Error exponents for finite-hypothesis channel identification

We consider the problem of designing optimal probing signals for finite-hypothesis testing. Equivalently, we cast the problem as the design of optimal channel input sequences for identifying a discrete channel under observation from a finite set of known channels. The optimality criterion that we employ is the exponent of the Bayesian probability of error. In our study, we consider a feedforward scenario where there is no feedback from the channel output to the signal selector at the channel input and a feedback scenario where the past channel outputs are revealed to the signal selector. In the feedforward scenario, only the type of the input sequence matters and our main result is an expression for the error exponent in terms of the limiting distribution of the input sequence. In the feedback case, we show that when discriminating between two channels, the optimal scheme in the first scenario is simultaneously the optimal time-invariant Markov feedback policy of any order.