Conjugate Bayesian Analysis of the Wald Model: On an Exact Drift-Rate Posterior

In cognitive psychology, simple response times are often modeled as the time required by a one-dimensional Wiener process with drift to first reach a given threshold. This stochastic process's first-passage time follows a Wald distribution, which is a specific parameterization of the inverse-Gaussian distribution. It can be shown that the Gaussian-Gamma distribution is a conjugate prior with respect to an inverse-Gaussian likelihood, albeit under a parameterization different from that of the Wald distribution. This leads to a posterior distribution that does not directly correspond to the core parameters of the Wiener process; that is, the drift-rate and the threshold parameter. While the marginal threshold posterior under a Gaussian-Gamma prior is relatively easy to derive and turns out to be a known distribution, this is not the case for the marginal drift-rate posterior. The present work addresses this issue by providing the exact marginal posterior distributions of the drift-rate parameter under a Gaussian-Gamma prior—something that has not yet been done in the literature. Unfortunately, the probability density function of this distribution cannot be expressed in terms of elementary functions. Thus, different methods of approximation are discussed as an expedient for time-critical applications.