This is a special post for short-form writing by Stuart Armstrong. Only they can create top-level comments. Comments here also appear on the Shortform Page and All Posts page.

This is a special post for short-form writing by Stuart Armstrong. Only they can create top-level comments. Comments here also appear on the Shortform Page and All Posts page.

Stuart_Armstrong's Shortform

4Stuart Armstrong

3Diffractor

Partial probability distribution

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

That's a Q that defines Q(A∣B) for some but not all A and B (with Q(A)=Q(A∣Ω) for Ω being the whole set of outcomes).

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

Suppose that Q(C∣D) is not defined. We can, however, say that Q(C∣D)=x is a logical implication of Q if all full extension P has P(C∣D)=x.

Eg: Q(A), Q(B), Q(A∪B) will logically imply the value of Q(A∩B).

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

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

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