# Example: Bayesian Neural Network¶

We demonstrate how to use NUTS to do inference on a simple (small) Bayesian neural network with two hidden layers.

import argparse
import os
import time

import matplotlib
import matplotlib.pyplot as plt
import numpy as np

from jax import vmap
import jax.numpy as jnp
import jax.random as random

import numpyro
from numpyro import handlers
import numpyro.distributions as dist
from numpyro.infer import MCMC, NUTS

matplotlib.use("Agg")  # noqa: E402

# the non-linearity we use in our neural network
def nonlin(x):
return jnp.tanh(x)

# a two-layer bayesian neural network with computational flow
# given by D_X => D_H => D_H => D_Y where D_H is the number of
# hidden units. (note we indicate tensor dimensions in the comments)
def model(X, Y, D_H, D_Y=1):
N, D_X = X.shape

# sample first layer (we put unit normal priors on all weights)
w1 = numpyro.sample("w1", dist.Normal(jnp.zeros((D_X, D_H)), jnp.ones((D_X, D_H))))
assert w1.shape == (D_X, D_H)
z1 = nonlin(jnp.matmul(X, w1))  # <= first layer of activations
assert z1.shape == (N, D_H)

# sample second layer
w2 = numpyro.sample("w2", dist.Normal(jnp.zeros((D_H, D_H)), jnp.ones((D_H, D_H))))
assert w2.shape == (D_H, D_H)
z2 = nonlin(jnp.matmul(z1, w2))  # <= second layer of activations
assert z2.shape == (N, D_H)

# sample final layer of weights and neural network output
w3 = numpyro.sample("w3", dist.Normal(jnp.zeros((D_H, D_Y)), jnp.ones((D_H, D_Y))))
assert w3.shape == (D_H, D_Y)
z3 = jnp.matmul(z2, w3)  # <= output of the neural network
assert z3.shape == (N, D_Y)

if Y is not None:
assert z3.shape == Y.shape

# we put a prior on the observation noise
prec_obs = numpyro.sample("prec_obs", dist.Gamma(3.0, 1.0))
sigma_obs = 1.0 / jnp.sqrt(prec_obs)

# observe data
with numpyro.plate("data", N):
# note we use to_event(1) because each observation has shape (1,)
numpyro.sample("Y", dist.Normal(z3, sigma_obs).to_event(1), obs=Y)

# helper function for HMC inference
def run_inference(model, args, rng_key, X, Y, D_H):
start = time.time()
kernel = NUTS(model)
mcmc = MCMC(
kernel,
num_warmup=args.num_warmup,
num_samples=args.num_samples,
num_chains=args.num_chains,
progress_bar=False if "NUMPYRO_SPHINXBUILD" in os.environ else True,
)
mcmc.run(rng_key, X, Y, D_H)
mcmc.print_summary()
print("\nMCMC elapsed time:", time.time() - start)
return mcmc.get_samples()

# helper function for prediction
def predict(model, rng_key, samples, X, D_H):
model = handlers.substitute(handlers.seed(model, rng_key), samples)
# note that Y will be sampled in the model because we pass Y=None here
model_trace = handlers.trace(model).get_trace(X=X, Y=None, D_H=D_H)
return model_trace["Y"]["value"]

# create artificial regression dataset
def get_data(N=50, D_X=3, sigma_obs=0.05, N_test=500):
D_Y = 1  # create 1d outputs
np.random.seed(0)
X = jnp.linspace(-1, 1, N)
X = jnp.power(X[:, np.newaxis], jnp.arange(D_X))
W = 0.5 * np.random.randn(D_X)
Y = jnp.dot(X, W) + 0.5 * jnp.power(0.5 + X[:, 1], 2.0) * jnp.sin(4.0 * X[:, 1])
Y += sigma_obs * np.random.randn(N)
Y = Y[:, np.newaxis]
Y -= jnp.mean(Y)
Y /= jnp.std(Y)

assert X.shape == (N, D_X)
assert Y.shape == (N, D_Y)

X_test = jnp.linspace(-1.3, 1.3, N_test)
X_test = jnp.power(X_test[:, np.newaxis], jnp.arange(D_X))

return X, Y, X_test

def main(args):
N, D_X, D_H = args.num_data, 3, args.num_hidden
X, Y, X_test = get_data(N=N, D_X=D_X)

# do inference
rng_key, rng_key_predict = random.split(random.PRNGKey(0))
samples = run_inference(model, args, rng_key, X, Y, D_H)

# predict Y_test at inputs X_test
vmap_args = (
samples,
random.split(rng_key_predict, args.num_samples * args.num_chains),
)
predictions = vmap(
lambda samples, rng_key: predict(model, rng_key, samples, X_test, D_H)
)(*vmap_args)
predictions = predictions[..., 0]

# compute mean prediction and confidence interval around median
mean_prediction = jnp.mean(predictions, axis=0)
percentiles = np.percentile(predictions, [5.0, 95.0], axis=0)

# make plots
fig, ax = plt.subplots(figsize=(8, 6), constrained_layout=True)

# plot training data
ax.plot(X[:, 1], Y[:, 0], "kx")
# plot 90% confidence level of predictions
ax.fill_between(
X_test[:, 1], percentiles[0, :], percentiles[1, :], color="lightblue"
)
# plot mean prediction
ax.plot(X_test[:, 1], mean_prediction, "blue", ls="solid", lw=2.0)
ax.set(xlabel="X", ylabel="Y", title="Mean predictions with 90% CI")

plt.savefig("bnn_plot.pdf")

if __name__ == "__main__":
assert numpyro.__version__.startswith("0.11.0")
parser = argparse.ArgumentParser(description="Bayesian neural network example")