Optimal sequence estimators for statistically unknown binary sources and channels

We consider an information source which is an i.i.d. binary sequence governed by unknown probability measures. The information sequence is transferred through a memoryless binary channel with unknown cross-over probabilities. The channel model also represents those cases in which an input quantizer is always used, so that the incoming information-bearing observations are threshold crossings of the observation process, and the unknown cross-over probabilities are associated with uncertainties concerning the signal-to-noise ratio. We derive and study the optimal (under a minimum error-probability criterion) sequence estimator (which utilizes the observed threshold crossings). The receiver is described by a practically implementable algorithm which involves a shortest path calculation, which is performed using the Viterbi algorithm, and appropriately incorporates the sufficient statistics of the unknown parameters. Its similarity to unsupervised decision directed learning procedures is noted.