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Perfect output feedback does not increase finite DMC capacity

Statement

For a finite DMC $W$ without input-cost constraints, the vanishing-average-error capacity with perfect causal output feedback equals its ordinary information capacity $C (W) = \max_{P_{X}} I (P_{X}; W)$ from Single-letter information capacity of a finite DMC. The feedback code convention is Fixed-length finite DMC code with perfect causal output feedback.

Facts & Assumptions

Given: A finite DMC $W$ , a finite-message feedback code, and its average error probability $P_{e}$ .

[F1]

Fixed-length finite DMC code with perfect causal output feedback: $X_{i} = f_{i} (M, Y^{i - 1})$ deterministically, and at each positive-probability history $(M = m, Y^{i - 1} = y^{i - 1})$ , the conditional law of $Y_{i}$ is $W (\cdot ∣ f_{i} (m, y^{i - 1}))$ .

[F2]

Single-letter information capacity of a finite DMC: $C (W)$ is the supremum of $I (P_{X}; W)$ over all input probability vectors.

[F3]

Random coding achieves every rate below finite DMC capacity: every rate below $C (W)$ is achievable by ordinary finite DMC codes.

[F4]

Finite-alphabet Fano inequality: for a finite message and finite observation, $H (M ∣ Y^{n}) \leq \ln 2 + P_{e} \ln M$ .

[F5]

Finite entropy and mutual-information chain rules: finite output entropy obeys the chain rule, and $I (M; Y^{n}) = \sum_{i} I (M; Y_{i} ∣ Y^{i - 1})$ .

[F6]

Finite conditional mutual information identities and equality: finite conditional mutual information is nonnegative, so conditioning cannot increase finite entropy.

[F7]

Finite mutual information: finite mutual information is the joint-to-product KL sum in nats.

[F8]

Finite conditional mutual information: finite conditional mutual information averages the conditional finite-law informations over positive-mass conditioning values.

Proof

1.1F1F3

Every no-feedback code is a feedback code whose maps ignore $Y^{i - 1}$ . By [F3], each rate below $C (W)$ is therefore feedback-achievable, giving $C_{f b} (W) \geq C (W)$ .

1.2F1F4algebra

Fix a length- $n$ feedback code with uniform message $M \in [M]$ . By [F4], $n R = H (M) \leq I (M; Y^{n}) + \ln 2 + P_{e} n R$ .

2.1F1F2F5F6F7F8step 1.2algebra

Expand the mutual information using [F5] and the conditional-information definition [F8]. Given $(M, Y^{i - 1})$ , $X_{i}$ is fixed and the DMC law makes $Y_{i}$ have law $W (\cdot ∣ X_{i})$ , so $H (Y_{i} ∣ M, Y^{i - 1}) = H (Y_{i} ∣ X_{i})$ . Also $H (Y_{i} ∣ Y^{i - 1}) \leq H (Y_{i})$ by [F6]. Hence $I (M; Y_{i} ∣ Y^{i - 1}) \leq H (Y_{i}) - H (Y_{i} ∣ X_{i}) = I (P_{X_{i}}; W) \leq C (W)$ by [F2]; the identity with entropy differences is the finite mutual-information identity [F7].

3.1F2F4F5step 1.1step 1.2step 2.1algebra∎

Summing step 2.1 in [F5] and substituting into step 1.2 gives $(1 - P_{e}) n R \leq n C (W) + \ln 2$ . Along any reliable sequence, divide by $n$ and let $n \to \infty$ to obtain $C_{f b} (W) \leq C (W)$ . Combining with step 1.1 proves equality.

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Sources

Thomas M. Cover and Joy A. Thomas, Elements of Information Theory, 2nd ed., Chapter 7, §7.12 (standard reference, not scraped)
Abbas El Gamal and Young-Han Kim, Network Information Theory, Lecture Notes 18, Feedback DMC (standard reference, not scraped)