Rényi Common Information for Doubly Symmetric Binary Sources

Abstract

In this note, we provide analytic expressions for the Rényi common information of orders in $(1,\infty )$ for the doubly symmetric binary source (DSBS). Until now, analytic expressions for the Rényi common information of all orders in $[0,\infty ]$ have been completely known for this source. We also consider the Rényi common information of all orders in $[-\infty ,0)$ and evaluate it for the DSBS. We provide a sufficient condition under which the Rényi common information of such orders coincides with Wyner's common information for the DSBS. Based on numerical analysis, we conjecture that there is a certain phase transition as the crossover probability increasing for the Rényi common information of negative orders for the DSBS. Our proofs are based on a lemma on splitting of the entropy and the analytic expression of relaxed Wyner's common information.

20 pages