Uniform polytope - Revision history

en>Tomruen at 23:11, 1 January 2015

2015-01-01T23:11:52Z

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	~~{{More footnotes\|date=December 2010}}~~		Friends contact him Royal. One of his preferred hobbies is playing crochet but he hasn't produced a dime with it. She works as a financial officer and she will not change it whenever quickly. Delaware is our birth location.<br><br>Also visit my weblog ... auto warranty ([http://jbcopy.com/node/430 click through the up coming webpage])
	~~In [[mathematics]], the '''disintegration theorem''' is a result in [[measure theory]] and [[probability theory]]~~. ~~It rigorously defines the idea of a non-trivial "restriction"~~ of ~~a [[measure (mathematics)\|measure]] to a [[measure zero]] subset of the [[measure space]] in question. It~~ is ~~related to the existence of [[conditioning (probability)\|conditional probability measures]]. In a sense, "disintegration" is the opposite process to the construction of a [[product measure]].~~

	~~==Motivation==~~
	Consider the unit square in the [[Euclidean plane]] '''R'''<sup>2</sup>, ''S'' = [0, 1] × [0, 1]. Consider the [[probability measure]] μ defined on ''S'' by the restriction of two-dimensional [[Lebesgue measure]] λ<sup>2</sup> to ''S''. That is, the probability of an event ''E'' ⊆ ''S'' is simply the area of ''E''. We assume '~~'E'' is~~ a ~~measurable subset of ''S''~~.

	~~Consider a one-dimensional subset of ''S'' such~~ as ~~the line segment ''L''<sub>''x''</sub> = {''x''} × [0, 1]. ''L''<sub>''x''</sub> has μ-measure zero; every subset of ''L''<sub>''x''</sub> is~~ a ~~μ-[[null set]]; since the Lebesgue measure space is a [[complete measure\|complete measure space]],~~

	~~:<math>E \subseteq L_{x} \implies \mu (E) = 0.</math>~~

	While true, this is somewhat unsatisfying. It would be nice to say that μ "restricted to" ''L''<sub>''x''</sub> is the one-dimensional Lebesgue measure λ<sup>1</sup>, rather than the [[trivial measure\|zero measure]]. The probability of a "two-dimensional" event ''E'' could then be obtained as an [[Lebesgue integration\|integral]] of the one-dimensional probabilities of the vertical "slices" ''E'' ∩ ''L''<sub>''x''</sub>: more formally, if μ<sub>''x''</sub> denotes one-dimensional Lebesgue measure on ''L''<sub>''x''</sub>, then

	~~:<math>\mu (E) = \int_{[0, 1]} \mu_{x} (E \cap L_{x}) \, \mathrm{d} x</math>~~

	~~for any "nice" ''E'' ⊆ ''S''. The disintegration theorem makes this argument rigorous in the context of measures on [[metric space]]s.~~

	~~==Statement of the theorem==~~
	~~(Hereafter, '''''P'''''(''X'')~~ will ~~denote the collection of [[Borel measure\|Borel]] probability measures on a [[metric space]] (''X'', ''d'').)~~

	~~Let ''Y'' and ''X'' be two [[Radon space]]s (i.e. [[separable space\|separable]] metric spaces on which every probability measure is a [[Radon measure]])~~. Let μ ∈ '''''P'''''(''Y''), let π : ''Y'' → ''X'' be a Borel-[[measurable function]], and let ν ∈ '''''P'''''(''X'') be the [[pushforward measure]] ν = π<sub>∗</sub>(μ) = μ ∘ π<sup>−1</sup>. Then there exists a ν-[[almost everywhere]] uniquely determined family of probability measures {μ<sub>''x''</sub>}<sub>''x''∈''X''</sub> ⊆ '''''P'''''(''Y'') such that
	* the function <math>x \mapsto \mu_{x}</math> is Borel measurable, in the sense that <math>x \mapsto \mu_{x} (B)</math> is ~~a Borel-measurable function for each Borel-measurable set ''B'' ⊆ ''Y'';~~
	* μ<sub>''x''</sub> "lives on" the [[fiber (mathematics)\|fiber]] π<sup>−1</sup>(''x''): for ν-[[almost all]] ''x'' ∈ ''X'',

	~~::<math>\mu_{x} \left( Y \setminus \pi^{-1} (x) \right) = 0,</math>~~

	~~:and so μ<sub>''x''</sub>(''E'') = μ<sub>''x''</sub>(''E'' ∩ π<sup>−1</sup>(''x''));~~

	* for every Borel-measurable function ''f'' : ''Y'' → [0, ∞],

	~~::<math>\int_{Y} f(y) \, \mathrm{d} \mu (y) = \int_{X} \int_{\pi^{-1} (x)} f(y) \, \mathrm{d} \mu_{x} (y) \mathrm{d} \nu (x)~~.<~~/math~~>

	~~:In particular, for any event ''E'' &sube; ''Y'', taking ''f'' to be the [[indicator function]] of ''E'',~~<~~ref name=Dellacherie_Meyer~~>~~{{cite book \| author=Dellacherie, C. & Meyer, P~~.-A. ~~\| title=Probabilities and potential\| publisher=North-Holland Mathematics Studies, North-Holland Publishing Co~~.~~, Amsterdam \| year=1978}}</ref>~~

	~~::<math>\mu (E) = \int_{X} \mu_{x} \left~~( ~~E \right) \, \mathrm{d} \nu (x).</math>~~

	~~==Applications==~~
	~~===Product spaces===~~
	~~The original example was a special case of the problem of product spaces, to which the disintegration theorem applies.~~

	~~When ''Y'' is written as a~~ [[Cartesian product]] ''Y'' = ''X''<sub>1</sub> × ''X''<sub>2</sub> and π<sub>''i''</sub> : ''Y'' → ''X''<sub>''i''</sub> is the natural [[projection (mathematics)\|projection]], then each fibre ''π''<sub>1</sub><sup>−1</sup>(''x''<sub>1</sub>) can be [[canonical form\|canonically]] identified with ''X''<sub>2</sub> and there exists a Borel family of probability measures <math>\{ \mu_{x_{1}} \}_{x_{1} \in X_{1}}</math> in '''''P'''''(''X''<sub>2</sub>) (which is (π<sub>1</sub>)<sub>∗</sub>(μ)-almost everywhere uniquely determined) such that

	:~~<math>\mu = \int_{X_{1}} \mu_{x_{1}} \, \mu \left(\pi_1^{-1}(\mathrm d x_1) \right)= \int_{X_{1}} \mu_{x_{1}} \, \mathrm{d} (\pi_{1})_{*} (\mu) (x_{1}),<~~/~~math>~~
	~~which is in particular~~
	~~:<math>\int_{X_1\times X_2} f(x_1,x_2)\, \mu(\mathrm d x_1,\mathrm d x_2) = \int_{X_1}\left( \int_{X_2} f(x_1,x_2) \mu(\mathrm d x_2\|x_1) \right) \mu\left( \pi_1^{-1}(\mathrm{d} x_{1})\right)<~~/~~math>~~
	~~and~~
	~~:<math>\mu(A \times B) = \int_A \mu\left(B\|x_1\right) \, \mu\left( \pi_1^{-1}(\mathrm{d} x_{1})\right)~~.</~~math>~~

	~~The relation to [[conditional expectation]] is given by the identities~~
	~~:<math>\operatorname E(f\|\pi_1)(x_1)= \int_{X_2} f(x_1,x_2) \mu(\mathrm d x_2\|x_1),</math>~~
	~~:<math>\mu(A\times B\|\pi_1)(x_1)= 1_A(x_1) \cdot \mu(B\| x_1).<~~/~~math>~~

	~~===Vector calculus===~~
	~~The disintegration theorem can also be seen as justifying the use of a "restricted" measure in [[vector calculus]]. For instance, in [[Stokes' theorem]] as applied to a [[vector field]] flowing~~ through ~~a [[compact space\|compact]] [[surface]] Σ ⊂ '''R'''<sup>3</sup>, it is implicit that~~ the "correct" measure on Σ is the disintegration of three-dimensional Lebesgue measure λ<sup>3</sup> on Σ, and that the disintegration of this measure on ∂Σ is the same as the disintegration of λ<sup>3</sup> on ∂Σ.<ref name=Ambrosio_Gigli_Savare>{{cite book \| author=Ambrosio, L., Gigli, N. & Savaré, G. \| title=Gradient Flows in Metric Spaces and in the Space of Probability Measures \| publisher=ETH Zürich, Birkhäuser Verlag, Basel \| year=2005 \| isbn=3-7643-2428-7 }}</ref>

	~~===Conditional distributions===~~
	The disintegration theorem can be applied to give a rigorous treatment of conditioning probability distributions in statistics, while avoiding purely abstract formulations of conditional probability.<ref name=Chang_Pollard>{{cite journal\|last=Chang\|first=J.T.\|coauthors=Pollard, D.\|title=Conditioning as disintegration\|journal=Statistica Neerlandica\| year=1997 \| volume=51\|issue=3\|url=http://www.stat.yale.edu/~jtc5/papers/ConditioningAsDisintegration.pdf\|doi=10.1111/1467-9574.00056\|pages=287}}</ref>

	~~==See also==~~
	*[[Joint probability distribution]]
	*[[Copula (statistics)]]
	*[[Conditional expectation]]

	~~==References==~~
	~~{{Reflist}}~~
	~~{{Use dmy dates\|date=December 2010}}~~

	~~[[Category:Theorems in measure theory]]~~
	~~[[Category:Probability theorems]]~~

en>Tomruen: /* Constructive summary */

2013-11-02T05:26:47Z

Constructive summary

← Older revision		Revision as of 07:26, 2 November 2013
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	~~Wilber Berryhill~~ is ~~what his wife loves~~ to ~~contact him and he totally loves this title~~. ~~For years she~~'~~s been residing in Kentucky but her husband wants them~~ to ~~move~~. ~~Invoicing~~ is ~~what I do~~ for a ~~living but I~~'~~ve usually wanted my personal business~~. ~~To climb~~ is ~~some thing I really enjoy doing~~.<br><br>~~Here~~ is ~~my website~~; ~~are psychics real~~ ([~~http~~://~~srncomm~~.~~com~~/~~blog~~/~~2014~~/08/25/~~relieve~~-that-~~stress~~-~~find~~-a-~~new~~-~~hobby~~/ ~~srncomm~~.~~com~~])		{{More footnotes\|date=December 2010}}
			In [[mathematics]], the '''disintegration theorem''' is a result in [[measure theory]] and [[probability theory]]. It rigorously defines the idea of a non-trivial "restriction" of a [[measure (mathematics)\|measure]] to a [[measure zero]] subset of the [[measure space]] in question. It is related to the existence of [[conditioning (probability)\|conditional probability measures]]. In a sense, "disintegration" is the opposite process to the construction of a [[product measure]].

			==Motivation==
			Consider the unit square in the [[Euclidean plane]] '''R'''<sup>2</sup>, ''S'' = [0, 1] × [0, 1]. Consider the [[probability measure]] μ defined on ''S'' by the restriction of two-dimensional [[Lebesgue measure]] λ<sup>2</sup> to ''S''. That is, the probability of an event ''E'' ⊆ ''S'' is simply the area of ''E''. We assume ''E'' is a measurable subset of ''S''.

			Consider a one-dimensional subset of ''S'' such as the line segment ''L''<sub>''x''</sub> = {''x''} × [0, 1]. ''L''<sub>''x''</sub> has μ-measure zero; every subset of ''L''<sub>''x''</sub> is a μ-[[null set]]; since the Lebesgue measure space is a [[complete measure\|complete measure space]],

			:<math>E \subseteq L_{x} \implies \mu (E) = 0.</math>

			While true, this is somewhat unsatisfying. It would be nice to say that μ "restricted to" ''L''<sub>''x''</sub> is the one-dimensional Lebesgue measure λ<sup>1</sup>, rather than the [[trivial measure\|zero measure]]. The probability of a "two-dimensional" event ''E'' could then be obtained as an [[Lebesgue integration\|integral]] of the one-dimensional probabilities of the vertical "slices" ''E'' ∩ ''L''<sub>''x''</sub>: more formally, if μ<sub>''x''</sub> denotes one-dimensional Lebesgue measure on ''L''<sub>''x''</sub>, then

			:<math>\mu (E) = \int_{[0, 1]} \mu_{x} (E \cap L_{x}) \, \mathrm{d} x</math>

			for any "nice" ''E'' ⊆ ''S''. The disintegration theorem makes this argument rigorous in the context of measures on [[metric space]]s.

			==Statement of the theorem==
			(Hereafter, '''''P'''''(''X'') will denote the collection of [[Borel measure\|Borel]] probability measures on a [[metric space]] (''X'', ''d'').)

			Let ''Y'' and ''X'' be two [[Radon space]]s (i.e. [[separable space\|separable]] metric spaces on which every probability measure is a [[Radon measure]]). Let μ ∈ '''''P'''''(''Y''), let π : ''Y'' → ''X'' be a Borel-[[measurable function]], and let ν ∈ '''''P'''''(''X'') be the [[pushforward measure]] ν = π<sub>∗</sub>(μ) = μ ∘ π<sup>−1</sup>. Then there exists a ν-[[almost everywhere]] uniquely determined family of probability measures {μ<sub>''x''</sub>}<sub>''x''∈''X''</sub> ⊆ '''''P'''''(''Y'') such that
			* the function <math>x \mapsto \mu_{x}</math> is Borel measurable, in the sense that <math>x \mapsto \mu_{x} (B)</math> is a Borel-measurable function for each Borel-measurable set ''B'' ⊆ ''Y'';
			* μ<sub>''x''</sub> "lives on" the [[fiber (mathematics)\|fiber]] π<sup>−1</sup>(''x''): for ν-[[almost all]] ''x'' ∈ ''X'',

			::<math>\mu_{x} \left( Y \setminus \pi^{-1} (x) \right) = 0,</math>

			:and so μ<sub>''x''</sub>(''E'') = μ<sub>''x''</sub>(''E'' ∩ π<sup>−1</sup>(''x''));

			* for every Borel-measurable function ''f'' : ''Y'' → [0, ∞],

			::<math>\int_{Y} f(y) \, \mathrm{d} \mu (y) = \int_{X} \int_{\pi^{-1} (x)} f(y) \, \mathrm{d} \mu_{x} (y) \mathrm{d} \nu (x).</math>

			:In particular, for any event ''E'' &sube; ''Y'', taking ''f'' to be the [[indicator function]] of ''E'',<ref name=Dellacherie_Meyer>{{cite book \| author=Dellacherie, C. & Meyer, P.-A. \| title=Probabilities and potential\| publisher=North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam \| year=1978}}</ref>

			::<math>\mu (E) = \int_{X} \mu_{x} \left( E \right) \, \mathrm{d} \nu (x).</math>

			==Applications==
			===Product spaces===
			The original example was a special case of the problem of product spaces, to which the disintegration theorem applies.

			When ''Y'' is written as a [[Cartesian product]] ''Y'' = ''X''<sub>1</sub> × ''X''<sub>2</sub> and π<sub>''i''</sub> : ''Y'' → ''X''<sub>''i''</sub> is the natural [[projection (mathematics)\|projection]], then each fibre ''π''<sub>1</sub><sup>−1</sup>(''x''<sub>1</sub>) can be [[canonical form\|canonically]] identified with ''X''<sub>2</sub> and there exists a Borel family of probability measures <math>\{ \mu_{x_{1}} \}_{x_{1} \in X_{1}}</math> in '''''P'''''(''X''<sub>2</sub>) (which is (π<sub>1</sub>)<sub>∗</sub>(μ)-almost everywhere uniquely determined) such that

			:<math>\mu = \int_{X_{1}} \mu_{x_{1}} \, \mu \left(\pi_1^{-1}(\mathrm d x_1) \right)= \int_{X_{1}} \mu_{x_{1}} \, \mathrm{d} (\pi_{1})_{*} (\mu) (x_{1}),</math>
			which is in particular
			:<math>\int_{X_1\times X_2} f(x_1,x_2)\, \mu(\mathrm d x_1,\mathrm d x_2) = \int_{X_1}\left( \int_{X_2} f(x_1,x_2) \mu(\mathrm d x_2\|x_1) \right) \mu\left( \pi_1^{-1}(\mathrm{d} x_{1})\right)</math>
			and
			:<math>\mu(A \times B) = \int_A \mu\left(B\|x_1\right) \, \mu\left( \pi_1^{-1}(\mathrm{d} x_{1})\right).</math>

			The relation to [[conditional expectation]] is given by the identities
			:<math>\operatorname E(f\|\pi_1)(x_1)= \int_{X_2} f(x_1,x_2) \mu(\mathrm d x_2\|x_1),</math>
			:<math>\mu(A\times B\|\pi_1)(x_1)= 1_A(x_1) \cdot \mu(B\| x_1).</math>

			===Vector calculus===
			The disintegration theorem can also be seen as justifying the use of a "restricted" measure in [[vector calculus]]. For instance, in [[Stokes' theorem]] as applied to a [[vector field]] flowing through a [[compact space\|compact]] [[surface]] Σ ⊂ '''R'''<sup>3</sup>, it is implicit that the "correct" measure on Σ is the disintegration of three-dimensional Lebesgue measure λ<sup>3</sup> on Σ, and that the disintegration of this measure on ∂Σ is the same as the disintegration of λ<sup>3</sup> on ∂Σ.<ref name=Ambrosio_Gigli_Savare>{{cite book \| author=Ambrosio, L., Gigli, N. & Savaré, G. \| title=Gradient Flows in Metric Spaces and in the Space of Probability Measures \| publisher=ETH Zürich, Birkhäuser Verlag, Basel \| year=2005 \| isbn=3-7643-2428-7 }}</ref>

			===Conditional distributions===
			The disintegration theorem can be applied to give a rigorous treatment of conditioning probability distributions in statistics, while avoiding purely abstract formulations of conditional probability.<ref name=Chang_Pollard>{{cite journal\|last=Chang\|first=J.T.\|coauthors=Pollard, D.\|title=Conditioning as disintegration\|journal=Statistica Neerlandica\| year=1997 \| volume=51\|issue=3\|url=http://www.stat.yale.edu/~jtc5/papers/ConditioningAsDisintegration.pdf\|doi=10.1111/1467-9574.00056\|pages=287}}</ref>

			==See also==
			*[[Joint probability distribution]]
			*[[Copula (statistics)]]
			*[[Conditional expectation]]

			==References==
			{{Reflist}}
			{{Use dmy dates\|date=December 2010}}

			[[Category:Theorems in measure theory]]
			[[Category:Probability theorems]]

66.31.177.106: /* Truncation operators */

2012-08-13T14:33:19Z

Truncation operators

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Wilber Berryhill is what his wife loves to contact him and he totally loves this title. For years she's been residing in Kentucky but her husband wants them to move. Invoicing is what I do for a living but I've usually wanted my personal business. To climb is some thing I really enjoy doing.<br><br>Here is my website; are psychics real ([http://srncomm.com/blog/2014/08/25/relieve-that-stress-find-a-new-hobby/ srncomm.com])