First passage time models describe the time it takes for a random process to exit a region of interest and are widely used across various scientific fields. Fast and accurate numerical methods for computing the likelihood function in these models are essential for efficient statistical inference. Specifically, in mathematical psychology, generalized drift diffusion models (GDDMs) are an important class of first passage time models that describe the latent psychological processes underlying simple decision-making scenarios. GDDMs model the joint distribution over choices and response times as the first hitting time of a one-dimensional stochastic differential equation (SDE) to possibly time-varying upper and lower boundaries. They are widely applied to extract parameters associated with distinct cognitive and neural mechanisms. However, current likelihood computation methods struggle with common scenarios where drift rates covary dynamically with exogenous covariates in each trial, such as in the attentional drift diffusion model (aDDM). In this work, we propose a fast and flexible algorithm for computing the likelihood function of GDDMs based on a large class of SDEs satisfying the Cherkasov condition. Our method divides each trial into discrete stages, employs fast analytical results to compute stage-wise densities, and integrates these to compute the overall trial-wise likelihood. Numerical examples demonstrate that our method not only yields accurate likelihood evaluations for efficient statistical inference, but also significantly outperforms existing approaches in terms of speed.

翻译：一阶通过时间模型描述了随机过程离开感兴趣区域所需的时间，在多个科学领域中得到广泛应用。计算这些模型中似然函数的快速且准确的数值方法对于高效统计推断至关重要。具体而言，在数学心理学中，广义漂移扩散模型（GDDMs）是一类重要的一阶通过时间模型，用于描述简单决策情境下潜在的心理过程。GDDMs 将选择与反应时间的联合分布建模为一维随机微分方程（SDE）首次到达可能随时间变化的上、下边界的时间。它们被广泛应用于提取与不同认知和神经机制相关的参数。然而，当前的似然计算方法在处理漂移率在每次试验中与外生协变量动态共变的常见场景时存在困难，例如在注意力漂移扩散模型（aDDM）中。在本工作中，我们提出了一种快速且灵活的算法，用于计算基于满足 Cherkasov 条件的一大类 SDE 的 GDDMs 的似然函数。我们的方法将每次试验划分为离散阶段，采用快速解析结果计算阶段密度，并通过积分这些密度来计算整个试验的似然。数值示例表明，我们的方法不仅能为高效统计推断提供准确的似然评估，而且在速度方面显著优于现有方法。