Jochen Voss

Interpreting the parameters of the diffusion model: An empirical validation

The diffusion model (Ratcliff, 1978) allows for the statistical separation of different components of a speeded binary decision process (decision threshold, bias, information uptake, and motor response). These components are represented by different parameters of the model. Two experiments were conducted to test the interpretational validity of the parameters. Using a color discrimination task, we investigated whether experimental manipulations of specific aspects of the decision process had specific effects on the corresponding parameters in a diffusion model data analysis (see Ratcliff, 2002; Ratcliff & Rouder, 1998; Ratcliff, Thapar, & McKoon, 2001, 2003). In support of the model, we found that (1) decision thresholds were higher when we induced accuracy motivation, (2) drift rates (i.e., information uptake) were lower when stimuli were harder to discriminate, (3) the motor components were increased when a more difficult form of response was required, and (4) the process was biased towardrewarded responses.

Fast-dm: A free program for efficient diffusion model analysis

In the present article, a flexible and fast computer program, calledfast-dm, for diffusion model data analysis is introduced. Fast-dm is free software that can be downloaded from the authors’ websites. The program allows estimating all parameters of Ratcliff ’s (1978) diffusion model from the empirical response time distributions of any binary classification task. Fast-dm is easy to use: it reads input data from simple text files, while program settings are specified by command0s in a control file. With fast-dm, complex models can be fitted, where some parameters may vary between experimental conditions, while other parameters are constrained to be equal across conditions. Detailed directions for use of fast-dm are presented, as well as results from three short simulation studies exemplifying the utility of fast-dm.

ANALYSIS OF SPDES ARISING IN PATH SAMPLING PART II: THE NONLINEAR CASE

In many applications, it is important to be able to sample paths of SDEs conditional on observations of various kinds. This paper studies SPDEs which solve such sampling problems. The SPDE may be viewed as an infinite-dimensional analogue of the Langevin equation used in finite-dimensional sampling. In this paper, conditioned nonlinear SDEs, leading to nonlinear SPDEs for the sampling, are studied. In addition, a class of preconditioned SPDEs is studied, found by applying a Green’s operator to the SPDE in such a way that the invariant measure remains unchanged; such infinite dimensional evolution equations are important for the development of practical algorithms for sampling infinite dimensional problems. The resulting SPDEs provide several significant challenges in the theory of SPDEs. The two primary ones are the presence of nonlinear boundary conditions, involving first order derivatives, and a loss of the smoothing property in the case of the pre-conditioned SPDEs. These challenges are overcome and a theory of existence, uniqueness and ergodicity is developed in sufficient generality to subsume the sampling problems of interest to us. The Gaussian theory developed in Part I of this paper considers Gaussian SDEs, leading to linear Gaussian SPDEs for sampling. This Gaussian theory is used as the basis for deriving nonlinear SPDEs which affect the desired sampling in the nonlinear case, via a change of measure.

MCMC methods for diffusion bridges

We present and study a Langevin MCMC approach for sampling nonlinear diffusion bridges. The method is based on recent theory concerning stochastic partial differential equations (SPDEs) reversible with respect to the target bridge, derived by applying the Langevin idea on the bridge pathspace. In the process, a Random-Walk Metropolis algorithm and an Independence Sampler are also obtained. The novel algorithmic idea of the paper is that proposed moves for the MCMC algorithm are determined by discretising the SPDEs in the time direction using an implicit scheme, parametrised by θ ∈ [0,1]. We show that the resulting infinite-dimensional MCMC sampler is well-defined only if θ = 1/2, when the MCMC proposals have the correct quadratic variation. Previous Langevin-based MCMC methods used explicit schemes, corresponding to θ = 0. The significance of the choice θ = 1/2 is inherited by the finite-dimensional approximation of the algorithm used in practice. We present numerical results illustrating the phenomenon and the theory that explains it. Diffusion bridges (with additive noise) are representative of the family of laws defined as a change of measure from Gaussian distributions on arbitrary separable Hilbert spaces; the analysis in this paper can be readily extended to target laws from this family and an example from signal processing illustrates this fact.

MAP estimators and their consistency in Bayesian nonparametric inverse problems

We consider the inverse problem of estimating an unknown function u from noisy measurements y of a known, possibly nonlinear, map

Assessing cognitive processes with diffusion model analyses: a tutorial based on fast-dm-30

\mathcal {G}

Approximations to the Stochastic Burgers Equation

applied to u. We adopt a Bayesian approach to the problem and work in a setting where the prior measure is specified as a Gaussian random field μ0. We work under a natural set of conditions on the likelihood which implies the existence of a well-posed posterior measure, μy. Under these conditions, we show that the maximum a posteriori (MAP) estimator is well defined as the minimizer of an Onsager–Machlup functional defined on the Cameron–Martin space of the prior; thus, we link a problem in probability with a problem in the calculus of variations. We then consider the case where the observational noise vanishes and establish a form of Bayesian posterior consistency for the MAP estimator. We also prove a similar result for the case where the observation of

Separating response-execution bias from decision bias: Arguments for an additional parameter in Ratcliff's diffusion model

\mathcal {G}(u)

Sampling conditioned diffusions

can be repeated as many times as desired with independent identically distributed noise. The theory is illustrated with examples from an inverse problem for the Navier–Stokes equation, motivated by problems arising in weather forecasting, and from the theory of conditioned diffusions, motivated by problems arising in molecular dynamics.

Some Fundamental Properties of a Multivariate von Mises Distribution

Diffusion models can be used to infer cognitive processes involved in fast binary decision tasks. The model assumes that information is accumulated continuously until one of two thresholds is hit. In the analysis, response time distributions from numerous trials of the decision task are used to estimate a set of parameters mapping distinct cognitive processes. In recent years, diffusion model analyses have become more and more popular in different fields of psychology. This increased popularity is based on the recent development of several software solutions for the parameter estimation. Although these programs make the application of the model relatively easy, there is a shortage of knowledge about different steps of a state-of-the-art diffusion model study. In this paper, we give a concise tutorial on diffusion modeling, and we present fast-dm-30, a thoroughly revised and extended version of the fast-dm software (Voss and Voss, 2007) for diffusion model data analysis. The most important improvement of the fast-dm version is the possibility to choose between different optimization criteria (i.e., Maximum Likelihood, Chi-Square, and Kolmogorov-Smirnov), which differ in applicability for different data sets.