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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 .

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.