#The model is based on the Bayen et al. 1996  model 6d 

## theta[s, 1] = D1
## theta[s, 2] = D2
##theta[s, 2] = D3 = D1
## theta[s, 3] = d1
## theta[s, 4]= d2
## theta[s, 5] = a
## theta[s, 6] = b
## theta[s, 5] = g = a


model{
  ## loop over items
  for (s in 1:n_items){    
    
    ## subtree 1 (shown complete): category probabilities
    
    Prob_C[s, 1] <- (theta[s, 1] * theta[s, 3]) + (theta[s, 1] * (1-theta[s, 3]) * (1-theta[s, 5])) + ((1-theta[s, 1]) * theta[s, 6] * (1-theta[s, 5]))
    
    Prob_C[s, 2] <- (theta[s, 1] * (1-theta[s, 3]) * theta[s, 5]) + ((1-theta[s, 1]) * theta[s, 6] * theta[s, 5]) 
    
    Prob_C[s, 3] <- ((1-theta[s, 1]) * (1-theta[s, 6]))
    
    
    ## Likelihood function for subtree 1:
    
    tree_complete[s, 1:3] ~ dmulti(Prob_C[s, 1:3], C[s])
    
    ## subtree 2 (shown halved): category probabilities
    
    Prob_H[s, 1] <- (theta[s, 2] * theta[s, 4]) + (theta[s, 2] * (1-theta[s, 4]) * theta[s, 5]) + ((1-theta[s, 2]) * theta[s, 6] * theta[s, 5])
    
    Prob_H[s, 2] <- (theta[s, 2] * (1-theta[s, 4]) * (1-theta[s, 5])) + ((1-theta[s, 2]) * theta[s, 6] * (1-theta[s, 5]))
    
    Prob_H[s, 3] <- ((1-theta[s, 2]) * (1-theta[s, 6]))
    
    ## Likelihood function for subtree 2:
    
    tree_halved[s, 1:3] ~ dmulti(Prob_H[s, 1:3], H[s])
    
    ## subtree 3 (no presenation (NEW)): category probabilities
    
    Prob_N[s, 1] <- theta[s, 1] + ((1- theta[s, 1]) *(1-theta[s, 6]))
    
    Prob_N[s, 2] <-  (1- theta[s, 1]) *theta[s, 6] * (1-theta[s, 5])
    
    Prob_N[s, 3] <- ((1- theta[s, 1]) *theta[s, 6] * theta[s, 5])
    
    ## Likelihood function for subtree 3:
    
    tree_new[s, 1:3] ~ dmulti(Prob_N[s, 1:3], N[s])
    
    
    for(p in 1:P) {
      #probit transformation of parameters
      theta[s, p] <- phi(theta_probit[s, p])
      
    }
    
    
    for (p in 1:P){
      #multivariate regression to predict mu of the distribution of probit transformed parameters
      theta_probit[s, p] <- b0[p] + b1[p] * x1[s]+ b2[p] * x2[s]+ xi_item[p]*delta_mu_raw[s,p]
    }
    # deviance of the stimulus parameters from the groups means
    delta_mu_raw[s, 1:P] ~ dmnorm(mu_delta_raw[1:P], tau_prec_mu[1:P,1:P]) 
    
    pred_complete[s, 1:3]  <-  Prob_C[s, 1:3]*C[s]
    pred_halved[s, 1:3]  <-  Prob_H[s, 1:3]*H[s]
    pred_new[s, 1:3]  <-  Prob_N[s, 1:3]*N[s]
    
    residualC[s, 1:3]  <- tree_complete[s, 1:3] - pred_complete[s, 1:3]
    residualH[s, 1:3]  <- tree_halved[s, 1:3] - pred_halved[s, 1:3]
    residualN[s, 1:3]  <- tree_new[s, 1:3] - pred_new[s, 1:3]
    sqC[s, 1:3]  <- pow(residualC[s, 1:3], 2)
    sqH[s, 1:3]  <- pow(residualH[s, 1:3], 2)
    sqN[s, 1:3]  <- pow(residualN[s, 1:3], 2)
    
    #posterior predictions
    postpred_complete[s, 1:3] ~ dmulti(Prob_C[s, 1:3], C[s])
    postpred_halved[s, 1:3] ~ dmulti(Prob_H[s, 1:3], H[s])
    postpred_new[s, 1:3] ~ dmulti(Prob_N[s, 1:3], N[s])
    
    residualC_post[s, 1:3]  <- postpred_complete[s, 1:3] - pred_complete[s, 1:3]
    residualH_post[s, 1:3]  <- postpred_halved[s, 1:3] - pred_halved[s, 1:3]
    residualN_post[s, 1:3]  <- postpred_new[s, 1:3] - pred_new[s, 1:3]
    
    sqC.post[s, 1:3]  <- pow(residualC_post[s, 1:3], 2)
    sqH.post[s, 1:3]  <- pow(residualH_post[s, 1:3], 2)
    sqN.post[s, 1:3]  <- pow(residualN_post[s, 1:3], 2)
    
  }
  
#priors For scaling factor and regression weights
  
  for (p in 1:P){
    xi_item[p] ~ dnorm(0,1)
    mu_delta_raw[p] <- 0
    b0[p] ~ dnorm(0,1)
    b1[p] ~ dnorm(0,1)
    b2[p] ~ dnorm(0,1)
    
  }
  
  
  # invers wishart: prior distribution of covariance matric of the multivariate distribution of the scaling factor   
  tau_prec_mu[1:P, 1:P] ~ dwish(W[1:P, 1:P], (P+1))
  sigma_mu[1:P, 1:P] <- inverse(tau_prec_mu[1:P, 1:P])
  
  fit  <- sum(sqC[,1:3]) + sum(sqH[,1:3]) + sum(sqN[,1:3])
  fit.new  <- sum(sqC.post[,1:3]) + sum(sqH.post[,1:3]) + sum(sqN.post[,1:3])
  test  <- step(fit.new-fit)
  bpvalue  <- mean(test)
  
  
  ## close model
}