Optimal remote estimation over Action Dependent Switching Channels: Managing workload and bias of a human operator

Consider a remote estimation system formed by a channel and an encoder that assesses a continuous random variable denoted as source. The internal structure of the channel has a finite state machine (FSM) whose state dictates the transmission characteristics. Each state of the FSM corresponds to a discrete memoryless channel (DMC). At each channel use, information is transmitted from the encoder to the channel output according to the DMC selected by the current FSM state. This class of channels is denoted as Action Dependent Switching Channel, or ADS. An action feedback policy maps the channel´s output into the input of the FSM. This paper investigates methods to analyze and design an action feedback policy and encoder that minimize the differential entropy of the source conditioned on the channel output. We show that there are optimal action feedback policies for which the input to the FSM is a deterministic sequence that does not depend on the channel output. We also provide additional structural results for the case when the FSM parametrizes a set of Binary Symmetric Channels (BSC) with differing crossover probabilities. Here, we consider that the ADS contains states of no transmission, which are modeled as a BSC crossover probability of one half. In this case, the FSM is also used to model channel degradation as a result of multiple transmissions, and it also allows for recovery when there are no transmissions. When the switching DMC is a BSC, the optimal encoder and action feedback policies can be computed separately. We also discuss the relevance of this model to applications in which the channel represents a human decision maker whose reliability and bias are affected by current and past outputs.