Fechnerian Scaling: Metric from Discriminability

该项目将研究控制感知辨别的基本规律，并将进一步发展广义技术标度理论。辨别一词指的是判断两个对象是否相同或不同的能力（在所有方面，在某一特定方面，或在属于同一类别的意义上）。这是生物体的基本认知能力之一，也是人工智能系统的基本要求之一。规则极小性原则已被提出作为相同-不同判断的基本性质。从本质上说，刺激Y与刺激X的区别最小，当且仅当X与Y的区别最小。这一直觉上看似合理且经经验证实的原则是广义技术尺度理论的基石，在该理论中，刺激之间的主观距离(即从感知者的角度来看的距离，无论是人类观察者，一群人，还是一个技术系统，或者“纸和铅笔”计算程序)是从判别概率计算出来的。然而，直到最近，这种计算的逻辑被假设为对于连续刺激空间（如颜色空间）和离散空间（如颜色名称空间）是不同的。该项目的一个目标是通过开发一种适用于所有可能的刺激空间的计算逻辑来弥补这种差异（“通用技术尺度”）。这一目的是通过一个新的数学概念——不相似函数来实现的，它的公理化理论推广了度量空间的公理化理论。除了规则极小性之外，判别概率通常还表现出另一种特性，称为非恒定自不相似。如果X与Y的可辨性最小，A与B的可辨性最小，则X与Y的可辨性通常与A与B的可辨性不同。正则极小性与非常数自异性的结合对于理解相同-不同比较具有重要意义。因此，这种结合与X和Y之间的主观距离与X与Y区分的概率单调相关的假设是不相容的（该假设是广泛使用的多维尺度技术的基础，当应用于区分概率时）。规则极小性和非恒定自异性也被证明与广泛使用的表现良好的瑟斯顿模型不相容，根据瑟斯顿模型，X和Y是不同还是相同的决定是基于X和Y的随机图像，每个刺激都“足够顺利地”影响其图像的分布。然而，瑟斯顿模型可以近似规则极小性，这使得确定经验数据中规则极小性的精度限制至关重要。现有的数据不能回答这个问题，本项目旨在通过特别设计的调整/匹配程序来调查这个问题。该项目的一个相关目标是通过分析和计算机模拟来研究，在实验误差范围内可以近似规则极小性的瑟斯顿模型的传统变体是否可以产生现实的判别概率函数。广义技术尺度和同异判断理论在教育评价和专业培训、应用统计学、人工认知系统和感知辅助的构建、大规模民意调查和消费者调查的分析等方面都有应用。该项目本质上是跨学科的，连接了行为、社会、数学和计算机科学。这个项目也是国际性的。研究人员来自不同的国家，在国际会议上组织的一系列专题讨论会和讲习班将加强其他合作。与此项目相关的主题将在普渡大学和奥尔登堡大学的研究人员教授的课程中使用。将努力吸引本科生参加该项目，特别强调妇女和少数民族的参与。

This project will investigate the fundamental laws governing perceptual discrimination and will further develop the theory of Generalized Fechnerian Scaling. The term discrimination refers to the ability to decide whether two objects are the same or different (in all respects, in a specified respect, or in the sense of belonging to one and the same category). This is one of the basic cognitive abilities of living organisms and one of the basic requirements of artificial intelligent systems. The principle of Regular Minimality has been proposed as a fundamental property of same-different judgments. It says, in essence, that stimulus Y is least discriminable from stimulus X if and only if X is least discriminable from Y. This intuitively plausible and empirically corroborated principle is the cornerstone of the theory of Generalized Fechnerian Scaling, in which subjective distances among stimuli (i.e., distances "from the point of view" of a perceiver, be it a human observer, group of people, a technical system, or a "paper-and-pencil" computational procedure) are computed from discrimination probabilities. Until recently, however, the logic of this computation was posited to be different for continuous stimulus spaces, such as a space of colors, and for discrete spaces, such as a space of color names. One aim of the project is to remedy this discrepancy by developing a computational logic which would apply to all possible stimulus spaces ("Universal Fechnerian Scaling"). This aim is achieved through a new mathematical concept, Dissimilarity Function, whose axiomatic theory generalizes that of metric spaces. In addition to Regular Minimality, discrimination probabilities typically exhibit another property, called Nonconstant Self-Dissimilarity. If X is least discriminable from Y, and so is A from B, the discriminability of X from Y is not generally the same as that of A from B. The conjunction of Regular Minimality with Nonconstant Self-Dissimilarity has important consequences for understanding same-different comparisons. Thus, this conjunction is incompatible with the hypothesis that the subjective distance between X and Y is monotonically related to the probability with which X is discriminated from Y (the hypothesis underlying, among other things, the widely used techniques of Multidimensional Scaling, when applied to discrimination probabilities). Regular Minimality and Nonconstant Self-Dissimilarity are also shown to be incompatible with the widely used well-behaved Thurstonian-type models, according to which the decision whether X and Y are different or the same is based on random images of X and Y, with each stimulus affecting its image's distribution "sufficiently smoothly." The Thurstonian-type models can, however, approximate Regular Minimality, which makes it critical to determine the limits of precision with which Regular Minimality holds in empirical data. Available data do not answer this question, and this project aims at investigating it by means of specially designed adjustment/matching procedures. A related aim of the project is to investigate, analytically and by means of computer simulations, whether the conventional variants of Thurstonian-type models which can approximate Regular Minimality within limits of experimental error can generate realistic discrimination probability functions.Generalized Fechnerian Scaling and theory of same-different judgments have applications in educational assessment and professional training, in applied statistics, in the construction of artificial cognitive systems and perceptual aids, and in the analysis of large-scale polls of public opinion and consumer surveys. The project is interdisciplinary in its nature, bridging behavioral, social, mathematical, and computer sciences. The project is also international. The investigators are from different countries, and other collaborations will be bolstered by a series of symposia and workshops organized at international conferences. Topics related to this project will be used in courses taught by the investigators at Purdue and Oldenburg Universities. An effort will be made to attract undergraduate students to participate in the project, with a special emphasis on the involvement of women and minorities.