Implementing Hierarchical Bayesian Regression with NumPyro

Overview

In this tutorial, we explore hierarchical Bayesian regression with NumPyro and walk through the entire workflow in a structured manner. We start by generating synthetic data, then we define a probabilistic model that captures both global patterns and group-level variations. Through each code snippet, we set up inference using NUTS, analyze posterior distributions, and perform posterior predictive checks.

Setting Up the Environment

Before diving into modeling, we set up our environment by installing NumPyro and importing the required libraries. This ensures our Colab session is fully equipped for hierarchical modeling.

try:
   import numpyro
except ImportError:
   !pip install -q "llvmlite>=0.45.1" "numpyro[cpu]" matplotlib pandas
 
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import jax
import jax.numpy as jnp
from jax import random
import numpyro
def generate_data(key, n_groups=8, n_per_group=40):
   # Function implementation here
 
key = random.PRNGKey(0)
df, truth = generate_data(key)

Generating Synthetic Data

Next, we generate synthetic hierarchical data mimicking real-world group-level variation. We prepare the data for processing in JAX:

def generate_data(key, n_groups=8, n_per_group=40):
   # Function implementation here
 
# Generate data
key = random.PRNGKey(0)
df, truth = generate_data(key)

Defining the Hierarchical Regression Model

We define our hierarchical regression model and launch the NUTS-based MCMC sampler:

def hierarchical_regression_model(x, group_idx, n_groups, y=None):
   # Function implementation here
 
nuts = NUTS(hierarchical_regression_model, target_accept_prob=0.9)
mcmc = MCMC(nuts, num_warmup=1000, num_samples=1000, num_chains=1)
mcmc.run(random.PRNGKey(1), x=x, group_idx=groups, n_groups=n_groups, y=y)
samples = mcmc.get_samples()

Analyzing Posterior Samples

We compute summaries of our posterior samples and conduct posterior predictive checks:

def param_summary(arr):
   # Function implementation here
 
for name in ["mu_alpha", "mu_beta", "sigma_alpha", "sigma_beta", "sigma_obs"]:
   m, lo, hi = param_summary(samples[name])
   print(f"{name}: mean={m:.3f}, HPDI=[{lo:.3f}, {hi:.3f}]")

Visualizing Results

Finally, we visualize the estimated group-level intercepts and slopes to compare them with true values:

alpha_g = np.asarray(samples["alpha_g"]).mean(axis=0)
beta_g = np.asarray(samples["beta_g"]).mean(axis=0)
 
fig, axes = plt.subplots(1, 2, figsize=(12, 4))
axes[0].bar(range(n_groups), alpha_g)
axes[0].axhline(truth["true_alpha"], linestyle="--")
plt.show()

Conclusion

We have implemented a comprehensive hierarchical Bayesian regression workflow using NumPyro. This process allows us to model hierarchical relationships effectively, leveraging the strengths of JAX-powered inference.