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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Backward-Algorithmus</span></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"><p>Der <b>Backward-Algorithmus</b> (auch Rückwärts-Algorithmus, Rückwärts-Prozedur) berechnet mit Hilfe von <b>Backward-Variablen</b> die <a href="Wahrscheinlichkeit" title="Wahrscheinlichkeit">Wahrscheinlichkeit</a>, in einem gegebenen <a href="Hidden-Markov-Modell" class="mw-redirect" title="Hidden-Markov-Modell">Hidden-Markov-Modell</a> (HMM) eine bestimmte Symbolsequenz zu beobachten.
Der Algorithmus verwendet die Programmiermethode der <a href="Dynamische_Programmierung" title="Dynamische Programmierung">dynamischen Programmierung</a>.
</p>

<div class="mw-heading mw-heading2"><h2 id="Markov-Modell">Markov-Modell</h2></div>
<p>Gegeben sei ein HMM <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 \lambda =(S;V;A;B;\pi )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>λ<!-- λ --></mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>S</mi>
<mo>;</mo>
<mi>V</mi>
<mo>;</mo>
<mi>A</mi>
<mo>;</mo>
<mi>B</mi>
<mo>;</mo>
<mi>π<!-- π --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda =(S;V;A;B;\pi )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c0a1e063a0203eeba91649504852be8cd859748.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.524ex; height:2.843ex;" alt="{\displaystyle \lambda =(S;V;A;B;\pi )}" loading="lazy"></span>, wobei
</p>
<ul><li><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
</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> die Menge der verborgenen Zustände,</li>
<li><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 V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> das <a href="Alphabet_(Informatik)" title="Alphabet (Informatik)">Alphabet</a> der beobachtbaren Symbole,</li>
<li><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 A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> die <a href="%C3%9Cbergangsmatrix" title="Übergangsmatrix">Übergangsmatrix</a>,</li>
<li><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 B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span> die Matrix der Emissionswahrscheinlichkeiten,</li>
<li><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 \pi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>π<!-- π --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"></span> die Anfangswahrscheinlichkeitsverteilung für die möglichen Anfangszustände,</li></ul>
<p>bezeichnet.
</p>
<div class="mw-heading mw-heading2"><h2 id="Aufgabenstellung_und_Backward-Variablen">Aufgabenstellung und Backward-Variablen</h2></div>
<p>Gegeben sei ein <a href="Wort_(Theoretische_Informatik)" class="mw-redirect" title="Wort (Theoretische Informatik)">Wort</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 {\boldsymbol {o}}=o_{1}o_{2}\dots o_{T}\in V^{*}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">o</mi>
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<mo>=</mo>
<msub>
<mi>o</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>o</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>…<!-- … --></mo>
<msub>
<mi>o</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<msup>
<mi>V</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
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</msup>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {o}}=o_{1}o_{2}\dots o_{T}\in V^{*}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5e6941a5773976db57f6601f589e8691a6f7f5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.648ex; height:2.676ex;" alt="{\displaystyle {\boldsymbol {o}}=o_{1}o_{2}\dots o_{T}\in V^{*}}" loading="lazy"></span>. Der Backward-Algorithmus berechnet nun <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({\boldsymbol {o}}|\lambda )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">o</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>λ<!-- λ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P({\boldsymbol {o}}|\lambda )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/778bda8a7a5c8f76557ea67236098d9364ed6ef7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.917ex; height:2.843ex;" alt="{\displaystyle P({\boldsymbol {o}}|\lambda )}" loading="lazy"></span>, also die Wahrscheinlichkeit, im vorhandenen Modell <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 \lambda }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>λ<!-- λ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span> tatsächlich die Beobachtung <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 {\boldsymbol {o}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">o</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {o}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/516a3553708e67304ca0f2f35e765f5e8b84bc2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.36ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {o}}}" loading="lazy"></span> zu machen.
</p><p>Dafür werden die <i>Backward-Variablen</i> <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 _{t}(i)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>i</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta _{t}(i)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a4715b46aa379ed37b3fe974dc7af79ab139898.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.753ex; height:2.843ex;" alt="{\displaystyle \beta _{t}(i)}" loading="lazy"></span> verwendet, sie bezeichnen die Wahrscheinlichkeit, das <a href="Wort_(Theoretische_Informatik)#Suffix" class="mw-redirect" title="Wort (Theoretische Informatik)">Suffix</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 o_{t+1}o_{t+2}\ldots o_{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>o</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>o</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msub>
<mo>…<!-- … --></mo>
<msub>
<mi>o</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle o_{t+1}o_{t+2}\ldots o_{T}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47b0e72e065fa0b65c3d5f7e938845d1cd95baa0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.123ex; height:2.009ex;" alt="{\displaystyle o_{t+1}o_{t+2}\ldots o_{T}}" loading="lazy"></span> zu beobachten, falls das HMM zum Zeitpunkt <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 1\leq t\leq T}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
<mo>≤<!-- ≤ --></mo>
<mi>t</mi>
<mo>≤<!-- ≤ --></mo>
<mi>T</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1\leq t\leq T}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/234966dc61d2f4dca4cb2efd5ca4d5737ec0c991.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.835ex; height:2.343ex;" alt="{\displaystyle 1\leq t\leq T}" loading="lazy"></span> im Zustand <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_{i}\in S}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>s</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<mi>S</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s_{i}\in S}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87a8178559b9f271a78c8287257e81fd924faef5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.23ex; height:2.509ex;" alt="{\displaystyle s_{i}\in S}" loading="lazy"></span> gewesen ist:
</p>
<dl><dd><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 _{t}(i)=P(o_{t+1}o_{t+2}\dotsc o_{T}|q_{t}=s_{i};\lambda )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>i</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>P</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>o</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>o</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo>+</mo>
<mn>2</mn>
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</msub>
<mo>…<!-- … --></mo>
<msub>
<mi>o</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>s</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>;</mo>
<mi>λ<!-- λ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta _{t}(i)=P(o_{t+1}o_{t+2}\dotsc o_{T}|q_{t}=s_{i};\lambda )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3149f85da74c7f6b316cbceff7697234065a4efd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.417ex; height:2.843ex;" alt="{\displaystyle \beta _{t}(i)=P(o_{t+1}o_{t+2}\dotsc o_{T}|q_{t}=s_{i};\lambda )}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Algorithmus">Algorithmus</h2></div>
<p>Die Backward-Variablen werden rekursiv bestimmt:
</p>
<dl><dt>Initialisierung</dt>
<dd><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 _{T}(i)=1,\qquad 1\leq i\leq \left|S\right|}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>i</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mspace width="2em"></mspace>
<mn>1</mn>
<mo>≤<!-- ≤ --></mo>
<mi>i</mi>
<mo>≤<!-- ≤ --></mo>
<mrow>
<mo>|</mo>
<mi>S</mi>
<mo>|</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta _{T}(i)=1,\qquad 1\leq i\leq \left|S\right|}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6457f11a7af280616e4f0a6bafe04dc7ad52a216.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.212ex; height:2.843ex;" alt="{\displaystyle \beta _{T}(i)=1,\qquad 1\leq i\leq \left|S\right|}" loading="lazy"></span></dd></dl>
<dl><dt>Rekursion</dt>
<dd><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 _{t}(i)=\sum _{j=1}^{\left|S\right|}b_{j}(o_{t+1})\cdot a_{ij}\cdot \beta _{t+1}(j),\qquad 1\leq i\leq \left|S\right|,\ 1\leq t<T}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>i</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>|</mo>
<mi>S</mi>
<mo>|</mo>
</mrow>
</mrow>
</munderover>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<msub>
<mi>o</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>j</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mspace width="2em"></mspace>
<mn>1</mn>
<mo>≤<!-- ≤ --></mo>
<mi>i</mi>
<mo>≤<!-- ≤ --></mo>
<mrow>
<mo>|</mo>
<mi>S</mi>
<mo>|</mo>
</mrow>
<mo>,</mo>
<mtext>&nbsp;</mtext>
<mn>1</mn>
<mo>≤<!-- ≤ --></mo>
<mi>t</mi>
<mo>&lt;</mo>
<mi>T</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta _{t}(i)=\sum _{j=1}^{\left|S\right|}b_{j}(o_{t+1})\cdot a_{ij}\cdot \beta _{t+1}(j),\qquad 1\leq i\leq \left|S\right|,\ 1\leq t&lt;T}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b052d285d5b3257e87e9ea4ebf1350792a2b9dbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:60.91ex; height:8.009ex;" alt="{\displaystyle \beta _{t}(i)=\sum _{j=1}^{\left|S\right|}b_{j}(o_{t+1})\cdot a_{ij}\cdot \beta _{t+1}(j),\qquad 1\leq i\leq \left|S\right|,\ 1\leq t<T}" loading="lazy"></span></dd></dl>
<dl><dt>Termination</dt>
<dd><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({\boldsymbol {o}}|\lambda )=\sum _{j=1}^{\left|S\right|}\pi _{j}\cdot b_{j}(o_{1})\cdot \beta _{1}(j)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">o</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>λ<!-- λ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>|</mo>
<mi>S</mi>
<mo>|</mo>
</mrow>
</mrow>
</munderover>
<msub>
<mi>π<!-- π --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<msub>
<mi>o</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>j</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P({\boldsymbol {o}}|\lambda )=\sum _{j=1}^{\left|S\right|}\pi _{j}\cdot b_{j}(o_{1})\cdot \beta _{1}(j)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d4140cbefb4678581e66a5065579f244eda4982f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:30.386ex; height:8.009ex;" alt="{\displaystyle P({\boldsymbol {o}}|\lambda )=\sum _{j=1}^{\left|S\right|}\pi _{j}\cdot b_{j}(o_{1})\cdot \beta _{1}(j)}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Komplexität"><span id="Komplexit.C3.A4t"></span>Komplexität</h2></div>
<p>Die Matrix <i>aller</i> Backward-Variablen braucht <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 O(|S|\cdot T)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>O</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>⋅<!-- ⋅ --></mo>
<mi>T</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle O(|S|\cdot T)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23903747f743a28d8b57afcc499e6e91be145e3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.691ex; height:2.843ex;" alt="{\displaystyle O(|S|\cdot T)}" loading="lazy"></span> Speicher, werden die Zwischenergebnisse im Anschluss nicht mehr verwendet, so reduziert sich der Platzbedarf auf <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 O(|S|)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>O</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle O(|S|)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47a605795e87b54fd697a78fd16c2e9517e3db01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.376ex; height:2.843ex;" alt="{\displaystyle O(|S|)}" loading="lazy"></span>, da nur mehr zwei Spalten der Länge <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|}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |S|}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/28d901e98a035ff4c0e37fe6dd8e750ece6c1f0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.793ex; height:2.843ex;" alt="{\displaystyle |S|}" loading="lazy"></span> benötigt werden, um die Werte 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 \beta _{t+1}(i)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>i</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta _{t+1}(i)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ba9626c8126314494d593ee2097fedd11e5df43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.854ex; height:2.843ex;" alt="{\displaystyle \beta _{t+1}(i)}" 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 _{t}(i)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>i</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta _{t}(i)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a4715b46aa379ed37b3fe974dc7af79ab139898.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.753ex; height:2.843ex;" alt="{\displaystyle \beta _{t}(i)}" loading="lazy"></span> in jedem Rekursionsschritt zu speichern.
</p><p>Für jede einzelne Variable wird über <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|}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |S|}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/28d901e98a035ff4c0e37fe6dd8e750ece6c1f0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.793ex; height:2.843ex;" alt="{\displaystyle |S|}" loading="lazy"></span> Zeilen summiert, also liegt die Laufzeit in <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 O(|S|^{2}\cdot T)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>O</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>S</mi>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>⋅<!-- ⋅ --></mo>
<mi>T</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle O(|S|^{2}\cdot T)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5fcfc23441a169454e26857e90be8e64b0d77065.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.745ex; height:3.343ex;" alt="{\displaystyle O(|S|^{2}\cdot T)}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Weitere_Anwendungen">Weitere Anwendungen</h2></div>
<p>Die Backward-Variablen <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 _{t}(i)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>i</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta _{t}(i)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a4715b46aa379ed37b3fe974dc7af79ab139898.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.753ex; height:2.843ex;" alt="{\displaystyle \beta _{t}(i)}" loading="lazy"></span> werden zusammen mit den <i><a href="Forward-Algorithmus#Aufgabenstellung_und_Forward-Variablen" title="Forward-Algorithmus">Forward-Variablen</a></i> <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 _{t}(i)=P(o_{1},o_{2},\ldots ,o_{t},q_{t}=s_{i}|\lambda )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>α<!-- α --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>i</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>P</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>o</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>o</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>o</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>s</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>λ<!-- λ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha _{t}(i)=P(o_{1},o_{2},\ldots ,o_{t},q_{t}=s_{i}|\lambda )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/25eec70a36306bba772dc7db544be824e8165975.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.996ex; height:2.843ex;" alt="{\displaystyle \alpha _{t}(i)=P(o_{1},o_{2},\ldots ,o_{t},q_{t}=s_{i}|\lambda )}" loading="lazy"></span> für den <a href="Baum-Welch-Algorithmus" title="Baum-Welch-Algorithmus">Baum-Welch-Algorithmus</a> zur Lösung des mit Hidden-Markov-Modellen gegebenen Lernproblems benötigt.
</p><p>Außerdem ermöglicht deren Kenntnis die Bestimmung der Wahrscheinlichkeit bei der Beobachtung 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 {\boldsymbol {o}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">o</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {o}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/516a3553708e67304ca0f2f35e765f5e8b84bc2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.36ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {o}}}" loading="lazy"></span> zu einem <i>festen</i> Zeitpunkt <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 t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span> im Zustand <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_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>s</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s_{i}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cfda82668232cbdc0874ed28ab8b6079420d1ffe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.89ex; height:2.009ex;" alt="{\displaystyle s_{i}}" loading="lazy"></span> gewesen zu sein, denn nach dem <a href="Satz_von_Bayes" title="Satz von Bayes">Satz von Bayes</a> gilt:
</p>
<dl><dd><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(q_{t}=s_{i}|{\boldsymbol {o}};\lambda )={\frac {\alpha _{t}(i)\cdot \beta _{t}(i)}{P({\boldsymbol {o}}|\lambda )}}}">
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<annotation encoding="application/x-tex">{\displaystyle P(q_{t}=s_{i}|{\boldsymbol {o}};\lambda )={\frac {\alpha _{t}(i)\cdot \beta _{t}(i)}{P({\boldsymbol {o}}|\lambda )}}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bded085071f6281655a4d47c46bdae02b31b8730.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.095ex; height:6.509ex;" alt="{\displaystyle P(q_{t}=s_{i}|{\boldsymbol {o}};\lambda )={\frac {\alpha _{t}(i)\cdot \beta _{t}(i)}{P({\boldsymbol {o}}|\lambda )}}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Forward-Algorithmus" title="Forward-Algorithmus">Forward-Algorithmus</a></li>
<li><a href="Viterbi-Algorithmus" title="Viterbi-Algorithmus">Viterbi-Algorithmus</a></li>
<li><a href="Baum-Welch-Algorithmus" title="Baum-Welch-Algorithmus">Baum-Welch-Algorithmus</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>Richard Durbin, Sean R. Eddy, Anders Krogh, Graeme Mitchison: <cite style="font-style:italic">Biological sequence analysis. Probabilistic models of proteins and nucleic acids</cite>. 11th printing, corrected 10th reprinting. Cambridge University Press, Cambridge u. a. 2006, ISBN 0-521-62971-3, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em">&nbsp;</span>59–60</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rfr_id=info:sid/de.wikipedia.org:Backward-Algorithmus&amp;rft.au=Richard+Durbin%2C+Sean+R.+Eddy%2C+Anders+Krogh%2C+...&amp;rft.btitle=Biological+sequence+analysis.+Probabilistic+models+of+proteins+and+nucleic+acids&amp;rft.date=2006&amp;rft.genre=book&amp;rft.isbn=0521629713&amp;rft.pages=59-60&amp;rft.place=Cambridge+u.+a.&amp;rft.pub=11th+printing%2C+corrected+10th+reprinting.+Cambridge+University+Press" style="display:none">&nbsp;</span></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>
<ul><li>Ernst G. Schukat-Talamazzini: <a rel="nofollow" class="external text" href="https://www.minet.uni-jena.de//fakultaet/schukat/MAS/Scriptum/lect05-HMM.pdf"><i>Spezielle Musteranalysesysteme.</i></a> (PDF, 1,3 MB). Vorlesung im Wintersemester 2012/2013 an der Universität Jena. Kapitel 5, Folie 34 ff.</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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