Two channels are equivalent if their maximum likelihood (ML) decoders coincide for every code. We show that this equivalence relation partitions the space of channels into a generalized hyperplane arrangement. With this, we define a coding distance between channels in terms of their ML-decoders which is meaningful from the decoding point of view, in the sense that the closer two channels are, the larger is the probability of them sharing the same ML-decoder. We give explicit formulas for these probabilities.
@cite_7: The mismatch capacity of a channel is the highest rate at which reliable communication is possible over the channel with a given (possibly suboptimal) decoding rule. This quantity has been studied extensively for single-letter decoding rules over discrete memoryless channels (DMCs). Here we extend the study to memoryless channels with general alphabets and to channels with memory with possibly non-single-letter decoding rules. We also study the wide-band limit, and, in particular, the mismatch capacity per unit cost, and the achievable rates on an additive-noise spread-spectrum system with single-letter decoding and binary signaling.
Mismatched decoding has an extensive literature ( @cite_7 has many relevant references on the subject). The approach, however, is essentially information theoretical, guided by the fundamental question of determining . This means that most of the work in the area aims to understand what is achievable asymptotically, for example, what are the achievable rates for families of channels with the input-output sets' size going to infinity. Those are very difficult questions and hence a significant part of the effort is directed to find bounds for those rates (and other significant invariants).