Monday, July 20, 2020

Complete conditional probabilities, infinitesimal probabilities and two easy Frankenstein facts

Suppose we have a complete finitely-additive conditional probability P(⋅|⋅) on some algebra F of events on a space Ω (i.e., P is a Popper function with all non-empty sets regular), so that P(A|B) is defined for every non-empty B.

Here’s a curious thing: there is a very large disconnect between how P behaves when conditioning on sets of zero measure and when conditioning on sets of non-zero measure.

Here’s one way to see the disconnect. Consider any other complete finitely additive conditional probability Q(⋅|⋅) on the same algebra F, and suppose that Q assigns unconditional measure zero to everything that P does: i.e., if P(A|Ω)=0, then Q(A|Ω)=0.

Then, Frankenstein-fashion, we can sew P and Q into a new conditional probability R where R(A|B)=P(A|B) if P(A|Ω)>0 and R(A|B)=Q(A|B) if P(A|Ω)=0. In other words, R behaves exactly like P when conditioning on non-zero measure sets and exactly like Q when conditioning on zero measure sets.

[To check that R is a conditional probability, the one non-trivial condition to check here is that R(A ∩ B|C)=R(A|C)R(B|A ∩ C). Suppose P(C|Ω)=0. Then the equality follows from the corresponding equality for Q. Suppose P(C|Ω)>0. If P(A ∩ C|Ω)>0 as well, then our equality follows from the corresponding equality for P. Suppose now that P(A ∩ C|Ω)=0. Then the equality to be demonstrated is equivalent to P(A ∩ B|C)=P(A|C)Q(B|A ∩ C). But if P(A ∩ C|Ω)=0, then P(A ∩ B|C)=0 and P(A|C)=0.]

There is an analogous Frankenstein fact for infinitesimal probabilities. Let P and Q be any two finitely-additive probabilities with values in some hyperreal field, and suppose that Q is tiny whenever P is tiny, where a hyperreal is “tiny” provided that it is zero or infinitesimal. Then there is a Frankenstein probability R = Std P + Inf Q, where Std x and Inf x are the standard and infinitesimal parts of a finite hyperreal. (The fact that Q is tiny when P is tiny is used to show that R is non-negative.) This R then has the same large-scale (i.e., standard scale) behavior as P and small-scale behavior as Q.

In other words, as we depart from classical probability, we get a nearly complete disconnect between small-scale and large-scale behavior.

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