In this paper we present a method for approximating the risks and Bayes risk associated with a Bayes decision procedure. Additionally, our method leads to approximating the least favorable distribution and the risk associated with the minimax decision procedure. We assume two states of nature (or classes of patterns) and multivariate probability density functions. Taylor series expansions are used, and an nth-order polynomial equation derived from such expansions provides an approximation to one of the least favorable probabilities. An application to a normally distributed random vector of observables is presented with numerical comparisons. The method can be generalized to cases having more than two states of nature by using Taylor series expansions in several variables.

For the problem of classifying an element (e.g., an unknown pattern) into one of two given categories where the associated observables are distributed according to one of two known multivariate normal populations having a common covariance matrix, it is shown that the minimum Bayes risk is a strict monotonic function of certain separability or statistical distance measures regardless of the a priori probabilities and the assigned loss function. However, for the associated conditional expected losses, strict monotonicity holds, if and only if a certain condition dependent on these probabilities and the given loss function is satisfied. These results remain valid for classification problems in which the observable can be transformed by a one-to-one differentiable mapping to normality.

For the average error probability Pe associated with the Bayes recognition procedures for two possible patterns, using no context, new upper and lower bounds and approximations are obtained. Results are given in terms of simple functions of feature "reliability" and a priori probabilities of the patterns. Two kinds of feature "reliability" are considered, i.e., distance between probability distributions and error probabilities without the use of a priori probabilities. Computational advantages offered by those bounds and approximations are pointed out. The question as to how close they are to Peis examined. In some special cases, they are perfect. Numerical examples show that the differences are in general about 5-10 percent, and comparisons with certain known results are quite favorable. Possible applications are discussed. Extension is also made to m possible patterns arranged in a hierarchy with two elements at each branching.

The general problem of the use of context in computer character recognition is briefly reviewed. For the special case where the context is generated by a two-state stationary Markov chain, upper bounds are obtained for the average error probability of an optimal recognition procedure, based on compound decision functions. These bounds are nonparametric and simple functions of the "differences" between: 1) the a priori and transition probabilities of the context, and 2) the distributions of the measurements used to identify the characters. Some justiflcations, applications to systems design, and illustrative examples are given. An improvement is also obtained on a previously derived upper bound for procedures using no context.

Communication systems are considered in which there exists an auxiliary channel over which the receiver can make requests for retransmission. In general, it is assumed that the auxiliary channel is noiseless and the main channel is discrete and memoryless. It is also assumed that the a priori probabilities of the message symbols are known. A sequential receiver strategy is introduced which at each step minimizes the probability of retransmission subject to the condition that the probability of incorrect decoding does not exceed a given limit. The strategy is easy to implement in the case where group codes are used as error correctors and detectors. Moreover, in this case, conditions are given for which the strategy terminates with probability I, regardless of whether it is truncated. Examples are given for illustration.

Upper bounds for the error probability of a Bayes decision function are derived in terms of the differences among the probability distributions of the features used in character recognition. Applications to feature selection and error reduction are discussed. It is shown that if a sufficient number of well-selected features is used, the error probability can be made arbitrarily small.

The use of statistical decision functions with computers for character recognition is investigated.The three eases considered are (1) where both the losses due to incorrect decisions and the a priori probability of the characters are known, (2) where the a priori probability is known, but the losses are not, and (3) the reverse of the second ease.For the first ease, Bayes decision functions are reviewed.A theorem about the advantage of using rejection is proved.For the second ease, minimum error and minimum rejection decision functions are defined and obtained.For the third ease, admissible, and a complete class of, decision functions are discussed.Illustrative ex'~mples are given.

Several methods for simplifying switching circuits using “don't-care” conditions are suggested. One of the methods uses the following procedure: Let the given circuit be represented by the truth function F . Let the given “don't-care” conditions be denoted by D = 0. (For example, if x 1 = 1 and x 2 = 0 represent the combination of input signals which never takes place, then D = x 1 x 2 .) Generate all the irredundant disjunctive and conjunctive forms which are equivalent to F whenever D = 0. From these forms, the simplest ones may then be chosen according to a given measure of simplicity. They correspond to the simplest two-level AND/OR or OR/AND switching circuits which, under the “don't-care” conditions, perform the same function as the given circuit. Several methods for generating the equivalent irredundant forms are suggested. They are generalizations of those due to Quine, Ghazala and Mott, and may be programmed for use on computers. Alternative simplification methods are also given.

A method of Quine for identifying the core prime implicants of a given truth function, without obtaining all its prime implicants, is generalized under the so-called ``don't care'' conditions. It is shown that our method is equivalent to, and sometimes an improvement of, a result of Roth. When all the prime implicants (under the don't care conditions) of a truth function are given, our method becomes a generalization of a result of Ghazala and is equivalent to another result of Roth. It is also pointed out that our method may be used, in a way similar to using Roth's, for simplifying truth functions.