Sunday, November 25, 2007

probability and public policy II: subjective bayes and the "fair coin"

[part one of this series here]

Subjective Bayes is the view that probabilistic statements reflect our state of uncertainty; as such, they are subjective, a mere reflection of a particular human's internal state of belief.

for example: Consider a hypothetical "fair coin": what does it mean to say this coin is "fair"? As it turns out, if we know the precise conditions under which a coin is flipped, we can confidently predict the outcome of a coin toss. The physics itself is relatively simple (comparatively), and if one knows the initial conditions one can predict the outcome (i.e. if one knows the physical properties of the coin and the details of the force which set it spinning (as well as air resistence, wind velocity, etc.), one can calculate precisely which side will land face up).

Consider briefly a simplified model where we assume that air resistance and wind play no role and that the mass of the coin is uniformly distributed. We can graph the outcome of a coin toss against the upward velocity at the time of release (V, in feet per second) and the angular velocity (ω, in revolutions per second). Assuming the coin starts heads up, we can treat grey as heads and white as tails:

Persi Diaconis, "A Place for Philosophy? The Rise of Modeling in Statistical Science," 1998.

The above analysis is from work by Persi Diaconis. He discusses where in the above graph the "typical" coin toss falls:
For a typical one-foot toss, experiments show that coins go up at about 5 m.p.h. and turn over 35-40 revolutions per second. In the units of the picture the velocity is concentrated at about 0.2 on the velocity scale. This is close to zero in the picture. Fortunately, the spin is concentrated at about 40 units up on the ω axis.

For additional details see "The Probability of Heads" by Joseph Keller.

The point here is just that the "probability," or fairness, in a coin toss is not a property of the physical setup, but rather of the observer's belief state about the outcome. Once we are privy to the initial conditions of the coin toss, there is no more uncertainty and thus no more "probability" or "chance" in the event.

Another way of thinking about subjective Bayes is in terms of one's willingness to bet upon an event the outcome of which is as yet undetermined. de Finetti used this thought experiment as the basis of his theory of probability and demonstrated that any set of odds which cannot be subjected to a Dutch Book is a valid probability distribution (a "Dutch Book" is a set of bets devised by the bookie such that you lose no matter what the outcome of the event is).

If probability is just a measure of one's uncertainty, how can we use it to interact with the world? To put it another way: how can I get my subjective probability assignments to match up with the "actual" proportion of outcomes due to the underlying mechanism at work? Or, if the coin isn't "fair, if I'm being cheated, how do I figure that out? The answers to these questions in our next installment.

next: Bayes' Rule

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