New perspectives on MAC feedback capacity using decentralized sequential active hypothesis testing paradigm

Abstract

The capacity of the MAC with feedback has been characterized through a multi-letter expression based on the work of Kramer. Except for the two-user Gaussian channel, this expression has resisted simplification; as a result there is no single-letter characterization for the capacity of the general discrete memoryless MAC (DM-MAC). In this paper we investigate connections between this problem and the problem of decentralized sequential active hypothesis testing (DSAHT). In this problem, two transmitting agents, each possessing a private message, are actively helping a third agent–and each other–to learn the message pair over a DM-MAC. The third agent (receiver) observes the noisy channel output, which is also available to the transmitting agents via noiseless feedback. We provide a characterization of the optimal transmission scheme for the DSAHT problem depending on an appropriately defined sufficient statistic. Returning to the problem of simplifying the multi-letter expression for the DM-MAC feedback capacity, we show that restricting attention to distributions induced by optimal transmission schemes for the DSAHT problem, without loss of optimality, transforms the capacity expression, so that it can be thought of as the average reward received by an appropriately defined stochastic dynamical system with time-invariant state space.