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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.
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 ...
The entropy of U is given by: H(U) , E[S(U)] = E log. 1. p(U) = E. log (p(U)) = X p(u) log p(u) u. (2) Where U represents all u values possible to the variable. The entropy is a property of the underlying distribution PU(u); u 2 U that measures the amount of randomness or surprise in the random variable. Lemma 3.
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. AI generated definition based on: Encyclopedia of Physical Science and Technology (Third Edition), 2003.
4 Μαρ 2022 · 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.
if X = Y , or if X = f(Y ) (so we know X given Y ), the conditional entropy is 0. On the other hand, if X and Y are independent, the conditional entropy is just H(X).
Conditional entropy. Let (X;Y) ∼ p. For x ∈ Supp(X), the random variable YSX = x is well defined. The entropy of Y conditioned on X, is defined by. H(YSX) ∶= E H(YSX = x) = E H(YSX) x←X X. Measures the uncertainty in Y given X. Let pX & pYSX be the marginal & conational distributions induced by p. H(YSX) = Q pX(x) ⋅ H(YSX = x) x∈X.
