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

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

Problem Statement
Drift Diffusion Models (DDMs) are widely used to model evidence accumulation in decision-making tasks. While standard DDMs with constant drift parameters admit closed-form likelihood solutions, many extensions—particularly the Attentional Drift Diffusion Model (aDDM)—incorporate drift rates that vary both within a trial (time-varying) and across trials (trial-varying). In the aDDM, the drift rate $\mu(t)$ depends on the participant's instantaneous visual fixation trajectory.

Evaluating the likelihood for such models is computationally prohibitive because the drift function is unique to every trial, preventing the sharing of computations across trials. To circumvent this, researchers frequently employ the Time-Averaged Drift Approximation (TADA). TADA simplifies the model by replacing the time-varying drift $\mu(t)$ with its temporal average over the trial duration $\tau$, effectively converting the problem into a standard DDM with a constant "effective" drift. This approximation allows for the use of analytical likelihood formulas, enabling fast parameter inference. However, the validity of TADA as a surrogate for the true aDDM, particularly regarding the consistency of its estimators, has not been rigorously established.

Methodology
The authors employ a combination of mathematical analysis and numerical simulation to evaluate the statistical properties of TADA.

1. Mathematical Proof of Inconsistency:
   The authors construct a simplified counterexample: a Brownian motion with a piecewise constant drift (drift $\mu$ for time $t \in [0, T]$ and 0 thereafter) hitting a one-sided upper boundary.

   • They derive the exact First Passage Time Density (FPTD) for this process, denoted $f_{aDDM}(\tau; \mu)$.
   • They derive the FPTD implied by TADA, denoted $f_{TADA}(\tau; \mu)$, which relies on the average drift $\alpha(\mu, \tau, T) = \frac{1}{\tau}\int_0^\tau \mu(t) dt$.
   • They demonstrate that $f_{TADA}$ is not a valid probability density function (it does not integrate to 1) because the "effective" drift depends on the stopping time $\tau$ itself, creating a circular dependency that violates the definition of a generative model.
   • Using the Strong Law of Large Numbers, they prove that the TADA maximum likelihood estimator ($\hat{\mu}_{TADA}$) converges almost surely to a value $\tilde{\mu}$ that is strictly greater than the true drift parameter $\mu$ (specifically, $\tilde{\mu} > \max\{\mu, b/T\}$), establishing statistical inconsistency.

2. Numerical Simulations:
   The authors validate their theoretical findings through three numerical experiments:

   • One-sided Boundary: Simulating the piecewise constant drift model to confirm that $\hat{\mu}_{TADA}$ converges to the biased theoretical limit rather than the true $\mu$, while the standard Maximum Likelihood Estimator (MLE) remains consistent.
   • Two-sided Boundaries: Simulating a DDM with alternating drifts between two boundaries. Using a new efficient algorithm (efpt) developed in parallel work (Liu et al., 2025) to compute exact likelihoods, they show that TADA systematically overestimates the magnitude of drift parameters.
   • Full aDDM: Simulating the full attentional model with gaze-dependent drifts. They estimate the attentional discount parameter $\eta$ (where $\eta=1$ implies no attentional effect and $\eta=0$ implies maximal effect). They compare TADA estimates against exact MLEs across varying sample sizes.

Key Results

• Statistical Inconsistency: The TADA estimator is mathematically proven to be inconsistent. As the sample size $n \to \infty$, the estimator does not converge to the true parameter value but to a biased limit.
• Systematic Bias: In numerical examples, TADA consistently overestimates drift magnitudes and systematically underestimates the attentional discount parameter $\eta$. For instance, when the true $\eta$ is low (strong attentional effects), TADA estimates can become negative, which is physically meaningless.
• Failure in Hypothesis Testing: The bias introduced by TADA is not uniform; it depends on nuisance parameters (e.g., boundary separation, drift scale) and covariate distributions. Consequently, when comparing two populations with different nuisance parameters, TADA can lead to qualitatively incorrect conclusions. The authors demonstrate scenarios where TADA incorrectly reverses the sign of the difference in attentional effects between two groups, leading to false positives or false negatives in likelihood ratio tests.
• Validity of Exact Methods: In contrast, the standard MLE (computed using the efpt algorithm) remains consistent and accurately recovers true parameters across all tested scenarios.

Significance and Claims
The paper claims that the widespread use of TADA in computational neuroscience and cognitive science poses a significant risk of biasing scientific conclusions. The authors argue that because TADA does not define a well-specified generative model (the drift depends on the future stopping time), its likelihood function is invalid, and its estimators are fundamentally flawed.

The significance of this work lies in:

1. Correcting a Common Practice: It challenges the assumption that TADA is a valid approximation for the aDDM, suggesting that previous studies relying on this approximation may have reported biased parameter estimates.
2. Highlighting Qualitative Risks: It demonstrates that the bias is not merely quantitative but can alter the direction of scientific conclusions regarding population differences.
3. Promoting Rigorous Alternatives: The authors advocate for the use of mathematically justified, fast likelihood computation methods (such as the efpt algorithm cited in their parallel work) rather than computationally convenient but statistically inconsistent approximations. They recommend re-examining existing results obtained via TADA.

The authors maintain a modest tone regarding the scope of their findings, noting that while their analysis invalidates the consistency of TADA, it does not necessarily invalidate all qualitative conclusions from prior studies, though it raises serious concerns about their reliability. They emphasize that the inconsistency arises specifically when the "effective" parameter depends on the first passage time itself, a condition common in time-dependent DDMs.