Gaussian one-armed bandit with both unknown parameters

We consider the one-armed bandit problem as applied to data processing. We assume that there are two alternative processing methods and efficiency of the second method is a priory unknown. During control process, one has to determine if the second method is more efficient than the first one and to provide a primary application of the most efficient method. The essential feature of considered approach is that the data is processed in batches and cumulative incomes in batches are used for the control. If the sizes of batches are large enough then according to the central limit theorem incomes in batches are approximately Gaussian. Also if the sizes of batches are large, one can estimate the variances of incomes during the processing initial batches and then use these estimates for the control. However, for batches of moderate sizes it is reasonable to estimate unknown variances throughout the control process. This optimization problem is described by Gaussian one-armed bandit with both unknown parameters. Given a prior distribution of unknown parameters of the second action, we derive a recursive Bellman-type equation for determining corresponding Bayesian strategy and Bayesian risk. Minimax strategy and minimax risk are searched for according to the main theorem of the game theory as Bayesian ones corresponding to the worst-case prior distribution.