# Example: Neural Transport¶

This example illustrates how to use a trained AutoBNAFNormal autoguide to transform a posterior to a Gaussian-like one. The transform will be used to get better mixing rate for NUTS sampler.

References:

1. Hoffman, M. et al. (2019), “NeuTra-lizing Bad Geometry in Hamiltonian Monte Carlo Using Neural Transport”, (https://arxiv.org/abs/1903.03704)
import argparse
import os

from matplotlib.gridspec import GridSpec
import matplotlib.pyplot as plt
import seaborn as sns

from jax import random
import jax.numpy as jnp
from jax.scipy.special import logsumexp

import numpyro
from numpyro import optim
from numpyro.diagnostics import print_summary
import numpyro.distributions as dist
from numpyro.distributions import constraints
from numpyro.infer import MCMC, NUTS, SVI, Trace_ELBO
from numpyro.infer.autoguide import AutoBNAFNormal
from numpyro.infer.reparam import NeuTraReparam

class DualMoonDistribution(dist.Distribution):
support = constraints.real_vector

def __init__(self):
super(DualMoonDistribution, self).__init__(event_shape=(2,))

def sample(self, key, sample_shape=()):
# it is enough to return an arbitrary sample with correct shape
return jnp.zeros(sample_shape + self.event_shape)

def log_prob(self, x):
term1 = 0.5 * ((jnp.linalg.norm(x, axis=-1) - 2) / 0.4) ** 2
term2 = -0.5 * ((x[..., :1] + jnp.array([-2.0, 2.0])) / 0.6) ** 2
pe = term1 - logsumexp(term2, axis=-1)
return -pe

def dual_moon_model():
numpyro.sample("x", DualMoonDistribution())

def main(args):
print("Start vanilla HMC...")
nuts_kernel = NUTS(dual_moon_model)
mcmc = MCMC(
nuts_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(random.PRNGKey(0))
mcmc.print_summary()
vanilla_samples = mcmc.get_samples()["x"].copy()

guide = AutoBNAFNormal(
dual_moon_model, hidden_factors=[args.hidden_factor, args.hidden_factor]
)
svi = SVI(dual_moon_model, guide, optim.Adam(0.003), Trace_ELBO())

print("Start training guide...")
svi_result = svi.run(random.PRNGKey(1), args.num_iters)
print("Finish training guide. Extract samples...")
guide_samples = guide.sample_posterior(
random.PRNGKey(2), svi_result.params, sample_shape=(args.num_samples,)
)["x"].copy()

print("\nStart NeuTra HMC...")
neutra = NeuTraReparam(guide, svi_result.params)
neutra_model = neutra.reparam(dual_moon_model)
nuts_kernel = NUTS(neutra_model)
mcmc = MCMC(
nuts_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(random.PRNGKey(3))
mcmc.print_summary()
zs = mcmc.get_samples(group_by_chain=True)["auto_shared_latent"]
print("Transform samples into unwarped space...")
samples = neutra.transform_sample(zs)
print_summary(samples)
zs = zs.reshape(-1, 2)
samples = samples["x"].reshape(-1, 2).copy()

# make plots

# guide samples (for plotting)
guide_base_samples = dist.Normal(jnp.zeros(2), 1.0).sample(
random.PRNGKey(4), (1000,)
)
guide_trans_samples = neutra.transform_sample(guide_base_samples)["x"]

x1 = jnp.linspace(-3, 3, 100)
x2 = jnp.linspace(-3, 3, 100)
X1, X2 = jnp.meshgrid(x1, x2)
P = jnp.exp(DualMoonDistribution().log_prob(jnp.stack([X1, X2], axis=-1)))

fig = plt.figure(figsize=(12, 8), constrained_layout=True)
gs = GridSpec(2, 3, figure=fig)

ax1.plot(svi_result.losses[1000:])
ax1.set_title("Autoguide training loss\n(after 1000 steps)")

ax2.contourf(X1, X2, P, cmap="OrRd")
sns.kdeplot(x=guide_samples[:, 0], y=guide_samples[:, 1], n_levels=30, ax=ax2)
ax2.set(
xlim=[-3, 3],
ylim=[-3, 3],
xlabel="x0",
ylabel="x1",
title="Posterior using\nAutoBNAFNormal guide",
)

sns.scatterplot(
x=guide_base_samples[:, 0],
y=guide_base_samples[:, 1],
ax=ax3,
hue=guide_trans_samples[:, 0] < 0.0,
)
ax3.set(
xlim=[-3, 3],
ylim=[-3, 3],
xlabel="x0",
ylabel="x1",
title="AutoBNAFNormal base samples\n(True=left moon; False=right moon)",
)

ax4.contourf(X1, X2, P, cmap="OrRd")
sns.kdeplot(x=vanilla_samples[:, 0], y=vanilla_samples[:, 1], n_levels=30, ax=ax4)
ax4.plot(vanilla_samples[-50:, 0], vanilla_samples[-50:, 1], "bo-", alpha=0.5)
ax4.set(
xlim=[-3, 3],
ylim=[-3, 3],
xlabel="x0",
ylabel="x1",
title="Posterior using\nvanilla HMC sampler",
)

sns.scatterplot(
x=zs[:, 0],
y=zs[:, 1],
ax=ax5,
hue=samples[:, 0] < 0.0,
s=30,
alpha=0.5,
edgecolor="none",
)
ax5.set(
xlim=[-5, 5],
ylim=[-5, 5],
xlabel="x0",
ylabel="x1",
title="Samples from the\nwarped posterior - p(z)",
)

ax6.contourf(X1, X2, P, cmap="OrRd")
sns.kdeplot(x=samples[:, 0], y=samples[:, 1], n_levels=30, ax=ax6)
ax6.plot(samples[-50:, 0], samples[-50:, 1], "bo-", alpha=0.2)
ax6.set(
xlim=[-3, 3],
ylim=[-3, 3],
xlabel="x0",
ylabel="x1",
title="Posterior using\nNeuTra HMC sampler",
)

plt.savefig("neutra.pdf")

if __name__ == "__main__":
assert numpyro.__version__.startswith("0.7.0")
parser = argparse.ArgumentParser(description="NeuTra HMC")