Log probability evaluates to log(0), i.e. negative infinity

Modeling
1

I am trying to write a bayesian APC model with second-order random walk priors assigned to age, cohort, and period effects in stan. However, after running the model, an 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.’
always appears. In my code, ‘log function’ is not used. So I cannot figure out the bugs in my code. Please help me to debug, I have been struggling in this program for quite a few days… Thank you!!

apc_model = "
data {
  int<lower=1> I; // Number of observations
  int<lower=0> cases [I] ; // Cases
  vector[I] pyr; // Person-years at risk
  vector[16] age; // Age group indicators
  vector[23] coh; // Cohort group indicators
  vector[9] per; // Period group indicators
}
parameters {
  vector<lower=-1000,upper=1000>[16] alpha; // Age effects
  vector<lower=-1000,upper=1000>[23] gamma; // Cohort effects
  vector<lower=-1000,upper=1000>[9] beta; // Period effects
  real<lower=1> sigma_a; // Standard deviation for age effects
  real<lower=1> sigma_c; // Standard deviation for cohort effects
  real<lower=1> sigma_p; // Standard deviation for period effects
}
transformed parameters {
  vector[16] lambda1;
  vector[23] lambda2;
  vector[9] lambda3;
  real tau_a = pow(sigma_a, -2);
  real tau_c = pow(sigma_c, -2);
  real tau_p = pow(sigma_p, -2);
  
  lambda1 = alpha .* age;
  lambda2 = coh .* gamma;
  lambda3 = per .* beta;
}

model {
  matrix[16, 9] mu;
  for (a in 1:16) {
    for (p in 1:9) {
      mu[a, p] = 100000 * exp(alpha[a] + beta[p] + gamma[p - a + 16]);
      // Likelihood
     }
  }
   cases ~ poisson(to_row_vector(mu));

  
  // Age effects
  alpha[1] ~ normal(0, sqrt(1e-6 * tau_a));
  alpha[2] ~ normal(0, sqrt(1e-6 * tau_a));
  for (a in 3:16) {
    alpha[a] ~ normal(2 * alpha[a - 1] - alpha[a - 2], sqrt(tau_a));
  }
  
  // Cohort effects
  gamma[1] ~ normal(0, sqrt(1e-6 * tau_c));
  gamma[2] ~ normal(0, sqrt(1e-6 * tau_c));
  for (c in 3:23) {
    gamma[c] ~ normal(2 * gamma[c - 1] - gamma[c - 2], sqrt(tau_c));
  }
  
  // Period effects
  beta[1] ~ normal(0, sqrt(1e-6 * tau_p));
  beta[2] ~ normal(0, sqrt(1e-6 * tau_p));
  for (p in 3:9) {
    beta[p] ~ normal(2 * beta[p - 1] - beta[p - 2], sqrt(tau_p));
  }
  
  // Priors for standard deviations
  sigma_a ~ uniform(1, 1000);
  sigma_c ~ uniform(1, 1000);
  sigma_p ~ uniform(1, 1000);
}"

The error message:

SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
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.
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.
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.
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.
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.
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.
[1] "Error : Initialization failed."
[1] "error occurred during calling the sampler; sampling not done"
2

Even though you are not explicitly computing the log of probability, Stan will compute the total likelihood as the sum of log of likelihoods (the alternative is computing product of likelihoods as is much more prone to numeric instabilities). So what is likely happening is the randomly-chosen parameters from the chain have very low probability which underflows to zero (and it’s associated negative infinity log); with that the chain cannot go anywhere – instead you need a finite probability, however small. The easiest way to test if that’s really what is happening is choosing some fixed parameter values (and later relaxing that to make sure chains are converging to the correct region). You can also use simulated data to try to test that, or see if there are any bugs in the implementation.

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