Informazioaren teorian baldintzazko entropia , baita ere nahastea deituta, kuantifikatzen du zenbatekoa den
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{\displaystyle \mathrm {H} (Y|X=x)}
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Probabilitate-funtzio edukiz,
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{\displaystyle p_{Y}{(y)}}
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{\displaystyle \mathrm {H} (Y):=\mathbb {E} [\operatorname {I} (Y)]}
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{\displaystyle \mathrm {H} (Y)=\sum _{i=1}^{n}{\mathrm {Pr} (Y=y_{i})\,\mathrm {I} (y_{i})}=-\sum _{i=1}^{n}{p_{Y}(y_{i})\log _{2}{p_{Y}(y_{i})}},}
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{\displaystyle \operatorname {I} (y_{i})}
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{\displaystyle \mathrm {H} (Y|X=x)=\mathbb {E} [\operatorname {I} (Y)|X=x]=-\sum _{i=1}^{n}{\Pr(Y=y_{i}|X=x)\log _{2}{\Pr(Y=y_{i}|X=x)}}.}
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{\displaystyle \mathrm {H} (Y|X)}
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{\displaystyle \mathrm {H} (Y|X=x)}
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zorizko aldagaiak emanez, bere imajinak
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{\displaystyle {\mathcal {X}}}
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{\displaystyle {\mathcal {Y}}}
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{\displaystyle Y}
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{\displaystyle \mathrm {H} (Y|X=x)}
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{\displaystyle p(x)}
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{\displaystyle {\begin{aligned}\mathrm {H} (Y|X)\ &\equiv \sum _{x\in {\mathcal {X}}}\,p(x)\,\mathrm {H} (Y|X=x)\\&=-\sum _{x\in {\mathcal {X}}}p(x)\sum _{y\in {\mathcal {Y}}}\,p(y|x)\,\log \,p(y|x)\\&=-\sum _{x\in {\mathcal {X}}}\sum _{y\in {\mathcal {Y}}}\,p(x,y)\,\log \,p(y|x)\\&=-\sum _{x\in {\mathcal {X}},y\in {\mathcal {Y}}}p(x,y)\log \,p(y|x)\\&=-\sum _{x\in {\mathcal {X}},y\in {\mathcal {Y}}}p(x,y)\log {\frac {p(x,y)}{p(x)}}.\\\end{aligned}}}
Oharra: Ulertzen da expresioa
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{\displaystyle 0log(0)}
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{\displaystyle 0log(c/0)}
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{\displaystyle c>0}
Baldintzazko entropia zero bada
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{\displaystyle \mathrm {H} (Y|X)=0}
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Baldintzazko entropia ausazko aldaigaia independentea bada
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{\displaystyle \mathrm {H} (Y|X)=\mathrm {H} (Y)}
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{\displaystyle {\begin{aligned}\mathrm {H} (Y|X)&=\sum _{x\in {\mathcal {X}},y\in {\mathcal {Y}}}p(x,y)\log \left({\frac {p(x)}{p(x,y)}}\right)\\[4pt]&=-\sum _{x\in {\mathcal {X}},y\in {\mathcal {Y}}}p(x,y)\log(p(x,y))+\sum _{x\in {\mathcal {X}},y\in {\mathcal {Y}}}{p(x,y)\log(p(x))}\\[4pt]&=\mathrm {H} (X,Y)+\sum _{x\in {\mathcal {X}}}p(x)\log(p(x))\\[4pt]&=\mathrm {H} (X,Y)-\mathrm {H} (X).\end{aligned}}}
Bayesen legeak baldintzazko entropiari buruz dio
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{\displaystyle \mathrm {H} (Y|X)\,=\,\mathrm {H} (X|Y)-\mathrm {H} (X)+\mathrm {H} (Y).}
Frogapena:
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{\displaystyle H(Y|X)=H(X,Y)-H(X)}
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{\displaystyle H(X|Y)=H(Y,X)-H(Y)}
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{\displaystyle H(X,Y)=H(Y,X)}
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{\displaystyle H(Y,X)=H(X|Y)+H(Y)}
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{\displaystyle H(Y|X)=(H(X|Y)+H(Y))-H(X)}
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{\displaystyle \mathrm {H} (Y|X)\,=\,\mathrm {H} (X|Y)-\mathrm {H} (X)+\mathrm {H} (Y).}
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{\displaystyle \mathrm {H} (Y|X,Z)\,=\,\mathrm {H} (Y|X).}
Edozein
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{\displaystyle {\begin{aligned}\mathrm {H} (Y|X)&\leq \mathrm {H} (Y)\,\\\mathrm {H} (X,Y)&=\mathrm {H} (X|Y)+\mathrm {H} (Y|X)+\operatorname {I} (X;Y),\qquad \\\mathrm {H} (X,Y)&=\mathrm {H} (X)+\mathrm {H} (Y)-\operatorname {I} (X;Y),\,\\\operatorname {I} (X;Y)&\leq \mathrm {H} (X),\,\end{aligned}}}
Non
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{\displaystyle \operatorname {I} (X;Y)}
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{\displaystyle \mathrm {H} (Y|X)=\mathrm {H} (Y)}
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{\displaystyle \mathrm {H} (X|Y)=\mathrm {H} (X)\,}
Naiz eta baldintza-espezifiko entropia
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{\displaystyle Eta(X|Y=y)}
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{\displaystyle Eta(X)}
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{\displaystyle Eta(X|Y)}
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{\displaystyle Eta(X)}
![{\displaystyle \mathrm {H} (Y):=\mathbb {E} [\operatorname {I} (Y)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f114631caeb95e508a74994486e35e972220b378)
![{\displaystyle \mathrm {H} (Y|X=x)=\mathbb {E} [\operatorname {I} (Y)|X=x]=-\sum _{i=1}^{n}{\Pr(Y=y_{i}|X=x)\log _{2}{\Pr(Y=y_{i}|X=x)}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcac23d77dcc6a01cc367830184893bd13eda9f9)
![{\displaystyle {\begin{aligned}\mathrm {H} (Y|X)&=\sum _{x\in {\mathcal {X}},y\in {\mathcal {Y}}}p(x,y)\log \left({\frac {p(x)}{p(x,y)}}\right)\\[4pt]&=-\sum _{x\in {\mathcal {X}},y\in {\mathcal {Y}}}p(x,y)\log(p(x,y))+\sum _{x\in {\mathcal {X}},y\in {\mathcal {Y}}}{p(x,y)\log(p(x))}\\[4pt]&=\mathrm {H} (X,Y)+\sum _{x\in {\mathcal {X}}}p(x)\log(p(x))\\[4pt]&=\mathrm {H} (X,Y)-\mathrm {H} (X).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2baddbd85be799629ae27dd729e3b067363aac06)
