Channel and question
- Input
- \(X\in\mathcal X\)
- Output
- \(Y\in\mathcal Y\)
- Law
- A fixed but unknown \(W_s(y|x)\) is selected from a known finite family \(\{W_s:s\in\mathcal S\}\).
- Quantity
- Compound-channel capacity \(C_{\mathrm{cmp}}\), measured in bits per channel use.
Criterion. Uniformly vanishing average error over the channel family.
- The same state s applies throughout the block.
- A single code must work uniformly for every channel in the family.
Current status
\[C_{\mathrm{cmp}}=\max_{P_X}\inf_{s\in\mathcal S} I(P_X,W_s)\]
| Result | Relation | Method | Year |
|---|---|---|---|
| Lower | \(C_{\mathrm{cmp}}\ge\max_{P_X}\inf_s I(P_X,W_s)\) | A common random code and universal decoding over the family. | 1959 |
| Upper | \(C_{\mathrm{cmp}}\le\max_{P_X}\inf_s I(P_X,W_s)\) | Every code must satisfy a converse for each member of the family. | 1959 |
Lean formalization
Canonical statementNone
Version 1 · Lean. Families of finite channels can reuse the shared DMC structure; uniform operational codes are missing.
Substantial proofs0 linked
No external Lean proof is registered. Proofs longer than roughly 50 lines or requiring problem-specific infrastructure should live in a dedicated repository and link back to this statement version.
References
- David Blackwell, Leo Breiman, and A. J. Thomasian (1959). The Capacity of a Class of Channels. Annals of Mathematical Statistics. DOI 10.1214/aoms/1177706106.
- Thomas M. Cover and Joy A. Thomas (2006). Elements of Information Theory. Wiley, second edition. DOI 10.1002/047174882X.
Discussion
Thread key: capacityatlas:compound-discrete-memoryless-channel