Saturday, August 20, 2022

A weird space for non-classical probability values

Consider the proper class V of formal expressions of the form xϵy where x is a non-negative real number that is permitted to be zero only if y = 0, y is a non-negative surreal number, and ϵ is a formal symbol to be thought of as “something very small”. (If we want to be rigorous, we let V be the set of ordered pairs (y,x).) Stipulate:

1. x = xϵ0 for real x

2. xϵy ≤ xϵy iff either (a) y > y or (b) y = y and x ≤ x

3. xϵy + xϵy equals (x+x′)ϵy if y = y′ and otherwise equals the greater of xϵy and xϵy

4. if xϵy ≤ xϵy and they’re not both zero, then (xϵy/xϵy) = (x/x′)ϵy − y

5. Std xϵy equals x if y = 0 and equals 0 othewise.

We can then define finitely-additive probabilities with values in V in the same way that we do so for reals, and we can then define conditional probabilities using the standard formula P(AB) = P(AB)/P(B).

Say that a V-valued probability P is regular iff 0 < P(A) whenever A is non-empty.

Now here is a fun fact. Given a V-valued probability P, we can define a real-valued full conditional probability as the standard part (Std) of P. Conversely, and less trivially, any real-valued full conditional probability can be obtained this way (this follows from the fact that any linear order can be embedded in the surreals).

So far this doesn’t mark any advantage of using V instead of hyperreals as the values of our probabilities. But there is an advantage. Specifically, if our probability space Ω is acted on by a supramenable group G of symmetries (any Abelian group is supramenable)—for instance, Ω might be a circle acted on by the group of rotations—then there is a V-valued regular G-invariant probability defined for all subsets of Ω. But if we have hyperreal (or surreal, for that matter) values, then the existence of a regular probability invariant under G requires significantly stricter conditions, ones that won’t be met in the case where Ω is the circle and G is rotations.

However, the advantage comes from the fact that V one to have a + b = a even though b > 0, so that one can have weak regularity—the condition that 0 < P(A) whenever A is nonempty—without strong regularity—the condition that P(A) < P(B) whenever A ⊂ B. If one wants strong regularity, using V instead of the hyperreals doesn’t have the same advantage.