A Simple Proof for the Optimality of Randomized Posterior Matching

TL;DR

This paper presents a simplified, transparent proof of the optimality of the fully sequential posterior matching scheme for general memoryless channels, extending previous results that were limited to discrete or non-sequential cases.

Contribution

It offers a new, concise proof of posterior matching optimality for all memoryless channels, using a random walk analysis of the RIFS decoder, broadening prior restricted results.

Findings

Proves the optimality of the sequential posterior matching scheme for general memoryless channels.

Uses a novel random walk analysis of the RIFS decoder to establish capacity achievement.

Simplifies previous complex proofs, making the result more accessible.

Abstract

Posterior matching (PM) is a sequential horizon-free feedback communication scheme introduced by the authors, who also provided a rather involved optimality proof showing it achieves capacity for a large class of memoryless channels. Naghshvar et al 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 Shannon-Jensen 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 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 (RIFS) decoder.