In this work , we use Martingale theory to derive formulas for the expected decision time, error rates , and first passage times associated with a multistage drift diffusion model, or a Wiener diffusion model with piecewise constant time-varying drift rates and decision boundaries. The model we study is a generalization of that considered in Ratcliff (1980) . The derivation relies on using the optional stopping theorem for properly chosen Martingales, thus obtaining formulae which may be used to compute performance metrics for a particular stage of the stochastic decision process. We also explicitly solve the case of a two stage diffusion model, and provide numerical demonstrations of the computations suggested by our analysis. Finally we present calculations that allow our techniques to approximate time-varying Ornstein-Uhlenbeck processes . By presenting these explicit formulae, we aim to foster the development of refined numerical methods and analytical techniques for studying diffusion decision processes with time-varying drift rates and thresholds .
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 .
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In this work
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
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Martingale
martingale
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expected
mean
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error rates
hitting probabilities
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times associated with a multistage drift diffusion model, or a Wiener diffusion model with piecewise constant
time (FPT) densities of a Wiener process with
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rates and decision boundaries. The model we study is a generalization of that considered in
between two time-varying absorbing boundaries. This model was first studied by
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. The derivation relies on using the optional stopping theorem for properly chosen Martingales, thus obtaining formulae which may be used to compute performance metrics for a particular stage of the stochastic decision
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
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explicitly solve the case of a two stage diffusion model, and provide numerical demonstrations of the computations suggested by our analysis. Finally we present calculations that allow our techniques to approximate time-varying Ornstein-Uhlenbeck processes . By presenting these explicit formulae, we aim to foster the development of refined numerical methods and analytical techniques for studying diffusion decision processes
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
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drift rates and thresholds
evidence