# BaseballΒΆ

```
import argparse
import numpy as onp
import jax.numpy as np
import jax.random as random
from jax.config import config as jax_config
from jax.scipy.special import logsumexp
import numpyro.distributions as dist
from numpyro.examples.datasets import BASEBALL, load_dataset
from numpyro.handlers import sample, seed, substitute, trace
from numpyro.hmc_util import initialize_model
from numpyro.mcmc import mcmc
"""
Original example from Pyro:
https://github.com/pyro-ppl/pyro/blob/dev/examples/baseball.py
Example has been adapted from [1]. It demonstrates how to do Bayesian inference using
NUTS (or, HMC) in Pyro, and use of some common inference utilities.
As in the Stan tutorial, this uses the small baseball dataset of Efron and Morris [2]
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 ([3]) to users who would like to get a
more comprehensive understanding of HMC and its variants, and to [4] 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.
[1] Carpenter B. (2016), ["Hierarchical Partial Pooling for Repeated Binary Trials"]
(http://mc-stan.org/users/documentation/case-studies/pool-binary-trials.html).
[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)
"""
# TODO: Remove broadcasting logic when support for `pyro.plate` is
# available.
#
# Note that the current manual broadcasting logic is designed to
# also work when the latent variables are batched along the leading
# axis. This is useful for doing vectorized predictions, i.e. getting
# the entire empirical distribution in one pass through the model.
# This broadcasting logic can be removed when support for plates is
# available.
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 (np.DeviceArray) at_bats: Number of at bats for each player.
:param (np.DeviceArray) hits: Number of hits for the given at bats.
:return: Number of hits predicted by the model.
"""
phi_prior = dist.Uniform(np.array([0.]), np.array([1.]))
phi = sample("phi", phi_prior)
return 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 (np.DeviceArray) at_bats: Number of at bats for each player.
:param (np.DeviceArray) hits: Number of hits for the given at bats.
:return: Number of hits predicted by the model.
"""
num_players = at_bats.shape[0]
phi_prior = dist.Uniform(np.zeros((num_players,)),
np.ones((num_players,)))
phi = sample("phi", phi_prior)
return 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 (np.DeviceArray) at_bats: Number of at bats for each player.
:param (np.DeviceArray) hits: Number of hits for the given at bats.
:return: Number of hits predicted by the model.
"""
num_players = at_bats.shape[0]
m = sample("m", dist.Uniform(np.array([0.]), np.array([1.])))
kappa = sample("kappa", dist.Pareto(np.array([1.5])))
shape = np.shape(kappa)[:np.ndim(kappa) - 1] + (num_players,)
phi_prior = dist.Beta(np.broadcast_to(m * kappa, shape),
np.broadcast_to((1 - m) * kappa, shape))
phi = sample("phi", phi_prior)
return 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 (np.DeviceArray) at_bats: Number of at bats for each player.
:param (np.DeviceArray) hits: Number of hits for the given at bats.
:return: Number of hits predicted by the model.
"""
num_players = at_bats.shape[0]
loc = sample("loc", dist.Normal(np.array([-1.]), np.array([1.])))
scale = sample("scale", dist.HalfCauchy(np.array([1.])))
shape = np.shape(loc)[:np.ndim(loc) - 1] + (num_players,)
alpha = sample("alpha", dist.Normal(np.broadcast_to(loc, shape),
np.broadcast_to(scale, shape)))
return sample("obs", dist.Binomial(at_bats, logits=alpha), obs=hits)
def run_inference(model, at_bats, hits, rng, args):
init_params, potential_fn, constrain_fn = initialize_model(rng, model, at_bats, hits)
hmc_states = mcmc(args.num_warmup, args.num_samples, init_params,
sampler='hmc', potential_fn=potential_fn, constrain_fn=constrain_fn)
return hmc_states
# TODO: Consider providing generic utilities for doing predictions
# and computing posterior log density
def predict(model, at_bats, hits, z, rng, player_names, train=True):
header = model.__name__ + (' - TRAIN' if train else ' - TEST')
model = substitute(seed(model, rng), z)
model_trace = trace(model).get_trace(at_bats)
predictions = model_trace['obs']['value']
print_results('=' * 30 + header + '=' * 30,
predictions,
player_names,
at_bats,
hits)
if not train:
model = substitute(model, z)
model_trace = trace(model).get_trace(at_bats, hits)
log_joint = 0.
for site in model_trace.values():
site_log_prob = site['fn'].log_prob(site['value'])
log_joint = log_joint + np.sum(site_log_prob.reshape(site_log_prob.shape[:1] + (-1,)),
-1)
log_post_density = logsumexp(log_joint) - np.log(np.shape(log_joint)[0])
print('\nPosterior log density: {:.2f}\n'.format(log_post_density))
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 = onp.quantile(preds, [0.25, 0.5, 0.75], axis=0)
print('\n', header, '\n')
print(header_format.format(*columns))
for i, p in enumerate(player_names):
print(row_format.format(p, at_bats[i], hits[i], *quantiles[:, i]), '\n')
def main(args):
jax_config.update('jax_platform_name', args.device)
_, 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, rng_predict = random.split(random.PRNGKey(i + 1))
zs = run_inference(model, at_bats, hits, rng, args)
predict(model, at_bats, hits, zs, rng_predict, player_names)
predict(model, season_at_bats, season_hits, zs, rng_predict, player_names, train=False)
if __name__ == "__main__":
parser = argparse.ArgumentParser(description="Baseball batting average using HMC")
parser.add_argument("-n", "--num-samples", nargs="?", default=3000, type=int)
parser.add_argument("--num-warmup", nargs='?', default=1500, type=int)
parser.add_argument('--device', default='cpu', type=str, help='use "cpu" or "gpu".')
args = parser.parse_args()
main(args)
```