Hi all,

I am trying to create a Stan model to perform parameter estimation of a basic SIR model using simulated data. However, when I run the model, I keep getting the following Informational Message:

‘The current Metropolis proposal is about to be rejected because of the following issue:

Exception: Max number of iterations exceeded (1000)’

The model is runs far more slowly than I feel it should, but does complete. However, it’s estimates are way off, and it finishes with a warning that 82.5% of the iterations ‘ended with a divergence’. I have tried increasing adapt_delta to 0.9, and reparameterizing my model but this hasn’t helped. Does anyone have any advice?

Here is my model:

```
import pystan
import numpy as np
import pandas as pd
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<lower = 0> y_init[3]; //Initial measured population
real<lower = 0> z_init[3]; //True initial population
real<lower = 0> y[M, 3]; //Measured population at measurement times
}
parameters{
real<lower = 0> theta[2]; //theta = {alpha, beta}
real<lower = 0> sigma; //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}] ~ gamma(1.5, 0.1);
sigma ~ gamma(2.3, 0.01);
for (k in 1:3) {
y_init[k] ~ normal(z_init[k], sigma);
y[ ,k] ~ normal(z[, k], sigma);
}
}
"""
sm = pystan.StanModel(model_code = model)
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)
tot = np.zeros(num_meas)
s[0], i[0], r[0] = y0
s_check, i_check, r_check = np.zeros(num_meas),np.zeros(num_meas),np.zeros(num_meas)
# s[1] = s[0] - alpha_true * s[0] * i[0] * N
# i[1] = i[0] + alpha_true * s[0] * i[0] * N - beta_true * i[0]
# r[1] = r[0] + beta_true * i[0]
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(0,0.015)),1)
i_noise[x] = max(0.0, i[x]+np.random.normal(0,0.015))
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]
tot[x] = s_noise[x] + i_noise[x] + r_noise[x]
y0 = [(N - initial_inf)/N ,initial_inf/N,0]
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,'z_init': y0, 'y':data_tr}
fit = sm.sampling(data = stan_data, chains = 2, iter = 1000, n_jobs=1, control = dict(adapt_delta=0.9))
```

Here is a copy-paste of my 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: 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: 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: Max number of iterations exceeded (1000). (in ‘unknown file name’ at

line 35)

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.

ecause of the following issue:

Exception: Max number of iterations exceeded (1000). (in ‘unknown file name’ at

line 35)

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.

ecause of the following issue:

Exception: Max number of iterations exceeded (1000). (in ‘unknown file name’ at

line 35)

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.

ecause of the following issue:

Exception: Max number of iterations exceeded (1000). (in ‘unknown file name’ at

line 35)

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.

ecause of the following issue:

Exception: Max number of iterations exceeded (1000). (in ‘unknown file name’ at

line 35)

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.

ecause of the following issue:

Exception: Max number of iterations exceeded (1000). (in ‘unknown file name’ at

line 35)

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.

ecause of the following issue:

Exception: Max number of iterations exceeded (1000). (in ‘unknown file name’ at

line 35)

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.

ecause of the following issue:

Exception: Max number of iterations exceeded (1000). (in ‘unknown file name’ at

line 35)

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.

ecause of the following issue:

Exception: Max number of iterations exceeded (1000). (in ‘unknown file name’ at

line 35)

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.

ecause of the following issue:

Exception: Max number of iterations exceeded (1000). (in ‘unknown file name’ at

line 35)

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.

ecause of the following issue:

Exception: Max number of iterations exceeded (1000). (in ‘unknown file name’ at

line 35)

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.

ecause of the following issue:

Exception: Max number of iterations exceeded (1000). (in ‘unknown file name’ at

line 35)

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.

ecause of the following issue:

Exception: Max number of iterations exceeded (1000). (in ‘unknown file name’ at

line 35)

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.

ecause of the following issue:

Exception: Max number of iterations exceeded (1000). (in ‘unknown file name’ at

line 35)

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: 1000 / 1000 [100%] (Sampling)

Elapsed Time: 730.557 seconds (Warm-up)

499.38 seconds (Sampling)

1229.94 seconds (Total)WARNING:pystan:825 of 1000 iterations ended with a divergence (82.5 %).

WARNING:pystan:Try running with adapt_delta larger than 0.9 to remove the diverg

ences.

Exectuion time 2773.0 secs

WARNING:pystan:Deprecation warning. PyStan plotting deprecated, use ArviZ librar

y (Python 3.5+).`pip install arviz`

;`arviz.plot_trace(fit)`

)