"""
Bernoulli Hidden Markov Model.
"""
from typing import NamedTuple, Optional, Tuple, Union
import jax.numpy as jnp
import jax.random as jr
import tensorflow_probability.substrates.jax.bijectors as tfb
import tensorflow_probability.substrates.jax.distributions as tfd
from jaxtyping import Array, Float
from dynamax.hidden_markov_model.models.abstractions import HMM, HMMEmissions
from dynamax.hidden_markov_model.models.initial import ParamsStandardHMMInitialState
from dynamax.hidden_markov_model.models.initial import StandardHMMInitialState
from dynamax.hidden_markov_model.models.transitions import ParamsStandardHMMTransitions
from dynamax.hidden_markov_model.models.transitions import StandardHMMTransitions
from dynamax.parameters import ParameterProperties, ParameterSet, PropertySet
from dynamax.types import Scalar
from dynamax.utils.utils import pytree_sum
class ParamsBernoulliHMMEmissions(NamedTuple):
"""Parameters for the Bernoulli emissions of an HMM."""
probs: Union[Float[Array, " emission_dim"], ParameterProperties]
class ParamsBernoulliHMM(NamedTuple):
"""Parameters for a Bernoulli Hidden Markov Model."""
initial: ParamsStandardHMMInitialState
transitions: ParamsStandardHMMTransitions
emissions: ParamsBernoulliHMMEmissions
class BernoulliHMMEmissions(HMMEmissions):
r"""Bernoulli emissions for an HMM.
:param num_states: number of discrete states $K$
:param emission_dim: number of conditionally independent emissions $N$
:param emission_prior_concentration0: $\gamma_0$
:param emission_prior_concentration1: $\gamma_1$
"""
def __init__(
self,
num_states: int,
emission_dim: int,
emission_prior_concentration1: Scalar = 1.1,
emission_prior_concentration0: Scalar = 1.1,
):
self.num_states = num_states
self.emission_dim = emission_dim
self.emission_prior_concentration0 = emission_prior_concentration0
self.emission_prior_concentration1 = emission_prior_concentration1
@property
def emission_shape(self) -> Tuple[int]:
"""Shape of the emission distribution."""
return (self.emission_dim,)
def distribution(self, params, state, inputs=None) -> tfd.Distribution:
"""Return the emission distribution for a given state."""
# This model assumes the emissions are a vector of conditionally independent
# Bernoulli observations. The `reinterpreted_batch_ndims` argument tells
# `tfd.Independent` that only the last dimension should be considered a "batch"
# of conditionally independent observations.
return tfd.Independent(tfd.Bernoulli(probs=params.probs[state]), reinterpreted_batch_ndims=1)
def log_prior(self, params) -> Float[Array, ""]:
"""Return the log prior probability of the emission parameters."""
prior = tfd.Beta(self.emission_prior_concentration1, self.emission_prior_concentration0)
return prior.log_prob(params.probs).sum()
def initialize(
self,
key: Array = jr.PRNGKey(0),
method="prior",
emission_probs: Optional[Float[Array, "num_states emission_dim"]] = None,
) -> Tuple[ParamsBernoulliHMMEmissions, ParamsBernoulliHMMEmissions]:
"""Initialize the emission parameters and their properties.
Args:
key: random number generator for sampling the parameters. Defaults to jr.PRNGKey(0).
method: method for initializing the parameters. Currently, only "prior" is allowed. Defaults to "prior".
emission_probs: manually specified emission probabilities. Defaults to None.
"""
if emission_probs is None:
if method.lower() == "prior":
prior = tfd.Beta(self.emission_prior_concentration1, self.emission_prior_concentration0)
emission_probs = prior.sample(seed=key, sample_shape=(self.num_states, self.emission_dim))
elif method.lower() == "kmeans":
raise NotImplementedError("kmeans initialization is not yet implemented!")
else:
raise Exception("invalid initialization method: {}".format(method))
else:
assert emission_probs.shape == (self.num_states, self.emission_dim)
assert jnp.all(emission_probs >= 0)
assert jnp.all(emission_probs <= 1)
# Add parameters to the dictionary
params = ParamsBernoulliHMMEmissions(probs=emission_probs)
props = ParamsBernoulliHMMEmissions(probs=ParameterProperties(constrainer=tfb.Sigmoid()))
return params, props
def collect_suff_stats(self, params, posterior, emissions, inputs=None):
"""Collect sufficient statistics for the emission parameters."""
expected_states = posterior.smoothed_probs
sum_x = jnp.einsum("tk, ti->ki", expected_states, jnp.where(jnp.isnan(emissions), 0, emissions))
sum_1mx = jnp.einsum("tk, ti->ki", expected_states, jnp.where(jnp.isnan(emissions), 0, 1 - emissions))
return (sum_x, sum_1mx)
def initialize_m_step_state(self, params, props):
"""Initialize the state for the M-step of the EM algorithm."""
return None
def m_step(self, params, props, batch_stats, m_step_state):
"""Perform the M-step of the EM algorithm."""
if props.probs.trainable:
sum_x, sum_1mx = pytree_sum(batch_stats, axis=0)
probs = tfd.Beta(
self.emission_prior_concentration1 + sum_x, self.emission_prior_concentration0 + sum_1mx
).mode()
params = params._replace(probs=probs)
return params, m_step_state
[docs]
class BernoulliHMM(HMM):
r"""An HMM with conditionally independent Bernoulli emissions.
Let $y_t \in \{0,1\}^N$ denote a binary vector of emissions at time $t$. In this model,
the emission distribution is,
$$p(y_t \mid z_t, \theta) = \prod_{n=1}^N \mathrm{Bern}(y_{tn} \mid \theta_{z_t,n})$$
$$p(\theta) = \prod_{k=1}^K \prod_{n=1}^N \mathrm{Beta}(\theta_{k,n}; \gamma_0, \gamma_1)$$
with $\theta_{k,n} \in [0,1]$ for $k=1,\ldots,K$ and $n=1,\ldots,N$ are the
*emission probabilities* and $\gamma_0, \gamma_1$ are their prior pseudocounts.
:param num_states: number of discrete states $K$
:param emission_dim: number of conditionally independent emissions $N$
:param initial_probs_concentration: $\alpha$
:param transition_matrix_concentration: $\beta$
:param transition_matrix_stickiness: optional hyperparameter to boost the concentration on the diagonal of the transition matrix.
:param emission_prior_concentration0: $\gamma_0$
:param emission_prior_concentration1: $\gamma_1$
"""
def __init__(
self,
num_states: int,
emission_dim: int,
initial_probs_concentration: Union[Scalar, Float[Array, " num_states"]] = 1.1,
transition_matrix_concentration: Union[Scalar, Float[Array, " num_states"]] = 1.1,
transition_matrix_stickiness: Scalar = 0.0,
emission_prior_concentration0: Scalar = 1.1,
emission_prior_concentration1: Scalar = 1.1,
):
self.emission_dim = emission_dim
initial_component = StandardHMMInitialState(num_states, initial_probs_concentration=initial_probs_concentration)
transition_component = StandardHMMTransitions(
num_states, concentration=transition_matrix_concentration, stickiness=transition_matrix_stickiness
)
emission_component = BernoulliHMMEmissions(
num_states,
emission_dim,
emission_prior_concentration0=emission_prior_concentration0,
emission_prior_concentration1=emission_prior_concentration1,
)
super().__init__(num_states, initial_component, transition_component, emission_component)
[docs]
def initialize(
self,
key: Array = jr.PRNGKey(0),
method: str = "prior",
initial_probs: Optional[Float[Array, " num_states"]] = None,
transition_matrix: Optional[Float[Array, "num_states num_states"]] = None,
emission_probs: Optional[Float[Array, "num_states emission_dim"]] = None,
) -> Tuple[ParameterSet, PropertySet]:
"""Initialize the model parameters and their corresponding properties.
You can either specify parameters manually via the keyword arguments, or you can have
them set automatically. If any parameters are not specified, you must supply a PRNGKey.
Parameters will then be sampled from the prior (if `method==prior`).
Note: in the future we may support more initialization schemes, like K-Means.
Args:
key: random number generator for unspecified parameters. Must not be None if there are any unspecified parameters. Defaults to jr.PRNGKey(0).
method: method for initializing unspecified parameters. Currently, only "prior" is allowed. Defaults to "prior".
initial_probs: manually specified initial state probabilities. Defaults to None.
transition_matrix: manually specified transition matrix. Defaults to None.
emission_probs: manually specified emission probabilities. Defaults to None.
Returns:
Model parameters and their properties.
"""
key1, key2, key3 = jr.split(key, 3)
params, props = dict(), dict()
params["initial"], props["initial"] = self.initial_component.initialize(
key1, method=method, initial_probs=initial_probs
)
params["transitions"], props["transitions"] = self.transition_component.initialize(
key2, method=method, transition_matrix=transition_matrix
)
params["emissions"], props["emissions"] = self.emission_component.initialize(
key3, method=method, emission_probs=emission_probs
)
return ParamsBernoulliHMM(**params), ParamsBernoulliHMM(**props)
