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 $hat\{Omega\}$; 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 $hat\{Omega\}$. To begin with, any event which was formerly represented by subset "A" of Ω is now represented by $hat\{A\}$ = "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 $hat\{F\}$ now be a set of subsets of $hat\{Omega\}$, 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. $hat\{F\}$ 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, $hat\{P\}$, based on a standard probability function "P":

:$hat\{P\}$("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 $hat\{P\}$ that $hat\{P\}$ ($hat\{A\}$) = "P"("A"). Thus $hat\{P\}$ = "P" over the domain of "P".

="P"("A" → "B") = "P"("B"|"A")=

Now comes the insight which 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 $hat\{P\}$ 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 $hat\{P\}$("A" → "B") = "P"("B"|"A") for any choice of "A", "B", and "P". But since $hat\{P\}$ = "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.