"""Poisson Hidden Markov Model."""
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 typing import Any, Dict, NamedTuple, Optional, Tuple, Union
from dynamax.hidden_markov_model.inference import HMMPosterior
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 IntScalar, Scalar
from dynamax.utils.utils import pytree_sum
class ParamsPoissonHMMEmissions(NamedTuple):
"""Parameters for the Poisson emissions of an HMM."""
rates: Union[Float[Array, "state_dim emission_dim"], ParameterProperties]
class PoissonHMMEmissions(HMMEmissions):
"""Poisson emissions for an HMM.
Args:
num_states: number of states in the HMM
emission_dim: dimensionality of the emission space
emission_prior_concentration: concentration parameter of the Gamma prior on the emission rates
emission_prior_rate: rate parameter of the Gamma prior on the emission rates
"""
def __init__(self,
num_states: int,
emission_dim: int,
emission_prior_concentration: Scalar = 1.1,
emission_prior_rate: Scalar = 0.1):
self.num_states = num_states
self.emission_dim = emission_dim
self.emission_prior_concentration = emission_prior_concentration
self.emission_prior_rate = emission_prior_rate
@property
def emission_shape(self) -> Tuple[int]:
"""Shape of the emission distribution."""
return (self.emission_dim,)
def initialize(self, key: Array=jr.PRNGKey(0),
method: str = "prior",
emission_rates: Optional[Float[Array, "num_states emission_dim"]] = None
) -> Tuple[ParamsPoissonHMMEmissions, ParamsPoissonHMMEmissions]:
"""Initialize the emission parameters.
Args:
key: random number generator for sampling from the prior. Defaults to jr.PRNGKey(0).
method: initialization method for the emission rates. Currently, only "prior" is allowed. Defaults to "prior".
emission_rates: manually specified emission rates. Defaults to None.
Returns:
Model parameters and their properties.
"""
# Initialize the emission probabilities
if emission_rates is None:
if method.lower() == "prior":
prior = tfd.Gamma(self.emission_prior_concentration, self.emission_prior_rate)
emission_rates = 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_rates.shape == (self.num_states, self.emission_dim)
assert jnp.all(emission_rates >= 0)
# Add parameters to the dictionary
params = ParamsPoissonHMMEmissions(rates=emission_rates)
props = ParamsPoissonHMMEmissions(rates=ParameterProperties(constrainer=tfb.Softplus()))
return params, props
def distribution(
self,
params: ParamsPoissonHMMEmissions,
state: IntScalar,
inputs: Optional[Array] = None
) -> tfd.Distribution:
"""Return the emission distribution for a given state."""
return tfd.Independent(tfd.Poisson(rate=params.rates[state]),
reinterpreted_batch_ndims=1)
def log_prior(self, params: ParamsPoissonHMMEmissions) -> Float[Array, ""]:
"""Return the log prior probability of the emission parameters."""
prior = tfd.Gamma(self.emission_prior_concentration, self.emission_prior_rate)
return prior.log_prob(params.rates).sum()
def collect_suff_stats(
self,
params: ParamsPoissonHMMEmissions,
posterior: HMMPosterior,
emissions: Float[Array, "num_timesteps emission_dim"],
inputs: Optional[Array] = None
) -> Dict[str, Float[Array, "..."]]:
"""Collect sufficient statistics for the emission parameters."""
expected_states = posterior.smoothed_probs
sum_w = jnp.einsum("tk->k", expected_states)[:, None]
sum_x = jnp.einsum("tk, ti->ki", expected_states, emissions)
return dict(sum_w=sum_w, sum_x=sum_x)
def initialize_m_step_state(self, params: ParamsPoissonHMMEmissions, props: ParamsPoissonHMMEmissions) -> None:
"""Initialize the state for the M-step."""
return None
def m_step(
self,
params: ParamsPoissonHMMEmissions,
props: ParamsPoissonHMMEmissions,
batch_stats: Dict[str, Float[Array, "..."]],
m_step_state: Any
) -> Tuple[ParamsPoissonHMMEmissions, Any]:
"""Perform the M-step for the emission parameters."""
if props.rates.trainable:
emission_stats = pytree_sum(batch_stats, axis=0)
post_concentration = self.emission_prior_concentration + emission_stats['sum_x']
post_rate = self.emission_prior_rate + emission_stats['sum_w']
rates = tfd.Gamma(post_concentration, post_rate).mode()
params = params._replace(rates=rates)
return params, m_step_state
class ParamsPoissonHMM(NamedTuple):
"""Parameters for a Poisson Hidden Markov Model."""
initial: ParamsStandardHMMInitialState
transitions: ParamsStandardHMMTransitions
emissions: ParamsPoissonHMMEmissions
[docs]
class PoissonHMM(HMM):
r"""An HMM with conditionally independent Poisson emissions.
Let $y_t \in \{0,1\}^N$ denote a vector of count emissions at time $t$. In this model,
the emission distribution is,
$$p(y_t \mid z_t, \theta) = \prod_{n=1}^N \mathrm{Po}(y_{tn} \mid \theta_{z_t,n})$$
$$p(\theta) = \prod_{k=1}^K \prod_{n=1}^N \mathrm{Ga}(\theta_{k,n}; \gamma_0, \gamma_1)$$
with $\theta_{k,n} \in \mathbb{R}_+$ for $k=1,\ldots,K$ and $n=1,\ldots,N$ are the
*emission rates* and $\gamma_0, \gamma_1$ are their prior concentration and rate, respectively.
: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_concentration: $\gamma_0$
:param emission_prior_rate: $\gamma_1$
"""
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_concentration: Scalar=1.1,
emission_prior_rate: Scalar=0.1):
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 = PoissonHMMEmissions(num_states, emission_dim, emission_prior_concentration=emission_prior_concentration, emission_prior_rate=emission_prior_rate)
super().__init__(num_states, initial_component, transition_component, emission_component)
[docs]
def initialize(self, key: Array=jr.PRNGKey(0),
method="prior",
initial_probs: Optional[Float[Array, " num_states"]]=None,
transition_matrix: Optional[Float[Array, "num_states num_states"]]=None,
emission_rates: 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_rates: 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_rates=emission_rates)
return ParamsPoissonHMM(**params), ParamsPoissonHMM(**props)
