Least-squares approximation of an improper correlation matrix by a proper one

An algorithm is presented for the best least-squares fitting correlation matrix approximating a given missing value or improper correlation matrix. The proposed algorithm is based upon a solution for Mosier's oblique Procrustes rotation problem offered by ten Berge and Nevels. A necessary and sufficient condition is given for a solution to yield the unique global minimum of the least-squares function. Empirical verification of the condition indicates that the occurrence of non-optimal solutions with the proposed algorithm is very unlikely. A possible drawback of the optimal solution is that it is a singular matrix of necessity. In cases where singularity is undesirable, one may impose the additional nonsingularity constraint that the smallest eigenvalue of the solution be , where is an arbitrary small positive constant. Finally, it may be desirable to weight the squared errors of estimation differentially. A generalized solution is derived which satisfies the additional nonsingularity constraint and also allows for weighting. The generalized solution can readily be obtained from the standard unweighted singular solution by transforming the observed improper correlation matrix in a suitable way.     

The comparison problem for approximating epistemic ideals

Some epistemologists think that the Bayesian ideals matter because we can approximate them. That is, our attitudes can be more or less close to the ones of our ideal Bayesian counterpart. In this paper, I raise a worry for this justification of epistemic ideals. The worry is this: In order to correctly compare agents to their ideal counterparts, we need to imagine idealized agents who have the same relevant information, knowledge, or evidence. However, there are cases in which one's ideal counterpart cannot have one's information, knowledge, or evidence. In these situations, agents cannot compare themselves to their ideal counterpart.     

An alternating least squares method for the weighted approximation of a symmetric matrix

Bailey and Gower examined the least squares approximationC to a symmetric matrixB, when the squared discrepancies for diagonal elements receive specific nonunit weights. They focussed on mathematical properties of the optimalC, in constrained and unconstrained cases, rather than on how to obtainC for any givenB. In the present paper a computational solution is given for the case whereC is constrained to be positive semidefinite and of a fixed rankr or less. The solution is based on weakly constrained linear regression analysis.The authors are obliged to John C. Gower for stimulating this research.     

Testing for independence between pairs of autocorrelated binomial data sequences

A problem arises in analyzing the existence of interdependence between the behavioral sequences of two individuals: tests involving a statistic such as chi-square assume independent observations within each behavioral sequence, a condition which may not exist in actual practice. Using Monte Carlo simulations of binomial data sequences, we found that the use of a chi-square test frequently results in unacceptable Type I error rates when the data sequences are autocorrelated. We compared these results to those from two other methods designed specifically for testing for intersequence independence in the presence of intrasequence autocorrelation. The first method directly tests the intersequence correlation using an approximation of the variance of the intersequence correlation estimated from the sample autocorrelations. The second method uses tables of critical values of the intersequence correlation computed by Nakamuraet al. (J. Am. Stat. Assoc., 1976,71, 214–222). Although these methods were originally designed for normally distributed data, we found that both methods produced much better results than the uncorrected chi-square test when applied to binomial autocorrelated sequences. The superior method appears to be the variance approximation method, which resulted in Type I error rates that were generally less than or equal to 5% when the level of significance was set at .05.     

A more powerful method for testing for agreement between a judge and a known standard

We assume that a judge's task is to categorize each ofN subjects into one ofr known classes. The design of primary interest is employed if the judge is presented withs groups, each containingr subjects, such that each group of sizer consists of exactly one subject of each of ther types. The probability distribution for the total number of correct choices is developed and used to test the null hypothesis that the judge is guessing in favor of the alternative that he or she is operating at a better than chance level. The power of the procedure is shown to be superior to two other procedures which appear in the literature.The authors are grateful for the suggestions of the referees and for computer funding provided by the Northeast Regional Data Center at the University of Florida.     

Refuting the unfolding-argument on the irrelevance of causal structure to consciousness

The unfolding argument (UA) was advanced as a refutation of prominent theories, which posit that phenomenal experience is determined by patterns of neural activation in a recurrent (neural) network (RN) structure. The argument is based on the statement that any input–output function of an RN can be approximated by an “equivalent” feedforward-network (FFN). According to UA, if consciousness depends on causal structure, its presence is unfalsifiable (thus non-scientific), as an equivalent FFN structure is behaviorally indistinguishable with regards to any behavioral test. Here I refute UA by appealing to computational theory and cognitive-neuroscience. I argue that a robust functional equivalence between FFN and RN is not supported by the mathematical work on the Universal Approximator theorem, and is also unlikely to hold, as a conjecture, given data in cognitive neuroscience; I argue that an equivalence of RN and FFN can only apply to static functions between input/output layers and not to the temporal patterns or to the network’s reactions to structural perturbations. Finally, I review data indicating that consciousness has functional characteristics, such as a flexible control of behavior, and that cognitive/brain dynamics reveal interacting top-down and bottom-up processes, which are necessary for the mediation of such control processes.     

贝叶斯决策理论对复杂运动决策中运动预期的启发——以网球和足球为例

对运动决策的研究是认知运动心理学的一个重要领域。运动预期被认为是运动决策的核心, 受到运动学和非运动学信息的影响。其中, 对运动预期研究的一个关键问题是探讨不同信息源对运动结果预期的贡献以及两者之间的相互作用。研究者运用贝叶斯决策理论解释运动预期中不同信息的整合过程, 分析运动员在复杂的竞赛情景中如何做出最佳决策, 尤其是对该理论在网球和足球领域的潜在应用进行分析。在不确定的情况下, 竞技体育中并非所有的选择、结果或概率都是已知的, 故有研究者认为概率论和经典的决策理论不能有效解决此类问题。然而新近提出的启发式近似, 为运动员在贝叶斯框架下如何快速做出选择提供了理论依据：首先, 在复杂和有时间压力的竞赛情景中, 启发式近似假设运动员依据竞赛中不同信息源的不确定程度, 很可能选择在运动学信息和情境先验之间进行切换启发式, 提高运动预期的效率。其次, 判断效用通过卷积效应影响两种信息源的整合, 降低情境先验的影响程度。     

Approximation of Value Efficient Solutions

We introduce an approximation set to the value efficient set in multiobjective problems under partial information on the decision maker's preferences modelled by a vector value function. We show monotonicity and convergence properties based on increasingly precise vector value functions with two components, which improve the approximation and might be a support to possible solution methods. © 1997 John Wiley & Sons, Ltd.