Time-Averaged Drift Approximations are Inconsistent for Inference in Drift Diffusion Models

Drift diffusion models (DDMs) have found widespread use in computational neuroscience, cognitive science, mathematical psychology as well as other fields. They model evidence accumulation in simple decision tasks as a stochastic process drifting towards decision barriers. In models where the drift is both time-varying within a trial and variable across trials, the high computational cost for accurate likelihood evaluation has often led to the use of a computationally convenient surrogate for parameter inference, the time-averaged drift approximation (TADA). In each trial, TADA assumes that the time-varying drift rate can be replaced by its temporal average throughout the trial. This approach enables fast parameter inference using analytical likelihood formulas for DDMs with constant drift. In this work, we show that such an estimator is inconsistent: it does not converge to the true drift, posing a risk of biasing scientific conclusions when parameter estimates are obtained by TADA and similar approximations. We provide an elementary proof of this inconsistency in what is perhaps the simplest possible setting: a Brownian motion with piecewise constant drift hitting a one-sided upper boundary. Furthermore, numerical examples based on an attentional DDM (aDDM) show that using TADA leads to systematic misestimation of attentional effects in decision making and can lead to false conclusions in scientific hypothesis testing.