# Example: analyzing baseball stats with MCMCΒΆ

View baseball.py on github

# Copyright (c) 2017-2019 Uber Technologies, Inc.

import argparse
import logging
import math

import pandas as pd
import torch

import pyro
from pyro.distributions import Beta, Binomial, HalfCauchy, Normal, Pareto, Uniform
from pyro.distributions.util import scalar_like
from pyro.infer import MCMC, NUTS, Predictive
from pyro.infer.mcmc.util import initialize_model, summary
from pyro.util import ignore_experimental_warning

"""
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)
"""

logging.basicConfig(format="%(message)s", level=logging.INFO)
DATA_URL = "https://d2hg8soec8ck9v.cloudfront.net/datasets/EfronMorrisBB.txt"

# ===================================
#               MODELS
# ===================================

def fully_pooled(at_bats, hits):
r"""
Number of hits in $K$ at bats for each player has a Binomial
distribution with a common probability of success, $\phi$.

:param (torch.Tensor) at_bats: Number of at bats for each player.
:param (torch.Tensor) hits: Number of hits for the given at bats.
:return: Number of hits predicted by the model.
"""
phi_prior = Uniform(scalar_like(at_bats, 0), scalar_like(at_bats, 1))
phi = pyro.sample("phi", phi_prior)
num_players = at_bats.shape[0]
with pyro.plate("num_players", num_players):
return pyro.sample("obs", Binomial(at_bats, phi), obs=hits)

def not_pooled(at_bats, hits):
r"""
Number of hits in $K$ at bats for each player has a Binomial
distribution with independent probability of success, $\phi_i$.

:param (torch.Tensor) at_bats: Number of at bats for each player.
:param (torch.Tensor) hits: Number of hits for the given at bats.
:return: Number of hits predicted by the model.
"""
num_players = at_bats.shape[0]
with pyro.plate("num_players", num_players):
phi_prior = Uniform(scalar_like(at_bats, 0), scalar_like(at_bats, 1))
phi = pyro.sample("phi", phi_prior)
return pyro.sample("obs", Binomial(at_bats, phi), obs=hits)

def partially_pooled(at_bats, hits):
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 (torch.Tensor) at_bats: Number of at bats for each player.
:param (torch.Tensor) hits: Number of hits for the given at bats.
:return: Number of hits predicted by the model.
"""
num_players = at_bats.shape[0]
m = pyro.sample("m", Uniform(scalar_like(at_bats, 0), scalar_like(at_bats, 1)))
kappa = pyro.sample(
"kappa", Pareto(scalar_like(at_bats, 1), scalar_like(at_bats, 1.5))
)
with pyro.plate("num_players", num_players):
phi_prior = Beta(m * kappa, (1 - m) * kappa)
phi = pyro.sample("phi", phi_prior)
return pyro.sample("obs", Binomial(at_bats, phi), obs=hits)

def partially_pooled_with_logit(at_bats, hits):
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 (torch.Tensor) at_bats: Number of at bats for each player.
:param (torch.Tensor) hits: Number of hits for the given at bats.
:return: Number of hits predicted by the model.
"""
num_players = at_bats.shape[0]
loc = pyro.sample("loc", Normal(scalar_like(at_bats, -1), scalar_like(at_bats, 1)))
scale = pyro.sample("scale", HalfCauchy(scale=scalar_like(at_bats, 1)))
with pyro.plate("num_players", num_players):
alpha = pyro.sample("alpha", Normal(loc, scale))
return pyro.sample("obs", Binomial(at_bats, logits=alpha), obs=hits)

# ===================================
#        DATA SUMMARIZE UTILS
# ===================================

def get_summary_table(
posterior,
sites,
player_names,
transforms={},
diagnostics=False,
group_by_chain=False,
):
"""
Return summarized statistics for each of the sites in the
traces corresponding to the approximate posterior.
"""
site_stats = {}

for site_name in sites:
marginal_site = posterior[site_name].cpu()

if site_name in transforms:
marginal_site = transforms[site_name](marginal_site)

site_summary = summary(
{site_name: marginal_site}, prob=0.5, group_by_chain=group_by_chain
)[site_name]
if site_summary["mean"].shape:
site_df = pd.DataFrame(site_summary, index=player_names)
else:
site_summary = {k: float(v) for k, v in site_summary.items()}
site_df = pd.DataFrame(site_summary, index=[0])
if not diagnostics:
site_df = site_df.drop(["n_eff", "r_hat"], axis=1)
site_stats[site_name] = site_df.astype(float).round(2)

return site_stats

def train_test_split(pd_dataframe):
"""
Training data - 45 initial at-bats and hits for each player.
Validation data - Full season at-bats and hits for each player.
"""
device = torch.Tensor().device
train_data = torch.tensor(
pd_dataframe[["At-Bats", "Hits"]].values, dtype=torch.float, device=device
)
test_data = torch.tensor(
pd_dataframe[["SeasonAt-Bats", "SeasonHits"]].values,
dtype=torch.float,
device=device,
)
first_name = pd_dataframe["FirstName"].values
last_name = pd_dataframe["LastName"].values
player_names = [
" ".join([first, last]) for first, last in zip(first_name, last_name)
]
return train_data, test_data, player_names

# ===================================
#       MODEL EVALUATION UTILS
# ===================================

def sample_posterior_predictive(model, posterior_samples, baseball_dataset):
"""
Generate samples from posterior predictive distribution.
"""
train, test, player_names = train_test_split(baseball_dataset)
at_bats = train[:, 0]
at_bats_season = test[:, 0]
logging.Formatter("%(message)s")
logging.info("\nPosterior Predictive:")
logging.info("Hit Rate - Initial 45 At Bats")
logging.info("-----------------------------")
# set hits=None to convert it from observation node to sample node
train_predict = Predictive(model, posterior_samples)(at_bats, None)
train_summary = get_summary_table(
train_predict, sites=["obs"], player_names=player_names
)["obs"]
train_summary = train_summary.assign(ActualHits=baseball_dataset[["Hits"]].values)
logging.info(train_summary)
logging.info("\nHit Rate - Season Predictions")
logging.info("-----------------------------")
with ignore_experimental_warning():
test_predict = Predictive(model, posterior_samples)(at_bats_season, None)
test_summary = get_summary_table(
test_predict, sites=["obs"], player_names=player_names
)["obs"]
test_summary = test_summary.assign(
ActualHits=baseball_dataset[["SeasonHits"]].values
)
logging.info(test_summary)

def evaluate_pointwise_pred_density(model, posterior_samples, baseball_dataset):
"""
Evaluate the log probability density of observing the unseen data (season hits)
given a model and posterior distribution over the parameters.
"""
_, test, player_names = train_test_split(baseball_dataset)
at_bats_season, hits_season = test[:, 0], test[:, 1]
trace = Predictive(model, posterior_samples).get_vectorized_trace(
at_bats_season, hits_season
)
# Use LogSumExp trick to evaluate $log(1/num_samples \sum_i p(new_data | \theta^{i}))$,
# where $\theta^{i}$ are parameter samples from the model's posterior.
trace.compute_log_prob()
post_loglik = trace.nodes["obs"]["log_prob"]
# computes expected log predictive density at each data point
exp_log_density = (post_loglik.logsumexp(0) - math.log(post_loglik.shape[0])).sum()
logging.info("\nLog pointwise predictive density")
logging.info("--------------------------------")
logging.info("{:.4f}\n".format(exp_log_density))

def main(args):
train, _, player_names = train_test_split(baseball_dataset)
at_bats, hits = train[:, 0], train[:, 1]
logging.info("Original Dataset:")
logging.info(baseball_dataset)

# (1) Full Pooling Model
# In this model, we illustrate how to use MCMC with general potential_fn.
init_params, potential_fn, transforms, _ = initialize_model(
fully_pooled,
model_args=(at_bats, hits),
num_chains=args.num_chains,
jit_compile=args.jit,
skip_jit_warnings=True,
)
nuts_kernel = NUTS(potential_fn=potential_fn)
mcmc = MCMC(
nuts_kernel,
num_samples=args.num_samples,
warmup_steps=args.warmup_steps,
num_chains=args.num_chains,
initial_params=init_params,
transforms=transforms,
)
mcmc.run(at_bats, hits)
samples_fully_pooled = mcmc.get_samples()
logging.info("\nModel: Fully Pooled")
logging.info("===================")
logging.info("\nphi:")
logging.info(
get_summary_table(
mcmc.get_samples(group_by_chain=True),
sites=["phi"],
player_names=player_names,
diagnostics=True,
group_by_chain=True,
)["phi"]
)
num_divergences = sum(map(len, mcmc.diagnostics()["divergences"].values()))
logging.info("\nNumber of divergent transitions: {}\n".format(num_divergences))
sample_posterior_predictive(fully_pooled, samples_fully_pooled, baseball_dataset)
evaluate_pointwise_pred_density(
fully_pooled, samples_fully_pooled, baseball_dataset
)

# (2) No Pooling Model
nuts_kernel = NUTS(not_pooled, jit_compile=args.jit, ignore_jit_warnings=True)
mcmc = MCMC(
nuts_kernel,
num_samples=args.num_samples,
warmup_steps=args.warmup_steps,
num_chains=args.num_chains,
)
mcmc.run(at_bats, hits)
samples_not_pooled = mcmc.get_samples()
logging.info("\nModel: Not Pooled")
logging.info("=================")
logging.info("\nphi:")
logging.info(
get_summary_table(
mcmc.get_samples(group_by_chain=True),
sites=["phi"],
player_names=player_names,
diagnostics=True,
group_by_chain=True,
)["phi"]
)
num_divergences = sum(map(len, mcmc.diagnostics()["divergences"].values()))
logging.info("\nNumber of divergent transitions: {}\n".format(num_divergences))
sample_posterior_predictive(not_pooled, samples_not_pooled, baseball_dataset)
evaluate_pointwise_pred_density(not_pooled, samples_not_pooled, baseball_dataset)

# (3) Partially Pooled Model
nuts_kernel = NUTS(partially_pooled, jit_compile=args.jit, ignore_jit_warnings=True)
mcmc = MCMC(
nuts_kernel,
num_samples=args.num_samples,
warmup_steps=args.warmup_steps,
num_chains=args.num_chains,
)
mcmc.run(at_bats, hits)
samples_partially_pooled = mcmc.get_samples()
logging.info("\nModel: Partially Pooled")
logging.info("=======================")
logging.info("\nphi:")
logging.info(
get_summary_table(
mcmc.get_samples(group_by_chain=True),
sites=["phi"],
player_names=player_names,
diagnostics=True,
group_by_chain=True,
)["phi"]
)
num_divergences = sum(map(len, mcmc.diagnostics()["divergences"].values()))
logging.info("\nNumber of divergent transitions: {}\n".format(num_divergences))
sample_posterior_predictive(
partially_pooled, samples_partially_pooled, baseball_dataset
)
evaluate_pointwise_pred_density(
partially_pooled, samples_partially_pooled, baseball_dataset
)

# (4) Partially Pooled with Logit Model
nuts_kernel = NUTS(
partially_pooled_with_logit, jit_compile=args.jit, ignore_jit_warnings=True
)
mcmc = MCMC(
nuts_kernel,
num_samples=args.num_samples,
warmup_steps=args.warmup_steps,
num_chains=args.num_chains,
)
mcmc.run(at_bats, hits)
samples_partially_pooled_logit = mcmc.get_samples()
logging.info("\nModel: Partially Pooled with Logit")
logging.info("==================================")
logging.info("\nSigmoid(alpha):")
logging.info(
get_summary_table(
mcmc.get_samples(group_by_chain=True),
sites=["alpha"],
player_names=player_names,
transforms={"alpha": torch.sigmoid},
diagnostics=True,
group_by_chain=True,
)["alpha"]
)
num_divergences = sum(map(len, mcmc.diagnostics()["divergences"].values()))
logging.info("\nNumber of divergent transitions: {}\n".format(num_divergences))
sample_posterior_predictive(
partially_pooled_with_logit, samples_partially_pooled_logit, baseball_dataset
)
evaluate_pointwise_pred_density(
partially_pooled_with_logit, samples_partially_pooled_logit, baseball_dataset
)

if __name__ == "__main__":
assert pyro.__version__.startswith("1.8.6")
parser = argparse.ArgumentParser(description="Baseball batting average using HMC")
"--jit", action="store_true", default=False, help="use PyTorch jit"
)
"--cuda", action="store_true", default=False, help="run this example in GPU"
)
args = parser.parse_args()

# work around the error "CUDA error: initialization error"
# see https://github.com/pytorch/pytorch/issues/2517
torch.multiprocessing.set_start_method("spawn")

pyro.set_rng_seed(args.rng_seed)
# Enable validation checks

# work around with the error "RuntimeError: received 0 items of ancdata"