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).

This is a special case of a crisp infradistribution: Q(A|B)=t is equivalent to Q(A∩B)=tQ(B), a linear equation in Q, so the set of all Q’s satisfying it is convex closed.

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.

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).

This is a special case of a crisp infradistribution: Q(A|B)=t is equivalent to Q(A∩B)=tQ(B), a linear equation in Q, so the set of all Q’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 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.