# Divergent transitions in my Cognitive Latent Variable Model - Caused by identifiability issues?

**URL:** <https://discourse.mc-stan.org/t/divergent-transitions-in-my-cognitive-latent-variable-model-caused-by-identifiability-issues/34043>\
**Category:** Modeling\
**Tags:** divergences, cognitive-science, fitting-issues\
**Created:** [February 6, 2024, 6:55am UTC](https://discourse.mc-stan.org/t/divergent-transitions-in-my-cognitive-latent-variable-model-caused-by-identifiability-issues/34043 "2024-02-06T06:55:36Z")\
**Posts on this page:** 2\
**Page:** 1

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**Author:** ![blam3](https://avatars.discourse-cdn.com/v4/letter/b/8491ac/32.png) [@blam3](https://discourse.mc-stan.org/u/blam3)\
**Post date:** [February 6, 2024, 6:55am UTC](https://discourse.mc-stan.org/t/divergent-transitions-in-my-cognitive-latent-variable-model-caused-by-identifiability-issues/34043/1 "2024-02-06T06:55:36Z")

</div>

I am trying to fit a cognitive/computational model and then regress the parameter values from the cognitive model onto a dependent variable in a regression. In other words, I am trying to simultaneously estimate a cognitive model and a multiple linear regression, in which the parameters from the cognitive models become predictors in a regression. I am doing this with cmdstanr. However, when I try to fit the model, I get a high number of divergent transitions:

```no-highlight
 Warning: 11310 of 16000 (71.0%) transitions ended with a divergence.
 See https://mc-stan.org/misc/warnings for details.

 All 6 chains finished successfully.
 Mean chain execution time: 967.8 seconds.
 Total execution time: 2177.3 seconds.

 Warning: 1911 of 12000 (16.0%) transitions ended with a divergence.
 See https://mc-stan.org/misc/warnings for details.

 Warning: 4908 of 12000 (41.0%) transitions hit the maximum treedepth limit of 10.
 See https://mc-stan.org/misc/warnings for details.

 Warning: 5 of 6 chains had an E-BFMI less than 0.2.
 See https://mc-stan.org/misc/warnings for details.

```

I have been reading about divergent transitions and before I try things like reparameterizing the model, I was wondering if anyone has other guesses about the causes of these divergent transitions? One concern I have is that the model is unidentified as I have not seen this type of model fit before in stan. I am sharing my stan code below and would appreciate any thoughts you may have. Thank you.

* * *

When including Stan code in your post it really helps if you make it as readable as possible by using Stan code chunks (```stan) with clear spacing and indentation. For example, use

```stan
functions {
    real SV(real alpha, real beta, real p, real A) {
    return (p - ( (beta*A) / 2))*pow(1.875, alpha);
  }
}

// Note about reward amount being 1.875: Since WTP is scaled to be 
// between 0 and 1, I'm scaling the reward amount so that it's still 
// 1.875x greater than the max WTP option -> 16 WTP/30 R = 1 WTP/1.875 R

data {
  int<lower=1> N; // num of subjects
  // Predictors
  vector[N] ones; // 1s for intercept in design matrix
  vector[N] ADI;
  vector<lower=0, upper=1>[N] sex;

  // K (which is also a predictor)
  int<lower=1> Tr; // max number of trials
  array[N] int<lower=1, upper=Tr> Tsubj; // num of trials for each subject
  array[N,Tr] real<lower=0> Red_Amount; // subjective prob of winning
  array[N,Tr] real<lower=0> Blue_Amount; // subjective prob of losing
  array[N,Tr] real<lower=0> A;
  array[N,Tr] real<lower=-1, upper=1> WTP; // Willingness to Pay
  
  // Dependent Variable
  vector[N] ext;
}

transformed data {
  // p = subjective probability of wining
  array[N,Tr] real<lower=0> p;
  
  for (i in 1:N) {
    for (t in 1:(Tsubj[i])) {
      if (Red_Amount[i, t] + Blue_Amount[i, t] == 0) {
        p[i,t] = 0;
  }
      else {
        p[i,t] = Red_Amount[i, t] / (Red_Amount[i, t] + Blue_Amount[i, t]);
    }
  }
}

  // for (i in 1:N) {
  // for (t in 1:(Tsubj[i])) {
  // p[i,t] = Red_Amount[i, t] / (Red_Amount[i, t] + Blue_Amount[i, t] + 0.00001);
  // }
  // }
}

parameters {
  // Regression parameters
  vector[3] alpha_regr;
  vector[3] beta_regr; // regression weights w/ ZS prior
  real log_sigma2_ext; // log of variance

  // Declare all parameters as vectors for vectorizing
  // Hyper(group)-parameters  
  vector[2] mu_pr;
  vector<lower=0>[2] sigma; // group level variance
  vector<lower=0>[N] WTP_sigma;

    
  // Subject-level raw parameters (for Matt trick)
  vector[N] z_alpha;
  vector[N] z_beta;
}

transformed parameters {
  // Transform subject-level raw parameters 
  vector[N] inter;
  
  // inverse logit transformation
  vector[N] alpha; // risk sens
  vector[N] beta; // ambiguity sens
  
  real<lower=0> sigma_ext;

  for (i in 1:N) {
    alpha[i] = Phi_approx(mu_pr[1] + sigma[1] * z_alpha[i]) * 2;
    beta[i] = mu_pr[2] + sigma[2] * z_beta[i];
  }
  
  // Compute interaction term
  for (i in 1:N) {
    inter[i] = ( ADI[i] - mean(ADI) ) * ( beta[i] - mean(beta) );
  }
  // // Individual mean
  // for (i in 1:N) {
  // mu_ext[i] = beta_0 + beta_lambda*lambda[i] + 
  // + beta_ADI*ADI[i] + beta_lambdaxADI*inter[i];
  // }
  
  // Transform sigma_ext from log scale
  sigma_ext = sqrt(exp(log_sigma2_ext));
  matrix[N, 3] Z; // Design matrix for intercept + covariates.
  
  // Covariates in regression
  Z[1:N, 1] = ones; // intercept
  Z[1:N, 2] = alpha; // risk-taking 
  Z[1:N, 3] = sex;
  
  matrix[N, 3] X;
  matrix[3, 3] XtX_inverse;
  cov_matrix[3] Sigma0;
  
  X[1:N, 1] = ADI;
  X[1:N, 2] = beta; // ambiguity sensitivity
  X[1:N, 3] = inter;
  
  // Relevant matrices
  XtX_inverse = inverse_spd(X'*X);
  
  // Prior covariance matrix of beta (given sigma_ext)
  Sigma0 = N*(sigma_ext^2)*XtX_inverse;
}

model {
  
  // hyper-parameters
  mu_pr ~ normal(0,1);
  sigma ~ normal(0,5);
  
  // Cognitive model priors
  z_alpha ~ normal(0,1);
  z_beta ~ normal(0, 1);
  WTP_sigma ~ student_t(3, 0, 1);

  // Regression priors
  sigma_ext ~ student_t(3, 0, 1);

  // Psychological model  
  for (i in 1:N) {
      // Define values
      real mu_WTP;
      
    for (t in 1:(Tsubj[i])) {
      mu_WTP = SV(alpha[i], beta[i], p[i,t], A[i,t]);
      WTP[i,t] ~ normal(mu_WTP, WTP_sigma[i]);
    }
  }
  
// Regression priors
  
  // Recall that uniform priors (on alphas) don't need to be specified.
  // Jeffreys prior on sigma2_ext <-> uniform prior on log_sigma2_ext
  // uniform (Jeffreys) prior on alpha (intercept)
  // Zellner-Siow prior on other regression parameters (multivariate Cauchy = multivariate t with df = 1)
  beta_regr ~ multi_student_t(1, [0, 0, 0], Sigma0);
  alpha_regr ~ uniform(-4,4);
  
  // Regression
  ext ~ normal(Z*alpha_regr + X*beta_regr, sigma_ext);

}

generated quantities {

  vector[N] mu_ext;
  vector[N] log_lik;
  
  log_lik = rep_vector(0, N);
  
  { // Local section
    
    for (i in 1:N) {
      // Getting conditional mean for PPC
      mu_ext[i] = Z[i,]*alpha_regr + X[i,]*beta_regr;
      
      // Log-likelihood for regression
      log_lik[i] = normal_lpdf(ext[i] | mu_ext[i], sigma_ext);
    }
  }
    // Generating predictions for posterior predictive check
    array[N] real pred_ext = normal_rng(mu_ext, sigma_ext);
}

} 

```

Here is more information about my model:

```no-highlight
Levy_clvm_fit <- Levy_clvm_mod$sample(
  data = data_list, 
  seed = 905, 
  chains = 8, 
  parallel_chains = 4,
  iter_warmup = 3000,
  iter_sampling = 2000,
  adapt_delta = 0.99,
  refresh = 10 # print update every 10 iters
)

```

I am using Jeffrey-Zellner-Siow priors. My cognitive model is from [Levy et al. (2010)](https://journals.physiology.org/doi/full/10.1152/jn.00853.2009?rfr_dat=cr_pub++0pubmed&url_ver=Z39.88-2003&rfr_id=ori%3Arid%3Acrossref.org) and I am regressing the “ambiguity aversion” parameter onto a dependent variable called externalizing. I appreciate any advice you might have to offer. Thanks!

---

<div class="post-metadata">

**Author:** ![stijn](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/stijn/32/105_2.png) [@stijn](https://discourse.mc-stan.org/u/stijn)\
**Post date:** [February 6, 2024, 8:21am UTC](https://discourse.mc-stan.org/t/divergent-transitions-in-my-cognitive-latent-variable-model-caused-by-identifiability-issues/34043/2 "2024-02-06T08:21:54Z")

</div>

> [@blam3](#):
>
> `alpha_regr ~ uniform(-4,4)`

```no-highlight
  vector[3] alpha_regr;

```

It’s likely not the only thing, but these two lines together can cause problems. It’s better to specify

```no-highlight
  vector<lower=-4, upper=4>[3] alpha_regr;

```

which makes the first statement obsolete.