Conditional Entropy - Chain Rule

Chain Rule

Assume that the combined system determined by two random variables X and Y has entropy, that is, we need bits of information to describe its exact state. Now if we first learn the value of, we have gained bits of information. Once is known, we only need bits to describe the state of the whole system. This quantity is exactly, which gives the chain rule of conditional probability:

Formally, the chain rule indeed follows from the above definition of conditional probability:

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

Read more about this topic:  Conditional Entropy

Famous quotes containing the words chain and/or rule:

    “Oft, in the stilly night, Ere Slumber’s chain has bound me, Fond Memory brings the light Of other days around me.”
    —Thomas Moore (1779–1852)

    “Men conceive themselves as morally superior to those with whom they differ in opinion. A Socialist who thinks that the opinions of Mr. Gladstone on Socialism are unsound and his own sound, is within his rights; but a Socialist who thinks that his opinions are virtuous and Mr. Gladstone’s vicious, violates the first rule of morals and manners in a Democratic country; namely, that you must not treat your political opponent as a moral delinquent.”
    —George Bernard Shaw (1856–1950)