A quadrature method for Bayesian sequential threshold estimation.

Abstract

For two-alternative forced-choice (2AFC) sequential threshold estimation, a finite Bayesian method (Emerson, 1986b) has been found to perform better than a closely related maximum-likelihood (ML) method. The comparison was especially apt because the two methods used all the same computations (Shelton, 1983) in maintaining and updating the representation of the log-likelihood function. This representation was in the form of a finite discrete array of about 50 equally spaced numerical elements in computer memory. The updating consisted of a series of offset lookups and additions from either of two precomputed arrays representing the log-likelihoods of correct and incorrect responses on a single trial. The difference between the ML and Bayesian methods was in the way the updated log-likelihood array was used to obtain the new estimate of the threshold after each trial. For the ML method, the array was merely scanned for its maximal element. For the Bayesian method, each element was exponentiated and the results were treated as unnormalized probabilities. These were then summed, with and without a multiplicative factor of the array index, to obtain the Bayesian normalizing constant and expected value of the threshold. These additional computations made the Bayesian method considerably slower than the ML method. Real-time use of that finite Bayesian method requires a compiled version of the program, rather than interpretive BASIC, and one of the faster PC-class computers. Statistically, the Bayesian estimates were generally better than the ML in that (1) there was no general negative bias, (2) biases toward the initial estimate decreased rapidly with increasing trials in the run, (3) variances were generally smaller, and (4) the biases and variances depended much less on the location of the true threshold in the assumed stimulus range.