Reliable deniable communication with channel uncertainty

Alice wishes to potentially communicate with Bob over a compound Binary Symmetric Channel while Willie listens in over a compound Binary Symmetric Channel that is noisier than Bob's. The channel noise parameters for both Bob and Willie are drawn according to uniform distribution over a range, but none of the three parties know their exact values. Willie's goal is to infer whether or not Alice is communicating with Bob. We show that Alice can send her messages reliably to Bob while ensuring that even whether or not she is actively communicating is deniable to Willie. We find the best rate at which Alice can communicate both deniably and reliably using Shannon's random coding and prove a converse.