Diffusion drift model (ddm) fitting issues with rstan

Dear Stan experts,

I’m now trying to apply the classical diffusion drift model (ddm) to my data (a binary-choice task; accept or reject), by using the hierarchical Bayesian estimation via rstan.

My model is based on the existent ddm examples (e.g., hBayesDM toolbox; and the wiener distribution doc, see: https://github.com/stan-dev/stan/issues/1576). The only modification I did here is that the drift rate (v) is modulated by trial-wise values (i.e., v = alphaP_S + betaP_O; see the model below for details).

Unfortunately, the estimation is quite poor (i.e., the rhat of the estimated parameters is much larger than 1; with n_sample = 3000, n_warmup = 2000, n_chain = 4). Also, there are always warnings about rejecting initial values at the beginning of the sampling.

Please find my Stan model code below:

data {
  int <lower=1> S; // number of subjects
  int <lower=1> T; // number of maxtrials among all sbjs
  int <lower=1, upper=T> Tsbj[S]; // numbers of valid trials per sbj
  int <lower=0> P_S[S,T]; // trial-wise payoff for S
  int <lower=0> P_O[S,T]; // trial-wise payoff for O
  int <lower=0,upper=1> IsAccept[S,T]; // 1 for accept, 0 for reject
  real <lower=0> DT[S,T]; // trial-wise decision time (in seconds)
  real minDT[S];       // minimum RT for each subject of the observed data 

parameters {
// Declare all parameters as vectors for vectorizing
  // Hyper(group)-parameters  
  vector[5] mu_p;  
  vector<lower=0>[5] sigma;
  // Subject-level raw parameters (for Mathematics trick)
  vector[S] alpha_pr;
  vector[S] beta_pr;
  vector[S] bound_pr; // boundary for acceptance
  vector[S] nonDT_pr; // non-decision time (in s)
  vector[S] SP_pr; // starting point or initial bias


transformed parameters {
  // Transform subject-level raw parameters 
  vector<lower=-20,upper=20>[S] alpha; // weights on the payoff for S [-20,20]
  vector<lower=-20,upper=20>[S] beta; // weights on the payoff for O [-20,20]
  vector<lower=0,upper=100>[S] bound; // boundary [0,20]
  vector<lower=0.1,upper=max(minDT)>[S] nonDT; // non-decision time [0.1,max(minRT)]; 0.1 s as the bilogical wall
  vector<lower=0,upper=1>[S] SP; // starting point or initial bias [0,1]
  for (i in 1:S) {
    alpha[i] = -20 + Phi_approx( mu_p[1] + sigma[1] * alpha_pr[i] )*40;
    beta[i] = -20 + Phi_approx( mu_p[2] + sigma[2] * beta_pr[i] )*40;
    bound[i] = Phi_approx( mu_p[3] + sigma[3] * bound_pr[i] )*100;
    nonDT[i]  = Phi_approx( mu_p[4] + sigma[4] * nonDT_pr[i] )*(minDT[i]-0.1) + 0.1;
    SP[i]  = Phi_approx( mu_p[5] + sigma[5] * SP_pr[i] );

model {
  // Hyperparameters
  mu_p  ~ normal(0, 1); 
  sigma ~ cauchy(0, 5);
  // individual parameters
  alpha_pr ~ normal(0, 1); 
  beta_pr ~ normal(0, 1);
  bound_pr ~ normal(0, 1);
  nonDT_pr ~ normal(0, 1);
  SP_pr ~ normal(0, 1);
  for (i in 1:S) {
    // Define values
    real U_Accept; // U_Accept is the utility for the accept option
    real U_Reject; // U_Reject is the utility for the reject option
    for (t in 1:(Tsbj[i])) {

      // Calculating utility (served as drift rate)
      U_Accept = alpha[i]*P_S[i,t] + beta[i]*P_O[i,t];
      U_Reject = 0;
     if (IsAccept[i,t] == 1) {
        DT[i,t] ~ wiener(bound[i], nonDT[i], SP[i], (U_Accept - U_Reject));
      } else {
        DT[i,t] ~ wiener(bound[i], nonDT[i], 1 - SP[i], -1*(U_Accept - U_Reject));

    } // end of trial 
  } //end of subjects
} // end of model section

I would like to make sure whether my code is correct (esp., the new drift rate). Thanks a lot for your help!


Dear Yang,

I hope I am not too late :D. The sampling issues you encounter have been solved by:

Ahn, Woo-Young, Nathaniel Haines, and Lei Zhang. 2016. “Revealing Neuro-Computational Mechanisms of Reinforcement Learning and Decision-Making with the hBayesDM Package.” bioRxiv . Cold Spring Harbor Laboratory. doi:10.1101/064287.

Check out their package and how they solved it.

I extended their framework to include trial-level estimates for inter-trial-variabilities. You can find my package here: https://github.com/Seneketh/StanDDM . You can also find my masters thesis here: http://fse.studenttheses.ub.rug.nl/19786/ , where I explain more details.

Let me know if you have any questions.

All the best,



This looks like awesome work @seneketh, thanks for your contributions!