Stuart_Armstrong's Shortform

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Stuart_Armstrong's Shortform

30th Sep 2019

1 min read

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Stuart_Armstrong's Shortform

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[-]Stuart_Armstrong5y*00

Partial probability distribution

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

That's a 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: $Q (A)$ , $Q (B)$ , $Q (A \cup B)$ will logically imply the value of $Q (A \cap B)$ .

[-]Diffractor4y30

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 $Q$ 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.

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