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arxiv: 2604.17964 · v1 · submitted 2026-04-20 · 💻 cs.IT · math.IT

Mismatch Capacity under Stochastic Decoding

Francesc Molina , Albert Guillen i Fabregas This is my paper

Pith reviewed 2026-05-10 04:15 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords mismatch capacitystochastic decodinginformation spectrumCsiszár-Narayan conjecturediscrete memoryless channelmismatched metricchannel capacity
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The pith

Mismatch capacity under stochastic decoding is the supremum over input sequences of the liminf in probability of normalized mismatched information densities.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper derives an information-spectrum formula for channel capacity when the decoder uses stochastic likelihood decoding with a fixed mismatched metric. It starts from Feinstein-style and Verdú-Han-style bounds on error probability and arrives at a capacity expression that is the direct analog of the matched Verdú-Han result. When the sequence of normalized mismatched information densities is uniformly integrable, the capacity is at most the limit of the sequence of expectations. The paper proves this upper bound is tight for discrete memoryless channels that use product decoding metrics, thereby showing the Csiszár-Narayan conjecture holds under stochastic decoding.

Core claim

The mismatch capacity is expressed as the supremum over all input distribution sequences of the limit inferior in probability of the sequence of normalized mismatched information densities. When the sequence is uniformly integrable, the capacity admits an upper bound as the limit of the corresponding sequence of expectations, and this bound is achievable for discrete-memoryless channels and product decoding metrics.

What carries the argument

The sequence of normalized mismatched information densities, whose liminf in probability, supremized over input distributions, gives the capacity.

If this is right

  • Feinstein- and Verdú-Han-style bounds on error probability extend directly to mismatched stochastic decoding.
  • The capacity formula is the mismatched counterpart of the Verdú-Han information-spectrum formula.
  • Uniform integrability yields an upper bound equal to the limit of the expectations of the normalized densities.
  • This upper bound equals the mismatch capacity for all discrete memoryless channels equipped with product decoding metrics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Stochastic decoding rules may allow capacity-achieving performance even when the metric is fixed and imperfect.
  • Similar spectrum characterizations could be derived for channels with memory or continuous alphabets under appropriate integrability conditions.
  • The result supplies a concrete computational path for mismatch capacity when the metric admits a product structure.

Load-bearing premise

The derivation assumes stochastic likelihood decoding with a fixed mismatched metric and invokes uniform integrability of the normalized mismatched information densities to obtain the tight upper bound.

What would settle it

A concrete discrete memoryless channel and product metric for which the achievable rate under stochastic decoding strictly exceeds the limit of the expected normalized mismatched information densities would disprove the claimed tightness.

read the original abstract

This manuscript investigates channel capacity under mismatched stochastic likelihood decoding. We derive Feinstein- and Verd\'u-Han-style bounds on the error probability coded communication. These are used to obtain a general information-spectrum formula for the channel capacity under mismatched stochastic decoding. The mismatch capacity formula is expressed as the supremum over all input distribution sequences of the limit inferior in probability of the sequence of normalized mismatched information densities. The resulting capacity formula is the mismatched analog of the channel capacity formula for the matched case by Verd\'u and Han. We also show that when the sequence of normalized mismatched information densities is uniformly integrable, the capacity formula admits an upper-bound as the limit of the corresponding sequence of expectations. This upper-bound is shown to be achievable for discrete-memoryless channels and product decoding metrics, showing that the Csisz\'ar-Narayan conjecture is tight for mismatched stochastic decoders.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript derives Feinstein- and Verdú-Han-style bounds on the error probability for coded communication under mismatched stochastic likelihood decoding. These are used to obtain a general information-spectrum formula for the mismatch capacity, expressed as the supremum over input distribution sequences of the limit inferior in probability of the normalized mismatched information densities. This is presented as the direct mismatched analog of the Verdú-Han formula. When the sequence of normalized mismatched information densities is uniformly integrable, the capacity admits an upper bound as the limit of the corresponding expectations; this bound is shown achievable for discrete-memoryless channels with product decoding metrics, establishing tightness of the Csiszár-Narayan conjecture for stochastic decoders.

Significance. If the derivations hold, the work provides a clean extension of the information-spectrum method to mismatched stochastic decoding, yielding a general capacity formula and a computable upper bound under uniform integrability. The explicit achievability proof for DMCs with product metrics is a notable strength, as it resolves the conjecture in the stochastic setting and confirms that the lim E[·] expression is tight without additional fitting parameters. This advances mismatched decoding theory by rigorously handling stochastic decoders while preserving the structure of the matched-case arguments.

minor comments (2)
  1. [§2] §2 (or the section defining the stochastic decoder): the likelihood-ratio test for the stochastic decoder should be written explicitly alongside the mismatched metric to make the transition from the deterministic case transparent.
  2. [capacity formula section] The uniform-integrability argument in the paragraph following the capacity formula is stated as a sufficient condition; a short remark on whether the condition is typically satisfied for common product metrics would help readers assess applicability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of the manuscript. The recommendation for minor revision is appreciated, and we note that the summary accurately captures the main contributions: the information-spectrum characterization of mismatch capacity under stochastic decoding and the tightness result for DMCs with product metrics. Since no specific major comments were raised, the responses below address the overall feedback provided.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives the general mismatch capacity formula as the supremum over input distribution sequences of the liminf-in-probability of normalized mismatched information densities, by adapting the standard Verdú-Han information-spectrum direct and converse arguments to the stochastic mismatched decoder. The uniform-integrability condition is explicitly introduced as a sufficient (not necessary) assumption that yields an upper bound via the limit of expectations; this bound is then shown achievable separately for DMC under product metrics. No equation reduces by construction to a fitted parameter, self-definition, or prior self-citation chain. The central claim rests on external, independently established information-spectrum techniques rather than internal renaming or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on standard information-theoretic concepts such as normalized information densities and limits in probability; no new free parameters or invented entities are introduced.

axioms (1)
  • standard math Standard definitions and properties of information densities and liminf in probability from information spectrum theory.
    The capacity formula is built directly on these established notions without additional justification in the abstract.

pith-pipeline@v0.9.0 · 5440 in / 1338 out tokens · 59129 ms · 2026-05-10T04:15:26.108223+00:00 · methodology

discussion (0)

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Reference graph

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