Note
Click here to download the full example code
Example: Bayesian Models of Annotation¶
In this example, we run MCMC for various crowdsourced annotation models in [1].
All models have discrete latent variables. Under the hood, we enumerate over (marginalize out) those discrete latent sites in inference. Those models have different complexity so they are great refererences for those who are new to Pyro/NumPyro enumeration mechanism. We recommend readers compare the implementations with the corresponding plate diagrams in [1] to see how concise a Pyro/NumPyro program is.
The interested readers can also refer to [3] for more explanation about enumeration.
The data is taken from Table 1 of reference [2].
Currently, this example does not include postprocessing steps to deal with “Label Switching” issue (mentioned in section 6.2 of [1]).
References:
Paun, S., Carpenter, B., Chamberlain, J., Hovy, D., Kruschwitz, U., and Poesio, M. (2018). “Comparing bayesian models of annotation” (https://www.aclweb.org/anthology/Q18-1040/)
Dawid, A. P., and Skene, A. M. (1979). “Maximum likelihood estimation of observer error‐rates using the EM algorithm”
“Inference with Discrete Latent Variables” (http://pyro.ai/examples/enumeration.html)
import argparse
import os
import numpy as np
from jax import nn, random, vmap
import jax.numpy as jnp
import numpyro
from numpyro import handlers
from numpyro.contrib.indexing import Vindex
import numpyro.distributions as dist
from numpyro.infer import MCMC, NUTS, Predictive
from numpyro.infer.reparam import LocScaleReparam
def get_data():
"""
:return: a tuple of annotator indices and class indices. The first term has shape
`num_positions` whose entries take values from `0` to `num_annotators - 1`.
The second term has shape `num_items x num_positions` whose entries take values
from `0` to `num_classes - 1`.
"""
# NB: the first annotator assessed each item 3 times
positions = np.array([1, 1, 1, 2, 3, 4, 5])
annotations = np.array(
[
[1, 1, 1, 1, 1, 1, 1],
[3, 3, 3, 4, 3, 3, 4],
[1, 1, 2, 2, 1, 2, 2],
[2, 2, 2, 3, 1, 2, 1],
[2, 2, 2, 3, 2, 2, 2],
[2, 2, 2, 3, 3, 2, 2],
[1, 2, 2, 2, 1, 1, 1],
[3, 3, 3, 3, 4, 3, 3],
[2, 2, 2, 2, 2, 2, 3],
[2, 3, 2, 2, 2, 2, 3],
[4, 4, 4, 4, 4, 4, 4],
[2, 2, 2, 3, 3, 4, 3],
[1, 1, 1, 1, 1, 1, 1],
[2, 2, 2, 3, 2, 1, 2],
[1, 2, 1, 1, 1, 1, 1],
[1, 1, 1, 2, 1, 1, 1],
[1, 1, 1, 1, 1, 1, 1],
[1, 1, 1, 1, 1, 1, 1],
[2, 2, 2, 2, 2, 2, 1],
[2, 2, 2, 1, 3, 2, 2],
[2, 2, 2, 2, 2, 2, 2],
[2, 2, 2, 2, 2, 2, 1],
[2, 2, 2, 3, 2, 2, 2],
[2, 2, 1, 2, 2, 2, 2],
[1, 1, 1, 1, 1, 1, 1],
[1, 1, 1, 1, 1, 1, 1],
[2, 3, 2, 2, 2, 2, 2],
[1, 1, 1, 1, 1, 1, 1],
[1, 1, 1, 1, 1, 1, 1],
[1, 1, 2, 1, 1, 2, 1],
[1, 1, 1, 1, 1, 1, 1],
[3, 3, 3, 3, 2, 3, 3],
[1, 1, 1, 1, 1, 1, 1],
[2, 2, 2, 2, 2, 2, 2],
[2, 2, 2, 3, 2, 3, 2],
[4, 3, 3, 4, 3, 4, 3],
[2, 2, 1, 2, 2, 3, 2],
[2, 3, 2, 3, 2, 3, 3],
[3, 3, 3, 3, 4, 3, 2],
[1, 1, 1, 1, 1, 1, 1],
[1, 1, 1, 1, 1, 1, 1],
[1, 2, 1, 2, 1, 1, 1],
[2, 3, 2, 2, 2, 2, 2],
[1, 2, 1, 1, 1, 1, 1],
[2, 2, 2, 2, 2, 2, 2],
]
)
# we minus 1 because in Python, the first index is 0
return positions - 1, annotations - 1
def multinomial(annotations):
"""
This model corresponds to the plate diagram in Figure 1 of reference [1].
"""
num_classes = int(np.max(annotations)) + 1
num_items, num_positions = annotations.shape
with numpyro.plate("class", num_classes):
zeta = numpyro.sample("zeta", dist.Dirichlet(jnp.ones(num_classes)))
pi = numpyro.sample("pi", dist.Dirichlet(jnp.ones(num_classes)))
with numpyro.plate("item", num_items, dim=-2):
c = numpyro.sample("c", dist.Categorical(pi))
with numpyro.plate("position", num_positions):
numpyro.sample("y", dist.Categorical(zeta[c]), obs=annotations)
def dawid_skene(positions, annotations):
"""
This model corresponds to the plate diagram in Figure 2 of reference [1].
"""
num_annotators = int(np.max(positions)) + 1
num_classes = int(np.max(annotations)) + 1
num_items, num_positions = annotations.shape
with numpyro.plate("annotator", num_annotators, dim=-2):
with numpyro.plate("class", num_classes):
beta = numpyro.sample("beta", dist.Dirichlet(jnp.ones(num_classes)))
pi = numpyro.sample("pi", dist.Dirichlet(jnp.ones(num_classes)))
with numpyro.plate("item", num_items, dim=-2):
c = numpyro.sample("c", dist.Categorical(pi))
# here we use Vindex to allow broadcasting for the second index `c`
# ref: http://num.pyro.ai/en/latest/utilities.html#numpyro.contrib.indexing.vindex
with numpyro.plate("position", num_positions):
numpyro.sample(
"y", dist.Categorical(Vindex(beta)[positions, c, :]), obs=annotations
)
def mace(positions, annotations):
"""
This model corresponds to the plate diagram in Figure 3 of reference [1].
"""
num_annotators = int(np.max(positions)) + 1
num_classes = int(np.max(annotations)) + 1
num_items, num_positions = annotations.shape
with numpyro.plate("annotator", num_annotators):
epsilon = numpyro.sample("epsilon", dist.Dirichlet(jnp.full(num_classes, 10)))
theta = numpyro.sample("theta", dist.Beta(0.5, 0.5))
with numpyro.plate("item", num_items, dim=-2):
# NB: using constant logits for discrete uniform prior
# (NumPyro does not have DiscreteUniform distribution yet)
c = numpyro.sample("c", dist.Categorical(logits=jnp.zeros(num_classes)))
with numpyro.plate("position", num_positions):
s = numpyro.sample("s", dist.Bernoulli(1 - theta[positions]))
probs = jnp.where(
s[..., None] == 0, nn.one_hot(c, num_classes), epsilon[positions]
)
numpyro.sample("y", dist.Categorical(probs), obs=annotations)
def hierarchical_dawid_skene(positions, annotations):
"""
This model corresponds to the plate diagram in Figure 4 of reference [1].
"""
num_annotators = int(np.max(positions)) + 1
num_classes = int(np.max(annotations)) + 1
num_items, num_positions = annotations.shape
with numpyro.plate("class", num_classes):
# NB: we define `beta` as the `logits` of `y` likelihood; but `logits` is
# invariant up to a constant, so we'll follow [1]: fix the last term of `beta`
# to 0 and only define hyperpriors for the first `num_classes - 1` terms.
zeta = numpyro.sample(
"zeta", dist.Normal(0, 1).expand([num_classes - 1]).to_event(1)
)
omega = numpyro.sample(
"Omega", dist.HalfNormal(1).expand([num_classes - 1]).to_event(1)
)
with numpyro.plate("annotator", num_annotators, dim=-2):
with numpyro.plate("class", num_classes):
# non-centered parameterization
with handlers.reparam(config={"beta": LocScaleReparam(0)}):
beta = numpyro.sample("beta", dist.Normal(zeta, omega).to_event(1))
# pad 0 to the last item
beta = jnp.pad(beta, [(0, 0)] * (jnp.ndim(beta) - 1) + [(0, 1)])
pi = numpyro.sample("pi", dist.Dirichlet(jnp.ones(num_classes)))
with numpyro.plate("item", num_items, dim=-2):
c = numpyro.sample("c", dist.Categorical(pi))
with numpyro.plate("position", num_positions):
logits = Vindex(beta)[positions, c, :]
numpyro.sample("y", dist.Categorical(logits=logits), obs=annotations)
def item_difficulty(annotations):
"""
This model corresponds to the plate diagram in Figure 5 of reference [1].
"""
num_classes = int(np.max(annotations)) + 1
num_items, num_positions = annotations.shape
with numpyro.plate("class", num_classes):
eta = numpyro.sample(
"eta", dist.Normal(0, 1).expand([num_classes - 1]).to_event(1)
)
chi = numpyro.sample(
"Chi", dist.HalfNormal(1).expand([num_classes - 1]).to_event(1)
)
pi = numpyro.sample("pi", dist.Dirichlet(jnp.ones(num_classes)))
with numpyro.plate("item", num_items, dim=-2):
c = numpyro.sample("c", dist.Categorical(pi))
with handlers.reparam(config={"theta": LocScaleReparam(0)}):
theta = numpyro.sample("theta", dist.Normal(eta[c], chi[c]).to_event(1))
theta = jnp.pad(theta, [(0, 0)] * (jnp.ndim(theta) - 1) + [(0, 1)])
with numpyro.plate("position", annotations.shape[-1]):
numpyro.sample("y", dist.Categorical(logits=theta), obs=annotations)
def logistic_random_effects(positions, annotations):
"""
This model corresponds to the plate diagram in Figure 5 of reference [1].
"""
num_annotators = int(np.max(positions)) + 1
num_classes = int(np.max(annotations)) + 1
num_items, num_positions = annotations.shape
with numpyro.plate("class", num_classes):
zeta = numpyro.sample(
"zeta", dist.Normal(0, 1).expand([num_classes - 1]).to_event(1)
)
omega = numpyro.sample(
"Omega", dist.HalfNormal(1).expand([num_classes - 1]).to_event(1)
)
chi = numpyro.sample(
"Chi", dist.HalfNormal(1).expand([num_classes - 1]).to_event(1)
)
with numpyro.plate("annotator", num_annotators, dim=-2):
with numpyro.plate("class", num_classes):
with handlers.reparam(config={"beta": LocScaleReparam(0)}):
beta = numpyro.sample("beta", dist.Normal(zeta, omega).to_event(1))
beta = jnp.pad(beta, [(0, 0)] * (jnp.ndim(beta) - 1) + [(0, 1)])
pi = numpyro.sample("pi", dist.Dirichlet(jnp.ones(num_classes)))
with numpyro.plate("item", num_items, dim=-2):
c = numpyro.sample("c", dist.Categorical(pi))
with handlers.reparam(config={"theta": LocScaleReparam(0)}):
theta = numpyro.sample("theta", dist.Normal(0, chi[c]).to_event(1))
theta = jnp.pad(theta, [(0, 0)] * (jnp.ndim(theta) - 1) + [(0, 1)])
with numpyro.plate("position", num_positions):
logits = Vindex(beta)[positions, c, :] - theta
numpyro.sample("y", dist.Categorical(logits=logits), obs=annotations)
NAME_TO_MODEL = {
"mn": multinomial,
"ds": dawid_skene,
"mace": mace,
"hds": hierarchical_dawid_skene,
"id": item_difficulty,
"lre": logistic_random_effects,
}
def main(args):
annotators, annotations = get_data()
model = NAME_TO_MODEL[args.model]
data = (
(annotations,)
if model in [multinomial, item_difficulty]
else (annotators, annotations)
)
mcmc = MCMC(
NUTS(model),
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), *data)
mcmc.print_summary()
posterior_samples = mcmc.get_samples()
predictive = Predictive(model, posterior_samples, infer_discrete=True)
discrete_samples = predictive(random.PRNGKey(1), *data)
item_class = vmap(lambda x: jnp.bincount(x, length=4), in_axes=1)(
discrete_samples["c"].squeeze(-1)
)
print("Histogram of the predicted class of each item:")
row_format = "{:>10}" * 5
print(row_format.format("", *["c={}".format(i) for i in range(4)]))
for i, row in enumerate(item_class):
print(row_format.format(f"item[{i}]", *row))
if __name__ == "__main__":
assert numpyro.__version__.startswith("0.8.0")
parser = argparse.ArgumentParser(description="Bayesian Models of Annotation")
parser.add_argument("-n", "--num-samples", nargs="?", default=1000, type=int)
parser.add_argument("--num-warmup", nargs="?", default=1000, type=int)
parser.add_argument("--num-chains", nargs="?", default=1, type=int)
parser.add_argument(
"--model",
nargs="?",
default="ds",
help='one of "mn" (multinomial), "ds" (dawid_skene), "mace",'
' "hds" (hierarchical_dawid_skene),'
' "id" (item_difficulty), "lre" (logistic_random_effects)',
)
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)