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.

Read more about this topic:  Conditional Entropy

Famous quotes containing the word definition:

    “The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
    —Ralph Waldo Emerson (1803–1882)

    “No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
    —Florence Nightingale (1820–1910)

    “I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
    —Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)