# Example: Baseball Batting Average¶

Original example from Pyro: https://github.com/pyro-ppl/pyro/blob/dev/examples/baseball.py

Example has been adapted from . It demonstrates how to do Bayesian inference using various MCMC kernels in Pyro (HMC, NUTS, SA), and use of some common inference utilities.

As in the Stan tutorial, this uses the small baseball dataset of Efron and Morris  to estimate players’ batting average which is the fraction of times a player got a base hit out of the number of times they went up at bat.

The dataset separates the initial 45 at-bats statistics from the remaining season. We use the hits data from the initial 45 at-bats to estimate the batting average for each player. We then use the remaining season’s data to validate the predictions from our models.

Three models are evaluated:

• Complete pooling model: The success probability of scoring a hit is shared amongst all players.

• No pooling model: Each individual player’s success probability is distinct and there is no data sharing amongst players.

• Partial pooling model: A hierarchical model with partial data sharing.

We recommend Radford Neal’s tutorial on HMC () to users who would like to get a more comprehensive understanding of HMC and its variants, and to  for details on the No U-Turn Sampler, which provides an efficient and automated way (i.e. limited hyper-parameters) of running HMC on different problems.

Note that the Sample Adaptive (SA) kernel, which is implemented based on , requires large num_warmup and num_samples (e.g. 15,000 and 300,000). So it is better to disable progress bar to avoid dispatching overhead.

References:

1. Carpenter B. (2016), “Hierarchical Partial Pooling for Repeated Binary Trials”.

2. Efron B., Morris C. (1975), “Data analysis using Stein’s estimator and its generalizations”, J. Amer. Statist. Assoc., 70, 311-319.

3. Neal, R. (2012), “MCMC using Hamiltonian Dynamics”, (https://arxiv.org/pdf/1206.1901.pdf)

4. Hoffman, M. D. and Gelman, A. (2014), “The No-U-turn sampler: Adaptively setting path lengths in Hamiltonian Monte Carlo”, (https://arxiv.org/abs/1111.4246)

import argparse
import os

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

import numpyro
import numpyro.distributions as dist
from numpyro.infer import HMC, MCMC, NUTS, SA, Predictive, log_likelihood

def fully_pooled(at_bats, hits=None):
r"""
Number of hits in $K$ at bats for each player has a Binomial
distribution with a common probability of success, $\phi$.

:param (jnp.ndarray) at_bats: Number of at bats for each player.
:param (jnp.ndarray) hits: Number of hits for the given at bats.
:return: Number of hits predicted by the model.
"""
phi_prior = dist.Uniform(0, 1)
phi = numpyro.sample("phi", phi_prior)
num_players = at_bats.shape
with numpyro.plate("num_players", num_players):
return numpyro.sample("obs", dist.Binomial(at_bats, probs=phi), obs=hits)

def not_pooled(at_bats, hits=None):
r"""
Number of hits in $K$ at bats for each player has a Binomial
distribution with independent probability of success, $\phi_i$.

:param (jnp.ndarray) at_bats: Number of at bats for each player.
:param (jnp.ndarray) hits: Number of hits for the given at bats.
:return: Number of hits predicted by the model.
"""
num_players = at_bats.shape
with numpyro.plate("num_players", num_players):
phi_prior = dist.Uniform(0, 1)
phi = numpyro.sample("phi", phi_prior)
return numpyro.sample("obs", dist.Binomial(at_bats, probs=phi), obs=hits)

def partially_pooled(at_bats, hits=None):
r"""
Number of hits has a Binomial distribution with independent
probability of success, $\phi_i$. Each $\phi_i$ follows a Beta
distribution with concentration parameters $c_1$ and $c_2$, where
$c_1 = m * kappa$, $c_2 = (1 - m) * kappa$, $m ~ Uniform(0, 1)$,
and $kappa ~ Pareto(1, 1.5)$.

:param (jnp.ndarray) at_bats: Number of at bats for each player.
:param (jnp.ndarray) hits: Number of hits for the given at bats.
:return: Number of hits predicted by the model.
"""
m = numpyro.sample("m", dist.Uniform(0, 1))
kappa = numpyro.sample("kappa", dist.Pareto(1, 1.5))
num_players = at_bats.shape
with numpyro.plate("num_players", num_players):
phi_prior = dist.Beta(m * kappa, (1 - m) * kappa)
phi = numpyro.sample("phi", phi_prior)
return numpyro.sample("obs", dist.Binomial(at_bats, probs=phi), obs=hits)

def partially_pooled_with_logit(at_bats, hits=None):
r"""
Number of hits has a Binomial distribution with a logit link function.
The logits $\alpha$ for each player is normally distributed with the
mean and scale parameters sharing a common prior.

:param (jnp.ndarray) at_bats: Number of at bats for each player.
:param (jnp.ndarray) hits: Number of hits for the given at bats.
:return: Number of hits predicted by the model.
"""
loc = numpyro.sample("loc", dist.Normal(-1, 1))
scale = numpyro.sample("scale", dist.HalfCauchy(1))
num_players = at_bats.shape
with numpyro.plate("num_players", num_players):
alpha = numpyro.sample("alpha", dist.Normal(loc, scale))
return numpyro.sample("obs", dist.Binomial(at_bats, logits=alpha), obs=hits)

def run_inference(model, at_bats, hits, rng_key, args):
if args.algo == "NUTS":
kernel = NUTS(model)
elif args.algo == "HMC":
kernel = HMC(model)
elif args.algo == "SA":
kernel = SA(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 or args.disable_progbar)
else True,
)
mcmc.run(rng_key, at_bats, hits)
return mcmc.get_samples()

def predict(model, at_bats, hits, z, rng_key, player_names, train=True):
header = model.__name__ + (" - TRAIN" if train else " - TEST")
predictions = Predictive(model, posterior_samples=z)(rng_key, at_bats)["obs"]
print_results(
"=" * 30 + header + "=" * 30, predictions, player_names, at_bats, hits
)
if not train:
post_loglik = log_likelihood(model, z, at_bats, hits)["obs"]
# computes expected log predictive density at each data point
exp_log_density = logsumexp(post_loglik, axis=0) - jnp.log(
jnp.shape(post_loglik)
)
# reports log predictive density of all test points
print(
"\nLog pointwise predictive density: {:.2f}\n".format(exp_log_density.sum())
)

def print_results(header, preds, player_names, at_bats, hits):
columns = ["", "At-bats", "ActualHits", "Pred(p25)", "Pred(p50)", "Pred(p75)"]
header_format = "{:>20} {:>10} {:>10} {:>10} {:>10} {:>10}"
row_format = "{:>20} {:>10.0f} {:>10.0f} {:>10.2f} {:>10.2f} {:>10.2f}"
quantiles = jnp.quantile(preds, jnp.array([0.25, 0.5, 0.75]), axis=0)
for i, p in enumerate(player_names):
print(row_format.format(p, at_bats[i], hits[i], *quantiles[:, i]), "\n")

def main(args):
_, fetch_train = load_dataset(BASEBALL, split="train", shuffle=False)
train, player_names = fetch_train()
_, fetch_test = load_dataset(BASEBALL, split="test", shuffle=False)
test, _ = fetch_test()
at_bats, hits = train[:, 0], train[:, 1]
season_at_bats, season_hits = test[:, 0], test[:, 1]
for i, model in enumerate(
(fully_pooled, not_pooled, partially_pooled, partially_pooled_with_logit)
):
rng_key, rng_key_predict = random.split(random.PRNGKey(i + 1))
zs = run_inference(model, at_bats, hits, rng_key, args)
predict(model, at_bats, hits, zs, rng_key_predict, player_names)
predict(
model,
season_at_bats,
season_hits,
zs,
rng_key_predict,
player_names,
train=False,
)

if __name__ == "__main__":
assert numpyro.__version__.startswith("0.10.0")
parser = argparse.ArgumentParser(description="Baseball batting average using MCMC")
"--algo", default="NUTS", type=str, help='whether to run "HMC", "NUTS", or "SA"'
)
"-dp",
"--disable-progbar",
action="store_true",
default=False,
help="whether to disable progress bar",
)
parser.add_argument("--device", default="cpu", type=str, help='use "cpu" or "gpu".')
args = parser.parse_args()

numpyro.set_platform(args.device)
numpyro.set_host_device_count(args.num_chains)

main(args)


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