Extended Gray–Wyner System With Complementary Causal Side Information

We establish the rate region of an extended Gray-Wyner system (EGW) for 2-DMS (X, Y) with two additional decoders having complementary causal side information. We show that the 5-D rate region of the EGW system is equivalent to the 3-D mutual information region consisting of the set of all triples of the form (I(X; U), I(Y; U), I(X, Y; U)) for some pU|X,Y. This correspondence greatly simplifies the exploration of the the extreme points of the rate region. In addition to the operationally significant extreme points of the original Gray-Wyner rate region, which include Wyner's common information, Gács-Körner common information, and the information bottleneck, the extreme points of the rate region for the EGW system also include the Körner graph entropy, the privacy funnel and excess functional information, as well as three new quantities of potential interest. We further show that projections of the mutual information region yield the rate regions for many settings involving a two discrete memoryless source (2-DMS), including lossless source coding with causal side information, distributed channel synthesis, and lossless source coding with a helper. To further motivate the mutual information region itself, we draw analogies between set operations and its extreme points. This allows us to find random variables that can be considered as intersection, difference, and symmetric difference of two random variables. Finally, we establish the rate regions for two related setups to the EGW system, namely, the noncausal EGW system and the lossy EGW system.