Abstract: In a multiple decision problem one has to choose the ``correct'' distribution out of a number of different distributions for an observation . When is a random sample, it is known that the minimum Bayes risk decays at exponential rate, which coincides with that of the minimax risk, and is determined by an information-type divergence between these distributions.
There are situations when it is desirable to allow new possible decisions. For example, if the data does not provide enough support to any of the models, one may want to allow a ``no-decision'' or ``rejection'' option. Another example of such a situation is the confidence estimation problem where the ``correct'' decisions correspond to one-point sets, and new non-standard actions are formed by subsets of the parameter space consisting of at least two elements.
In the version of the multiple decision problem with augmented action space, we derive the optimal exponential rate of the minimum Bayes risk, and show that it coincides with the mentioned information-type divergence in the classical multiple decision problem. However, the component of the Bayes risk corresponding to the error occurring when the decision belongs to the standard action space may decrease at a faster exponential rate. In a binomial example the accuracy of two asymptotic formulas for the risks containing oscillating (diverging) factors is compared.
Key words and phrases: Bayes risk, binomial distribution, Chernoff theorem, decision space, error probability, loss function, probabilities of large deviations.