Analytical methods in the theory of psychophysical discrimination I: Inequalities, convexity and integration of just noticeable differences

We re-examine the theoretical status of Fechner's Mathematical Auxiliary Principle [Fechner, G. T. (1889). Elemente der psychophysik. Leipzig: Breitkopf und Härtel] which underlies Fechner's method of constructing a sensory scale by integrating just noticeable differences (jnds). That “Principle” has been roundly criticized [Luce, R. D., & Edwards, W. (1958). The derivation of subjective scales from just noticeable differences. Psychological Review, 65, 227-237] as being inconsistent with the very basis of Fechner's psychophysical theory, and indeed this is the case. In important papers Pfanzagl [(1962). Über die stochastische Fundierung des psychophysischen Gesetzes (On the stochastic foundations of the psychophysical law). Biometrische Zeitschrift, 4, 1-14] and Krantz [(1971). Integration of just noticeable differences. Journal of Mathematical Psychology, 8, 591-599] resurrected Fechner's method; their analysis showed that the sensory scale could be written as the limit of a sequence of integrals, each of the form suggested by the auxiliary principle. In this work, we investigate the properties of a typical member of the Krantz-Pfanzagl sequence of integrals; we do so with the view to obtaining useful approximations to the true scale. Weber's inequality [Falmagne, J.-Cl. (1977). Note: Weber's inequality and Fechner's problem. Journal of Mathematical Psychology, 16, 267-271] plays an important role in our developments. That inequality, and other inequalities of a similar nature, allows us to place bounds on the error incurred by approximating a true scale u by an integral of jnds. Under appropriate conditions these bounds are sufficiently tight that the relative error is very small over the entire stimulus domain. We illustrate our theoretical results with a number of examples.