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#REDIRECT [[Conditional event algebra]]
A '''Goodman–Nguyen–Van Fraassen algebra''' is a type of [[Conditional event algebra|conditional event algebra]] (CEA) that embeds the standard [[Boolean algebra (structure)|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|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&mdash;so that ''F'' is the set of possible events&mdash;consider an infinite [[Cartesian product]] of the form ''E''<sub>1</sub> × ''E''<sub>2</sub> ×  … × ''E''<sub>''n''</sub> × Ω × Ω × Ω ×  …, where ''E''<sub>1</sub>, ''E''<sub>2</sub>, … ''E''<sub>''n''</sub> are members of ''F''. Such a product specifies the set of all infinite sequences whose first element is in ''E''<sub>1</sub>, whose second element is in ''E''<sub>2</sub>, …, and whose ''n''th element is in ''E''<sub>''n''</sub>, and all of whose elements are in Ω. Note that one such product is the one where ''E''<sub>1</sub> = ''E''<sub>2</sub> = … = ''E''<sub>''n''</sub> = Ω, i.e., the set Ω × Ω ×  Ω ×  Ω × …. Designate this set as <math>\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>\hat{\Omega}</math>. To begin with, any event which was formerly represented by subset ''A'' of Ω is now represented by <math>\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>\hat{F}</math> now be a set of subsets of <math>\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 [[Algebra of sets|Boolean operations]]. <math>\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>\hat{P}</math>, based on a standard [[Probability measure|probability function]] ''P'': 

:<math>\hat{P}</math>(''E''<sub>1</sub> × ''E''<sub>2</sub> × … ''E''<sub>''n''</sub> × Ω × Ω × Ω × …) = ''P''(''E''<sub>1</sub>)⋅''P''(''E''<sub>2</sub>)⋅ … ⋅''P''(''E''<sub>''n''</sub>)⋅''P''(Ω)⋅''P''(Ω)⋅''P''(Ω)⋅ … = ''P''(''E''<sub>1</sub>)⋅''P''(''E''<sub>2</sub>)⋅ … ⋅''P''(''E''<sub>''n''</sub>), since ''P''(Ω) = 1.

It follows from the definition of <math>\hat{P}</math> that <math>\hat{P}</math> (<math>\hat{A}</math>) = ''P''(''A''). Thus <math>\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&nbsp;–&nbsp;''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 [[Taylor series|Maclaurin series expansion]], 1 + ''P''(''A''′) + ''P''(''A''′)<sup>2</sup> …. Therefore, ''P''(''B''|''A'') = ''P''(''A'' ∩ ''B'') + ''P''(''A''′)''P''(''A'' ∩ ''B'') + ''P''(''A''′)<sup>2</sup>''P''(''A'' ∩ ''B'') + ….

The right side of the equation is exactly the expression for the probability <math>\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>\hat{P}</math>(''A'' → ''B'') = ''P''(''B''|''A'') for any choice of ''A'', ''B'', and ''P''. But since <math>\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.

{{DEFAULTSORT:Goodman-Nguyen-Van Fraassen algebra}}
[[Category:Abstract algebra]]
[[Category:Conditional probability]]

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