Goodman-Nguyen-van Fraassen algebra
Encyclopedia
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.
of the form E1 × E2 × … × En × Ω × Ω × Ω × …, where E1, E2, … En are members of F. Such a product specifies the set of all infinite sequences whose first element is in E1, whose second element is in E2, …, and whose nth element is in En, and all of whose elements are in Ω. Note that one such product is the one where E1 = E2 = … = En = 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:
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.
, based on a standard probability function P:
(E1 × E2 × … En × Ω × Ω × Ω × …) = P(E1)⋅P(E2)⋅ … ⋅P(En)⋅P(Ω)⋅P(Ω)⋅P(Ω)⋅ … = P(E1)⋅P(E2)⋅ … ⋅P(En), since P(Ω) = 1.
It follows from the definition of
that 
(
) = P(A). Thus 
= P over the domain of P.
, 1 + P(A′) + P(A′)2 …. Therefore, P(B|A) = P(A ∩ B) + P(A′)P(A ∩ B) + P(A′)2P(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.
Conditional event algebra
A conditional event algebra is an algebraic structure whose domain consists of logical objects described by statements of forms such as "If A, then B," "B, given A," and "B, in case A." Unlike the standard Boolean algebra of events, a CEA allows the defining of a probability function, P, which...
(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 productCartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of the form E1 × E2 × … × En × Ω × Ω × Ω × …, where E1, E2, … En are members of F. Such a product specifies the set of all infinite sequences whose first element is in E1, whose second element is in E2, …, and whose nth element is in En, and all of whose elements are in Ω. Note that one such product is the one where E1 = E2 = … = En = 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 operationsBoolean operation
Boolean operation or Boolean operator may refer to one of the following related meanings.*Boolean function*an operation in a Boolean algebra; in particular:**operations over the algebra of sets: union , intersection , etc....
.
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:(E1 × E2 × … En × Ω × Ω × Ω × …) = P(E1)⋅P(E2)⋅ … ⋅P(En)⋅P(Ω)⋅P(Ω)⋅P(Ω)⋅ … = P(E1)⋅P(E2)⋅ … ⋅P(En), 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 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 expansionTaylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
, 1 + P(A′) + P(A′)2 …. Therefore, P(B|A) = P(A ∩ B) + P(A′)P(A ∩ B) + P(A′)2P(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.