them. This is why people come to read blogs like this one: to glean some truth out of the game of deception that is Wall Street. But how we describe events determines how valuable the lessons we learn from today eventually become.

In trading, ways of describing the world come in two forms; there are those that try to provide discrete causal explanations for a particular market event -- this is known as determinism. These explanations are consistent in logic and view each event as a unique combination of forces. The other form -- probability -- takes a different approach, considering each event as the summation of a number of random variables. These probabilistic explanations are less satisfying on the surface. They make no causal claims about today's event, they merely say that there was "a 60% chance of the market rising today given treasury bond yields and what the nikkei did yesterday." Unlike determinism, the probabilistic mind makes no attempt at causality, he merely counts the past and extrapolates into the future, knowing all the while that relationships are bound to change. The probabilist makes use of causation, but not as a one way street. He knows that when oil goes up there is a good chance airlines are going down. But he does not say that airlines went down BECAUSE of oil.

A determinist and a probabilist will view the same event in different ways. Suppose we flipped a coin fifty times without knowing whether the coin was fair or not. If the results show 30 heads and 20 tails the determinist might conclude that the coin was indeed biased to come up heads. Or he might say that there was an error in the flipping method that we used. He has no way to find out the quality of his theory, as it is generated ad hoc to explain the 30 heads. The probabilist will say that if the coin was fair he knows there is a 4.19% chance of getting 30 heads exactly and moreover, we would only get 30 heads or more about 10.1% of the time (see chart). He will do another statistical test and will most likely conclude that he too thinks the coin to be unfair and biased towards heads.

Risk Affine Market Wisdom Cumulative Binomial Distribution with 50 trials @ p = 0.5

If both parties are not allowed to examine the true state of the coin (as is often the case in the market if we replace "coin" with "company") then the two approaches can produce numerous different explanations and not have any real idea which is true. The determinist could say that since he couldn't examine the coin it could be any of his explanations that led to the bias. The probabilist will tell you that on one hand it could happen even if the coin was far, but on the other an outcome with greater than 30 heads would only happen 10% of the time, but on the other hand...

This example should highlight both the usefulness and the pitfalls of using either approach by itself. The solution comes in the form of a judicious combination of the two philosophies and an iterative approach to model building. One must use both models together while using how the models respond to repeated experiments; each time refining the models to reflect new information.

If we further assume that our two coin commentators can repeat the fifty tosses, they can use their combined knowledge to come up with a better explanation of events. Suppose the coins again show a bias with 30 heads and 20 tails. The determinist will be cock sure about his explanation. The probabilist will say that the odds of that happening twice (if unbiased) are 0.16%. He will calculate that the probability of the 30/20 split is maximized when the probability of each coin toss going heads is about 60%. After two experiments, both approaches conclude that the coin is likely biased.

Determinism also provides a useful tool for the imagination of the probabilist, and visa versa. Combing the cannon of possible causal explanations may lead one to examine a relationship that would have never been considered under either approach alone. Likewise, knowing that market participants follow probabilities intensely could lead to a deterministic explanation if the numbers change rapidly.

Through a combination of these two approaches one can overcome a great deal of the downsides inherent in a single strategy. Take this lesson to heart, and always remember that its never as complicated or as simple as it looks.


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