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In information theory, the conditional entropy quantifies the amount of information needed to describe the outcome of a random variable given that the value of another random variable is known. Here, information is measured in shannons, nats, or hartleys.
Definition The conditional entropy of X given Y is. H(X|Y ) = −. p(x, y) log p(x|y) = −E[ log(p(x|y)) ] (5) The conditional entropy is a measure of how much uncertainty remains about the random variable.
If \((X, Y) \sim p(x, y)\), the conditional entropy \(H(Y \mid X)\) is defined as \[\begin{split}H(Y \mid X) & = \sum_{x \in \mathcal{X}} p(x) H(Y \mid X = x) \\ & = - \sum_{x \in \mathcal{X}} p(x) \sum_{y \in \mathcal{Y}} p(y \mid x) \log p(y \mid x) \\ & = - \sum_{x \in \mathcal{X}}\sum_{y \in \mathcal{Y}} p(x, y) \log p(y \mid x) \\ & = - E ...
conditional information, and relative entropy (discrimination, Kullback-Leibler information), along with the limiting normalized versions of these quantities such as entropy rate and information rate.
Conditional entropy refers to the uncertainty about a variable Y when another variable X is known. It can also be understood as the expected number of bits required to describe Y when both the encoder and decoder have knowledge of X.
Summary. Entropy measures the amount of information in a random variable or the length of the message required to transmit the outcome; joint entropy is the amount of information in two (or more) random variables; conditional entropy is the amount of information in one random variable given we already know the other.
4 Μαρ 2022 · Conditional Entropies. There are two steps to understand conditional entropies. The first is the uncertainty of a random variable caused by a single outcome only. Suppose we have the same random variables and defined earlier in joint entropies. Let's denote as the conditional probability of when event happened.
