The de Finetti definition of comparative probabilities in terms of conditional probabilities

Suppose we have a full conditional probability P(A∣B) defined for all pairs of events (stipulating that P(A∣⌀) = 1 if we wish). Two methods have been proposed for defining a probability comparison using conditional probabilities:

1. Pruss: A ⪅ B iff P(A∣A∪B) ≤ P(B∣A∪B).

2. De Finetti: A ⪅ B iff P(A−B∣AΔB) ≤ P(B−A∣AΔB), where AΔB = (A−B) ∪ (B−A) is the symmetric difference.

In a footnote in a paper, I wrote about the de Finetti ordering: “This ordering has the advantage that if A is a proper subset of B, then A < B, but it is somewhat harder to prove transitivity”.

Well, that was an understatement! It’s not just harder to prove transitivity: it’s impossible.

Define:

• Ω: all integers

• E₀: non-negative even integers

• E: positive even integers

• D: positive odd integers.

Let P be a full conditional probability such that:

3. P(E₀∣E₀∪D) = 1/2 = P(D∣E₀∪D)

4. P(D∣D∪E) = 1/2 = P(E|D∪E).

It is easy to see from (3) and (4) that because E₀ and D are disjoint, and so are D and E, then by the de Finetti definition (2) we have E₀ ⪅ D and D ⪅ E. (For disjoint A and B, the de Finetti definition of A ⪅ B is equivalent to the Pruss definition.) However, E₀ − E = {0}, E − E₀ = ⌀, and E₀ΔE = {0}, so P(E−E₀∣E₀ΔE) = 0 while P(E₀−E∣E₀ΔE) = 1, and thus we cannot have E₀ ⪅ E.

The only question is whether there actually is a full conditional probability satisfying (3) and (4). If there is, then the de Finetti comparison is not transitive in our case.

There is such a full conditional probability. Let Q_n(A) =  ∣ A ∩ [−n,n] ∣ /(2n+1), where $B$ is the cardinality of a set B. Then Q_n is a probability. Let Q be a limit of the Q_n along an ultrafilter. This is a finitely additive hyperreal probability which is non-zero for all non-empty sets. Define P(A∣B) as the standard part of Q(A∩B)/Q(B) for B non-empty. This is a full conditional probability. Moreover, P(A∣B) = lim Q_n(A∩B)/Q_n(B) whenever the latter limit is defined. That limit is defined in the cases of the events involved in (3) and (4), and it is easy to evaluate the limits and see that (3) and (4) are true.

I don’t know what I was thinking when I wrote that footnote. My guess is that I had in my mind a proof sketch that doesn’t work (I have some idea what that might have been).

Whew! I noticed this afternoon that Theorem 1 of this paper of mine was incompatible with the transitivity of the de Finetti comparison (2). This made me really worried that my Theorem 1 was false. But since the de Finetti comparison isn’t transitive, I can relax.

This raises an interesting potential research problem. The Pruss definition of ⪅ does not satisfy the additivity axiom for comparative probabilities, namely that if C is disjoint from A ∪ B, then A ≲ B if and only if A ∪ C ≲ B ∪ C (it only preserves the left-to-right implication). The de Finetti definition does satisfy the additivity axiom, which is what I liked about it.

I suspect there is no good definition of comparative probabilities in terms of full conditional probabilities that satisfies the additivity axiom. (One reason for this intuition has to do with the fact that in Figure 1 here there are entries with a Yes in column 3 and a No in column 5.)

So, I now wonder: Is there some good combination of a definition of comparative probabilities in terms of full conditional probabilities with some weakened version of the additivity axiom?