Bayes estimators whose range has convex hull of zero prior probability

It is shown that in a general statistical decision problem Bayes procedures under convex loss can possess the following property: The induced prior measure of the convex hull of the range of the corresponding Bayes rule is equal to zero. A stronger statement holds in the case of binomial distributions.     

On the empirical Bayes approach to multiple decision problems with sequential components

The empirical Bayes approach to multiple decision problems with a sequential decision problem as the component is studied. An empirical Bayesm-truncated sequential decision procedure is exhibited for general multiple decision problems. With a sequential component, an empirical Bayes sequential decision procedure selects both a stopping rule function and a terminal decision rule function for use in the component. Asymptotic results are presented for the convergence of the Bayes risk of the empirical Bayes sequential decision procedure.     

Primal and dual linear decision rules in stochastic and robust optimization

Linear stochastic programming provides a flexible toolbox for analyzing real-life decision situations, but it can become computationally cumbersome when recourse decisions are involved. The latter are usually modeled as decision rules, i.e., functions of the uncertain problem data. It has recently been argued that stochastic programs can quite generally be made tractable by restricting the space of decision rules to those that exhibit a linear data dependence. In this paper, we propose an efficient method to estimate the approximation error introduced by this rather drastic means of complexity reduction: we apply the linear decision rule restriction not only to the primal but also to a dual version of the stochastic program. By employing techniques that are commonly used in modern robust optimization, we show that both arising approximate problems are equivalent to tractable linear or semidefinite programs of moderate sizes. The gap between their optimal values estimates the loss of optimality incurred by the linear decision rule approximation. Our method remains applicable if the stochastic program has random recourse and multiple decision stages. It also extends to cases involving ambiguous probability distributions.     

A Note on the Best Invariant Estimator of a Distribution Function Under the Kolmogorov-Smirnov Loss

For the invariant decision problem of estimating a continuous distribution function with the Kolmogorov-Smirnov loss within the class of proper– distribution functions, it is proved that the sample distribution function is the best invariant estimator only for the sample size n = 1 and 2. Further it is shown that the best invariant estimator is minimax. Exact jumps of the best invariant estimator are derived for n 4.     

Time-based detection of changes to multivariate patterns

Detection of changes to multivariate patterns is an important topic in a number of different domains. Modern data sets often include categorical and numerical data and potentially complex in-control regions. Given a flexible, robust decision rule for this environment that signals based on an individual observation vector, an important issue is how to extend the rule to incorporate time-based information. A decision rule can be learned to detect shifts through artificial data that transforms the problem to one of supervised learning. Then class probability ratios are derived from a relationship to likelihood ratios to form the basis for time-weighted updates of the monitoring scheme.     

Optimality of the Expert Rule Under Partial Information

We study the uncertain dichotomous choice model. In this model a group of decision makers is required to select one of two alternatives. The applications of this model are relevant to a wide variety of areas, such as medicine, management and banking. The decision rule may be the simple majority rule; however, it is also possible to assign more weight to the opinion of members known to be more qualified. The extreme example of such a rule is the expert decision rule. We are concerned with the probability of the expert rule to be optimal. Our purpose is to investigate the behaviour of this probability as a function of the group size for several rather general types of distributions. One such family of distributions is that where the density function of the correctness probability is a polynomial (on the interval [1/2,1]). Our main result is an explicit formula for the probability in question. This contains formerly known results as very special cases.     

On the risk-equivalence of two methods of randomization in statistics

For statistical decision problems, there are two well-known methods of randomization: on the one hand, randomization by means of mixtures of nonrandomized decision functions (randomized decision rules) in the game “statistician against nature,” on the other hand, randomization by means of randomized decision functions. In this paper, we consider the problem of risk-equivalence of these two procedures, i.e., imposing fairly general conditions on a nonsequential decision problem, it is shown that to each randomized decision rule, there is a randomized decision function with uniformly the same risk, and vice versa. The crucial argument is based on rewriting risk-equivalence in terms of Choquet's integral representation theorem. It is shown, in addition, that for certain special cases that do not fulfill the assumptions of the Main Theorem, risk-equivalence holds at least partially.     

Multiple lot-sizing decisions in a two-stage production with an interrupted geometric yield and non-rigid demand

In a production system with random yield, it may be more cost effective to release lots multiple times towards fulfilling a customer order. Such a decision, called the multiple lot-sizing problem, has been investigated in various contexts. This paper proposes an efficient algorithm for solving a new multiple lot-sizing problem defined in the context of a two-stage production system with non-rigid demand when its process yields are governed by interrupted geometric distributions. We formulate this problem as a dynamic program (DP) and develop lemmas to solve it. However, solving such a DP may be computationally extensive, particularly for large-scale cases with a high yield. Therefore, this study proposes an efficient algorithm for resolving computational issues. This algorithm is designed to reduce the DP network into a much simpler algorithm by combining a group of DP branches into a single one. Extensive experiments were carried out. Results indicate that the proposed reduction algorithm is quite helpful for practitioners dealing with large-scale cases characterized by high-yield.     

A consistent talmudic rule for division problems with?multiple references

We consider an extension of the classic division problem with claims, division problems with multiple references. We show that the theory of cooperative games is able to provide a single-valued allocation rule for this class of problems. Moreover, this rule can be related with the Talmud rule for the classic division problem, as in Aumann and Maschler (J. Econ. Theory 36, 195?C213, 1985). Finally, we establish the consistency and other basic properties of the rule.     

Expected utility for nonstochastic risk

Stochastic random phenomena considered in von Neumann–Morgenstern utility theory constitute only a part of all possible random phenomena (Kolmogorov, 1986). We show that any sequence of observed consequences generates a corresponding sequence of frequency distributions, which in general does not have a single limit point but a non-empty closed limit set in the space of finitely additive probabilities. This approach to randomness allows to generalize the expected utility theory in order to cover decision problems under nonstochastic random events. We derive the maxmin expected utility representation for preferences over closed sets of probability measures. The derivation is based on the axiom of preference for stochastic risk, i.e. the decision maker wishes to reduce a set of probability distributions to a single one. This complements Gilboa and Schmeidler’s (1989) consideration of the maxmin expected utility rule with objective treatment of multiple priors.     

大型群体决策问题中的简单多数规则（3个评选对象场合）

一个群体决策问题取决于两个因素,一个是群体决策的规则,另一个是投票。当选定群体决策规则时,一个群体决策问题由投票完全决定,此时,群体决策问题与投票之间一一对应。简单多数规则是个简单且被广泛采用的群体决策规则,但它有缺陷,我们可举出些群体决策问题使用简单多数规则没法从投票得到最后群体决策的结果。这里我们将给出一个简单多数规则的有趣性质,即在3个评选对象场合,使用简单多数规则没法从投票得到最后群体决策结果的n个评选人的群体决策问题的个数与所有n个评选人的群体决策问题的个数之比当评选人个数n趋向无穷时趋于零,这说明3个评选对象的大型群体决策场合,简单多数规则的缺陷不严重。     

f-Dissimilarity of Several Distributions in Testing Statistical Hypotheses

Various problems in statistics have been treated by the decision rule, based on the concept of distance between distributions. The aim of this paper is to give an approach for testing statistical hypotheses, using a general class of dissimilarity measures among k 2 distributions. The test statistics are obtained by the replacement, in the expression of the dissimilarity measure, of the unknown parameters by their maximum likelihood estimators. The asymptotic distributions of the resulting test statistics are investigated and the results are applied to multinomial and multivariate normal populations.     

Hit or miss: A decision support system framework for signing new musical talent

In the music industry, the process of signing new musical talent is one of the most complex decision-making problems. The decision, which is generally made by an artist and repertoire (A&R) team, involves consideration of various quantitative and qualitative criteria, and usually results in a low success rate. We conducted a series of mental model interviews with the aim of developing a decision support framework for A&R teams. This framework was validated by creating a decision support system that utilises multi-criteria decision analysis to support decision-making. Our framework and subsequent implementation of the decision support system involving decision rule and weighted sum methods show an improvement in the ability to analyse and decide on greater amounts of talent. This paper serves as a building block for developing systems to aid in this complex decision-making problem.     

渐近最优的经验贝叶斯决策

本文获得了刻度指数族变量带误差情形下的贝叶斯决策,且利用解卷积的核方法构造出了经验贝叶斯决策.在适当的条件下,证明了经验贝叶斯决策的渐近最优性.     

Prefix classes of krom formulae with identity

Two small classes of first order formulae without function symbols but with identity, in prenex conjunctive normal form with all disjunctions binary, are shown to have a recursively unsolvable decision problem, whereas for another such class an algorithm is developed which solves the decision problem of that class. This solves the prefix problem for such classes of formulae except for the Gödel-Kalmàr-Schütte case.     

A locally most powerful invariant test for the equality of means associated with covariate discriminant analysis

In this paper, the authors propose a locally most powerful invariant test for the equality of means in the presence of covariate variables. Also the null and nonnull distributions associated with the above test are developed. This problem arises in covariate discriminant analysis and has been treated by various authors, notably Cochran and Bliss 1948, Ann. Math. Statist.19, 151–176 and Rao 1949, Sankhyã9 343–366; 1966. The test derived here locally dominates in power the tests proposed so far. It is also shown that the Cochran-Bliss test is uniformly most powerful in the class of conditional invariant tests.     

Approximate credal network updating by linear programming with applications to decision making

Credal nets are probabilistic graphical models which extend Bayesian nets to cope with sets of distributions. An algorithm for approximate credal network updating is presented. The problem in its general formulation is a multilinear optimization task, which can be linearized by an appropriate rule for fixing all the local models apart from those of a single variable. This simple idea can be iterated and quickly leads to accurate inferences. A transformation is also derived to reduce decision making in credal networks based on the maximality criterion to updating. The decision task is proved to have the same complexity of standard inference, being NP^PP-complete for general credal nets and NP-complete for polytrees. Similar results are derived for the E-admissibility criterion. Numerical experiments confirm a good performance of the method.     

Condorcet Winners for Public Goods

In this work, we consider a public facility allocation problem decided through a voting process under the majority rule. A location of the public facility is a majority rule winner if there is no other location in the network where more than half of the voters would have been closer to than the majority rule winner. We develop fast algorithms for interesting cases with nice combinatorial structures. We show that the computing problem and the decision problem in the general case, where the number of public facilities is more than one and is considered part of the input size, are all NP-hard. Finally, we discuss majority rule decision making for related models.     

Bi-objective optimisation with multiple decision-makers: a convex approach to attain majority solutions

An interactive approach for solving bi-objective optimisation problems with multiple decision-makers in a context where the decisions are made via the strict majority voting rule is proposed. An adequate use of projection and decomposition techniques leads to a hierarchical algorithm in which the upper level is represented by a relaxed version of the bi-objective problem in the objective space. The feasibility of the partial consensus obtained in the upper (decision) level by the strict majority voting rule is tested in the lower (analysis) level of the algorithm. Some properties of the relaxed bi-objective problem allow that different preference structures of the decision-makers can be represented by appropriate multiobjective methods. The paper includes numerical examples that illustrate the characteristics of the approach proposed.     

群体多目标决策联合有效解类的几何特性

群体多目标决策是群体决策和多目标决策的一个交叉研究领域,借助供选方案的有效数,文[1]引进了群体多目标决策问题的联合有效解类概念,并且建立了这些解类的K－T最优条件,本文研究这类解的几何特性,得到若干基本的必要条件一充分条件。