Hello, I have been following the bayesian hierarchical tutorial here and this forum question here in trying to solve my problem but I cannot seem to figure out what I am doing wrong and I figure someone here would probably be able to figure it out.

For my example, let us assume that I am trying to model whether a student applies for a spot at a university. My data generating process looks like this.

```
import uuid
import numpy as np
import pandas as pd
from scipy.special import expit, logit
treatment_effect_magnitude = 0
n_students = 200
n_universities = 10
prob_treat = 0.5
intercept = -0.25
n_students = int(n_students)
n_universities = int(n_universities)
# create a list of students
student_ids = [str(uuid.uuid4()) for _ in range(n_students)]
# create a list of universities
uni_ids = [str(uuid.uuid4()) for _ in range(n_universities)]
# mu_i is the affinity of student i to apply for any uni
mu_i = np.random.normal(0, 0.2, n_students)
# zeta_j is the affinity of uni j to receive an apply from any student
zeta_j = np.random.normal(0, 0.1, n_universities)
# assign users to a flight
treatment = np.random.binomial(1, prob_treat, n_students)
data = []
for student_i, student_id in enumerate(student_ids):
for uni_j, uni_id in enumerate(uni_ids):
# generate linear predictor
eta = intercept + treatment_effect_magnitude * treatment[student_i] + mu_i[student_i] + zeta_j[uni_j]
# generate the probability that the student will apply for uni j
prob_apply = expit(eta)
# generate outcome
y = np.random.binomial(1, prob_apply, 1)[0]
# store row of data for later model fitting
data.append(dict(
y = y,
treatment = 'control' if treatment[student_i] == 0 else 'treatment',
student_id = student_id,
uni_id = uni_id,
))
data = pd.DataFrame(data)
```

I can then fit it using `bambi`

and it looks like the correct answer, like so.

```
import bambi
import arviz as az
formula = 'y ~ 0 + treatment + (1|uni_id) + (1|student_id)'
model = bambi.Model(formula, data, family = 'bernoulli')
# build graph
model.build()
model.graph()
fitted_model= model.fit(
method = 'mcmc',
draws = 1000,
tune = 1000,
cores = 4,
chains = 4,
include_mean = False,
omit_offsets = False
)
# cut down list of variables summary
az.summary(fitted_model, var_names = ['treatment','1|uni_id_sigma','1|student_id_sigma'])
# full list of variables
az.summary(fitted_model)
```

but when I try to fit with numpyro I am getting very different results and I think I am doing something wrong. I suspect I have messed something up with specifying the random component or I am encoding the variables incorrectly. If you could help it would be appreciated.My model is below.

```
import numpyro
from numpyro.infer import MCMC, NUTS, Predictive
import numpyro.distributions as dist
from jax import random
from sklearn.preprocessing import LabelEncoder
assert numpyro.__version__.startswith("0.9.0")
# define model
def numpyro_model(student_id, uni_id, treatment_id, y_obs):
sigma_student = numpyro.sample("sigma_student", dist.HalfNormal(5))
sigma_uni = numpyro.sample("sigma_uni", dist.HalfNormal(5))
unique_student_ids = np.unique(student_id)
n_students = len(unique_student_ids)
unique_uni_IDs = np.unique(uni_id)
n_unis = len(unique_uni_IDs)
unique_treatment_IDs = np.unique(treatment_id)
n_treatment = len(unique_treatment_IDs)
with numpyro.plate("plate_i", n_students):
offset_student = numpyro.sample("offset_student", dist.Normal(0.0, 1.0))
alpha_student_i = offset_student * sigma_student
with numpyro.plate("plate_j", n_unis):
offset_uni = numpyro.sample("offset_uni", dist.Normal(0.0, 1.0))
beta_uni_j = offset_uni * sigma_uni
with numpyro.plate("plate_f", n_treatment):
treatment_k = numpyro.sample("treatment_k", dist.Normal(0, 5))
eta = alpha_student_i[student_id] + beta_uni_j[uni_id] + treatment_k[treatment_id]
with numpyro.plate("data", len(y_obs)):
y_est = numpyro.sample("y_est", dist.Bernoulli(logits=eta), obs=y_obs)
# run model
le = LabelEncoder()
data["student_id_encoded"] = le.fit_transform(data["student_id"].values)
data["uni_id_encoded"] = le.fit_transform(data["uni_id"].values)
data["treatment_id_encoded"] = le.fit_transform(data["treatment"].values)
y_obs = data["y"].values
uni_id_encoded = data["uni_id_encoded"].values
student_id_encoded = data["student_id_encoded"].values
treatment_id_encoded = data["treatment_id_encoded"].values
nuts_kernel = NUTS(numpyro_model)
mcmc = MCMC(nuts_kernel, num_samples=3000, num_warmup=3000, num_chains = 4)
rng_key = random.PRNGKey(0)
mcmc.run(rng_key, student_id_encoded, uni_id_encoded, treatment_id_encoded,y_obs)
posterior_samples = mcmc.get_samples()
# summary of results.
import arviz as az
data_az = az.from_numpyro(mcmc)
az.summary(data_az)
```

so if you could help that would be appreciated. I am also trying to figure out how I can increase the speed of the fitting process which is why I am trying to get it to work on numpyro instead of bambi (pymc3). It is okay at the example data size but when I am working with my real data set it is extremely slow, so if you have any tips for speed increases that would also be appreciated.