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Finite conditional entropy
Definition
Let be a joint probability vector on finite alphabets . It is the weight function of the canonical finite probability space from Finite probability spaces, outcome weights, events, and event probabilities; its coordinate maps are finite random elements in the sense of Random elements and real random variables. Write . For each with , set ; this is a probability vector by Marginals and positive-mass conditionals of a finite joint law are probability vectors. Define the conditional entropy in nats by
where is the finite entropy of Finite distributions, entropy, divergence, and stochastic matrices. Terms for zero-mass are omitted. The base- version replaces by .
Depends on
- Information units and logarithm-base convention
- Marginals and positive-mass conditionals of a finite joint law are probability vectors
- Finite distributions, entropy, divergence, and stochastic matrices
- Finite probability spaces, outcome weights, events, and event probabilities
- Random elements and real random variables
Used by
- Strict asymptotic separation need not give zero finite-block error Counterexample
- Finite-alphabet conditional entropy given an observation Definition
- Finite-alphabet IID multisource Definition
- Information leakage and equivocation for finite observations Definition
- Jointly typical blocks for a finite IID multisource Definition
- Lossless transmission of two correlated IID sources over a DM-MAC Definition
- A finite information-bottleneck privacy model Example
- A finite-alphabet source-channel rate budget Example
- Berger-Tung inner bound contains the Slepian-Wolf region Example
- Binary symmetric source: Slepian–Wolf corners and common part Example
- Capacity of the binary erasure channel Example
- Capacity of the binary symmetric channel Example
- Independent and deterministic finite information Example
- Lossless Wyner-Ziv endpoint is the Slepian-Wolf rate Example
- An average-cost DMC block input has mutual information at most n C_I(Gamma) Lemma
- Capacity-achieving output laws bound every row divergence Lemma
- Conditional block law factors across IID coordinates Lemma
- Conditional type packing bounds for source-indexed DM-MAC codebooks Lemma
- Conditional type-class cardinality bound Lemma
- Conditional typical-set fan cardinality bound Lemma
- Constrained DMC information capacity is attained, nondecreasing, and concave Lemma
- Continuity of finite conditional entropy in the joint law Lemma
- Finite DMC information capacity is attained Lemma
- Finite joint-typicality packing bound for independent random bins Lemma
- Finite random-codebook covering and selected-codeword law Lemma
- Joint typicality probability tends to one for a finite IID multisource Lemma
- The n-letter mutual information of a DMC is at most nC Lemma
- Cover-El Gamal-Salehi sufficient condition for correlated sources over a DM-MAC Theorem
- Finite conditional mutual information identities and equality Theorem
- Finite entropy and mutual-information chain rules Theorem
- Finite information quantities convert by the logarithm base Theorem
- Finite-alphabet Berger-Tung multiterminal lossy inner bound Theorem
- Finite-alphabet conditional entropy is version-invariant and obeys its chain rule Theorem
- Independent-private-message DM-MAC capacity region Theorem
- Physically degraded two-user DM broadcast capacity region Theorem
Dependency tree · two levels
17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Robert M. Gray, Entropy and Information Theory, corrected first edition (standard reference, not scraped)
- Yury Polyanskiy and Yihong Wu, Information Theory: From Coding to Learning, author prepublication (standard reference, not scraped)