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The magical number seven plus or minus two: some limits on our capacity for processing information
01 Jan 1956-
TL;DR: The theory provides us with a yardstick for calibrating the authors' stimulus materials and for measuring the performance of their subjects, and the concepts and measures provided by the theory provide a quantitative way of getting at some of these questions.
Abstract: First, the span of absolute judgment and the span of immediate memory impose severe limitations on the amount of information that we are able to receive, process, and remember. By organizing the stimulus input simultaneously into several dimensions and successively into a sequence or chunks, we manage to break (or at least stretch) this informational bottleneck. Second, the process of recoding is a very important one in human psychology and deserves much more explicit attention than it has received. In particular, the kind of linguistic recoding that people do seems to me to be the very lifeblood of the thought processes. Recoding procedures are a constant concern to clinicians, social psychologists, linguists, and anthropologists and yet, probably because recoding is less accessible to experimental manipulation than nonsense syllables or T mazes, the traditional experimental psychologist has contributed little or nothing to their analysis. Nevertheless, experimental techniques can be used, methods of recoding can be specified, behavioral indicants can be found. And I anticipate that we will find a very orderly set of relations describing what now seems an uncharted wilderness of individual differences. Third, the concepts and measures provided by the theory of information provide a quantitative way of getting at some of these questions. The theory provides us with a yardstick for calibrating our stimulus materials and for measuring the performance of our subjects. In the interests of communication I have suppressed the technical details of information measurement and have tried to express the ideas in more familiar terms; I hope this paraphrase will not lead you to think they are not useful in research. Informational concepts have already proved valuable in the study of discrimination and of language; they promise a great deal in the study of learning and memory; and it has even been proposed that they can be useful in the study of concept formation. A lot of questions that seemed fruitless twenty or thirty years ago may now be worth another look. In fact, I feel that my story here must stop just as it begins to get really interesting. And finally, what about the magical number seven? What about the seven wonders of the world, the seven seas, the seven deadly sins, the seven daughters of Atlas in the Pleiades, the seven ages of man, the seven levels of hell, the seven primary colors, the seven notes of the musical scale, and the seven days of the week? What about the seven-point rating scale, the seven categories for absolute judgment, the seven objects in the span of attention, and the seven digits in the span of immediate memory? For the present I propose to withhold judgment. Perhaps there is something deep and profound behind all these sevens, something just calling out for us to discover it. But I suspect that it is only a pernicious, Pythagorean coincidence.
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TL;DR: Ajzen, 1985, 1987, this article reviewed the theory of planned behavior and some unresolved issues and concluded that the theory is well supported by empirical evidence and that intention to perform behaviors of different kinds can be predicted with high accuracy from attitudes toward the behavior, subjective norms, and perceived behavioral control; and these intentions, together with perceptions of behavioral control, account for considerable variance in actual behavior.
65,095 citations
TL;DR: The nearest neighbor decision rule assigns to an unclassified sample point the classification of the nearest of a set of previously classified points, so it may be said that half the classification information in an infinite sample set is contained in the nearest neighbor.
Abstract: The nearest neighbor decision rule assigns to an unclassified sample point the classification of the nearest of a set of previously classified points. This rule is independent of the underlying joint distribution on the sample points and their classifications, and hence the probability of error R of such a rule must be at least as great as the Bayes probability of error R^{\ast} --the minimum probability of error over all decision rules taking underlying probability structure into account. However, in a large sample analysis, we will show in the M -category case that R^{\ast} \leq R \leq R^{\ast}(2 --MR^{\ast}/(M-1)) , where these bounds are the tightest possible, for all suitably smooth underlying distributions. Thus for any number of categories, the probability of error of the nearest neighbor rule is bounded above by twice the Bayes probability of error. In this sense, it may be said that half the classification information in an infinite sample set is contained in the nearest neighbor.
12,243 citations
13 Aug 2016
TL;DR: In this article, the authors propose LIME, a method to explain models by presenting representative individual predictions and their explanations in a non-redundant way, framing the task as a submodular optimization problem.
Abstract: Despite widespread adoption, machine learning models remain mostly black boxes. Understanding the reasons behind predictions is, however, quite important in assessing trust, which is fundamental if one plans to take action based on a prediction, or when choosing whether to deploy a new model. Such understanding also provides insights into the model, which can be used to transform an untrustworthy model or prediction into a trustworthy one. In this work, we propose LIME, a novel explanation technique that explains the predictions of any classifier in an interpretable and faithful manner, by learning an interpretable model locally varound the prediction. We also propose a method to explain models by presenting representative individual predictions and their explanations in a non-redundant way, framing the task as a submodular optimization problem. We demonstrate the flexibility of these methods by explaining different models for text (e.g. random forests) and image classification (e.g. neural networks). We show the utility of explanations via novel experiments, both simulated and with human subjects, on various scenarios that require trust: deciding if one should trust a prediction, choosing between models, improving an untrustworthy classifier, and identifying why a classifier should not be trusted.
11,104 citations
TL;DR: It is proposed that cognitive control stems from the active maintenance of patterns of activity in the prefrontal cortex that represent goals and the means to achieve them, which provide bias signals to other brain structures whose net effect is to guide the flow of activity along neural pathways that establish the proper mappings between inputs, internal states, and outputs needed to perform a given task.
Abstract: ▪ Abstract The prefrontal cortex has long been suspected to play an important role in cognitive control, in the ability to orchestrate thought and action in accordance with internal goals. Its neural basis, however, has remained a mystery. Here, we propose that cognitive control stems from the active maintenance of patterns of activity in the prefrontal cortex that represent goals and the means to achieve them. They provide bias signals to other brain structures whose net effect is to guide the flow of activity along neural pathways that establish the proper mappings between inputs, internal states, and outputs needed to perform a given task. We review neurophysiological, neurobiological, neuroimaging, and computational studies that support this theory and discuss its implications as well as further issues to be addressed
10,943 citations
Cites background from "The magical number seven plus or mi..."
...It may also be important to distinguish the capacity limits of cognitive control from those of short-term storage of item information (e.g. verbal or visual short term memory) (Miller 1956, Baddeley 1986)....
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TL;DR: This paper reviewed the evidence for multistore theories of memory and pointed out some difficulties with the approach and proposed an alternative framework for human memory research in terms of depth or levels of processing.
8,195 citations
References
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TL;DR: In this paper, an articulatory analysis of 16 English consonants was performed over voice communication systems with frequency distortion and with random masking noise. The listeners were forced to guess at every sound and a count was made of all the different errors that resulted when one sound was confused with another.
Abstract: Sixteen English consonants were spoken over voice communication systems with frequency distortion and with random masking noise. The listeners were forced to guess at every sound and a count was made of all the different errors that resulted when one sound was confused with another. With noise or low‐pass filtering the confusions fall into consistent patterns, but with high‐pass filtering the errors are scattered quite randomly. An articulatory analysis of these 16 consonants provides a system of five articulatory features or “dimensions” that serve to characterize and distinguish the different phonemes: voicing, nasality, affrication, duration, and place of articulation. The data indicate that voicing and nasality are little affected and that place is severely affected by low‐pass and noisy systems. The indications are that the perception of any one of these five features is relatively independent of the perception of the others, so that it is as if five separate, simple channels were involved rather than a single complex channel.
1,842 citations
"The magical number seven plus or mi..." refers background in this paper
...I am personally much interested in this new approach (15), and I regret that there is not time to discuss it here....
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TL;DR: It is suggested that some property of a collection of objects makes it possible for a person to say that one of these groups is greater-than, lessthan, or equal-to the other group.
Abstract: Suppose that there are two collections or groups of objects-coins, trees, beans, or aircraft-and we do not know how many objects there are. Suppose further that for some reason we cannot count the number of objects in either group. Still, some property of each group makes it possible for a person to say that one of these groups is greater-than, lessthan, or equal-to the other group. It is this property of a collection of objects that we define as numerousness.l We might say that numerousness is that property of a group of objects which we can discriminate, without counting, under instruction to judge how many objects the group contains. We shall wish to modify this definition later as a result of the experiments reported in this paper, but it is adequate for the present discussion of the problem. The judgment of 'numerousness' may be made in several different ways: (a) it may be comparative-more numerous or less numerous, larger or smaller, etc.; (b) or it may be 'absolute.' There is one special form that the absolute judgment of numerousness can take. It is called the direct reporting of number. In this method of reporting, a numeral is assigned to represent how many things there are in any given collection of objects. After a brief look-so brief that counting is impossible-we say 10, 23, or 250 to indicate that we estimate that the group contained 10, 23, or 250 members.
1,023 citations
TL;DR: The experiments reported in this paper are part of a study of the conditions which affect the reproduction of visually perceived form as discussed by the authors, which is an effort has been made to control by experimental means the nature of the change in form.
Abstract: The experiments reported in this paper are part of a study of the conditions which affect the reproduction of visually perceived form. Previous investigators such as G. E. Muller, F. Wulf, and J. J. Gibson 8 have pointed out various factors which are important in bringing about the changes that occur in reproduced forms when they are compared with the forms as originally presented. In the present study an effort has been made to control by experimental means the nature of the change in form. The directive agency used was language, or, more precisely, those processes of the organism that are initiated by language. Historically, Muller treats of the changes that take place after the passage of time in the reproduction of perceived forms in terms which include such concepts as 'convergence' or 'blurring' of the characteristic features of patterns. F. Wulf criticizes this view and proposes, on the basis of experimental study, to describe the observed changes in terms of such specific tendencies as 'sharpening,' 'levelling' and what may be termed the 'equilibrium tendency' in the structure. He remarks, however, that in some instances his subjects spontaneously identified the presented forms with 1 Muller, G. E., Zur Analyse der GedachtnistStigkeit und dee Vorstellungsverlaufes, III. Teil, Zeitsckriftfur Psyehologie, Erganzungsband, 1913, 8, 1-567. 1 Wulf, F., tJber die Veranderung von Vorstellungen (Gedachtnis und Gestalt) (Beitrage zur Psyehologie der Gestalt), Psychologische Forsckung, 1921, x, 333-373. * Gibson, J. J., The reproduction of visually perceived forms, / . Exper. Psychol.,
394 citations
"The magical number seven plus or mi..." refers methods in this paper
...The well-known experiment by Carmichael, Hogan, and Walter (3) on the influence that names have on the recall of visual figures is one demonstration of the process....
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TL;DR: In this article, it was shown that the ability of listeners to identify (and name) tones presented in isolation is relatively poor when the frequency of a single tone (in the frequency range 100-8000 cps) is varied in equal logarithmic steps (and when the sound level is arbitrarily varied to reduce loudness cues).
Abstract: Whereas the ear's sensitivity for detecting a difference in frequency between two tones is remarkably acute, the ability of listeners to identify (and name) tones presented in isolation is relatively poor. When the frequency of a single tone (in the frequency range 100–8000 cps) is varied in equal logarithmic steps (and when the sound level is arbitrarily varied to reduce loudness cues), the amount of information transferred is about 2.3 bits. (The equivalent proficiency of response for an informational transfer of 2.3 bits is perfect identification among only 5 tones.) The amount of information that can be transferred is, within rather wide limits, independent of the number of tones and the range of frequencies employed.
344 citations
"The magical number seven plus or mi..." refers background in this paper
...Data from Pollack (17, 18) on the amount of information that is transmitted by listeners who make absolute judgments of auditory pitch....
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...A third example is provided by Pollack (18), who asked listeners to judge both the loudness and the pitch of pure tones....
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200 citations
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