On some convergence properties of ´learning with a probabilistic teacher´ algorithms (Corresp.)

Satisfactory proof does not yet exist for the consistency of "learning with a probabilistic teacher" estimators, which are a class of randomized decision-directed estimators for adaptive multihypothesis decision making. Since a number of computer simulations described in the published literature indicate that the algorithms are convergent, we take as our starting point the assumption that this is generally true and develop an equation for use in determining for general distributions whether convergence is to the true parameter value. Using a numerical solution, we show that for the example of two one-dimensional Gaussian hypotheses and one unknown mean (the example appearing in the paper introducing the algorithm), if the estimator is convergent, convergence is to the true parameter value. Our formulation should be of help in constructing a more complete solution to the convergence problem and may be of use in investigating the consistency of other adaptive decision-making algorithms.