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What is chain rule in information theory?
With equality if and only if the Xi are independent. Proof: By the chain rule of entropies: We know that the conditional entropy of a random variable X given another random variable Y is zero if and only if X is a function of Y. Hence we can estimate X from Y with zero probability of error if and only if H(X|Y) = 0.
What is chain rule for entropy?
Chain rule for entropy • Last time, simple chain rule H(X, Y ) = H(X) + H(Y |X) • No matter how we play with chain rule, we get the same answer. H(X, Y ) = H(X) + H(Y |X) = H(Y ) + H(X|Y ) “entropy of two experiments” Dr.
What is information theory information rate?
Information Rate : R = rH. Here R is the information rate. H is the Entropy or average information. And r is the rate at which messages are generated. Information rate R is represented in average number of bits of information per second.
How do you prove entropy is non negative?
- Note that relative entropy can be infinite if νi = 0 where µi = 0. To show the utility of relative entropy we prove the following theorem.
- Theorem 2 (Non-negativity). The relative entropy is non-negative and is zero if and only if ν = µ.
- Proof. First, note that log(x) ≤ x − 1 for all x > 0. Thus. H(µ||ν) = −
Does conditioning reduce entropy?
Conditioning reduces entropy H(X|Y ) ≤ H(X) with equality of and only if X and Y are independent. Knowing another random variable Y reduces (on average) the uncertainty of variable X. X → Y → Z implies that Z → Y → X.
What are chain rules and inequalities in information theory?
This time { Chain rules { Jensen’s inequality { Log-sum inequality { Concavity of entropy { Convex/concavity of mutual information Dr. Yao Xie, ECE587, Information Theory, Duke University
This time { Chain rules { Jensen’s inequality { Log-sum inequality { Concavity of entropy { Convex/concavity of mutual information Dr. Yao Xie, ECE587, Information Theory, Duke University Logic order of things
How to prove the chain rule for entropy?
Chain rule for entropy Entropy for a collection of RV’s is the sum of the conditional entropies More generally:H(X1;X2; ;Xn) = ∑n i=1H(XijXi1; ;X1) Proof:
When does Jensen’s inequality hold for a convex function?
Dr. Yao Xie, ECE587, Information Theory, Duke University 14 Theorem. (Jensen’s inequality) If f is a convex function, Ef(X)\ f(EX): If f strictly convex, equality holds when X=constant: