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=== Bayes' rule 贝叶斯法则 ===

=== Bayes' rule 贝叶斯法则 ===

[[Bayes' rule]] for conditional entropy states

[[Bayes' rule]] for conditional entropy states

:<math>\Eta(Y|X) \,=\, \Eta(X|Y) - \Eta(X) + \Eta(Y).</math>

:<math>H(Y|X) \,=\, H(X|Y) - H(X) + H(Y).</math>

''Proof.'' <math>\Eta(Y|X) = \Eta(X,Y) - \Eta(X)</math> and <math>\Eta(X|Y) = \Eta(Y,X) - \Eta(Y)</math>. Symmetry entails <math>\Eta(X,Y) = \Eta(Y,X)</math>. Subtracting the two equations implies Bayes' rule.

''Proof.'' <math>H(Y|X) = H(X,Y) - H(X)</math> and <math>H(X|Y) = H(Y,X) - H(Y)</math>. Symmetry entails <math>H(X,Y) = H(Y,X)</math>. Subtracting the two equations implies Bayes' rule.

If <math>Y</math> is [[Conditional independence|conditionally independent]] of <math>Z</math> given <math>X</math> we have:

If <math>Y</math> is [[Conditional independence|conditionally independent]] of <math>Z</math> given <math>X</math> we have:

:<math>\Eta(Y|X,Z) \,=\, \Eta(Y|X).</math>

:<math>H(Y|X,Z) \,=\, H(Y|X).</math>

=== Other properties 其他性质 ===

=== Other properties 其他性质 ===