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The Generalized Universal Law of Generalization
TL;DR: In this paper, the authors show that the universal law of generalization holds with probability going to one-provided the confusion probabilities are computable, and they also give a mathematically more appealing form.
Abstract: It has been argued by Shepard that there is a robust psychological law that relates the distance between a pair of items in psychological space and the probability that they will be confused with each other. Specifically, the probability of confusion is a negative exponential function of the distance between the pair of items. In experimental contexts, distance is typically defined in terms of a multidimensional Euclidean space-but this assumption seems unlikely to hold for complex stimuli. We show that, nonetheless, the Universal Law of Generalization can be derived in the more complex setting of arbitrary stimuli, using a much more universal measure of distance. This universal distance is defined as the length of the shortest program that transforms the representations of the two items of interest into one another: the algorithmic information distance. It is universal in the sense that it minorizes every computable distance: it is the smallest computable distance. We show that the universal law of generalization holds with probability going to one-provided the confusion probabilities are computable. We also give a mathematically more appealing form
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TL;DR: It is argued that the top-down approach to modeling cognition yields greater flexibility for exploring the representations and inductive biases that underlie human cognition.
464 citations
TL;DR: A novel, principled and unified technique for pattern analysis and generation that ensures computational efficiency and enables a straightforward incorporation of domain knowledge will be presented and has the potential to reduce computational time significantly.
Abstract: The advent of multiple-point geostatistics (MPS) gave rise to the integration of complex subsurface geological structures and features into the model by the concept of training images Initial algorithms generate geologically realistic realizations by using these training images to obtain conditional probabilities needed in a stochastic simulation framework More recent pattern-based geostatistical algorithms attempt to improve the accuracy of the training image pattern reproduction In these approaches, the training image is used to construct a pattern database Consequently, sequential simulation will be carried out by selecting a pattern from the database and pasting it onto the simulation grid One of the shortcomings of the present algorithms is the lack of a unifying framework for classifying and modeling the patterns from the training image In this paper, an entirely different approach will be taken toward geostatistical modeling A novel, principled and unified technique for pattern analysis and generation that ensures computational efficiency and enables a straightforward incorporation of domain knowledge will be presented In the developed methodology, patterns scanned from the training image are represented as points in a Cartesian space using multidimensional scaling The idea behind this mapping is to use distance functions as a tool for analyzing variability between all the patterns in a training image These distance functions can be tailored to the application at hand Next, by significantly reducing the dimensionality of the problem and using kernel space mapping, an improved pattern classification algorithm is obtained This paper discusses the various implementation details to accomplish these ideas Several examples are presented and a qualitative comparison is made with previous methods An improved pattern continuity and data-conditioning capability is observed in the generated realizations for both continuous and categorical variables We show how the proposed methodology is much less sensitive to the user-provided parameters, and at the same time has the potential to reduce computational time significantly
287 citations
TL;DR: How and why the P-Cognition thesis may be overly restrictive is explained, risking the exclusion of veridical computational-level theories from scientific investigation, and an argument is made to replace the Tractable Cognition thesis by the FPT-Cognitive thesis as an alternative formalization.
215 citations
TL;DR: This paper presents a series of analyses of phonological cues and distributional cues and their potential for distinguishing grammatical categories of words in corpus analyses and indicates that phonological and Distributional cues contribute differentially towards grammatical categorisation.
205 citations
TL;DR: It is suggested that, for the vast majority of classic findings in cognitive science, embodied cognition offers no scientifically valuable insight and is also unable to adequately address the basic experiences of cognitive life.
Abstract: In recent years, there has been rapidly growing interest in embodied cognition, a multifaceted theoretical proposition that (1) cognitive processes are influenced by the body, (2) cognition exists in the service of action, (3) cognition is situated in the environment, and (4) cognition may occur without internal representations. Many proponents view embodied cognition as the next great paradigm shift for cognitive science. In this article, we critically examine the core ideas from embodied cognition, taking a "thought exercise" approach. We first note that the basic principles from embodiment theory are either unacceptably vague (e.g., the premise that perception is influenced by the body) or they offer nothing new (e.g., cognition evolved to optimize survival, emotions affect cognition, perception-action couplings are important). We next suggest that, for the vast majority of classic findings in cognitive science, embodied cognition offers no scientifically valuable insight. In most cases, the theory has no logical connections to the phenomena, other than some trivially true ideas. Beyond classic laboratory findings, embodiment theory is also unable to adequately address the basic experiences of cognitive life.
125 citations
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TL;DR: In this article, Cartwright argues that despite their great explanatory power these laws do not describe reality, but describe highly idealized objects in models, and that the correct account of explanation in science is not the traditional covering law view, but the "simulacrum" account.
Abstract: Nancy Cartwright argues for a novel conception of the role of fundamental scientific laws in modern natural science. If we attend closely to the manner in which theoretical laws figure in the practice of science, we see that despite their great explanatory power these laws do not describe reality. Instead, fundamental laws describe highly idealized objects in models. Thus, the correct account of explanation in science is not the traditional covering law view, but the ‘simulacrum’ account. On this view, explanation is a matter of constructing a model that may employ, but need not be consistent with, a theoretical framework, in which phenomenological laws that are true of the empirical case in question can be derived. Anti-realism about theoretical laws does not, however, commit one to anti-realism about theoretical entities. Belief in theoretical entities can be grounded in well-tested localized causal claims about concrete physical processes, sometimes now called ‘entity realism’. Such causal claims provide the basis for partial realism and they are ineliminable from the practice of explanation and intervention in nature.
696 citations
TL;DR: Here Shepard's theory is recast in a more general Bayesian framework and it is shown how this naturally extends his approach to the more realistic situation of generalizing from multiple consequential stimuli with arbitrary representational structure.
Abstract: Shepard has argued that a universal law should govern generalization across different domains of perception and cognition, as well as across organisms from different species or even different planets. Starting with some basic assumptions about natural kinds, he derived an exponential decay function as the form of the universal generalization gradient, which accords strikingly well with a wide range of empirical data. However, his original formulation applied only to the ideal case of generalization from a single encountered stimulus to a single novel stimulus, and for stimuli that can be represented as points in a continuous metric psychological space. Here we recast Shepard's theory in a more general Bayesian framework and show how this naturally extends his approach to the more realistic situation of generalizing from multiple consequential stimuli with arbitrary representational structure. Our framework also subsumes a version of Tversky's set-theoretic model of similarity, which is conventionally thought of as the primary alternative to Shepard's continuous metric space model of similarity and generalization. This unification allows us not only to draw deep parallels between the set-theoretic and spatial approaches, but also to significantly advance the explanatory power of set-theoretic models.
681 citations
606 citations
01 Jan 1997
TL;DR: The Journal of Symbolic Logic as discussed by the authors presents a thorough treatment of the subject with a wide range of illustrative applications, such as the randomness of finite objects or infinite sequences, Martin-Loef tests for randomness, information theory, computational learning theory, the complexity of algorithms, and thermodynamics of computing.
Abstract: The book is outstanding and admirable in many respects. ... is necessary reading for all kinds of readers from undergraduate students to top authorities in the field. Journal of Symbolic Logic Written by two experts in the field, this is the only comprehensive and unified treatment of the central ideas and their applications of Kolmogorov complexity. The book presents a thorough treatment of the subject with a wide range of illustrative applications. Such applications include the randomness of finite objects or infinite sequences, Martin-Loef tests for randomness, information theory, computational learning theory, the complexity of algorithms, and the thermodynamics of computing. It will be ideal for advanced undergraduate students, graduate students, and researchers in computer science, mathematics, cognitive sciences, philosophy, artificial intelligence, statistics, and physics. The book is self-contained in that it contains the basic requirements from mathematics and computer science. Included are also numerous problem sets, comments, source references, and hints to solutions of problems. New topics in this edition include Omega numbers, KolmogorovLoveland randomness, universal learning, communication complexity, Kolmogorov's random graphs, time-limited universal distribution, Shannon information and others.
593 citations
TL;DR: A sequence distance that works on unaligned sequences using the information theoretical concept of Kolmogorov complexity and a program to estimate this distance is presented.
Abstract: Motivation: Traditional sequence distances require an alignment and therefore are not directly applicable to the problem of whole genome phylogeny where events such as rearrangements make full length alignments impossible. We present a sequence distance that works on unaligned sequences using the information theoretical concept of Kolmogorov complexity and a program to estimate this distance. Results: We establish the mathematical foundations of our distance and illustrate its use by constructing a phylogeny of the Eutherian orders using complete unaligned mitochondrial genomes. This phylogeny is consistent with the commonly accepted one for the Eutherians. A second, larger mammalian dataset is also analyzed, yielding a phylogeny generally consistent with the commonly accepted one for the mammals. Availability: The program to estimate our sequence distance, is available at http://www.cs.cityu.edu.hk/ ∼cssamk/gencomp/GenCompress1.htm. The distance matrices used to generate our phylogenies are available at http://www.math.uwaterloo.ca/∼mli/distance.html
564 citations