Goodman–Nguyen–van Fraassen algebra
A Goodman–Nguyen–van Fraassen algebra is a type of conditional event algebra (CEA) that embeds the standard Boolean algebra of unconditional events in a larger algebra which is itself Boolean. The goal (as with all CEAs) is to equate the conditional probability P(A ∩ B) / P(A) with the probability of a conditional event, P(A → B) for more than just trivial choices of A, B, and P.
Construction of the algebra
Given set Ω, which is the set of possible outcomes, and set F of subsets of Ω—so that F is the set of possible events—consider an infinite Cartesian product of the form E_{1} × E_{2} × … × E_{n} × Ω × Ω × Ω × …, where E_{1}, E_{2}, … E_{n} are members of F. Such a product specifies the set of all infinite sequences whose first element is in E_{1}, whose second element is in E_{2}, …, and whose nth element is in E_{n}, and all of whose elements are in Ω. Note that one such product is the one where E_{1} = E_{2} = … = E_{n} = Ω, i.e., the set Ω × Ω × Ω × Ω × …. Designate this set as ; it is the set of all infinite sequences whose elements are in Ω.
A new Boolean algebra is now formed, whose elements are subsets of . To begin with, any event which was formerly represented by subset A of Ω is now represented by = A × Ω × Ω × Ω × ….
Additionally, however, for events A and B, let the conditional event A → B be represented as the following infinite union of disjoint sets:
- [(A ∩ B) × Ω × Ω × Ω × …] ∪
- [A′ × (A ∩ B) × Ω × Ω × Ω × …] ∪
- [A′ × A ′ × (A ∩ B) × Ω × Ω × Ω × …] ∪ ….
The motivation for this representation of conditional events will be explained shortly. Note that the construction can be iterated; A and B can themselves be conditional events.
Intuitively, unconditional event A ought to be representable as conditional event Ω → A. And indeed: because Ω ∩ A = A and Ω′ = ∅, the infinite union representing Ω → A reduces to A × Ω × Ω × Ω × ….
Let now be a set of subsets of , which contains representations of all events in F and is otherwise just large enough to be closed under construction of conditional events and under the familiar Boolean operations. is a Boolean algebra of conditional events which contains a Boolean algebra corresponding to the algebra of ordinary events.
Definition of the extended probability function
Corresponding to the newly constructed logical objects, called conditional events, is a new definition of a probability function, , based on a standard probability function P:
- (E_{1} × E_{2} × … E_{n} × Ω × Ω × Ω × …) = P(E_{1})⋅P(E_{2})⋅ … ⋅P(E_{n})⋅P(Ω)⋅P(Ω)⋅P(Ω)⋅ … = P(E_{1})⋅P(E_{2})⋅ … ⋅P(E_{n}), since P(Ω) = 1.
It follows from the definition of that ( ) = P(A). Thus = P over the domain of P.
P(A → B) = P(B|A)
Now comes the insight that motivates all of the preceding work. For P, the original probability function, P(A′) = 1 – P(A), and therefore P(B|A) = P(A ∩ B) / P(A) can be rewritten as P(A ∩ B) / [1 – P(A′)]. The factor 1 / [1 – P(A′)], however, can in turn be represented by its Maclaurin series expansion, 1 + P(A′) + P(A′)^{2} …. Therefore, P(B|A) = P(A ∩ B) + P(A′)P(A ∩ B) + P(A′)^{2}P(A ∩ B) + ….
The right side of the equation is exactly the expression for the probability of A → B, just defined as a union of carefully chosen disjoint sets. Thus that union can be taken to represent the conditional event A→ B, such that (A → B) = P(B|A) for any choice of A, B, and P. But since = P over the domain of P, the hat notation is optional. So long as the context is understood (i.e., conditional event algebra), one can write P(A → B) = P(B|A), with P now being the extended probability function.
References
Bamber, Donald, I. R. Goodman, and H. T. Nguyen. 2004. "Deduction from Conditional Knowledge". Soft Computing 8: 247–255.
Goodman, I. R., R. P. S. Mahler, and H. T. Nguyen. 1999. "What is conditional event algebra and why should you care?" SPIE Proceedings, Vol 3720.