Conditional Entropy - Definition

Definition

If is the entropy of the variable conditioned on the variable taking a certain value, then is the result of averaging over all possible values that may take.

Given discrete random variable with support and with support, the conditional entropy of given is defined as:

\begin{align} H(Y|X)\ &\equiv \sum_{x\in\mathcal X}\,p(x)\,H(Y|X=x)\\ &{=}\sum_{x\in\mathcal X}p(x)\sum_{y\in\mathcal Y}\,p(y|x)\,\log\, \frac{1}{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)} {p(x,y)}. \\ \end{align}


Note: The supports of X and Y can be replaced by their domains if it is understood that should be treated as being equal to zero.

if and only if the value of is completely determined by the value of . Conversely, if and only if and are independent random variables.

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