# Rejecting initial value - Gradient evaluated at the initial value is not finite

**URL:** <https://discourse.mc-stan.org/t/rejecting-initial-value-gradient-evaluated-at-the-initial-value-is-not-finite/36298>\
**Category:** Modeling\
**Tags:** fitting-issues, specification\
**Created:** [August 18, 2024, 7:33am UTC](https://discourse.mc-stan.org/t/rejecting-initial-value-gradient-evaluated-at-the-initial-value-is-not-finite/36298 "2024-08-18T07:33:08Z")\
**Posts on this page:** 2\
**Page:** 1

<div class="post-metadata">

**Author:** ![seanufo](https://avatars.discourse-cdn.com/v4/letter/s/71e660/32.png) [@seanufo](https://discourse.mc-stan.org/u/seanufo)\
**Post date:** [August 18, 2024, 7:33am UTC](https://discourse.mc-stan.org/t/rejecting-initial-value-gradient-evaluated-at-the-initial-value-is-not-finite/36298/1 "2024-08-18T07:33:08Z")

</div>

I’m trying to fit a hierachical model described by this paper, [A Bayesian spatio-temporal model of COVID-19 spread in England | Scientific Reports](https://www.nature.com/articles/s41598-024-60964-0#:~:text=The%20Bayesian%20model%20outperformed%20the,inconsistent%20clustering%20patterns%20over%20time). Yet I can’t seem to figure out this message:  
Chain 1: Rejecting initial value:  
Chain 1: Gradient evaluated at the initial value is not finite.  
Chain 1: Stan can’t start sampling from this initial value.  
Chain 1:  
Chain 1: Initialization between (-2, 2) failed after 100 attempts.  
Chain 1: Try specifying initial values, reducing ranges of constrained values, or reparameterizing the model.

I’ve tried initialising everything to 0 at the start, and printing each of the parameters, but none of them are `nan`, so I’m not too sure what’s going on. Model code is below. Any help would be greatly appreciated.

```stan
data {
  int<lower=0> s; // number of spatial locations
  int<lower=0> t; // number of time points
  int<lower=0> k; // number of covariates
  
  vector<lower=0>[t*s] N; // population
  
  int<lower=0> y[t*s]; // response data
  matrix[s, k] X; // dependent data
  
  cov_matrix[s] big_sigma_w;
}

parameters {
  vector[k] beta; // covariates
  vector[s] w[t]; // spatial random effect

  vector[s] xi_start; // first value of xi
  
  real<lower=0> sigma_w; // Spatial variance parameter
  real<lower=-1, upper=1> alpha; // Temporal dependence parameter
}

transformed parameters {
  
  // Transform
  vector[s] N_vec[t];
  for (i in 1:t) {
    N_vec[i] = to_vector(N[((i - 1) * s + 1):(i * s)]);
  }
  
  vector[s] xi[t]; // spatio-temporal random effect
  xi[1] = xi_start; // Assign first value

  if (t > 1){
    for (i in 2:t){
      xi[i] = alpha * xi[i-1] + w[i];
    }
  }
  
  vector[s] log_theta[t]; // infection risk
  vector[s] mu[t]; // mean
  
  for(i in 1:t){
    log_theta[i] = X * beta + xi[i];
    mu[i] = N_vec[i] .* exp(log_theta[i]);
  }
  
  vector[t*s] mu_flat;
  
  for (i in 1:t) {
    mu_flat[((i - 1) * s + 1):(i * s)] = mu[i];
  }
  
}

model {
  
  // Penalized complexity prior for sigma_w using Half-Normal
  // This ensures p(sigma_w >1) = 0.01
  sigma_w ~ normal(0, 0.43); // Adjust the standard deviation as needed

  // // Apply Beta prior to bound_alpha, which is transformed to alpha
  // // This ensures p(alpha > 0) = 0.9
  // bound_alpha ~ beta(2, 20);
  
  xi_start ~ normal(0, sigma_w^2 / (1 - alpha^2));
  
  beta ~ normal(0, 1000);
  
  w ~ multi_normal(rep_vector(0, s), sigma_w^2*big_sigma_w);
  
  
  for (i in 1:t*s){
    y[i] ~ poisson(mu_flat[i]);
  }
  
  for (i in 1:t*s){
    print("mu_flat[", i, "] = ", mu_flat[i]);
    print("y[", i, "] = ", y[i]);
  }
  
  // for(i in 1:k){
  // print("beta[", i, "]", beta[k]);
  // }
  
  // for( i in 1:t){
  // print("w[", i, "]", w[i]);
  // }
  
  // print("sigma_w = ", sigma_w);
  // print("alpha = ", alpha);
  
  
}

generated quantities {
  vector[t*s] fitted_values;

  for (i in 1:t*s) {
    fitted_values[i] = poisson_rng(mu_flat[i]);
  }
}

```

Trying with the full dataset gives this error:  
Chain 1: Rejecting initial value:  
Chain 1: Log probability evaluates to log(0), i.e. negative infinity.  
Chain 1: Stan can’t start sampling from this initial value.

Thanks all

---

<div class="post-metadata">

**Author:** ![martinmodrak](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/martinmodrak/32/133_2.png) [@martinmodrak](https://discourse.mc-stan.org/u/martinmodrak)\
**Post date:** [August 22, 2024, 5:07am UTC](https://discourse.mc-stan.org/t/rejecting-initial-value-gradient-evaluated-at-the-initial-value-is-not-finite/36298/2 "2024-08-22T05:07:43Z")

</div>

The problem is reported with the gradient, which may be problematic even if the values all look good. Unfortunately, debugging gradients is not really possible in any direct way.

I don’t see anything _obviously_ wrong with the model. So I think your best bet is to try to simplify the model to see which part introduces the issues (e.g. by replacing some parameters with constants/moving them to data)

Best of luck with your model!