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<title>Suffiziente σ-Algebra</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Suffiziente σ-Algebra</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Eine suffiziente σ-Algebra ist ein spezielles Mengensystem in der <a href="Mathematische_Statistik" title="Mathematische Statistik">mathematischen Statistik</a>, das verwendet wird, um die Kompression von Daten ohne Informationsverlust mittels <a href="Suffiziente_Statistik" title="Suffiziente Statistik">suffizienter Statistiken</a> zu formalisieren.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

Gegeben sei ein <a href="Statistisches_Modell" title="Statistisches Modell">statistisches Modell</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {A}},{\mathcal {P}})}">

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<annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {A}},{\mathcal {P}})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38cfacb98ecdaf3e55a8d5f3060473578107ce91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.162ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {A}},{\mathcal {P}})}" loading="lazy"> sowie eine <a href="Teil-%CF%83-Algebra" class="mw-redirect" title="Teil-σ-Algebra">Teil-σ-Algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}\subset {\mathcal {A}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}\subset {\mathcal {A}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4101d8ce49e204d9a8ff668ee6e5f54a5ce6ef52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.494ex; height:2.343ex;" alt="{\displaystyle {\mathcal {S}}\subset {\mathcal {A}}}" loading="lazy">. Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} _{P}(\cdot |{\mathcal {S}})}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} _{P}(\cdot |{\mathcal {S}})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35b0b1c68545a9155e2c12bd1b9f54c67bb5083c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.645ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} _{P}(\cdot |{\mathcal {S}})}" loading="lazy"> der <a href="Bedingter_Erwartungswert" title="Bedingter Erwartungswert">bedingte Erwartungswert</a> gegeben <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> unter Verwendung des Wahrscheinlichkeitsmaßes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy">. Die σ-Algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> heißt dann suffizient für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10d6ec962de5797ba4f161c40e66dca74ae95cc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}" loading="lazy">, wenn für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathcal {A}}}">

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<annotation encoding="application/x-tex">{\displaystyle A\in {\mathcal {A}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6e69647797be244cf2ebc28ecd61fafba8790c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.487ex; height:2.343ex;" alt="{\displaystyle A\in {\mathcal {A}}}" loading="lazy"> eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy">-<a href="Messbare_Funktion" title="Messbare Funktion">messbare Funktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{A}}">

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<annotation encoding="application/x-tex">{\displaystyle f_{A}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4df88573ffd17201a3c77be3bee49037c3f9aa4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.604ex; height:2.509ex;" alt="{\displaystyle f_{A}}" loading="lazy"> existiert, so dass

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<annotation encoding="application/x-tex">{\displaystyle f_{A}=\operatorname {E} _{P}(\mathbf {1} _{A}|{\mathcal {S}}){\text{ für alle }}P\in {\mathcal {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46a0e20ded43415924a63898982725239f9e1494.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:30.062ex; height:3.343ex;" alt="{\displaystyle f_{A}=\operatorname {E} _{P}(\mathbf {1} _{A}|{\mathcal {S}}){\text{ für alle }}P\in {\mathcal {P}}}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Bemerkungen">Bemerkungen</h2></div>

Ein Defizit des Suffizienzbegriffes ist, dass wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}_{1},{\mathcal {S}}_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}_{1},{\mathcal {S}}_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eb201c855085df284abab915dca4457457acb997.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.96ex; height:2.509ex;" alt="{\displaystyle {\mathcal {S}}_{1},{\mathcal {S}}_{2}}" loading="lazy"> σ-Algebren sind mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}_{1}\subset {\mathcal {S}}_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}_{1}\subset {\mathcal {S}}_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9e611581e0a85a33c7beccd805821d6848ae2a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.024ex; height:2.509ex;" alt="{\displaystyle {\mathcal {S}}_{1}\subset {\mathcal {S}}_{2}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}_{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31dbf01a888be8dd4184319517189418a78a1a59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.463ex; height:2.509ex;" alt="{\displaystyle {\mathcal {S}}_{1}}" loading="lazy"> suffizient ist (bezüglich einer vorgegebenen Verteilungsklasse), dann folgt im Allgemeinen nicht, dass auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e0470c26679126290f674631a2022e5b71e3df7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.463ex; height:2.509ex;" alt="{\displaystyle {\mathcal {S}}_{2}}" loading="lazy"> suffizient ist. Das würde man aber intuitiv erwarten, denn wenn schon die kleinere σ-Algebra ausreichend ist, um eine verlustfreie Datenkompression zu ermöglichen, dann sollte dies ebenso für die größere gelten, da sie ja die kleinere enthält, in der alle Informationen von Belang schon vorhanden sind. Zu beachten ist hier, dass die Datenkompression hier dem Weglassen der Mengen aus der größeren σ-Algebra entspricht.

Formell lässt sich dieses Defizit wie folgt einsehen: ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}_{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31dbf01a888be8dd4184319517189418a78a1a59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.463ex; height:2.509ex;" alt="{\displaystyle {\mathcal {S}}_{1}}" loading="lazy"> suffizient, so gilt laut Definition des bedingten Erwartungswertes

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<annotation encoding="application/x-tex">{\displaystyle \int _{S}f_{A}\,\mathrm {d} P=P(A\cap S)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24fe235a5390360bc308781881538f00d5025904.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.479ex; height:5.676ex;" alt="{\displaystyle \int _{S}f_{A}\,\mathrm {d} P=P(A\cap S)}" loading="lazy"></dd></dl>

für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\in {\mathcal {S}}_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle S\in {\mathcal {S}}_{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/598e636d594b0ffe463a9626bcfcb19f5818330a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.803ex; height:2.509ex;" alt="{\displaystyle S\in {\mathcal {S}}_{1}}" loading="lazy">, aber eben nicht notwendigerweise für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\in {\mathcal {S}}_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle S\in {\mathcal {S}}_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7361fc16d9dd4199fc72ab9e3f758da48b8c3a1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.803ex; height:2.509ex;" alt="{\displaystyle S\in {\mathcal {S}}_{2}}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Erläuterung">Erläuterung</h2></div>

Klar wird die Bedeutung des Begriffes, wenn man die Wahrscheinlichkeitsmaße aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10d6ec962de5797ba4f161c40e66dca74ae95cc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> einschränkt. Dann gilt

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<annotation encoding="application/x-tex">{\displaystyle P(A)=\int P(A|{\mathcal {S}})\mathrm {d} P|_{\mathcal {S}}=\int f_{A}\mathrm {d} P|_{\mathcal {S}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35fcc4f10d0f2cdcf3664ce6eb7479f03f2dedf7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.642ex; height:5.676ex;" alt="{\displaystyle P(A)=\int P(A|{\mathcal {S}})\mathrm {d} P|_{\mathcal {S}}=\int f_{A}\mathrm {d} P|_{\mathcal {S}}}" loading="lazy">.</dd></dl>

Da aber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{A}}">

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<annotation encoding="application/x-tex">{\displaystyle f_{A}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4df88573ffd17201a3c77be3bee49037c3f9aa4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.604ex; height:2.509ex;" alt="{\displaystyle f_{A}}" loading="lazy"> nicht von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\in {\mathcal {P}}}">

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<annotation encoding="application/x-tex">{\displaystyle P\in {\mathcal {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb727fbfb3d66abc8c9a7e7b649323f87b85a98d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.29ex; height:2.176ex;" alt="{\displaystyle P\in {\mathcal {P}}}" loading="lazy"> abhängt, können sich die Wahrscheinlichkeitsmaße nur dann unterscheiden, wenn schon deren Einschränkungen auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> verschieden sind. Damit sind alle möglichen Informationen, welche die Wahrscheinlichkeitsmaße aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10d6ec962de5797ba4f161c40e66dca74ae95cc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}" loading="lazy"> liefern können, bereits in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> enthalten.

<div class="mw-heading mw-heading2"><h2 id="Stabilität_bezüglich_Operationen">Stabilität bezüglich Operationen</h2></div>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}^{*}\subset {\mathcal {S}}\subset {\mathcal {A}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4171fe09c187a0025360fbd6cf79ea9529bc2d4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.164ex; height:2.343ex;" alt="{\displaystyle {\mathcal {S}}^{*}\subset {\mathcal {S}}\subset {\mathcal {A}}}" loading="lazy"> und ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> suffizient für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10d6ec962de5797ba4f161c40e66dca74ae95cc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}" loading="lazy">, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}^{*}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b132e99314f544fb15ad2aa2dee2b225731c441.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.572ex; height:2.343ex;" alt="{\displaystyle {\mathcal {S}}^{*}}" loading="lazy"> genau dann suffizient für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {A}},{\mathcal {P}})}">

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<annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {A}},{\mathcal {P}})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38cfacb98ecdaf3e55a8d5f3060473578107ce91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.162ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {A}},{\mathcal {P}})}" loading="lazy">, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}^{*}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b132e99314f544fb15ad2aa2dee2b225731c441.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.572ex; height:2.343ex;" alt="{\displaystyle {\mathcal {S}}^{*}}" loading="lazy"> suffizient ist für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {S}},{\mathcal {P}}|_{\mathcal {S}})}">

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<annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {S}},{\mathcal {P}}|_{\mathcal {S}})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/78f953289a450c3f967299eed2ea1b5cf7d30324.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.685ex; height:3.009ex;" alt="{\displaystyle (\Omega ,{\mathcal {S}},{\mathcal {P}}|_{\mathcal {S}})}" loading="lazy">.</li>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}_{\mathcal {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e256a327961b142a39b750ec00a22a81d331eccf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.062ex; width:3.404ex; height:2.843ex;" alt="{\displaystyle {\mathcal {N}}_{\mathcal {P}}}" loading="lazy"> die Mengen aller <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10d6ec962de5797ba4f161c40e66dca74ae95cc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}" loading="lazy">-Nullmengen. Sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}_{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31dbf01a888be8dd4184319517189418a78a1a59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.463ex; height:2.509ex;" alt="{\displaystyle {\mathcal {S}}_{1}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e0470c26679126290f674631a2022e5b71e3df7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.463ex; height:2.509ex;" alt="{\displaystyle {\mathcal {S}}_{2}}" loading="lazy"> suffizient und ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}_{\mathcal {P}}\subset {\mathcal {S}}_{1}}">

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</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}_{\mathcal {P}}\subset {\mathcal {S}}_{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee26ccea14b01aad776bba57e813e663ebf57d4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.062ex; width:8.965ex; height:2.843ex;" alt="{\displaystyle {\mathcal {N}}_{\mathcal {P}}\subset {\mathcal {S}}_{1}}" loading="lazy">, so ist auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}_{1}\cap {\mathcal {S}}_{2}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}_{1}\cap {\mathcal {S}}_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59049d12e985838f48876b3e57edf8ed45d7d75f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.508ex; height:2.509ex;" alt="{\displaystyle {\mathcal {S}}_{1}\cap {\mathcal {S}}_{2}}" loading="lazy"> suffizient.</li>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> suffizient und ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c298ed828ff778065aeb5f0f305097f55bb9ae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.311ex; height:2.176ex;" alt="{\displaystyle {\mathcal {E}}}" loading="lazy"> eine <a href="Abz%C3%A4hlbar_erzeugte_%CF%83-Algebra" class="mw-redirect" title="Abzählbar erzeugte σ-Algebra">abzählbar erzeugte σ-Algebra</a>, so ist auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ({\mathcal {S}},{\mathcal {E}})}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma ({\mathcal {S}},{\mathcal {E}})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac1feefc8a7d6dfcade9e029962d588acf5d416c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.976ex; height:2.843ex;" alt="{\displaystyle \sigma ({\mathcal {S}},{\mathcal {E}})}" loading="lazy"> suffizient. Daraus folgt direkt, dass abzählbar erzeugte Ober-σ-Algebren von suffizienten σ-Algebren wieder suffizient sind.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Suffizienz_und_dominierte_Verteilungsklassen">Suffizienz und dominierte Verteilungsklassen</h2></div>

Mittels des <a href="Satz_von_Halmos-Savage" title="Satz von Halmos-Savage">Satzes von Halmos-Savage</a> lassen sich für <a href="Dominierte_Verteilungsklasse" title="Dominierte Verteilungsklasse">dominierte Verteilungsklassen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10d6ec962de5797ba4f161c40e66dca74ae95cc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}" loading="lazy"> einige stärkere Aussagen zeigen:

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> suffizient und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}\subset {\mathcal {A}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}\subset {\mathcal {A}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4101d8ce49e204d9a8ff668ee6e5f54a5ce6ef52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.494ex; height:2.343ex;" alt="{\displaystyle {\mathcal {S}}\subset {\mathcal {A}}}" loading="lazy">. Dann ist jede σ-Algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}^{*}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b132e99314f544fb15ad2aa2dee2b225731c441.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.572ex; height:2.343ex;" alt="{\displaystyle {\mathcal {S}}^{*}}" loading="lazy"> mit</li></ul>

</mrow>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}\subset {\mathcal {S}}^{*}\subset {\mathcal {A}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/95451fe18d215d560c12ae14042868c2ab998272.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.164ex; height:2.343ex;" alt="{\displaystyle {\mathcal {S}}\subset {\mathcal {S}}^{*}\subset {\mathcal {A}}}" loading="lazy"></dd></dl>

<dl><dd>ebenfalls suffizient.</dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> ist genau dann suffizient bezüglich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}">

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10d6ec962de5797ba4f161c40e66dca74ae95cc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}" loading="lazy">, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> suffizient bezüglich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}^{*}:=\{P_{i},P_{j}\}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}^{*}:=\{P_{i},P_{j}\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103fafcd80c5b0800c41af74647de58f024cc193.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.582ex; height:3.009ex;" alt="{\displaystyle {\mathcal {P}}^{*}:=\{P_{i},P_{j}\}}" loading="lazy"> ist für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i},P_{j}\in {\mathcal {P}}}">

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<annotation encoding="application/x-tex">{\displaystyle P_{i},P_{j}\in {\mathcal {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c13a09e7845b8071539cda65ea1d4741097ef8f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.272ex; height:2.843ex;" alt="{\displaystyle P_{i},P_{j}\in {\mathcal {P}}}" loading="lazy">.</li>

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<annotation encoding="application/x-tex">{\displaystyle i=1,2}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/608b3c5e448c465889913a88a105e38e7316fba7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.26ex; height:2.509ex;" alt="{\displaystyle i=1,2}" loading="lazy"> die Verteilungsklassen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/926699acbd8706e62b9393080b217fe89b3790c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.417ex; height:2.509ex;" alt="{\displaystyle {\mathcal {P}}_{i}}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega _{i},{\mathcal {A}}_{i})}">

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<annotation encoding="application/x-tex">{\displaystyle (\Omega _{i},{\mathcal {A}}_{i})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/94da0f1031128175c3e8bbd7f44b26660626b265.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.975ex; height:2.843ex;" alt="{\displaystyle (\Omega _{i},{\mathcal {A}}_{i})}" loading="lazy"> dominiert und ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}_{i}\subset {\mathcal {A}}_{i}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}_{i}\subset {\mathcal {A}}_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/075709ef7ca0c161a8359e399650d64bac6982ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.961ex; height:2.676ex;" alt="{\displaystyle {\mathcal {S}}_{i}\subset {\mathcal {A}}_{i}}" loading="lazy"> suffizient, so ist auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}_{1}\otimes {\mathcal {S}}_{2}}">

<msub>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}_{1}\otimes {\mathcal {S}}_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8be86357a0a71781a586be1ade83ce9ef41582da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.766ex; height:2.509ex;" alt="{\displaystyle {\mathcal {S}}_{1}\otimes {\mathcal {S}}_{2}}" loading="lazy"> suffizient bezüglich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}_{1}\otimes {\mathcal {P}}_{2}}">

<msub>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}_{1}\otimes {\mathcal {P}}_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5bc85d92d940f7cd6b527363a277b8e0de0cb892.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.184ex; height:2.509ex;" alt="{\displaystyle {\mathcal {P}}_{1}\otimes {\mathcal {P}}_{2}}" loading="lazy">.</li></ul>

Eine weitere Möglichkeit zur Überprüfung der Suffizienz einer σ-Algebra bei Vorliegen einer dominierten Verteilungsklasse ist das <a href="Neyman-Kriterium" title="Neyman-Kriterium">Neyman-Kriterium</a>.

<div class="mw-heading mw-heading2"><h2 id="Verwandte_Begriffe">Verwandte Begriffe</h2></div>

Der bekannteste Begriff, der sich aus der Suffizienz einer σ-Algebra ableiten lässt, ist die <a href="Suffiziente_Statistik" title="Suffiziente Statistik">suffiziente Statistik</a>. Eine Statistik <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> heißt suffizient, wenn die von ihr erzeugte σ-Algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (T)}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma (T)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9888a5496892a8c482ad465f48b839102187bd1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.775ex; height:2.843ex;" alt="{\displaystyle \sigma (T)}" loading="lazy"> suffizient ist.

Eine Abwandlung des hier behandelten Suffizienzbegriffes ist die <a href="Starke_Suffizienz" title="Starke Suffizienz">starke Suffizienz</a>, die mittels <a href="Markow-Kern" class="mw-redirect" title="Markow-Kern">Markow-Kernen</a> definiert wird. Auf <a href="Borel'scher_Raum" class="mw-redirect mw-disambig" title="Borel'scher Raum">borelschen Räumen</a> stimmen die beiden Begriffe überein. Eine Verstärkung der Suffizienz ist die <a href="Minimalsuffizienz" title="Minimalsuffizienz">Minimalsuffizienz</a>: eine σ-Algebra ist minimalsuffizient, wenn sie bis auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10d6ec962de5797ba4f161c40e66dca74ae95cc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}" loading="lazy">-Nullmengen in jeder weiteren suffizienten σ-Algebra enthalten ist. Demnach ist eine minimalsuffiziente σ-Algebra die maximal mögliche Datenreduktion.

Ein ebenfalls verwandter, aber gegenläufiger Begriff ist der einer <a href="Vollst%C3%A4ndigkeit_(Statistik)" title="Vollständigkeit (Statistik)">vollständigen Verteilungsklasse</a>. Dies ist eine Verteilungsklasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10d6ec962de5797ba4f161c40e66dca74ae95cc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}" loading="lazy">, so dass auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}(\Omega ,{\mathcal {S}},{\mathcal {P}})}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}(\Omega ,{\mathcal {S}},{\mathcal {P}})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/17c457a9b5ce2f925c76aac9466c337da104b268.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.355ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}(\Omega ,{\mathcal {S}},{\mathcal {P}})}" loading="lazy"> alle Funktionen unterschieden werden können.

Gegenteil des Suffizienzbegriffs ist die <a href="Verteilungsfreiheit" title="Verteilungsfreiheit">Verteilungsfreiheit</a>. Sie formalisiert, dass eine σ-Algebra keine Informationen trägt bzw. dass eine Statistik keine Informationen überträgt.

Eine Verbindung von Suffizienz, Vollständigkeit und Verteilungsfreiheit schlagen die drei <a href="S%C3%A4tze_von_Basu" title="Sätze von Basu">Sätze von Basu</a>.

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>Ludger Rüschendorf: <cite style="font-style:italic">Mathematische Statistik</cite>. Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-642-41997-3">10.1007/978-3-642-41997-3</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Suffiziente+%CF%83-Algebra&rft.au=Ludger+R%C3%BCschendorf&rft.btitle=Mathematische+Statistik&rft.date=2014&rft.doi=10.1007%2F978-3-642-41997-3&rft.genre=book&rft.isbn=9783642419966&rft.place=Berlin+Heidelberg&rft.pub=Springer+Verlag" style="display:none"> </li></ul></div><div><div class="zim-footer">

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