Statistical information and uncertainty: A critique of applications in experimental psychology

Entropy

Volumn 12, Issue 4, 2010, Pages 720-771

  Laming, Donald ^a  

^a University of Cambridge   (United Kingdom)

Author keywords

Bayes theorem; Category judgment; Channel capacity; Choice reaction times; Entropy; Hick's law; Information theory; Signal detection; Uncertainty; Weber's law

Indexed keywords

EID: 77953493176     PISSN: None     EISSN: 10994300     Source Type: Journal    
DOI: 10.3390/e12040720     Document Type: Article

References (138)

1. Miller, G.A.; Frick, F.C. Statistical behavioristics and sequences of responses. Psychol. Rev. 1949, 56, 311-324.
2. Shannon, C.E. A mathematical theory of communication. Bell Labs Tech. J. 1948, 27, 379-423; 623-656.
3. Garner, W.R. Uncertainty and Structure as Psychological Concepts; Wiley: New York, NY, USA, 1962.
4. Cherry, E.C. Information Theory. Presented at Information Theory Symposium, London, UK, 12-16 September, 1955; Butterworths: London, UK, 1956.
5. Cherry, E.C. Information Theory. Presented at Information Theory Symposium, London, UK, 29 August- 2 September, 1960; Butterworths: London, UK, 1961.
6. Jackson, W. Report of proceedings, symposium on information theory, London, England, September 1950. IEEE Trans. Inf. Theory 1953, PGIT-1, 1-218.
7. Jackson, W. Communication Theory. Presented at Applications of Communication Theory Symposium, London, UK, 22-26 September, 1952; Butterworths: London, UK, 1953.
8. Quastler, H. Essays on the Use of Information Theory in Biology; University of Illinois Press: Urbana, Illinois, USA, 1953.
9. Quastler, H. Information theory in psychology: problems and methods. In Proceedings of a conference on the estimation of information flow, Monticello, IL, USA, 5-9 July, 1954; and related papers; The Free Press: Glencoe, Illinois, USA, 1955.
10. Luce, R.D. Whatever happened to information theory? Rev. Gen. Psychol. 2103, 7, 183-188.
11. Tanner, W.P. Jr.; Swets, J.A. A decision-making theory of visual detection. Psychol Rev. 1954, 61, 401-409.
12. Peterson, W.W.; Birdsall, T.G.; Fox, W.C. The theory of signal detectability. IEEE Trans. Inf. Theory 1954, PGIT-4, 171-212.
13. Neyman, J.; Pearson, E.S. On the problem of the most efficient tests of statistical hypotheses. Philos. Trans. R. Soc. Lond. A 1933, 231, 289-337.
14. Laming, D. The antecedents of signal-detection theory. A commentary on D.J. Murray, A perspective for viewing the history of psychophysics. Behav. Brain Sci. 1993, 16, 151-152.
15. The term 'ideal observer' has been used in two different ways. Green and Swets (Ch. 6 [16]) use it to denote a model of the information contained in the physical stimulus, borrowing ideas and mathematics from Peterson, Birdsall and Fox [12]. Peterson, Birdsall and Fox do not themselves use the term in this sense, but do refer to 'Siegert's "Ideal observer's" criteria', which adds to their analyses of physical stimuli criteria that maximize the expectation of a correct decision. More recently (e.g., [17]), the term has been used to denote the selection of a criterion to maximize the expectation of a correct decision, without necessarily there being a prior model of the physical stimulus. To avoid confusion, I shall not use this term hereafter.
16. Green, D.M.; Swets, J.A. Signal Detection Theory and Psychophysics; Wiley: New York, NY, USA, 1966.
17. Kersten, D.; Mamassian, P.; Yuille, A. Object perception as bayesian inference. Annu. Rev. Psychol. 2104, 55, 271-304.
18. Edwards, W.; Lindeman, H.; Savage, L.J. Bayesian statistical inference for psychological research. Psychol. Rev. 1963, 70, 193-242.
19. Green, D.M. Psychoacoustics and detection theory. J. Acoust. Soc. Am. 1960, 32, 1189-1213.
20. Norwich, K.H. Information, Sensation and Perception; Academic Press: San Diego, CA, USA, 1993.
21. Braida, L.D.; Durlach, N.I. Intensity perception. II. Resolution in one-interval paradigms. J. Acoust. Soc. Am. 1972, 51, 483-502.
22. Laming, D. Statistical information, uncertainty, and Bayes' theorem: Some applications in experimental psychology. In Symbolic and Quantitative Approaches to Reasoning with Uncertainty (Lecture notes in Artificial Intelligence, Volume 2143); Benferhat, S., Besnard, P., Eds.; Springer-Verlag: Berlin, Germany, 2001; pp. 635-646.
23. Kullback, S. Information Theory and Statistics; Wiley: New York, NY, USA, 1959.
24. Wilkes, S.S. The large-sample distribution of the likelihood ratio for testing composite hypotheses. Ann. Math. Stat. 1938, 9, 60-62.
25. Wilkes, S.S. The likelihood test of independence in contingency tables. Ann. Math. Stat. 1935, 6, 190-196.
26. Swets, J.A.; Tanner, W.P. Jr.; Birdsall, T.G. Decision processes in perception. Psychol. Rev. 1961, 68, 301-340.
27. Laming, D. Mathematical Psychology; Academic Press: London, UK, 1973.
28. Shannon, C.E. Communication in the presence of noise. Pro. Inst. Radio Eng. 1949, 37, 10-21.
29. A distinction is needed between uncertainty (Equation 13) in a psychological context and the same formula in a purely physical context. The interpretation of this formula is not the same in all contexts. To preserve this distinction I shall use 'uncertainty' in psychological contexts and 'entropy' in relation to the second law of thermodynamics (below).
30. i., and the mean information averaged over the set of messages is Formula 13 (p. 7 [23]). But these hypotheses are not of any practical interest.
31. Jaynes, E.T. Information theory and statistical mechanics. Phys. Rev. 1957, 106, 620-630; 108, 171-190.
32. Landau, L.D.; Lifshitz, E.M. Statistical Physics; Sykes, J.B., Kearsley, M.J., Eds.; Pergamon Press: Oxford, UK, 1968.
33. Aczél, J.; Daróczy, Z. On Measures of Information and Their Characterizations; Academic Press: New York, NY, USA, 1975.
34. Laming, D. Spatial frequency channels. In Vision and Visual Dysfunction; Kulikowski, J.J., Walsh, V., Murray, I.J., Eds.; Macmillan: London, UK, 1991; Volume 5.
35. Good, I.J. Probability and the weighting of evidence; Griffin: London, UK, 1950.
36. Krech, D; Crutchfield, R.S. Theory and Problems of Social Psychology; McGraw-Hill: New York, NY, USA, 1948.
37. Sherif, M.; Hovland, C.I. Social Judgment; Yale University Press: New Haven, CN, USA, 1961.
38. Hick, W.E. On the rate of gain of information. Q.J. Exp. Psychol. 1952, 4, 11-26.
39. Merkel, J. Die zietlichen Verhältnisse der Willensthätigkeit. Philosophische Studien 1885, 2, 73-127.
40. Kirchner, W.K. Age differences in short-term retention of rapidly changing information. J. Exp. Psychol. 1958, 55, 352-358.
41. Bricker, P.D. Information measurement and reaction time: A review. In Information Theory in Psychology: Problems and Methods, Proceedings of a conference on the estimation of information flow, Monticello, IL, USA, 5-9 July, 1954; and related papers; The Free Press: Glencoe, IL, USA, 1955; pp. 350-359.
42. Laming, D. Information Theory of Choice-Reaction Times; Academic Press: London, UK, 1968.
43. Leonard, J.A. Tactual choice reactions: I.Q.J. Exp. Psychol. 1959, 11, 76-83.
44. Christie, L.S.; Luce, R.D. Decision structure and time relations in simple choice behavior. Bull. Math. Biophys. 1956, 18, 89-111.
45. Laming, D. A new interpretation of the relation between choice-reaction time and the number of equiprobable alternatives. Br. J. Math. Stat. Psychol. 1966, 19, 139-149.
46. Townsend, J.T.; Ashby, F.G. The Stochastic Modeling of Elementary Psychological Processes; Cambridge University Press: Cambridge, UK, 1983.
47. Hake, H.W.; Garner, W.R. The effect of presenting various numbers of discrete steps on scale reading accuracy. J. Exp. Psychol. 1951, 42, 358-66.
48. Chapanis, A.; Halsey, R.M. Absolute judgments of spectrum colors. J. Psychol. 1956, 42, 99-103.
49. Eriksen, C.W.; Hake, H.W. Multidimensional stimulus differences and, accuracy of discrimination. J. Exp. Psychol. 1955, 50, 153-160.
50. Muller, P.F., Jr.; Sidorsky, R.C.; Slivinske, A.J.; Alluisi, E.A.; Fitts, P.M. The symbolic coding of information on cathode ray tubes and similar displays. USAF WADC Technical Report 1955, No. 55-37.
51. Johnson-Laird, P.N.; Wason, P.C. A theoretical analysis of insight into a reasoning task. Cogn. Psychol. 1970, 1, 134-148.
52. Oaksford, M.; Chater, N. A rational analysis of the selection task as optimal data selection. Psychol. Rev. 1994, 101, 608-631.
53. Oaksford, M.; Chater, N. Optimal data selection: Revision, review and re-evaluation. Psychon. Bull. Rev. 2003, 10, 289-318.
54. Oaksford, M.; Chater, N. Bayesian rationality: The Probabilistic Approach to Reasoning; Oxford University Press: Oxford, UK, 2007.
55. Oaksford, M.; Chater, N.; Grainger, B. Probabilistic effects in data selection. Thinking and Reasoning 1999, 5, 193-243.
56. Oberauer, K.; Wilhelm, O.; Dias, R.R. Bayesian rationality for the Wason selection task? A test of optimal data selection theory. Thinking and Reasoning 1999, 5, 115-144.
57. 1 generates the so-called 'ravens paradox' [58,59], in which the observation of a pink flamingo (anything that is neither black nor a raven) is asserted, erroneously, to confirm the hypothesis that all ravens are black. Observations of black ravens (of course) and of non-black non-ravens jointly militate against the hypothesis that color and ravenhood are independent in nature. But confirmation requires the further assumption that independence and 'all ravens are black' are mutually exhaustive hypotheses, and that is patently false.
58. Mackie, J.L. The paradox of confirmation. Br. J. Philos. Sci. 1963, 13, 265-277.
59. Oaksford, M.; Chater, N. Rational explanation of the selection task. Psychol. Rev. 1996, 103, 381-391.
60. Klauer, K.C. On the normative justification for information gain in Wason's selection task. Psychol. Rev. 1999, 106, 215-222.
61. Birnbaum, M.H. Tests of branch splitting and branch-splitting independence in Allais paradoxes with positive and mixed consequences. Organ. Behav. Hum. Decis. Process 2007, 102, 154-173.
62. Birnbaum, M.H.; Birnbaum, H. Causes of Allais common consequence paradoxes: An experimental dissection. J. Math. Psychol. 2004, 48, 87-106.
63. Fryback; Goodman; Edwards report two 'laboratory' experiments conducted in a Las Vegas casino with patrons of the casino. In these experiments participants typically won or lost ±$30 of their own money.
64. Fryback, D.G.; Goodman, B.C.; Edwards, W. Choices among bets by Las Vegas gamblers: Absolute and contextual effects. J. Exp. Psychol. 1973, 98, 271-278.
65. Wagenaar, W.A. Paradoxes of Gambling Behavior; Lawrence Erlbaum: Hove, UK, 1988.
66. Von Neumann, J.; Morgenstern, O. Theory of Games and Economic Behavior; Princeton University Press: Princeton, NJ, USA, 1944.
67. Allais, M. Le Comportement de l'Homme Rationnel devant le Risque: Critique des Postulats et Axiomes de l'Ecole Americaine. Econometrica 1953, 21, 503-546.
68. Kahneman, D.; Tversky, A. Prospect theory: an analysis of decision under risk. Econometrica 1979, 47, 263-291.
69. Luce, R.D; Ng, C.T.; Marley, A.A.J.; Aczél, J. Utility of gambling II: Risk, Paradoxes, and Data. J. Econ. Theory 2008, 36, 165-187.
70. Luce, R.D; Ng, C.T.; Marley, A.A.J.; Aczél, J. Utility of gambling I: Entropy-modified linear weighted utility. J. Econ. Theory 2008, 36, 1-33.
71. In this context Formula 13 does not permit the same kind of psychological interpretation (transmission through an ideal communications system) as previous examples. Instead, the formula is derived from a set of individually testable behavioral axioms.
72. 2 jointly and all the other outcomes. This is substantially the same set of constraints as previously (p. 11 above).
73. Laming, D. Human Judgment: The Eye of the Beholder; Thomson Learning: London, UK, 2004.
74. Used here in an everyday, not a technical, sense.
75. Laming, D. Understanding Human Motivation: What Makes People Tick? Blackwells: Malden, MA, USA, 2004.
76. Wagenaar, W.A. Subjective randomness and the capacity to generate information. In Attention and performance III, Proceedings of a symposium on attention and performance, Soesterberg,, The Netherlands, August 4-8, 1969; Sanders, A.F., Ed.; Acta Psychol. (Amst) 1970, 33, 233-242.
77. Cronbach, L.J. On the non-rational application of information measures in psychology. In Information theory in psychology: problems and methods, Proceedings of a conference on the estimation of information flow, Monticello, IL, USA, July 5-9, 1954; and related papers; The Free Press: Glencoe, IL, USA, 1955; pp. 14-30.
78. Linker, E.; Moore, M.E.; Galanter, E. Taste thresholds, detection models, and disparate results. J. Exp. Psychol. 1964, 67, 59-66.
79. Mountcastle, V.B.; Talbot, W.H.; Sakata, H.; Hyvärinen, J. Cortical neuronal mechanisms in flutter-vibration studied in unanesthetized monkeys. Neuronal periodicity and frequency discrimination, J. Neurophysiol. 1969, 32, 452-484.
80. Nachmias, J.; Steinman, R.M. Brightness and discriminability of light flashes. Vision Res. 1965, 5, 545-557.
81. Semb, G. The detectability of the odor of butanol. Percept. Psychophys. 1968, 4, 335-340.
82. Viemeister, N.F. Intensity discrimination: Performance in three paradigms. Percept. Psychophys. 1971, 8, 417-419.
83. Brown, W. The judgment of difference. Publ. Psychol. 1910, 1, 1-71.
84. Hanna, T.E.; von Gierke, S.M.; Green, D.M. Detection and intensity discrimination of a sinusoid. J. Acoust. Soc. Am. 1986, 80, 1335-1340.
85. Leshowitz, B.; Taub, H.B.; Raab, D.H. Visual detection of signals in the presence of continuous and pulsed backgrounds. Percept. Psychophys. 1968, 4, 207-213.
86. Laming, D. Fechner's Law: Where does the log transform come from? In Fechner Day 2001; Sommerfeld, E., Kompass, R., Lachmann, T., Eds.; Pabst: Lengerich, Germany, 2001; pp. 36-41.
87. Laming, D. Fechner's Law: Where does the log transform come from? Seeing and Perceiving 2010, in press.
88. McBurney, D.H.; Kasschau, R.A.; Bogart, L.M. The effect of adaptation on taste jnds. Percept. Psychophys. 1967, 2, 175-178.
89. Stone, H.; Bosley, J.J. Olfactory discrimination and Weber's Law. Percept. Mot. Skills 1965, 20, 657-665.
90. Ernst, M.O.; Banks, M.S. Humans integrate visual and haptic information in a statistically optimal fashion. Nature 2002, 415, 429-433.
91. Hamer, R.D.; Verrillo, R.T.; Zwislocki, J.J. Vibrotactile masking of Pacinian and non-Pacinian channels. J. Acoust. Soc. Am. 1983, 73, 1293-303.
92. Harris, J.D. The effect of sensation-levels on intensive discrimination of noise. Am. J. Psychol. 1950, 63, 409-421.
93. Schutz, H.G.; Pilgrim, F.J. Differential sensitivity in gustation. J. Exp. Psychol. 1957, 54, 41-48.
94. Stone, H. Determination of odor difference limens for three compounds. J. Exp. Psychol. 1963, 66, 466-473.
95. Laming, D. The discrimination of smell and taste compared with other senses. Chem. Ind. 1987, 12-18.
96. de Vries, H. The quantum character of light and its bearing upon threshold of vision, the differential sensitivity and visual acuity of the eye. Physica. 1943, 10, 553-564.
97. Rose, A. The sensitivity performance of the human eye on an absolute scale. J. Opt. Soc. Am. 1948, 38, 196-208.
98. van Nes, F.L.; Bouman, M.A. Spatial modulation transfer in the human eye. J. Opt. Soc. Am. 1967, 57, 401-406.
99. Fullerton, G.S.; Cattell, J.McK. On the perception of small differences. Publications of the University of Pennsylvania, Philosophical Series, No. 2, 1892.
100. Laming, D. Sensory Analysis; Academic Press: London, UK, 1986.
101. Laming, D. On the limits of visual perception. In Vision and Visual Dysfunction; Kulikowski, J.J., Walsh, V., Murray, I.J., Eds.; Macmillan: London, UK, 1991; Volume 5.
102. Graham, C.H.; Kemp, E.H. Brightness discrimination as a function of the duration of the increment in intensity. J. Gen. Physiol. 1938, 21, 635-650.
103. Bartlett, N.R. The discrimination of two simultaneously presented brightnesses. J. Exp. Psychol. 1942, 31, 380-392.
104. Steinhardt, J. Intensity discrimination in the human eye. I. The relation of ΔI/I to intensity. J. Gen. Physiol. 1936, 20, 185-209.
105. Cornsweet, T.N.; Pinsker, H.M. Luminance discrimination of brief flashes under various conditions of adaptation. J. Physiol. 1965, 176, 294-310.
106. Rodieck, R.W.; Stone, J. Analysis of receptive fields of cat retinal ganglion cells. J. Neurophysiol. 1965, 28, 833-849.
107. Ditchburn, R.W. Eye-Movements and Visual Perception; Oxford University Press: Oxford, UK, 1973.
108. Yarbus, A.L. Eye Movements and Vision, from the 1965 Russian Edition.; Haigh, B.T., Ed.; Plenum Press: New York, NY, USA, 1967.
109. Laming, D. Précis of Sensory Analysis and A reexamination of Sensory Analysis. Behav. Brain Scis. 1988, 11, 275-296; 316-339.
110. Barlow, H.B. Optic nerve impulses and Weber's Law. Cold Spring Harb. Symp. Quant. Biol. 1965, 30, 539-546.
111. Grossberg, S. The quantized geometry of visual space: The coherent computation of depth, form and lightness. Behav. Brain Sci. 1983, 6, 625-692.
112. If this idea be applied to an increment added to a continuous background of luminance L, it neglects entirely the relation of threshold to size of increment, which changes with increase in luminance. If it be applied to a discrimination between separate luminances, where Weber's Law holds the most accurately, it requires the luminance L+ΔL to be scaled as though it were luminance L.
113. van Nes, F.L. Experimental studies in spatiotemporal contrast transfer by the human eye. Ph.D Thesis, University of Utrecht, Utrecht, The Netherlands, 1968.
114. Laming, D. Contrast sensitivity. In Vision and Visual Dysfunction; Kulikowski, J.J., Walsh, V., Murray, I.J., Eds.; Macmillan: London, UK, 1991; Volume 5.
115. Legge, G.E. Spatial frequency masking in human vision: Binocular interactions. J. Opt. Soc. Am. 1979, 69, 838-847.
116. Campbell, F.W.; Kulikowski, J.J. Orientational selectivity of the human visual system. J. Physiol. 1966, 187, 437-445.
117. Hubel, D.H.; Wiesel, T.N. Functional architecture of macaque monkey visual cortex. Proc. R. Soc. Lond B Biol. Sci. 1977, 198, 1-59.
118. Watson, A.B. Summation of grating patches indicates many types of detector at one retinal location. Vision Res. 1982, 22, 17-25.
119. Campbell, F.W.; Cooper, G.F.; Enroth-Cugell, C. The spatial selectivity of the visual cells of the cat. J. Physiol. 1969, 203, 223-235.
120. Maffei, L.; Fiorentini, A. The visual cortex as a spatial frequency analyzer. Vision Res. 1973, 13, 1255-1267.
121. Robson, J.G.; Tolhurst, D.J.; Freeman, R.D.; Ohzawa, I. Simple cells in the visual cortex of the cat can be narrowly tuned for spatial frequency. Vis. Neurosci. 1988, 1, 415-419.
122. Roitman, J.D.; Shadlen, M.N. Response of neurons in the lateral intraparietal area during a combined visual discrimination reaction time task. J. Neuros. 2002, 22, 9475-9489.
123. Shadlen, M.N.; Newsome, W.T. Neural basis of a perceptual decision in the parietal cortex (area LIP) of the rhesus monkey. J. Neurophysiol. 2001, 86, 1916-1936.
124. Britten, K.H.; Shadlen, M.N.; Newsome, W.T.; Movshon, J.A. Responses of neurons in macaque MT to stochastic motion signals. Vis. Neurosci. 1993, 10, 1157-1169.
125. Britten, K.H.; Newsome, W.T. Tuning bandwidths for near-threshold stimuli in area MT. J. Neurophysiol. 1998, 80, 762-770.
126. Stone, M. Models for choice-reaction time. Psychometrika 1960, 25, 251-260.
127. Edwards, W. Subjective probabilities inferred from decisions. Psychol. Rev. 1962, 69, 109-135.
128. Wald, A. Sequential Analysis; Wiley: New York, NY, USA, 1947.
129. Thomas, E.A.C.; Legge, D. Probability matching as a basis for detection and recognition decisions. Psychol. Rev. 1970, 77, 65-72.
130. Tanner, W.P. Jr.; Swets, J.A.; Green, D.M. Some general properties of the hearing mechanism. University of Michigan: Electronic Defense Group, Technical Report, No. 30, 1956.
131. Tanner, T.A.; Rauk, J.A.; Atkinson, R.C. Signal recognition as influenced by information feedback. J. Math. Psychol. 1970, 7, 259-274.
132. Laming, D. Choice-reaction performance following an error. Acta Psychol. (Amst) 1979, 43, 199-224.
133. The unbiased estimate of functional relationship depends on the relative variances attributed to the two sets of variables (Ch. 29, esp. p. 404 [134]). One or the other regression line is correct when one of those variances is zero. Since neither variance can be negative, the unbiased estimate must lies between the two regression lines.
134. Kendall, M.G.; Stuart, A. Advanced Theory of Statistics, 4th Edition; Griffin: London, UK, 1979.
135. Torgerson, W.S. Theory and Methods of Scaling; Wiley: New York, NY, USA, 1958.
136. Laming, D. The Measurement of Sensation; Oxford University Press: Oxford, UK, 1997.
137. Laming, D. Screening cervical smears. Br. J. Psychol. 1995, 86, 507-516.
138. Laming, D. Reconciling Fechner and Stevens? Behav. Brain Sci. 1991, 14, 188-191.