# Stuart_Armstrong comments on Stuart_Armstrong’s Shortform

• Partial probability distribution

A concept that’s useful for some of my research: a partial probability distribution.

That’s a that defines for some but not all and (with for being the whole set of outcomes).

This is a partial probability distribution iff there exists a probability distribution that is equal to wherever is defined. Call this a full extension of .

Suppose that is not defined. We can, however, say that is a logical implication of if all full extension has .

Eg: , , will logically imply the value of .

• This is a special case of a crisp infradistribution: is equivalent to , a linear equation in , so the set of all ’s satisfying it is convex closed.

• Thanks! That’s useful to know.

• Sounds like a special case of crisp infradistributions (ie, all partial probability distributions have a unique associated crisp infradistribution)

Given some , we can consider the (nonempty) set of probability distributions equal to where is defined. This set is convex (clearly, a mixture of two probability distributions which agree with about the probability of an event will also agree with about the probability of an event).

Convex (compact) sets of probability distributions = crisp infradistributions.