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Finite-DMC hypothesis-testing meta-converse

Statement

Let a finite-DMC code have $M \geq 1$ messages and average error $ϵ$ . For any output probability vector $Q_{Y^{n}}$ , let $P_{M Y^{n}} (m, y^{n}) = M^{- 1} W^{n} (y^{n} ∣ x_{m}^{n})$ be the true joint law and let ${\tilde{Q}}_{M Y^{n}} (m, y^{n}) = M^{- 1} Q_{Y^{n}} (y^{n})$ . With the type-II testing function $β_{ϵ}$ of Finite type-II testing function with explicit orientation, which constrains rejection under its first argument, one has $β_{ϵ} (P_{M Y^{n}}, {\tilde{Q}}_{M Y^{n}}) \leq \frac{1}{M} .$

Facts & Assumptions

Given: A deterministic length- $n$ code with $M$ messages, average error $ϵ$ , and any output probability vector $Q_{Y^{n}}$ .

[F1]

Finite type-II testing function with explicit orientation: $β_{ϵ} (P, Q)$ is the infimum of $E_{Q} (1 - ϕ)$ over tests $0 \leq ϕ \leq 1$ with $E_{P} ϕ \leq ϵ$ ; the first law is the null and the second is the alternative.

[F2]

Fixed-length codes for a finite DMC and Average and maximal error of a fixed-length code: the decoder is a deterministic map $d : Y^{n} \to [M]$ , and the average error is the uniform-message error probability.

[F3]

Finite discrete memoryless channels and The DMC product laws are stochastic kernels: $W^{n} (\cdot ∣ x_{m}^{n})$ is a probability vector for every codeword.

Proof

1.1F1F2F3givenalgebra

Define $ϕ (m, y^{n}) = 1 {d (y^{n}) \neq m}$ . Under the true law $P_{M Y^{n}}$ , $E_{P} ϕ = ϵ$ , so $ϕ$ is feasible in [F1]. Under $\tilde{Q}$ , its type-II error is $E_{\tilde{Q}} (1 - ϕ) = \frac{1}{M} \sum_{m = 1}^{M} \sum_{y^{n}} Q_{Y^{n}} (y^{n}) 1 {d (y^{n}) = m} = \frac{1}{M},$ since for each output exactly one message is decoded. No likelihood ratio is used, so zeros in $Q_{Y^{n}}$ cause no support problem.

2.1F1step 1.1algebra∎

The infimum over all feasible tests in [F1] is at most the value of the particular test in step 1.1. This gives the claimed bound for every auxiliary output law, including singular ones.

Depends on

Used by

Exact beta value for a one-use BSC meta-converse test Example

Dependency tree · two levels

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Sources

Yury Polyanskiy and Yihong Wu, Information Theory: From Coding to Learning, Chapter 22 §22.3 (standard reference, not scraped)
Yury Polyanskiy, H. Vincent Poor, and Sergio Verdú, Channel Coding Rate in the Finite Blocklength Regime (standard reference, not scraped)