Optimal decision functions for two noise states

• Published in: Information and control 1967, Vol.10 (5), p.485-498
• Main Authors: Mulholland, R.G., Joshi, A.K., Chu, J.T.
• Format: Article
• Language: English
• ISSN: 0019-9958; 1878-2981
• DOI: 10.1016/S0019-9958(67)91179-5

A decision problem is considered in which the outcome of a random variable X is to be identified from the outcome of a variable Y. Due to noise, Y is subject to random variations and is itself a random variable. The probability distribution of Y is determined by the outcome of X and that of a random variable Z which is independent of X. The variable Z takes its values from the set če:italic>z k : k = 1, 2⩽ce:italic>, and the outcome of Y is said to be observed in the noise state k when Z = z k . In general, it is assumed that the a priori probabilities of all possible outcomes of X are known, but the a priori probabilities of noise states 1 and 2 are unknown. In response to an outcome of Y, the observer can either attempt to identify the outcome of X or else he can reject, i.e., refuse to attempt identification. Hence, to any decision function that the observer might use, there corresponds a probability of error and a probability of rejection for each noise state. Decision functions are introduced which minimize the probability of rejection in state 1 among all decision functions for which the probability of error in state 2 does not exceed a prescribed limit β. Among all such decision functions, it is shown how to construct those which are best in the sense that they provide the least probability of error in state 2. A special class of problems is considered in which these best decision functions provide probabilities of error in state 1 not exceeding the least probability of error that can be obtained in that state without the aid of rejection. In addition, it is shown that in some cases they also minimize the probability of rejection in state 2 among all decision functions for which the probability of error in the same state does not exceed β. Some possible applications in the areas of pattern recognition and communications have been stated.