Srivastava Vaibhav, Feng Samuel F, Cohen Jonathan D, Leonard Naomi Ehrich, Shenhav Amitai
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ, USA.
Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI, USA.
J Math Psychol. 2017 Apr;77:94-110. doi: 10.1016/j.jmp.2016.10.001. Epub 2016 Nov 9.
Research in psychology and neuroscience has successfully modeled decision making as a process of noisy evidence accumulation to a decision bound. While there are several variants and implementations of this idea, the majority of these models make use of a noisy accumulation between two absorbing boundaries. A common assumption of these models is that decision parameters, e.g., the rate of accumulation (drift rate), remain fixed over the course of a decision, allowing the derivation of analytic formulas for the probabilities of hitting the upper or lower decision threshold, and the mean decision time. There is reason to believe, however, that many types of behavior would be better described by a model in which the parameters were allowed to vary over the course of the decision process. In this paper, we use martingale theory to derive formulas for the mean decision time, hitting probabilities, and first passage time (FPT) densities of a Wiener process with time-varying drift between two time-varying absorbing boundaries. This model was first studied by Ratcliff (1980) in the two-stage form, and here we consider the same model for an arbitrary number of stages (i.e. intervals of time during which parameters are constant). Our calculations enable direct computation of mean decision times and hitting probabilities for the associated multistage process. We also provide a review of how martingale theory may be used to analyze similar models employing Wiener processes by re-deriving some classical results. In concert with a variety of numerical tools already available, the current derivations should encourage mathematical analysis of more complex models of decision making with time-varying evidence.
心理学和神经科学领域的研究已成功地将决策制定建模为一个向决策边界累积有噪声证据的过程。虽然这个想法有多种变体和实现方式，但这些模型中的大多数都利用了两个吸收边界之间的有噪声累积。这些模型的一个常见假设是，决策参数，例如累积速率（漂移率），在决策过程中保持固定，从而可以推导出关于达到上决策阈值或下决策阈值的概率以及平均决策时间的解析公式。然而，有理由相信，许多类型的行为可以通过一个允许参数在决策过程中变化的模型得到更好的描述。在本文中，我们使用鞅理论来推导在两个随时间变化的吸收边界之间具有时变漂移的维纳过程的平均决策时间、击中概率和首次通过时间（FPT）密度的公式。这个模型最初由拉特克利夫（1980）以两阶段形式进行研究，在这里我们考虑任意数量阶段（即参数保持恒定的时间间隔）的相同模型。我们的计算能够直接计算相关多阶段过程的平均决策时间和击中概率。我们还通过重新推导一些经典结果，回顾了如何使用鞅理论来分析采用维纳过程的类似模型。与现有的各种数值工具相结合，当前的推导应该会鼓励对具有时变证据的更复杂决策模型进行数学分析。
