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A Simple Proof for the Optimality of Randomized Posterior Matching

机译：后验随机匹配最优性的简单证明

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摘要

Posterior matching (PM) is a sequential horizon-free feedback communication scheme introduced by the authors, who also provided a rather involved optimality proof, showing that it achieves capacity for a large class of memoryless channels. Naghshvar considered a non-sequential variation of PM with a fixed number of messages and a random decision-time, and gave a simpler proof establishing its optimality via a novel extrinsic Jensen–Shannon divergence argument. Another simpler optimality proof was given by Li and El Gamal, who considered a fixed-rate fixed block-length variation of PM with an additional randomization. Both these works also provided error exponent bounds. However, their simpler achievability proofs apply only to discrete memoryless channels, and are restricted to a non-sequential setup with a fixed number of messages. In this paper, we provide a short and transparent proof for the optimality of the fully sequential randomized horizon-free PM scheme over general memoryless channels. Borrowing the key randomization idea of Li and El Gamal, our proof is based on analyzing the random walk behavior of the shrinking posterior intervals induced by a reversed iterated function system decoder.
机译：后验匹配（PM）是作者引入的一种顺序无视域反馈通信方案，他们还提供了相当复杂的最优性证明，表明它可以实现大容量无记忆通道的容量。 Naghshvar考虑了具有固定消息数量和随机决策时间的PM的非顺序变化，并通过新颖的外部Jensen-Shannon发散论证给出了确定其最优性的更简单证明。 Li和El Gamal给出了另一个更简单的最优性证明，他们考虑了PM的固定速率固定块长变化和附加随机化。这些工作还提供了误差指数范围。但是，它们更简单的可实现性证明仅适用于离散的无内存通道，并且仅限于具有固定数量消息的非顺序设置。在本文中，我们为一般无记忆通道上完全顺序随机无视域PM方案的最优性提供了简短透明的证明。借用Li和El Gamal的关键随机化思想，我们的证明是基于分析由反向迭代函数系统解码器导致的后收缩间隔的随机行走行为。

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