# Copyright Contributors to the Pyro project.

# SPDX-License-Identifier: Apache-2.0

"""

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 references 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:**

1. 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/)

2. Dawid, A. P., and Skene, A. M. (1979).

"Maximum likelihood estimation of observer error‐rates using the EM algorithm"

3. "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

import numpyro.distributions as dist

from numpyro.infer import MCMC, NUTS, Predictive

from numpyro.infer.reparam import LocScaleReparam

from numpyro.ops.indexing import Vindex

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])

# fmt: off

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]])

# fmt: on

# 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), infer={"enumerate": "parallel"})

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), infer={"enumerate": "parallel"})

# 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):

c = numpyro.sample(

"c",

dist.DiscreteUniform(0, num_classes - 1),

infer={"enumerate": "parallel"},

)

with numpyro.plate("position", num_positions):

s = numpyro.sample(

"s",

dist.Bernoulli(1 - theta[positions]),

infer={"enumerate": "parallel"},

)

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), infer={"enumerate": "parallel"})

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), infer={"enumerate": "parallel"})

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), infer={"enumerate": "parallel"})

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.key(0), *data)

mcmc.print_summary()

posterior_samples = mcmc.get_samples()

predictive = Predictive(model, posterior_samples, infer_discrete=True)

discrete_samples = predictive(random.key(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))

# %%

# .. note::

# In the above inference code, we marginalized the discrete latent variables `c`

# hence `mcmc.get_samples(...)` does not include samples of `c`. We then utilize

# `Predictive(..., infer_discrete=True)` to get posterior samples for `c`, which

# is stored in `discrete_samples`. To merge those discrete samples into the `mcmc`

# instance, we can use the following pattern::

#

# chain_discrete_samples = jax.tree.map(

# lambda x: x.reshape((args.num_chains, args.num_samples) + x.shape[1:]),

# discrete_samples)

# mcmc.get_samples().update(discrete_samples)

# mcmc.get_samples(group_by_chain=True).update(chain_discrete_samples)

#

# This is useful when we want to pass the `mcmc` instance to `arviz` through

# `arviz.from_numpyro(mcmc)`.

if __name__ == "__main__":

assert numpyro.__version__.startswith("0.21.0")

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)