On the theory of Weber fractions

Abstract

The Weber fraction is treated as part of an information theoretical view of perception. In this theory of sensory perception, subjective magnitude is related to the information transmissible from stimulus to perceiver. The derived psychophysical law can be approximated as a power or logarithmic law, depending on conditions. The mathematical expression for the Weber fraction is obtained as a derivative of the psychophysical law. One of many interesting relationships derived here is that the product of the Stevens exponent with log,o stimulus range, for many sensory modalities, is equal to the maximum information transmitted per stimulus (in bits, measured in a test of categorical judgments) multiplied by 0.6. This number is a derived constant and is not measured from the data. 1978, 1981a, 1981b, 1982, 1983, 1984). The reader who is familiar with even the first and fourth of these papers may wish to proceed immediately to the section on Derivation of the Weber Fraction from the Psychophysical Law.Since the subject does not perform flawlessly, H. is not equal to the transmitted information, Hm , but must generally be diminished by the "stimulus equivocation," H•.The equivocation may be calculated from a generalization of Equation 1, which allows for the presence of conditional probabilities. Thus we have If H; is measured in bits of information, then 2 H .. gives us a measure of the maximum number of categories that can be identified without error. This value is largely independent of the number of categories (beyond a certain minimum) into which the'stimulus continuum was divided. Henceforth, we shall refer to H; as the "information content of the stimulus." H; was found to assume a relatively constant value across a number of sensory modalities. We shall take its mean value to be about 2.5 bits of information per stimulus. The Information Content of a Steady Stimulus From an Experiment on Categorical JudgmentsIt was not many years after the appearance of the seminal papers by Shannon (1948) that information theory was applied in the analysis of experiments on perception. Gamer and Hake (1951) demonstrated how to calculate H m , the maximum information transmissible by a steady stimulus, from the stimulus-response matrix generated when a subject makes categorical judgments of a number of stimuli.From the known probabilities, pi' governing the administration of stimuli in each category, the stimulus information, H., may be calculated from the equation In this paper, we shall develop a general theory of the Weber fraction. We shall obtain a mathematical expression for ..::lei> lei> that accounts for the observed shape of the curve in Figure 1; and we shall be able to predict the parameters of the curve on theoretical grounds. As an unexpected bonus from the theory, we shall be able to derive a general mathematical relationship connecting the Stevens exponent, n, with the range of physical magnitudes, eI>, spanned by a given modality. The theoretical function for the Weber fraction that we shall obtain is derived qu...

Cultural transmission, copying errors, and the generation of variation in material culture and the archaeological record

“…Research in the psychological sciences has helped to identify the empirical limits of the acuity of human sensory systems, especially our ability to perceive differences (Coren et al, 1994;Norwich, 1983;Palmer, 1988). Due to physiological constraints humans have great difficulty observing differences below certain threshold values.…”

On the fundamental nature of perception

“…The explicit mathematical evaluation of the perceptual H-function has been done elsewhere (Norwich, 1977(Norwich, , 1987, but with the advantage of hindsight, and armed with some knowledge of the physical properties of the H-function, we can make an educated guess about its approximate mathematical structure.…”

Reaction times reveal the contribution of the different receptor components in luminance perception

“…On theoretical (Norwich, 1987;Norwich, Sebum, & Axelrad, 1989) and empirical (Bonnet, 1992a(Bonnet, , 1992bWard & Davidson, 1997) grounds, it can be assumed that the parameter a of the Pieron function is specific for a given sensory modality and presumably for each receptor type. Moreover, some authors (Baird, 1970;Baird & 359 Copyright 1997 Psychonomic Society, Inc. , 1978;Link, 1992;Teghtsoonian, 1971) suggested that the exponent ofthe Stevens function would be related to discriminability, or, in other terms, to the Weber ratio.…”

“…Similarities between cell latencies and reaction time (RT) to luminance suggest that this relationship may be determined by retinal processes. Ifthe exponent of the Pieron function is specific to a given sensory modality, as assumed by some authors (e.g., Bonnet, 1992aBonnet, , 1992bNorwich, 1987), it should reflect receptor activities. Consequently, functions with different exponents should fit data for different luminance ranges.…”