# 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 *n*th 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 [math]\displaystyle{ \hat{\Omega} }[/math]; 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 [math]\displaystyle{ \hat{\Omega} }[/math]. To begin with, any event which was formerly represented by subset *A* of Ω is now represented by [math]\displaystyle{ \hat{A} }[/math] = *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 [math]\displaystyle{ \hat{F} }[/math] now be a set of subsets of [math]\displaystyle{ \hat{\Omega} }[/math], 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. [math]\displaystyle{ \hat{F} }[/math] 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, [math]\displaystyle{ \hat{P} }[/math], based on a standard probability function *P*:

- [math]\displaystyle{ \hat{P} }[/math](
*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 [math]\displaystyle{ \hat{P} }[/math] that [math]\displaystyle{ \hat{P} }[/math] ([math]\displaystyle{ \hat{A} }[/math]) = *P*(*A*). Thus [math]\displaystyle{ \hat{P} }[/math] = *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 [math]\displaystyle{ \hat{P} }[/math] 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 [math]\displaystyle{ \hat{P} }[/math](*A* → *B*) = *P*(*B*|*A*) for any choice of *A*, *B*, and *P*. But since [math]\displaystyle{ \hat{P} }[/math] = *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.