Frank Knight's Favorite Issue
Risk and uncertainty: a Bayesian perspective:
Statistical Modeling, Causal Inference, and Social Science: p=1/2 or E(p)=1/2, or, the boxer vs. the wrestler: In Bayesian inference, all uncertainty is represented by probability distributions. I remember in grad school discussing the puzzle of distinguishing the following two probabilities:
- p1 = the probability that a particular coin will land "heads" in its next flip;
- p2 = the probability that the world's greatest boxer would defeat the world's greatest wrestler in a fight to the death.
The first of these probabilities is essentially exactly 1/2. Let us suppose, for the sake of argument, that the second probability is also 1/2. Or, to put it more formally, suppose that we have uncertainty about p2, thus a prior distribution, p(p2), and that the mean of this prior distribution, E(p2), equals 1/2.
The paradox: In Bayesian inference, p1 = p2 = 1/2. Which doesn't seem quite right, since we know p1 much more than we know p2. More generally, it seems a problem with representation-of-uncertainty-by-probability. To put it another way, the integral of a probability is a probability, and once we've integrated out the uncertainty in p2, it's just plain 1/2.
Resolution of the paradox: The resolution of the paradox is that probabilities, and decisions, do not take place in a vacuum. If the only goal were to make a statement, or a bet, about the outcome of the coin flip or the boxing/wrestling match, then yes, p=1/2 is what you can say. But the events occur within a context. In particular, the coin flip probability p1 remains at 1/2, pretty much no matter what information you provide (before the actual flipping occurs, of course). In contrast, one could imagine gathering lots of information (such as in the photo above) that would refine one's beliefs about p2. "Uncertainty in p2" corresponds to potential information we could learn that would tell us something about p2.
Uncertainty is a measure of our ignorance. Risk is what remains when we know everything that can be known.
