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This article develops multiple-choice test scoring rules, concentrating on Bayes rules and their frequency theory analogs, empirical Bayes rules. Conditions are given for empirical Bayes estimates to lie in the probability simplex. The misinformation model is considered in detail. It is shown that ranking by raw scores is equivalent to ranking by Bayes scores when the loss function increases with error and the sampling distribution has the monotone likelihood ratio property. Application of the techniques is made to data from a multiple-choice test given to students of an elementary statistics course.

SUMMARY The null hypothesis of row-column independence in a two-way contingency table can be expressed as a constraint on the parameters in various standard statistical sampling models. Four sampling models are considered, which are related by nested conditioning. By having the prior distribution in any one model induce the prior distribution in each further conditioned model, it is shown that the Bayes factors for independence will factorize, and thereby expose the evidence residing in the marginal row and column of the table. Bounds on the marginal Bayes factors justify, in a weak sense, Fisher's practice of conditioning. A general theorem is given for factorized Bayes factors from a factorized likelihood function.

During production, storage and shipment, a crop of feed grain is often contaminated with admixtures of other grains or foreign material. Failure to meet the contractural limits of purity can result in monetary penalties or may prevent completion of the sales transaction. Presently, purity is established by human inspectors. They subject a carefully selected sample to 100 per cent visual inspection. This paper describes part of the development of a high speed, automatic sorting and grading procedure. A recursive learning pattern classification scheme is described which yields an overall accuracy of about 98 per cent.

Summary The data that are used in constructing empirical Bayes estimates can properly be regarded as arising in a two-stage sampling scheme. In this setting it is possible to modify the conventional parameter estimates so that a reduction in expected squared error is effected. In the empirical Bayes approach this is done through the use of Bayes's theorem. The alternative approach proposed in this paper specifies a class of modified estimates and then seeks to identify that member of the class which yields the minimum squared error. One advantage of this approach relative to the empirical Bayes approach is that certain problems involving multiple parameters are easily overcome. Further, it permits the use of relatively efficient methods of non-parametric estimation, such as those based on quantiles or ranks; this has not been achieved by empirical Bayes methods.

Compromises between Bayesian and non-Bayesian significance testing are exemplified by examining distributions of criteria for multinominal equiprobability. They include Pearson's X2, the likelihood-ratio, the Bayes factor F, and a statistic G that previously arose from a Bayesian model by “Type II Maximum Likelihood.” Its asymptotic distribution, implied by the theory of the “Type II Likelihood Ratio,” is remarkably accurate into the extreme tail. F too can be treated as a non-Bayesian criterion and is almost equivalent to G. The relationship between F and its own tail area sheds further light on the relationship between Bayesian and “Fisherian” significance.

This paper presents the mathematical derivation of a Bayes solution for the symmetric multiple comparisons problem presented earlier by Waller and Duncan [4]. Because the earlier paper details the model and assumptions on which the solution is based, only the functions and results needed to complete the development are stated here. Typically the data for a multiple comparisons problem consists of n independent normally distibuted t r ea tmen t means (~, z2, " " , ~ ) which est imate n t rue means (pl, p~, . . . , p~) and an independent estimate, say 8~E with f~ degrees of freedom, of the error variance a~. Fur ther , we suppose tha t each mean ~ is based on r replications. The problem is then to choose for each of the h = n ( n 1 ) ordered pairs of means ei ther decision d~: ~,>;~j or decision d o , j : /~ not ranked relative to/J j . However, Duncan [2] shows tha t under the assumptions of symmet ry and an additive loss model, the Bayes rule for the full multiple comparisons problem is the simultaneous application of the Bayes rule to select ei ther d~ or d[~ for the means (/~,/~2) to all h pairs of means. Thus our major concern is to obtain the Bayes rule for selecting one of the decisions

Two nonparametric methods to estimate the Bayes risk using classified sample sets are described and compared. The first method uses the nearest neighbor error rate as an estimate to bound the Bayes risk. The second method estimates the Bayes decision regions by applying Parzen probability-density function estimates and counts errors made using these regions. This estimate is shown to be asymptotically consistent in mean square. The results of experiments with these estimators on simulated and empirical data imply that the estimators both have acceptable small-sample properties; however, small-sample convergence of both estimators depends strongly on the choice of metric and local area or window size in the measurement space.

Alterations in probability densities produced by iterative application of Bayes' rule are analyzed. Computational requirements for finding a sequence of a posteriori densities remain reasonable despite a growing sample size if and only if a sufficient statistic expressible as a vector of fixed dimension exists. The existence of such a sufficient statistic also insures existence of reproducing a priori probability densities (a priori probability densities insuring that the a posteriori densities are in the same family). The theory is applied to find a class of sequential Bayes estimators for a Gaussian covariance matrix, and to treat a variety of adaptive "Bayesian learning" schemes in a unified manner.

A statistical decision model based on regression and Bayesian classification techniques is designed to help predict the most favorable drilling targets in regional exploration programs. The procedure is applied to a 225 square mile portion of the southwest Wisconsin zinc area hitherto little explored. The analysis led to the selection of 22 targets with high success probabilities (0.80 or better) and high regression estimates.The validity of the method depends on the quality of field data and on a careful selection of relevant factors rather than the nature of the field techniques used for the coverage. The aim of the method is to assist the exploration management in making objective decisions fully consistent with the initial goals of the programs.

A result of Doob regarding consistency of Bayes estimators is extended to a large class of Bayes decision procedures in which the loss functions are not necessarily convex. Rather weak conditions are given under which the Bayes procedures are consistent. One set involves restrictions on the a priori distribution and follows an example in which the choice of a priori distribution determines whether the Bayes estimators are consistent. Another example shows that the maximum likelihood estimators may be consistent when the Bayes estimators are not. However, the conditions given are of an essentially weaker nature than those established for consistency of maximum likelihood estimators.

A 2-stage classification model is presented in which the first stage is a quick, computerized Bayes rule decision device, and the second a slow, but perfectly accurate, classifier. A stationary stream of elements or objects to be classified into one of several mutually exclusive categories is fed into the model. The conditional probabilities associated with the Bayes device are assumed unknown at the outset, except up to an initial probability distribution. The a posteriori probabilities from the first stage are treated as information that can speed up or slow down the processing time in the second stage. The latter, after a delay time, feeds back accurate classification information to the first stage to update the conditional probabilities. It is shown that, as the classification process unfolds, any updating scheme that causes the Bayes classifier ultimately to learn the true values of the conditional probabilities also minimizes the expected processing time in the second stage. The learning rate of the system is discussed as a function of the updating scheme. An example of a simple system is presented and the learning rate is derived specifically for that case.

A smooth empirical Bayes estimator is derived for the intensity parameter (hazard rate) in the Poisson distribution as used in life testing. The reliability function is also estimated either by using the empirical Bayes estimate of the parameter, or by obtaining the expectation of the reliability function. The behavior of the empirical Bayes procedure is studied through Monte Carlo simulation in which estimates of mean-squared errors of the empirical Bayes estimators are compared with those of conventional estimators such as minimum variance unbiased or maximum likelihood. Results indicate a significant reduction in mean-squared error of the empirical Bayes estimators over the conventional variety.

Maritz (1966) and Lemon & Krutchkoff (1969) each describe discrete empirical Bayes smoothing techniques. These techniques essentially attempt to approximate the prior distribution function. Here a continuous smoothing technique which is based on a smooth and continuous approximation to the prior density function is presented. Results from a Monte Carlo study of the Poisson distribution are reported which show that the continuous smoothing technique has desirable small-sample properties. Some comparisons with discrete smoothing techniques are also made.

It is shown that for a class of cost functions and a class of a priori distributions on the parameters to be estimated, the risks associated with the Bayes and maximum-likelihood estimates of the signal parameters become equal as the number of observations becomes large. This implies that the optimality properties usually attributed to Bayes estimates can be attributed in some cases to maximum-likelihood estimates as well. The work is an extension of known results to the case in which the statistical description of the observed process may be nonstationary.

In point estimation with a squared-error loss function the Bayes estimator is the posterior mean. In the empirical Bayes approach we must construct a consistent sequence of estimators for this posterior mean using past experience. This construction is done here for four general families of conditional distributions which include as special cases: the Poisson, the gamma, the normal and the uniform.

An algorithm is presented for calculating recognition error when applying pattern vectors to an optimum Bayes' classifier. The pattern vectors are assumed to come from two classes whose populations have Gaussian statistics with unequal covariance matrices and arbitrary a priori probabilities. The quadratic discriminant function associated with a Bayes' classifier is used as a one-dimensional random variable from which the probability of error is calculated, once the distribution of the discriminant function is obtained.

SUMMARY A number of established methods of empirical Bayes estimation for the Poisson distribution are surveyed, and their performances are compared. A new method of smooth empirical Bayes estimation is presented, and its performance is also studied. The question of assessing the effectiveness of the empirical Bayes approach in practice is examined.

Traditionally, discriminant functions have been used to classify individuals according to a linear function of the individuals’ measured characteristics. This article develops a Bayesian classification procedure which explicitly takes into account the costs of misclassification and attribute measurement. The article presents the results of a computer program designed to determine the optimal number of characteristics to measure and discusses the results’ implications for consumer research.

A study is made of an empirical Bayes estimation problem in which the set of parameter values is a realization of a stationary Markov chain. The Bayes rule is derived as a function of the transition probabilities of the Markov chain. The performance of empirical Bayes rules, which use estimates of these transition probabilities based on the observations, is investigated by simulation. Two other rules are shown to be competitors of the empirical Bayes rules in some particular cases. These are a two-step rule which uses only the (i -1 )th, ith and (i + 1)th observations when making the ith decision, and a rule based on the mode of the posterior distribution of the parameter sequence.

It is shown that for n-valued conditionally independent features a large family of classifiers can be expressed as an (n?1)st-degree polynomial discriminant function. The usefulness of the polynomial expansion is discussed and demonstrated by considering the first-order Minkowski metric, the Euclidean distance, and Bayes' classifiers for the ternary-feature case. Finally, some interesting side observations on the classifiers are made with respect to optimality and computational requirements.

An empirical Bayes procedure is introduced for M -ary detection in a statistically undefined noise background. The Bayes decision rule, which depends on a set of nuisance parameters, is formed first from sample qualitative information so that the requirement of known noise distribution can be relaxed. The rule is then approximated by estimating the nuisance parameters. The resulting empirical procedure possesses the desirable property of consistency and is simple to implement.

(1973). Some Comments on “Bayes” Estimators. The American Statistician: Vol. 27, No. 1, pp. 12-14.

In this paper the asymptotic behaviors of the Bayes test and Bayes risk are studied in both the one-sided and two-sided testing problems, where the independent observations are taken from one member of a one-parameter exponential family. Precise asymptotic expressions are found which show the Bayes procedure to be relatively insensitive to the prior distribution as the sample size increases to infinity.

There are now a vast array of heterogeneous cloud service resources, which makes it difficult to identify suitable services for the various types of cloud users. A classification of cloud service resources would help users find suitable cloud services more easily. We propose such a classification strategy, which has two parts. First, we improve the original naive Bayesian classification algorithm, designing a Bayesian classification algorithm based on feature similarity. Second, to improve the efficiency of the classification algorithm, we design a parallel programming model using the Hadoop platform. Simulation results show that the proposed classification strategy is feasible and effective, improving not only the resource classification accuracy but also greatly enhancing the processing efficiency for large-scale cloud service resources

The construction of proper Bayes tests which are invariant under non-compact groups has been treated in Kiefer and Schwartz (1965) and Schwartz (1966, 1969). These papers are limited to testing problems concerning exponential families' $\operatorname{pdf}$'s. The present note shows, by means of two simple examples, that the same methods can sometimes be applied more generally. One example considers estimation of the ratio of the mean to the variance for the normal distribution. The second example considers testing problems in which the pdf's have an exponential factor as well as a non-exponential factor.

This paper discusses methods of estimating the probability of error for the Bayes' classifier which must be designed and tested with a finite number of classified samples. The expected difference between estimators is discussed. A simplified algorithm to compute Lachenbruch's method is proposed for multivariate normal distributions with unequal covariance matrices. Also, the variances of the likelihood ratios are given so as to compare them with the differences between the estimates. The discussion is extended to nonparametric classifiers by using the Parzen approximation for the density functions. Experimental results are shown for both parametric and nonparametric cases.

This chapter presents a problem of which a sound mathematical basis for dealing with the problem of probability of hypotheses was suggested by Thomas Bayes. The aim is to make an inference on the unknown probability of deuce by means of the known outcome of 600 (= n ) trials. The theory is known under various headings — probability of hypotheses, inverse probability, and probability of inference. The chapter also presents the idea leading to Bayes' solution of this problem and Bayes' solution. It also discusses analogies of Bayes' distribution to Bernoulli's distribution. The chapter presents the Bernoulli's distribution with the relative number of successes z = x / n , as the independent variable. The most significant analogy between the Bayes' distribution and the Bernoulli's distribution is the fact that the variance goes to zero as n increases indefinitely. This means that with increasing n, both distributions become more and more concentrated around the point whose abscissa is, at the same time, the mean value and the location of the maximum. The chapter also discusses the implications of this concentration in the case of the Bayes' problem for general p 0 ( x ).

A model is presented for evaluating potential effectiveness of a Bayesian classification system using the expected value of the posterior probability for true classifications as an evaluation metric. For a given set of input parameters, the value of this complex metric is predictable from a simply computed row variance metric. Prediction equations are given for several representative sets of input parameters.

A two stage classification model is presented in which the first stage is a quick computerizd Bayes Rule decision device, and the second is a slow, but perfectly accurate, classifier. A stationary stream of elements or objects to be classified into one of several mutually exclusive categories is fed into the model. The conditional probabilities associated with the Bayes device are assumed unknown at the outset, except up to an initial probability distribution. The a posteriori probabilities from the first stage are treated as information that can speed up or slow down the processing time in the second stage. The latter, after a delay time, feeds back accurate classification information to the first stage to update the conditional probabilities. It is shown that, as the classification process unfolds, any updating scheme that causes the Bayes Classifier ultimately to "learn" the true values of the conditional probabilities also minimizes the expected processing time in the second stage. The learning rate of the system is discussed as a function of the updating scheme. An example of a simple system is presented and the learning rate is derived specifically for that case.

Let $X$ be a normal random variable with variance one and mean either $\pm a/2$, where $a$ is a given positive constant. Let $C_m(m \geqq 0)$ denote the class of all two-stage rules with first sample size of $m$ for deciding, after two successive samples of independent observations on $X$, which of the two mean values is correct. This paper investigates the class of Bayes rules in $C_m$, parametrized by a priori probabilities on the hypotheses, and simple wrong decision losses. The cost per observation is taken throughout to be unity. Section 1 gives some general properties of Bayes rules in $C_m$ for decisions between any two continuous densities for $X$; Sections 2, 3, and 4 concern the densities specified above. Section 2 consists of a detailed development of Bayes second sample size properties in terms of the Bayes parameters and first sample outcomes. For example, Theorem 2.1 gives non-trivial lower and upper bounds for positive values of the Bayes second sample size corresponding to any fixed value for the minimum wrong decision loss. In Section 3, sufficient conditions are given under which the losses may be chosen so as to obtain Bayes rules with preassigned invariant error probabilities. (Invariance is taken with respect to changes in the prior probabilities.) It is shown how this result leads to rules which minimize the maximum expected sample size among rules in $C_m$ with error probabilities less than or equal to specified values. An illustrative example is considered for the case when these specified bounds are equal. The selection of an optimum first sample size for this example is treated in Section 4. The resulting rule has the above described good property among all two-stage rules (of any first sample size) subject to these bounds. Tables are included giving optimum first and second sample sizes and the values of auxiliary functions when this common bound on the error probabilities is .05 and .01.

Given the usual Bayesian scenario for the estimation problem, with quadratic loss function and the assumption of finite posterior risk, it is shown that the Bayes risk of a rule, δ, can be calculated in the following manner: The result is computationally expedient when the investigation of the behavior of a class of rules requires the use of a computer.

For reference to the article under discussion, see this Bibliography Vol. 34, No. 6, 02 E70-17081

This work evaluates and compares different supervised learning algorithms using a costsensitive approach to find a model that classifies legal rules related to pesticides as prohibitions and permissions. The naive Bayes classifier achieves the best results and it would be applicable because it doesn't misclassify prohibitions as permissions.

For reference to the articles under discussion, see this Bibliography Vol. 34, No. 6 02 E70-17081