Invariance of Popper functions under symmetries

Popper functions are primitive finitely additive conditional probabilities—i.e., P(A|B) is the fundamental quantity, and P(A) is the defined quantity. Now, in some situations we have expect our probabilities to be invariant under some group G of symmetries. For instance, if we're shooting an idealized dart at a circular target and aiming at the center, our idealized method of shooting might be rotationally invariant so that the probability of hitting some region A will be the same as the probability of hitting rA where r is some rotation about the center. (In real life, this need not be so. For instance, in archery, one might have bigger error in the vertical direction than in the horizontal direction, or vice versa, depending on one's skills.) We might also think that similarly there is invariance under reflections about lines through the center.

With unconditional probabilities, we can just formulate these invariance condition as: P(gA)=P(A) for all symmetries g in G and all (measurable) regions A. But how to formulate this for conditional probabilities?

There are two natural definitions:

• P is weakly invariant if and only if P(A|B)=P(gA|gB) for all A, B and g.
• P is strongly invariant if and only if (a) whenever A∪gA⊆B, we have P(A|B)=P(gA|B) and (b) whenever A⊆B∩gB, we have P(A|B)=P(A|gB).

Fact: Conditions (a) and (b) in the definition of strong invariance are equivalent.

Personally, I find weak invariance to be the more intuitive condition, though strong invariance has some intuitive pull. It's an interesting question how the two are related.

One interesting special case is where G is generated by symmetries of finite order. A symmetry g has finite order provided that there is a finite number n such that gⁿ is the identity—i.e., applying it n times gets you back to where you started. For instance, rotation by an angle of 360/n degrees where n is a non-zero integer has finite order—you do this |n| times and you're back where you started. And all reflections have finite order.
Fact: If every symmetry in G can be written as a combination of symmetries of finite order, then weak invariance implies strong invariance.

For instance, while most rotations in the plane don't have finite order (only ones by a rational-number angle do), any rotation in the plane can be generated by combining two reflections. Thus, in our circular target case, where we are looking at invariance under reflections and rotations, weak invariance implies strong invariance.

Armstrong in this paper claims that weak invariance implies strong invariance in general (Prop. 1.3). Unfortunately Armstrong's proof is incomplete. And well it might be. For yesterday I came up with a super-simple case showing:
Fact: Weak invariance does not imply strong invariance.

It would be interesting to characterize cases where weak invariance does imply strong invariance. Two general cases are known to me. One is where the symmetries are generated by symmetries of finite order. The second is where the conditional probabilities are defined by the ratio formula starting with a regular probability (one that assigns non-zero probability to each empty set).

Popper functions and infinite sequences of heads

Williamson gave a lovely argument that infinitesimals can't capture the probability of an infinite sequence of heads in fair independent trials: Let H_n be the event that we have heads in each of the trials n,n+1,n+2,.... Then, P(H₁)=(1/2)P(H₂). But P(H₁)=P(H₂) since they're both just the probability of getting an infinite sequence of heads. Thus, P(H₁)=(1/2)P(H₁) and so P(H₁) is zero, not infinitesimal.

It turns out that a somewhat similar result holds for Popper functions as well. For technical reasons, I need a bidirectionally infinite sequence of coin tosses, one for each integer (positive, zero or negative). Our probability space Ω of infinite sequences will then be the set of all functions from the integers Z to {H,T}. Let G be the group of transformations of Ω generated by translations defined by reflections on Z. In other words, G is generated by the transformations R_a, where a is an integer or a half-integer, and (R_as)(n)=s(2a−n) for any sequence s in Ω.

Let F be any G-invariant field in Ω that contains all the H_n. A very plausible symmetry condition on the Popper function on Ω representing the double sequence of heads then is:

1. For any g in G and any A and B in F, P(A|B)=P(gA|gB).

In other words, if we flip the sequences around, we don't change the probabilities. E.g., the probability of getting heads on tosses 2,3,4,5,... conditionally on B is the same as the probability of getting heads on tosses 1,0,-1,-2,... conditionally on R_1.5B. This symmetry condition is related to Williamson's symmetry assumption that P(H₂)=P(H₁).

A second obvious condition is that the probability of getting heads on tosses 1,2,3,... given that one has heads on 2,3,... is equal to the probability of getting heads on toss 1, i.e., is 1/2:

2. P(H₁|H₂)=1/2.

Theorem: There is no Popper function on F satisfying (1) and (2).