The likelihood function of additive learning models: Sufficient conditions for strict log-concavity and uniqueness of maximum

Abstract

Luce introduced a family of learning models in which response probabilities are a function of some underlying continuous real variable. This variable can be represented as an additive function of the parameters of these learning models. Additive learning models have also been applied to signal-detection data. There are a wide variety of problems of contemporary psychophysics for which the assumption of a continuum of sensory states seems appropriate, and this family of learning models has a natural extension to such problems. One potential difficulty in the application of such models to data is that estimation of parameters requires the use of numerical procedures when the method of maximum likelihood is used. Given a likelihood function generated from an additive model, this paper gives sufficient conditions for log-concavity and strict log-concavity of the likelihood function. If a likelihood function is strictly log-concave, then any local maximum is a unique global maximum, and any solution to the likelihood equations is the unique global maximum point. These conditions are quite easy to evaluate in particular cases, and hence, the results should be quite useful. Some applications to Luce's beta model and to the signal-detection learning models of Dorfman and Biderman are presented.