# Issue with Beta Priors in ODE model

**URL:** <https://discourse.mc-stan.org/t/issue-with-beta-priors-in-ode-model/13114>\
**Category:** Modeling\
**Created:** [February 13, 2020, 5:54pm UTC](https://discourse.mc-stan.org/t/issue-with-beta-priors-in-ode-model/13114 "2020-02-13T17:54:38Z")\
**Posts on this page:** 3\
**Page:** 1

<div class="post-metadata">

**Author:** ![Debbie](https://avatars.discourse-cdn.com/v4/letter/d/90ced4/32.png) [@Debbie](https://discourse.mc-stan.org/u/Debbie)\
**Post date:** [February 13, 2020, 5:54pm UTC](https://discourse.mc-stan.org/t/issue-with-beta-priors-in-ode-model/13114/1 "2020-02-13T17:54:38Z")

</div>

Hello,

I am trying to use PyStan to perform parameter estimation on an SIR model. When I try to sample, I get

> Runtime Error: Exception: Max number of iterations exceeded (1000)

I think the program is taking issue with my beta priors. Apparently the ‘random variable’ is not between 1 and 0. But I don’t understand what random variable it is talking about. Where is it getting these numbers from? And why are they not between 1 and 0? Any advice would be greatly appreciated.

Here is my code:

```stan
import pystan
import numpy as np

model = """
functions{
real[] dz_dt(real t, real[] z, real[] theta, 
              real[] x_r, int[] x_i){
    real N = 1000;
    real s = z[1];
    real i = z[2];
    real r = z[3];
        
    real alpha = theta[1];
    real beta = theta[2];
        
    real ds_dt = -alpha * s * i * N;
    real di_dt = alpha * s * i * N- beta * i;
    real dr_dt = beta * i;
    return {ds_dt, di_dt, dr_dt};
}
}

              
data{
     int<lower = 0> M; //Number of measurements
     real ts[M]; //Measurement times > 0
     real y_init[3]; //Initial measured population proportions
     real<lower = 0> y[M, 3]; //Measured population proportion at measurement times
}

parameters{
    real<lower = 0> theta[2]; //theta = {alpha, beta}
    real<lower = 0> z_init[3]; //True initial population proportion
    real<lower = 0> sigma[3]; //error scale
}

transformed parameters{
    real z[M, 3]
    = integrate_ode_rk45(dz_dt, z_init, 0.0, ts, theta,
                         rep_array(0.0,0), rep_array(0,0),
                         1e-6, 1e-5, 1e3);
}

model{
     theta[{1,2}] ~ normal(0.005, 1);
     sigma ~ normal(0.01, 0.5);
     z_init[1] ~ beta(5, 1.2);
     z_init[2] ~ beta(1.5, 3);
     z_init[3] ~ beta(1.5, 5);
     for (k in 1:3) {
             y_init[k] ~ normal(z_init[k], sigma[k]);
             y[,k] ~ normal(z[, k], sigma[k]);
     }
}
"""
sm = pystan.StanModel(model_code = model)

#######Data Creation##############
num_meas = 100 #Number of measurements
M = num_meas - 1 #Number of measurements minus initial condition
t = np.arange(1,M+1)

alpha_true = 0.001
beta_true = 0.09

initial_inf = 1
N = 1000

y0 = [(N - initial_inf)/N ,initial_inf/N,0]

s,i,r= np.zeros(num_meas), np.zeros(num_meas), np.zeros(num_meas)
s_noise = np.zeros(num_meas)
i_noise = np.zeros(num_meas)
r_noise = np.zeros(num_meas)
s[0], i[0], r[0] = y0

for x in range(M):
    s[x+1] = s[x] - alpha_true * s[x] * i[x] * N
    i[x+1] = i[x] + alpha_true * s[x] * i[x] * N - beta_true * i[x]
    r[x+1] = r[x] + beta_true * i[x]
    
for x in range(M+1):
    s_noise[x] = min(max(0.0, s[x]+np.random.normal(0,0.01)),1) 
    i_noise[x] = max(0.0, i[x]+np.random.normal(0,0.02))
    if s_noise[x] + i_noise[x] > 1:
        i_noise[x] = 1 - s_noise[x]
    r_noise[x] = 1 - s_noise[x] - i_noise[x]

data = np.array([s_noise[1:num_meas],i_noise[1:num_meas],r_noise[1:num_meas]])
data_tr = data.transpose()
stan_data = {'M': M, 'ts':t, 'y_init': y0, 'y':data_tr}

######Run MCMC###########
fit = sm.sampling(data = stan_data, chains = 4, iter = 1000, n_jobs=1)

```

And here is my Error Traceback:

> If this warning occurs sporadically, such as for highly constrained variable typ  
> es like covariance matrices, then the sampler is fine,  
> but if this warning occurs often then your model may be either severely ill-cond  
> itioned or misspecified.
> 
> Informational Message: The current Metropolis proposal is about to be rejected b  
> ecause of the following issue:  
> Exception: beta\_lpdf: Random variable is 1.00135, but must be less than or equal  
> to 1 (in ‘unknown file name’ at line 44)
> 
> If this warning occurs sporadically, such as for highly constrained variable typ  
> es like covariance matrices, then the sampler is fine,  
> but if this warning occurs often then your model may be either severely ill-cond  
> itioned or misspecified.
> 
> Informational Message: The current Metropolis proposal is about to be rejected b  
> ecause of the following issue:  
> Exception: beta\_lpdf: Random variable is 1.02215, but must be less than or equal  
> to 1 (in ‘unknown file name’ at line 44)
> 
> If this warning occurs sporadically, such as for highly constrained variable typ  
> es like covariance matrices, then the sampler is fine,  
> but if this warning occurs often then your model may be either severely ill-cond  
> itioned or misspecified.
> 
> Informational Message: The current Metropolis proposal is about to be rejected b  
> ecause of the following issue:  
> Exception: beta\_lpdf: Random variable is 1.03049, but must be less than or equal  
> to 1 (in ‘unknown file name’ at line 44)
> 
> If this warning occurs sporadically, such as for highly constrained variable typ  
> es like covariance matrices, then the sampler is fine,  
> but if this warning occurs often then your model may be either severely ill-cond  
> itioned or misspecified.
> 
> Informational Message: The current Metropolis proposal is about to be rejected b  
> ecause of the following issue:  
> Exception: beta\_lpdf: Random variable is 1.01304, but must be less than or equal  
> to 1 (in ‘unknown file name’ at line 44)
> 
> If this warning occurs sporadically, such as for highly constrained variable typ  
> es like covariance matrices, then the sampler is fine,  
> but if this warning occurs often then your model may be either severely ill-cond  
> itioned or misspecified.
> 
> Informational Message: The current Metropolis proposal is about to be rejected b  
> ecause of the following issue:  
> Exception: beta\_lpdf: Random variable is 1.0421, but must be less than or equal  
> to 1 (in ‘unknown file name’ at line 44)
> 
> If this warning occurs sporadically, such as for highly constrained variable typ  
> es like covariance matrices, then the sampler is fine,  
> but if this warning occurs often then your model may be either severely ill-cond  
> itioned or misspecified.
> 
> Informational Message: The current Metropolis proposal is about to be rejected b  
> ecause of the following issue:  
> Exception: beta\_lpdf: Random variable is 1.0984, but must be less than or equal  
> to 1 (in ‘unknown file name’ at line 44)
> 
> If this warning occurs sporadically, such as for highly constrained variable typ  
> es like covariance matrices, then the sampler is fine,  
> but if this warning occurs often then your model may be either severely ill-cond  
> itioned or misspecified.
> 
> Informational Message: The current Metropolis proposal is about to be rejected b  
> ecause of the following issue:  
> Exception: beta\_lpdf: Random variable is 1.11171, but must be less than or equal  
> to 1 (in ‘unknown file name’ at line 44)
> 
> If this warning occurs sporadically, such as for highly constrained variable typ  
> es like covariance matrices, then the sampler is fine,  
> but if this warning occurs often then your model may be either severely ill-cond  
> itioned or misspecified.
> 
> Informational Message: The current Metropolis proposal is about to be rejected b  
> ecause of the following issue:  
> Exception: Max number of iterations exceeded (1000). (in ‘unknown file name’ at  
> line 35)
> 
> If this warning occurs sporadically, such as for highly constrained variable typ  
> es like covariance matrices, then the sampler is fine,  
> but if this warning occurs often then your model may be either severely ill-cond  
> itioned or misspecified.
> 
> Informational Message: The current Metropolis proposal is about to be rejected b  
> ecause of the following issue:  
> Exception: beta\_lpdf: Random variable is 1.01083, but must be less than or equal  
> to 1 (in ‘unknown file name’ at line 44)
> 
> If this warning occurs sporadically, such as for highly constrained variable typ  
> es like covariance matrices, then the sampler is fine,  
> but if this warning occurs often then your model may be either severely ill-cond  
> itioned or misspecified.
> 
> Informational Message: The current Metropolis proposal is about to be rejected b  
> ecause of the following issue:  
> Exception: beta\_lpdf: Random variable is 1.10168, but must be less than or equal  
> to 1 (in ‘unknown file name’ at line 44)
> 
> If this warning occurs sporadically, such as for highly constrained variable typ  
> es like covariance matrices, then the sampler is fine,  
> but if this warning occurs often then your model may be either severely ill-cond  
> itioned or misspecified.
> 
> Informational Message: The current Metropolis proposal is about to be rejected b  
> ecause of the following issue:  
> Exception: beta\_lpdf: Random variable is 1.00218, but must be less than or equal  
> to 1 (in ‘unknown file name’ at line 44)
> 
> If this warning occurs sporadically, such as for highly constrained variable typ  
> es like covariance matrices, then the sampler is fine,  
> but if this warning occurs often then your model may be either severely ill-cond  
> itioned or misspecified.
> 
> Informational Message: The current Metropolis proposal is about to be rejected b  
> ecause of the following issue:  
> Exception: beta\_lpdf: Random variable is 1.01742, but must be less than or equal  
> to 1 (in ‘unknown file name’ at line 44)
> 
> If this warning occurs sporadically, such as for highly constrained variable typ  
> es like covariance matrices, then the sampler is fine,  
> but if this warning occurs often then your model may be either severely ill-cond  
> itioned or misspecified.
> 
> Informational Message: The current Metropolis proposal is about to be rejected b  
> ecause of the following issue:  
> Exception: beta\_lpdf: Random variable is 1.0002, but must be less than or equal  
> to 1 (in ‘unknown file name’ at line 44)
> 
> If this warning occurs sporadically, such as for highly constrained variable typ  
> es like covariance matrices, then the sampler is fine,  
> but if this warning occurs often then your model may be either severely ill-cond  
> itioned or misspecified.
> 
> Informational Message: The current Metropolis proposal is about to be rejected b  
> ecause of the following issue:  
> Exception: beta\_lpdf: Random variable is 1.08993, but must be less than or equal  
> to 1 (in ‘unknown file name’ at line 44)
> 
> If this warning occurs sporadically, such as for highly constrained variable typ  
> es like covariance matrices, then the sampler is fine,  
> but if this warning occurs often then your model may be either severely ill-cond  
> itioned or misspecified.
> 
> Informational Message: The current Metropolis proposal is about to be rejected b  
> ecause of the following issue:  
> Exception: Max number of iterations exceeded (1000). (in ‘unknown file name’ at  
> line 35)
> 
> If this warning occurs sporadically, such as for highly constrained variable typ  
> es like covariance matrices, then the sampler is fine,  
> but if this warning occurs often then your model may be either severely ill-cond  
> itioned or misspecified.
> 
> Informational Message: The current Metropolis proposal is about to be rejected b  
> ecause of the following issue:  
> Exception: beta\_lpdf: Random variable is 3.02119, but must be less than or equal  
> to 1 (in ‘unknown file name’ at line 45)
> 
> If this warning occurs sporadically, such as for highly constrained variable typ  
> es like covariance matrices, then the sampler is fine,  
> but if this warning occurs often then your model may be either severely ill-cond  
> itioned or misspecified.
> 
> Informational Message: The current Metropolis proposal is about to be rejected b  
> ecause of the following issue:  
> Exception: beta\_lpdf: Random variable is 1.01776, but must be less than or equal  
> to 1 (in ‘unknown file name’ at line 44)
> 
> If this warning occurs sporadically, such as for highly constrained variable typ  
> es like covariance matrices, then the sampler is fine,  
> but if this warning occurs often then your model may be either severely ill-cond  
> itioned or misspecified.
> 
> Informational Message: The current Metropolis proposal is about to be rejected b  
> ecause of the following issue:  
> Exception: beta\_lpdf: Random variable is 1.00703, but must be less than or equal  
> to 1 (in ‘unknown file name’ at line 44)
> 
> If this warning occurs sporadically, such as for highly constrained variable typ  
> es like covariance matrices, then the sampler is fine,  
> but if this warning occurs often then your model may be either severely ill-cond  
> itioned or misspecified.
> 
> Informational Message: The current Metropolis proposal is about to be rejected b  
> ecause of the following issue:  
> Exception: beta\_lpdf: Random variable is 1.00346, but must be less than or equal  
> to 1 (in ‘unknown file name’ at line 44)
> 
> If this warning occurs sporadically, such as for highly constrained variable typ  
> es like covariance matrices, then the sampler is fine,  
> but if this warning occurs often then your model may be either severely ill-cond  
> itioned or misspecified.
> 
> Informational Message: The current Metropolis proposal is about to be rejected b  
> ecause of the following issue:  
> Exception: Max number of iterations exceeded (1000). (in ‘unknown file name’ at  
> line 35)
> 
> If this warning occurs sporadically, such as for highly constrained variable typ  
> es like covariance matrices, then the sampler is fine,  
> but if this warning occurs often then your model may be either severely ill-cond  
> itioned or misspecified.
> 
> Iteration: 300 / 1000 [30%] (Warmup)  
> Informational Message: The current Metropolis proposal is about to be rejected b  
> ecause of the following issue:  
> Exception: beta\_lpdf: Random variable is 1.00189, but must be less than or equal  
> to 1 (in ‘unknown file name’ at line 44)
> 
> If this warning occurs sporadically, such as for highly constrained variable typ  
> es like covariance matrices, then the sampler is fine,  
> but if this warning occurs often then your model may be either severely ill-cond  
> itioned or misspecified.
> 
> Informational Message: The current Metropolis proposal is about to be rejected b  
> ecause of the following issue:  
> Exception: beta\_lpdf: Random variable is 1.00208, but must be less than or equal  
> to 1 (in ‘unknown file name’ at line 44)
> 
> If this warning occurs sporadically, such as for highly constrained variable typ  
> es like covariance matrices, then the sampler is fine,  
> but if this warning occurs often then your model may be either severely ill-cond  
> itioned or misspecified.
> 
> Informational Message: The current Metropolis proposal is about to be rejected b  
> ecause of the following issue:  
> Exception: beta\_lpdf: Random variable is 1.00331, but must be less than or equal  
> to 1 (in ‘unknown file name’ at line 44)
> 
> If this warning occurs sporadically, such as for highly constrained variable typ  
> es like covariance matrices, then the sampler is fine,  
> but if this warning occurs often then your model may be either severely ill-cond  
> itioned or misspecified.
> 
> Informational Message: The current Metropolis proposal is about to be rejected b  
> ecause of the following issue:  
> Exception: beta\_lpdf: Random variable is 1.00026, but must be less than or equal  
> to 1 (in ‘unknown file name’ at line 44)
> 
> If this warning occurs sporadically, such as for highly constrained variable typ  
> es like covariance matrices, then the sampler is fine,  
> but if this warning occurs often then your model may be either severely ill-cond  
> itioned or misspecified.
> 
> Iteration: 400 / 1000 [40%] (Warmup)  
> Iteration: 500 / 1000 [50%] (Warmup)  
> Iteration: 501 / 1000 [50%] (Sampling)  
> Iteration: 600 / 1000 [60%] (Sampling)  
> Iteration: 700 / 1000 [70%] (Sampling)  
> Iteration: 800 / 1000 [80%] (Sampling)  
> Iteration: 900 / 1000 [90%] (Sampling)  
> Iteration: 1000 / 1000 [100%] (Sampling)
> 
> Elapsed Time: 580.341 seconds (Warm-up)  
> 103.124 seconds (Sampling)  
> 683.465 seconds (Total)
> 
> Rejecting initial value:  
> Error evaluating the log probability at the initial value.  
> Exception: beta\_lpdf: Random variable is 3.19315, but must be less than or equal  
> to 1 (in ‘unknown file name’ at line 44)
> 
> Rejecting initial value:  
> Error evaluating the log probability at the initial value.  
> Exception: beta\_lpdf: Random variable is 4.03353, but must be less than or equal  
> to 1 (in ‘unknown file name’ at line 45)
> 
> Rejecting initial value:  
> Error evaluating the log probability at the initial value.  
> Exception: beta\_lpdf: Random variable is 5.94499, but must be less than or equal  
> to 1 (in ‘unknown file name’ at line 46)
> 
> Unrecoverable error evaluating the log probability at the initial value.  
> Exception: Max number of iterations exceeded (1000). (in ‘unknown file name’ at  
> line 35)
> 
> Traceback (most recent call last):  
> File “C:\Users\dms228\SIR\_norm\_txt.py”, line 101, in   
> fit = sm.sampling(data = stan\_data, chains = 4, iter = 1000, n\_jobs=1)  
> File “C:\Users\dms228\AppData\Local\Continuum\anaconda3\envs\stan\_env\lib\site  
> -packages\pystan\model.py”, line 955, in sampling  
> ret\_and\_samples = \_map\_parallel(call\_sampler\_star, call\_sampler\_args, n\_jobs  
> )  
> File “C:\Users\dms228\AppData\Local\Continuum\anaconda3\envs\stan\_env\lib\site  
> -packages\pystan\model.py”, line 151, in \_map\_parallel  
> map\_result = list(map(function, args))  
> File “stanfit4anon\_model\_3d2c4c1db96e4b77eec6e73ad0ea2184\_587230830382530664.p  
> yx”, line 373, in stanfit4anon\_model\_3d2c4c1db96e4b77eec6e73ad0ea2184\_5872308303  
> 82530664.\_call\_sampler\_star  
> File “stanfit4anon\_model\_3d2c4c1db96e4b77eec6e73ad0ea2184\_587230830382530664.p  
> yx”, line 406, in stanfit4anon\_model\_3d2c4c1db96e4b77eec6e73ad0ea2184\_5872308303  
> 82530664.\_call\_sampler  
> RuntimeError: Exception: Max number of iterations exceeded (1000). (in ‘unknown  
> file name’ at line 35)

---

<div class="post-metadata">

**Author:** ![bbbales2](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/bbbales2/32/77_2.png) [@bbbales2](https://discourse.mc-stan.org/u/bbbales2)\
**Post date:** [February 13, 2020, 7:18pm UTC](https://discourse.mc-stan.org/t/issue-with-beta-priors-in-ode-model/13114/2 "2020-02-13T19:18:03Z")

</div>

> [@Debbie](#):
>
> real\<lower = 0\> z\_init[3];

The constraints need to be specified separately from the distributions. So putting a beta on z\_init doesn’t constrain it to be between 0 and 1 and so it can try to go outside that range (at which point the likelihood cannot be evaluated and you get errors).

Try:

```no-highlight
real<lower = 0, upper = 1> z_init[3];

```

---

<div class="post-metadata">

**Author:** ![Debbie](https://avatars.discourse-cdn.com/v4/letter/d/90ced4/32.png) [@Debbie](https://discourse.mc-stan.org/u/Debbie)\
**Post date:** [February 14, 2020, 2:56pm UTC](https://discourse.mc-stan.org/t/issue-with-beta-priors-in-ode-model/13114/3 "2020-02-14T14:56:20Z")

</div>

Ah, makes sense. Thanks Ben!