Hi all. I am trying to fit some data generated using an exponential function with added noises using PyStan. Attached below is the Stan code as well as the dummy data in the Python code I’ve been using for fitting purposes.
exp_func_fit = '''
data {
int<lower=0> N;
real<lower=0> t[N];
real<lower=0> y[N];
}
transformed data{}
parameters{
real theta;
real sigma;
}
transformed parameters{
real<lower=0> e_func[N];
for (n in 1:N) {
e_func[n] = 1  exp((theta * t[n]));
}
}
model{
theta ~ normal(1, 0.05);
sigma ~ normal(0.2, 0.02);
y ~ normal(e_func, sigma);
}
generated quantities {
// Simulate data from posterior predictive distribution
//real<lower=0> y_tilde[N];
real y_tilde[N];
y_tilde[1] = 0;
for (n in 2:N) {
y_tilde[n] = normal_rng((1  exp((theta * t[n]))), sigma);
}
}
'''
sm = stan.StanModel(model_code=exp_func_fit)
pickle.dump(sm, open('exp_func_fit.pkl','wb'))
sm = pickle.load(open('exp_func_fit.pkl', 'rb'))
N = 21
t = [(x * 0.5) for x in range(0, N*2)]
A = np.array([0. , 0.35644848, 0.58936862, 0.70920895, 0.54697252,
0.8042943 , 0.80233711, 0.99545308, 1.54664198, 1.42830176,
0.79185403, 0.85867071, 1.1133023 , 0.87957971, 1.17921721,
0.86249417, 0.60621984, 0.83061225, 0.98675397, 0.82198135,
1.18468728, 1.09527567, 0.85214074, 1.11789476, 0.8759278 ,
1.00827872, 0.68980416, 0.85304309, 1.15551683, 1.33109103,
1.06333705, 0.98227956, 0.73697556, 1.28526364, 1.21358908,
1.14064481, 0.98691966, 0.92229333, 0.61856001, 1.25956421,
0.43418382, 0.9630146 ])
sam_fit_func = dict(N=len(t),t=t,y=A)
fit = sm.sampling(data=sam_fit_func, seed=12345)
The result of the sampling is as shown below, with the scatter plot being the data, the navy plots representing part of the samples and the other lines in the middle as the posterior means of the 4 chains:
I have mainly two issues here:

Suppose I know that the data can’t be less than zero. If I set
real<lower=0> y_tilde[N];
in thegenerated quantities
block, I get thrown some error indicating
"RuntimeError: Exception: validate transformed params: y_tilde[i_0__] is 0.778627, but must be greater than or equal to 0 (in ‘unknown file name’ at line 36)
"
and it just get stuck. While it has not caused anything to break but even the restriction in thetransformed parameters
spewed some exception error in the terminal. Is this just simply a restriction issue or I have to sort of improve the model? 
Since I know at point 0 I will get y=0 so I’ve hardcoded it to be zero in the samples as well. Is there a better way to do this since this has messed up the sampling and I get complaints like:
“WARNING:pystan:n_eff / iter below 0.001 indicates that the effective sample size has likely been overestimated
WARNING:pystan:Rhat above 1.1 or below 0.9 indicates that the chains very likely have not mixed”
but I know it’s where t=0 that’s causing the problem and other parameters have converged well.
I thank you for your help in advance!