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

TL;DR

This paper demonstrates that the widely used time-averaged drift approximation in drift diffusion models is inconsistent and can bias parameter estimates, especially in models with time-varying drift rates, affecting scientific conclusions.

Contribution

It provides a formal proof of the inconsistency of TADA and illustrates its impact on parameter estimation in decision-making models.

Findings

TADA is mathematically inconsistent and does not converge to true drift values.

Using TADA systematically biases estimates of attention effects in decision models.

The proof is demonstrated in a simple Brownian motion setting.

Abstract

Drift diffusion models (DDMs) have found widespread use in computational neuroscience and other fields. They model evidence accumulation in simple decision tasks as a stochastic process drifting towards a decision barrier. In models where the drift rate is both time-varying within a trial and variable across trials, the high computational cost for accurate likelihood evaluation has led to the common use of a computationally convenient surrogate for parameter inference, the time-averaged drift approximation (TADA). In each trial, the 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 drawn from parameter estimates produced by TADA and similar surrogates. 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, we conduct numerical examples with an attentional DDM (aDDM) to show that the use of TADA systematically misestimates the effect of attention in decision making.