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 trade-off 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.