To investigate which model best described the data, we computed the Bayesian evidence E  m or probability of the model given the data for each model, using the Laplace approximation ( Kass and Raftery, 1995): equation(Equation 6) Em≈logp(θˆm)+logp(c1:T|θˆm)+12Gmlog2π−12log|Hm|.This quantity, like the Bayesian Information Criterion ( Schwarz, 1978), which can be derived from it via a further approximation) scores each model according to its fit to the data, penalized for overfitting due to optimizing the models’ parameters. Here, θˆm are the best fitting MAP parameters, p(θˆm) is the value of the prior

on the MAP parameters, p(c1:T|θˆm) is the likelihood of the series of observed choices on trials 1-T, Gm is the number of parameters in CP-868596 order the model m, and |Hm| is the determinant of the Hessian matrix of the second derivatives of the negative log posterior with respect to the parameters, evaluated at the MAP estimate. This Bayesian evidence can then be used to compare models of different complexity by correctly

penalizing models for their differing VE-822 solubility dmso (effective) number of free parameters. Having computed this score separately for each subject and model, to compare the fits at the population level, we used the random-effects Bayesian model selection procedure (Stephan et al., 2009), in which model identity is taken as a random effect—i.e., each subject might instantiate a different model—and the relative proportions of each model across the population are estimated. From these, we derive the exceedance probability XPm, i.e., the posterior probability, given the data, that a particular model m is the most common model in the group. To assess evidence for dose-dependent effects of the DAT1 polymorphism on any of the model parameters of the best-fitting model, we used Jonckheere-Terpstra for ordered alternatives, a nonparametric test due to non-Gaussianity of the parameters. Significance is reported at a very strict Bonferroni-corrected

significance level of 0.0083 (2 genes × 3 parameters). For completeness, we also tested until whether fitted parameter values in the losing model differed with DAT1 genotype. To assess whether the model could replicate the behavioral findings, we generated trial-by-trial choices using the fitted parameters of the best fitting model. We then analyzed these choices in the same way as the original data, again using robust regression analyses. We thank Sabine Kooijman for logistic support; Angelien Heister, Remco Makkinje, and Marlies Naber for genotyping; and Bradley Doll, Sean Fallon, Michael Frank, Guillaume Sescousse, and Jennifer Cook for insightful discussions and feedback. This work makes use of the Brain Imaging Genetics (BIG) database, first established in Nijmegen, the Netherlands, in 2007.