Closed-Form Approximation of Drift Diffusion Response Time for Parameter Estimation

Jiseob Kim, Yung-Kyun Noh, Mario Fific, and Byoung-Tak Zhang

Abstract

We consider the Drift Diffusion Model (DDM) for a Two-Alternative Forced Choice (2AFC) in human decision making. DDM uses a computational model which has a diffusing random variable and two thresholds for alternative choices; a decision is made by a random variable diffusing up and down according to the information coming in with noise and finally reaching one of two thresholds. The model explains the time-accuracy trade-off of human decision making as well as the shape of the empirical distribution of response times (RTs), while the model is designed simply using a few parameters: diffusing rates towards each direction, number of diffusing states, and the thresholds.

Unfortunately, estimating the parameters from the empirical data of RTs is difficult. A standard maximum-likelihood method cannot be applied because the predicted distribution is complex, and the standard gradient descent optimization is intractable. In this work, we investigate how the RT distribution can be approximated using a simple closed-form equation, and we utilize the approximated distribution to estimate the parameters. In particular, we interpret how our derived equations are relaxed into the Skellam process, assuming discrete mental-space, and show how the RT distribution can be approximated using this process.

We first show how the RT distribution in 2AFC can be approximated using a one-threshold system. In Skellam processes, the RT density function can be obtained as a simple closed form. Next, we show that the exact 2AFC RT density function can be represented as one eigenvector problem and the solution for maximum-likelihood parameter estimation can be approximated. In various experiments, we show that the proposed methods are very reliable even with a small number of samples.