Mismatch Capacity under Stochastic Decoding

This manuscript investigates channel capacity under mismatched stochastic likelihood decoding. We derive Feinstein- and Verd\'u-Han-style bounds on the error probability coded communication. These are used to obtain a general information-spectrum formula for the channel capacity under mismatched stochastic decoding. The mismatch capacity formula is expressed as the supremum over all input distribution sequences of the limit inferior in probability of the sequence of normalized mismatched information densities. The resulting capacity formula is the mismatched analog of the channel capacity formula for the matched case by Verd\'u and Han. We also show that when the sequence of normalized mismatched information densities is uniformly integrable, the capacity formula admits an upper-bound as the limit of the corresponding sequence of expectations. This upper-bound is shown to be achievable for discrete-memoryless channels and product decoding metrics, showing that the Csisz\'ar-Narayan conjecture is tight for mismatched stochastic decoders.