The boxer, the wrestler, and the coin flip: a paradox of Bayesian inference, robust Bayes, and belief functions

My entry on the boxer and the wrestler sparked some interesting discussion here and here. In order to understand the distinction between randomness and uncertainty in a probability distribution, one has to embed that probabability into a larger structure with potentially more information. As Aki Vehtari pointed out, Tony O’Hagan made this point in a nice article published last year.

Dempster-Shafer not a solution. Neither is robust Bayes

Anyway, one of the other comments on my post alluded to belief functions (Dempster and Shafer’s theory of upper and lower probabilities) as a solution to the boxer/wrestler paradox. Actually, though, the boxer/wrestler thing was part of something that Augustine Kong and I came up with 15 years ago as a counterexample, or paradox, for Bayesian inference, robust Bayes, and also belief functions. For this particular example, the answer given by belief functions doesn’t make much sense.

Here’s my recent paper on the topic, and here’s the abstract to the paper:

Bayesian inference requires all unknowns to be represented by probability distributions, which awkwardly implies that the probability of an event for which we are completely ignorant (e.g., that the world’s greatest boxer would defeat the world’s greatest wrestler) must be assigned a particular numerical value such as 1/2, as if it were known as precisely as the probability of a truly random event (e.g., a coin flip).

Robust Bayes and belief functions are two methods that have been proposed to distinguish ignorance and randomness. In robust Bayes, a parameter can be restricted to a range, but without a prior distribution, yielding a range of potential posterior inferences. In belief functions (also known as the Dempster-Shafer theory), probability mass can be assigned to subsets of parameter space, so that randomness is represented by the probability distribution and uncertainty is represented by large subsets, within which the model does not attempt to assign probabilities.

Through a simple example involving a coin flip and a boxing/wrestling match, we illustrate difficulties with pure Bayes, robust Bayes, and belief functions. In short: pure Bayes does not distinguish ignorance and randomness; robust Bayes allows ignorance to spread too broadly, and belief functions inappropriately collapse to simple Bayesian models.

One Comment

1. anon says:

P(Boxer wins) = 0
P(Wrestler wins) = 1

Check out some of the "Ultimate
Fighting Videos": the wrestler
ALWAYS beats the boxer.