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en>I dream of horses: Reverted edits by 111.93.183.194 (talk): editing tests (HG)

2014-08-25T13:42:24Z

Reverted edits by 111.93.183.194 (talk): editing tests (HG)

← Older revision		Revision as of 15:42, 25 August 2014
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	A '''Goodman–Nguyen–van Fraassen algebra''' is a type of [[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]] ''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''.		Greetings. The author's name is Phebe and she feels comfortable when individuals use the complete title. Managing individuals is his occupation. To gather coins is 1 of the issues I love most. California is where her home is but she needs to move because of her family.<br><br>Feel free to visit my weblog :: [http://Checkmates.Co.za/index.php?do=/profile-120229/info/ Checkmates.Co.za]

	~~==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''<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 [[Boolean operation\|the familiar 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 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 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 [[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:Boolean algebra]]~~
	~~[[Category:Probability theory]~~]

en>MrOllie: rm non-WP:RS cite to someone's personal site

2013-09-06T20:03:28Z

rm non-WP:RS cite to someone's personal site

← Older revision		Revision as of 22:03, 6 September 2013
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	~~Nice~~ to ~~satisfy you~~, ~~I am Marvella Shryock~~. ~~Years ago we moved to Puerto Rico~~ and ~~my family enjoys it~~. It'~~s not a common factor but what she likes doing~~ is ~~foundation jumping~~ and now ~~she~~ is ~~attempting~~ to ~~earn money with it~~. ~~I utilized~~ to be ~~unemployed but now I am~~ a ~~librarian~~ and the ~~wage has been truly fulfilling~~.<br><br>~~Also visit my website~~: ~~std home test~~ ([~~http:~~//~~nuvem~~.tk/~~altergalactica~~/~~AliceedMaurermy~~ you ~~can try these out~~])		A '''Goodman–Nguyen–van Fraassen algebra''' is a type of [[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]] ''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''<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 [[Boolean operation\|the familiar 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 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 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 [[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:Boolean algebra]]
			[[Category:Probability theory]]

64.173.205.74 at 19:41, 24 March 2011

2011-03-24T19:41:31Z

New page

Nice to satisfy you, I am Marvella Shryock. Years ago we moved to Puerto Rico and my family enjoys it. It's not a common factor but what she likes doing is foundation jumping and now she is attempting to earn money with it. I utilized to be unemployed but now I am a librarian and the wage has been truly fulfilling.<br><br>Also visit my website: std home test ([http://nuvem.tk/altergalactica/AliceedMaurermy you can try these out])