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<title>Maximum-Entropie-Methode</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Maximum-Entropie-Methode</span></h1>

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<p>Die <b>Maximum-Entropie-Methode</b> oder <b>MEM</b> ist eine Methode der <a href="Bayessche_Statistik" title="Bayessche Statistik">Bayesschen Statistik</a>, die erlaubt, trotz mangelhafter problemspezifischer Information eine <a href="A-priori-Wahrscheinlichkeit" title="A-priori-Wahrscheinlichkeit">A-priori-Wahrscheinlichkeit</a> zuzuweisen. Sie ersetzt frühere Ansätze, wie etwa das von <a href="Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Laplace</a> formulierte „<a href="Prinzip_vom_unzureichenden_Grund" class="mw-redirect" title="Prinzip vom unzureichenden Grund">Prinzip vom unzureichenden Grunde</a>“.

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<div class="mw-heading mw-heading2"><h2 id="Ursprung_und_Vorgehensweise">Ursprung und Vorgehensweise</h2></div>

<p>Die Methode wurde 1957 von <a href="Edwin_Thompson_Jaynes" title="Edwin Thompson Jaynes">Edwin Thompson Jaynes</a> in Anlehnung an Methoden der <a href="Statistische_Mechanik" title="Statistische Mechanik">statistischen Mechanik</a> und der <a href="Claude_Elwood_Shannon" class="mw-redirect" title="Claude Elwood Shannon">Shannonschen</a> <a href="Informationstheorie" title="Informationstheorie">Informationstheorie</a> eingeführt.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Die Grundidee des Maximum-Entropie-Verfahrens ist es, in Situationen ohne spezifische Informationen, die Unsicherheit der <a href="A-priori-Wahrscheinlichkeit" title="A-priori-Wahrscheinlichkeit">A-priori-Wahrscheinlichkeiten</a> so zu maximieren, dass keine willkürlichen Annahmen über die gegebene Situation gemacht werden müssen. Die Maximum-Entropie-Methode legt sich so wenig wie möglich fest. Jaynes zufolge<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> ist dies aber nur der letzte Schritt, um nach Einfüllen aller vorhandenen Information etwaige noch bestehende Lücken zu schließen.

</p><p>In der statistischen Physik bedeutet dies: „<i>Unter allen Zuständen eines physikalischen Systems, die kompatibel mit dem vorhandenen Wissen über das System sind, ist jener zu wählen, welcher die Entropie maximiert.</i>“

</p><p>Die Methode wird zur optimalen Extraktion von <a href="Information" title="Information">Information</a> aus <a href="Rauschen_(Physik)" title="Rauschen (Physik)">verrauschten</a> <a href="Signal" title="Signal">Signalen</a> in Abhängigkeit vom <a href="Signal-Rausch-Verh%C3%A4ltnis" title="Signal-Rausch-Verhältnis">Signal-Rausch-Verhältnis</a> verwendet. Sie findet auch in der <a href="Fourieranalyse" class="mw-redirect" title="Fourieranalyse">Spektralanalyse</a> und der digitalen <a href="Bildverarbeitung" title="Bildverarbeitung">Bildverarbeitung</a> Anwendung.

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

<p><a href="Entropie_(Informationstheorie)" title="Entropie (Informationstheorie)">Entropie</a> ist ein Maß für den <a href="Informationsgehalt" title="Informationsgehalt">Informationsgehalt</a> einer <a href="Zufallsvariable" title="Zufallsvariable">Zufallsvariable</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>. Ein weniger wahrscheinliches Ergebnis vermittelt mehr Informationen als ein wahrscheinlicheres Ergebnis. Entropie ist also ein Maß für die Unsicherheit eines Ergebnisses. Wenn das Ziel darin besteht, eine möglichst „unwissende“ <a href="Wahrscheinlichkeitsverteilung" class="mw-redirect" title="Wahrscheinlichkeitsverteilung">Wahrscheinlichkeitsverteilung</a> zu finden, sollte die Entropie folglich maximal sein. Formal ist Entropie wie folgt definiert:

</p><p>Wenn <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> eine <a href="Diskrete_Zufallsvariable" class="mw-redirect" title="Diskrete Zufallsvariable">diskrete Zufallsvariable</a> mit der <a href="Wahrscheinlichkeitsverteilung" class="mw-redirect" title="Wahrscheinlichkeitsverteilung">Wahrscheinlichkeitsverteilung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X=x_{i})=p_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle P(X=x_{i})=p_{i}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d37f7dae0906fe81cc750127acabaa88646a8263.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.83ex; height:2.843ex;" alt="{\displaystyle P(X=x_{i})=p_{i}}" loading="lazy"></span> ist, dann ist die <a href="Entropie_(Informationstheorie)" title="Entropie (Informationstheorie)">Entropie</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> definiert als

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {H} (X)=-\sum _{i}p_{i}\log _{2}p_{i}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4219478523066864b5061161beb60c9d7748a85f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.307ex; height:5.509ex;" alt="{\displaystyle \mathrm {H} (X)=-\sum _{i}p_{i}\log _{2}p_{i}}" loading="lazy"></span></dd></dl>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> eine <a href="Stetige_Zufallsvariable" class="mw-redirect" title="Stetige Zufallsvariable">stetige Zufallsvariable</a> mit der <a href="Wahrscheinlichkeitsdichtefunktion" title="Wahrscheinlichkeitsdichtefunktion">Wahrscheinlichkeitsdichte</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8cb7afced134ef75572e5314a5d278c2d644f438.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.398ex; height:2.843ex;" alt="{\displaystyle p(x)}" loading="lazy"></span> ist, dann ist die <a href="Differentielle_Entropie" title="Differentielle Entropie">differentielle Entropie</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> definiert als<sup id="cite_ref-:0_3-0" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {H} (X)=-\int _{-\infty }^{+\infty }p(x)\log p(x)\,\mathrm {d} x}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/335e9dd22ba9b29f4db70c8236f93529ce59e5f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.203ex; height:6.176ex;" alt="{\displaystyle \mathrm {H} (X)=-\int _{-\infty }^{+\infty }p(x)\log p(x)\,\mathrm {d} x}" loading="lazy"></span></dd></dl>

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

<p>Ein Ansatz zur Berechnung der <a href="Entropie_(Informationstheorie)" title="Entropie (Informationstheorie)">Entropie</a> wurde von Graham Wallis vorgeschlagen. Es sind Informationen gegeben, die <a href="Wahrscheinlichkeit" title="Wahrscheinlichkeit">Wahrscheinlichkeiten</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1},\ldots ,p_{m}}">

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<annotation encoding="application/x-tex">{\displaystyle p_{1},\ldots ,p_{m}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92e43e4cbe8115165ca3559393f90caee2e86a75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.336ex; height:2.009ex;" alt="{\displaystyle p_{1},\ldots ,p_{m}}" loading="lazy"></span> verschiedenen <a href="Zufallsvariable" title="Zufallsvariable">Zufallsvariablen</a> zuweisen. Die Gesamtwahrscheinlichkeit wird unter ihnen aufgeteilt, also gilt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{m}p_{i}=1}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{m}p_{i}=1}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1e97b8ea32a6a56d44c469047f91a8829a66693a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:9.972ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{m}p_{i}=1}" loading="lazy"></span>.

</p><p>Wählt man einige <a href="Ganze_Zahl" title="Ganze Zahl">ganze Zahlen</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>, die wesentlich größer als <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span> sind, und nimmt an, man hat <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> kleine Mengen von <a href="Wahrscheinlichkeit" title="Wahrscheinlichkeit">Wahrscheinlichkeiten</a>, jeweils von der Größe <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ={\tfrac {1}{n}}}">

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</mfrac>

</mstyle>

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \delta ={\tfrac {1}{n}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0daa3a6d285a7fd14bc228c30ccf775fbeeb05fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.969ex; height:3.343ex;" alt="{\displaystyle \delta ={\tfrac {1}{n}}}" loading="lazy"></span>, um sie auf richtige Weise zu verteilen. Angenommen, man soll diese Mengen unter <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span> Möglichkeiten zufällig verteilen. Wenn man diese Wahrscheinlichkeiten so verteilt, dass jede Box die gleiche Wahrscheinlichkeit hat, erhält man, dass das <a href="Zufallsexperiment" title="Zufallsexperiment">Zufallsexperiment</a> folgende Wahrscheinlichkeiten hat: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}=n_{i}\cdot \delta ={\tfrac {n_{i}}{n}}}">

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<annotation encoding="application/x-tex">{\displaystyle p_{i}=n_{i}\cdot \delta ={\tfrac {n_{i}}{n}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/29755b3032b795541d5607d4be6af1cd4b462856.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:15.625ex; height:3.343ex;" alt="{\displaystyle p_{i}=n_{i}\cdot \delta ={\tfrac {n_{i}}{n}}}" loading="lazy"></span>. Die Wahrscheinlichkeit, dass dies geschehen wird, ist die <a href="Multinomialverteilung" title="Multinomialverteilung">Multinomialverteilung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {n!}{n_{1}!\cdot \dots \cdots n_{m}!}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {n!}{n_{1}!\cdot \dots \cdots n_{m}!}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b1150d2754928a2299a04c6790e42714ab2cbd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.548ex; height:5.843ex;" alt="{\displaystyle {\frac {n!}{n_{1}!\cdot \dots \cdots n_{m}!}}}" loading="lazy"></span>.

</p><p>Für große <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> folgt aus der <a href="Stirlingformel" title="Stirlingformel">Stirlingformel</a>

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<annotation encoding="application/x-tex">{\displaystyle n!\sim {\sqrt {2\pi n}}\;\left({\frac {n}{\mathcal {e}}}\right)^{n}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/69691a2256e8693f0dd73ae8ebea7d11fd9a4d86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.835ex; height:4.843ex;" alt="{\displaystyle n!\sim {\sqrt {2\pi n}}\;\left({\frac {n}{\mathcal {e}}}\right)^{n}}" loading="lazy"></span></dd></dl>

<p><a href="Logarithmieren" class="mw-redirect" title="Logarithmieren">Logarithmieren</a> ergibt

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<annotation encoding="application/x-tex">{\displaystyle \log(n!)\sim \log \left({\sqrt {2\pi n}}\right)+n\log \left({\tfrac {n}{e}}\right)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2daca0ca5560850656d7b27ba133c114888d5e01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.394ex; height:3.343ex;" alt="{\displaystyle \log(n!)\sim \log \left({\sqrt {2\pi n}}\right)+n\log \left({\tfrac {n}{e}}\right)}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle \log(n!)\sim \log \left({\sqrt {2\pi n}}\right)+n\log(n)-n}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2235b06c97c230b7ec3e25a6721e51de71ffaab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.881ex; height:3.343ex;" alt="{\displaystyle \log(n!)\sim \log \left({\sqrt {2\pi n}}\right)+n\log(n)-n}" loading="lazy"></span></dd></dl>

<p>Nimmt man den Logarithmus von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> und ersetzt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log(n!)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \log(n!)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0f429974f9df24d255f098a432b0c4f1cde90a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.823ex; height:2.843ex;" alt="{\displaystyle \log(n!)}" loading="lazy"></span> durch die Näherung der <a href="Stirlingformel" title="Stirlingformel">Stirlingformel</a>, erhält man schließlich die Definition der <a href="Informationsentropie" class="mw-redirect" title="Informationsentropie">Informationsentropie</a>, wie sie durch den <a href="Satz_von_Shannon" class="mw-redirect" title="Satz von Shannon">Satz von Shannon</a> abgeleitet wird:<sup id="cite_ref-:0_3-1" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\sum _{i=1}^{m}p_{i}\log _{2}p_{i}=\mathrm {H} (p_{1},\ldots ,p_{m})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0e5a96c5a06adceeccb218de1b58f8e31b6e083a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.573ex; height:6.843ex;" alt="{\displaystyle -\sum _{i=1}^{m}p_{i}\log _{2}p_{i}=\mathrm {H} (p_{1},\ldots ,p_{m})}" loading="lazy"></span></dd></dl>

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

<p>Anstatt die Einschränkungsgleichungen zu verwenden, um die Anzahl der Unbekannten zu verringern, kann man die Anzahl der Unbekannten erhöhen. Man definiert die <a href="Lagrange-Multiplikator" title="Lagrange-Multiplikator">Lagrange-Multiplikatoren</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"></span> und dann die Lagrange-Funktion

</p>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L=S-(\alpha -\log _{2}e)\left(\sum _{i}p(A_{i})-1\right)-\beta \left(\sum _{i}g(A_{i})p(A_{i})-G\right)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ba24617700b9016d23f96e85249a1e4db000577.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:65.629ex; height:7.509ex;" alt="{\displaystyle L=S-(\alpha -\log _{2}e)\left(\sum _{i}p(A_{i})-1\right)-\beta \left(\sum _{i}g(A_{i})p(A_{i})-G\right)}" loading="lazy"></span></dd></dl>

<p>wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{2}e\approx 1{,}4427}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \log _{2}e\approx 1{,}4427}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5605dfd5681a53b6c75adc265d4f84f7acc392d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.054ex; height:2.676ex;" alt="{\displaystyle \log _{2}e\approx 1{,}4427}" loading="lazy"></span>. Der Lagrange-Multiplikator <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> wird wie <a href="Entropie_(Informationstheorie)" title="Entropie (Informationstheorie)">Entropie</a> in <a href="Bit" title="Bit">Bit</a> gemessen, und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"></span> wird in Bit pro Einheit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> gemessen. Wenn <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"></span> in Joule pro Kelvin ausgedrückt wird und <a href="Nat%C3%BCrlicher_Logarithmus" class="mw-redirect" title="Natürlicher Logarithmus">natürliche Logarithmen</a> in der Entropiedefinition verwendet werden, ist die Formel für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span> etwas anders:

</p>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L=S-k_{B}(\alpha -1)\left(\sum _{i}p(A_{i})-1\right)-k_{B}\beta \left(\sum _{i}g(A_{i})p(A_{i})-G\right)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5e68cb7a12cad91bc02b1704565ad96fcc6668b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:66.677ex; height:7.509ex;" alt="{\displaystyle L=S-k_{B}(\alpha -1)\left(\sum _{i}p(A_{i})-1\right)-k_{B}\beta \left(\sum _{i}g(A_{i})p(A_{i})-G\right)}" loading="lazy"></span></dd></dl>

<p>und die Einheiten für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"></span> sind nicht mehr in Bits: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> ist dimensionslos und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"></span> wird mit dem Inversen der Einheiten von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> ausgedrückt.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Anwendungen_in_den_Wirtschaftswissenschaften">Anwendungen in den Wirtschaftswissenschaften</h2></div>

<p>Ein relativ neues Anwendungsgebiet der MEM stellt die Makroökonomik dar. Im Rahmen der <a href="%C3%96konophysik" title="Ökonophysik">ökonophysikalischen Strömung</a>, die abseits des wirtschaftswissenschaftlichen Mainstreams verschiedene Methoden der statistischen Mechanik auf die Modellierung der Wirtschaft anwendet, kam es zur Verwendung der MEM.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Anwendungen_in_der_Ökologie"><span id="Anwendungen_in_der_.C3.96kologie"></span>Anwendungen in der Ökologie</h2></div>

<p>In der <a href="Biogeographie" title="Biogeographie">Biogeographie</a> wird die Maximum-Entropie-Methode zur <a href="Modell" title="Modell">Modellierung</a> von <a href="Verbreitungsgebiet" title="Verbreitungsgebiet">Verbreitungsgebieten</a> verwendet. Ein Beispiel dafür ist die Software <a href="Maxent_(Software)" title="Maxent (Software)">Maxent</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p>

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

<ul><li>Edwin Thompson Jaynes: <cite style="font-style:italic">Information Theory and Statistical Mechanics</cite>. In: <cite style="font-style:italic">The Physical Review</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>106</span>, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>4</span>, 15. Mai 1957, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>620–630</span> (<a rel="nofollow" class="external text" href="http://bayes.wustl.edu/etj/articles/theory.1.pdf">bayes.wustl.edu</a> [PDF]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Maximum-Entropie-Methode&rft.atitle=Information+Theory+and+Statistical+Mechanics&rft.au=Edwin+Thompson+Jaynes&rft.date=1957-05-15&rft.genre=journal&rft.issue=4&rft.jtitle=The+Physical+Review&rft.pages=620-630&rft.volume=106" style="display:none"> </span></li>

<li>Nailong Wu: <cite style="font-style:italic">The Maximum Entropy Method</cite>. Springer, Berlin 1997, ISBN 978-3-540-61965-9.<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:Maximum-Entropie-Methode&rft.au=Nailong+Wu&rft.btitle=The+Maximum+Entropy+Method&rft.date=1997&rft.genre=book&rft.isbn=9783540619659&rft.place=Berlin&rft.pub=Springer" style="display:none"> </span></li></ul>

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

<ul><li><a rel="nofollow" class="external text" href="http://cmm.cit.nih.gov/maxent/">Eine Einführung (auf Englisch)</a></li>

<li><a rel="nofollow" class="external text" href="http://aips2.nrao.edu/docs/glossary/m.html#Maximum_Entropy_Method">Kurzerklärung (auf Englisch)</a></li></ul>

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

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Edwin Thompson Jaynes: <cite style="font-style:italic">Information Theory and Statistical Mechanics</cite>. In: <cite style="font-style:italic">The Physical Review</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>106</span>, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>4</span>, 15. Mai 1957, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>620–630</span> (<a rel="nofollow" class="external text" href="http://bayes.wustl.edu/etj/articles/theory.1.pdf">bayes.wustl.edu</a> [PDF]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Maximum-Entropie-Methode&rft.atitle=Information+Theory+and+Statistical+Mechanics&rft.au=Edwin+Thompson+Jaynes&rft.date=1957-05-15&rft.genre=journal&rft.issue=4&rft.jtitle=The+Physical+Review&rft.pages=620-630&rft.volume=106" style="display:none"> </span></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Persi Diaconis: <cite style="font-style:italic">A Frequentist Does This, A Bayesian That</cite>. In: <cite style="font-style:italic">SIAM News</cite>. 13. März 2004 (<a rel="nofollow" class="external text" href="http://www.siam.org/news/news.php?id=81">volltext</a> [abgerufen am 28. Dezember 2007]).<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:Maximum-Entropie-Methode&rft.atitle=A+Frequentist+Does+This%2C+A+Bayesian+That&rft.au=Persi+Diaconis&rft.btitle=SIAM+News&rft.date=2004-03-13&rft.genre=book" style="display:none"> </span></span>

</li>

<li id="cite_note-:0-3"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-:0_3-0">a</a></sup> <sup><a href="#cite_ref-:0_3-1">b</a></sup></span> <span class="reference-text">Michael Franke: <a rel="nofollow" class="external text" href="https://michael-franke.github.io/intro-data-analysis/the-maximum-entropy-principle.html">The Maximum Entropy Principle</a></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">Massachusetts Institute of Technology: <a rel="nofollow" class="external text" href="https://mtlsites.mit.edu/Courses/6.050/2003/notes/chapter10.pdf">Principle of Maximum Entropy</a></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Duncan K. Foley: <i><style data-mw-deduplicate="TemplateStyles:r261891140">

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</style><a rel="nofollow" class="external text" href="https://web.archive.org/web/20060908212638/http://homepage.newschool.edu/~foleyd/stateqnotes.pdf">Statistical Equilibrium in Economics: Method, Interpretation, and an Example</a> (<span class="webarchiv-memento"><a href="Webarchivierung#Begrifflichkeiten" title="Webarchivierung">Memento</a></span> vom 8. September 2006 im </i><a href="Internet_Archive" title="Internet Archive">Internet Archive</a><i>)</i> In: <i>XII Workshop on „General Equilibrium: Problems, Prospects and Alternatives“</i> 07-1999 New School University, New York.</span>

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<li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text">Steven J. Phillips, Miroslav Dudík, Robert E. Schapire (2006): <i>Maximum entropy modeling of species geographic distributions</i>. Ecological Modelling 190, 231-259. <a rel="nofollow" class="external text" href="http://www.cs.princeton.edu/~schapire/papers/ecolmod.pdf">pdf</a></span>

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