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18.3 Wiener First Passage Time Distribution

18.3.1 Probability Density Function

If \(\alpha \in \mathbb{R}^+\), \(\tau \in \mathbb{R}^+\), \(\beta \in [0, 1]\) and \(\delta \in \mathbb{R}\), then for \(y > \tau\), \[ \text{Wiener}(y|\alpha, \tau, \beta, \delta) = \frac{\alpha^3}{(y-\tau)^{3/2}} \exp \! \left(- \delta \alpha \beta - \frac{\delta^2(y-\tau)}{2}\right) \sum_{k = - \infty}^{\infty} (2k + \beta) \phi \! \left(\frac{2k \alpha + \beta}{\sqrt{y - \tau}}\right) \] where \(\phi(x)\) denotes the standard normal density function; see (Feller 1968), (Navarro and Fuss 2009).

18.3.2 Sampling Statement

y ~ wiener(alpha, tau, beta, delta)

Increment target log probability density with wiener_lpdf(y | alpha, tau, beta, delta) dropping constant additive terms.

18.3.3 Stan Functions

real wiener_lpdf(reals y | reals alpha, reals tau, reals beta, reals delta)
The log of the Wiener first passage time density of y given boundary separation alpha, non-decision time tau, a-priori bias beta and drift rate delta

18.3.4 Boundaries

Stan returns the first passage time of the accumulation process over the upper boundary only. To get the result for the lower boundary, use \[ \text{wiener}(y | \alpha, \tau, 1 - \beta, - \delta) \] For more details, see the appendix of Vandekerckhove and Wabersich (2014).


Feller, William. 1968. An Introduction to Probability Theory and Its Applications. Vol. 1. 3. Wiley, New York.

Navarro, Danielle J, and Ian G Fuss. 2009. “Fast and Accurate Calculations for First-Passage Times in Wiener Diffusion Models.” Journal of Mathematical Psychology 53 (4): 222–30.

Vandekerckhove, Joachim, and Dominik Wabersich. 2014. “The RWiener Package: An R Package Providing Distribution Functions for the Wiener Diffusion Model.” The R Journal 6/1.