Joint communication and state sensing under logarithmic loss

We study a basic joint communication and sensing setting where a transmitter wishes to convey a message to a receiver and simultaneously estimate its channel state through generalized feedback. We assume that the sensing goal is to obtain a soft estimate (i.e. probability distribution) of the state, instead of a point estimate as often assumed; and we use the logarithmic loss distortion as a fidelity measure. Focusing on the memoryless setting with an i.i.d. state, we characterize the optimal tradeoff between the communication rate and the sensing distortion. In doing so, we shed light on the operational meaning of the so-called sensing mutual information utility function.