Omega

Prof Alan Hajek (Australian National University)

Probability theory is the dominant approach to modeling uncertainty. We begin with a set of possibilities or outcomes, usually designated ‘Ω’. We then assign probabilities—real numbers between 0 and 1 inclusive—to subsets of Ω. Nearly all of the action in the mathematics and philosophy of probability for over three and a half centuries has concerned the probabilities: their axiomatization, their associated theorems, and their interpretation.

I want instead to put Ω in the spotlight. Ω is a set of possibilities, but which possibilities? While the probability calculus constrains our numerical assignments, and its interpretation guides us further regarding them, we are entirely left to our own devices regarding Ω. What makes one Ω better than another? Its members are typically not exhaustive—but which possibilities should be excluded? Its members are typically not maximally fine-grained—but how refined should they be? I will discuss both philosophical and practical problems with the construction of a good Ω. I will offer some desirable features that a given Ω might have, and some heuristics for coming up with an Ω that has them, and for improving an Ω that we already have.