tfp.substrates.numpy.distributions.HiddenMarkovModel

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Hidden Markov model distribution.

Inherits From: Distribution

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Main aliases

tfp.experimental.substrates.numpy.distributions.HiddenMarkovModel

tfp.substrates.numpy.distributions.HiddenMarkovModel(
    initial_distribution, transition_distribution, observation_distribution,
    num_steps, validate_args=False, allow_nan_stats=True,
    time_varying_observation_distribution=False, name='HiddenMarkovModel'
)

The HiddenMarkovModel distribution implements a (batch of) hidden Markov models where the initial states, transition probabilities and observed states are all given by user-provided distributions. This model assumes that the transition matrices are fixed over time.

In this model, there is a sequence of integer-valued hidden states: z[0], z[1], ..., z[num_steps - 1] and a sequence of observed states: x[0], ..., x[num_steps - 1]. The distribution of z[0] is given by initial_distribution. The conditional probability of z[i + 1] given z[i] is described by the batch of distributions in transition_distribution. For a batch of hidden Markov models, the coordinates before the rightmost one of the transition_distribution batch correspond to indices into the hidden Markov model batch. The rightmost coordinate of the batch is used to select which distribution z[i + 1] is drawn from. The distributions corresponding to the probability of z[i + 1] conditional on z[i] == k is given by the elements of the batch whose rightmost coordinate is k. Similarly, the conditional distribution of z[i] given x[i] is given by the batch of observation_distribution. When the rightmost coordinate of observation_distribution is k it gives the conditional probabilities of x[i] given z[i] == k. The probability distribution associated with the HiddenMarkovModel distribution is the marginal distribution of x[0],...,x[num_steps - 1].

Examples

tfd = tfp.distributions

# A simple weather model.

# Represent a cold day with 0 and a hot day with 1.
# Suppose the first day of a sequence has a 0.8 chance of being cold.
# We can model this using the categorical distribution:

initial_distribution = tfd.Categorical(probs=[0.8, 0.2])

# Suppose a cold day has a 30% chance of being followed by a hot day
# and a hot day has a 20% chance of being followed by a cold day.
# We can model this as:

transition_distribution = tfd.Categorical(probs=[[0.7, 0.3],
                                                 [0.2, 0.8]])

# Suppose additionally that on each day the temperature is
# normally distributed with mean and standard deviation 0 and 5 on
# a cold day and mean and standard deviation 15 and 10 on a hot day.
# We can model this with:

observation_distribution = tfd.Normal(loc=[0., 15.], scale=[5., 10.])

# We can combine these distributions into a single week long
# hidden Markov model with:

model = tfd.HiddenMarkovModel(
    initial_distribution=initial_distribution,
    transition_distribution=transition_distribution,
    observation_distribution=observation_distribution,
    num_steps=7)

# The expected temperatures for each day are given by:

model.mean()  # shape [7], elements approach 9.0

# The log pdf of a week of temperature 0 is:

model.log_prob(tf.zeros(shape=[7]))

References

[1] https://en.wikipedia.org/wiki/Hidden_Markov_model

Args
initial_distribution	A Categorical-like instance. Determines probability of first hidden state in Markov chain. The number of categories must match the number of categories of transition_distribution as well as both the rightmost batch dimension of transition_distribution and the rightmost batch dimension of observation_distribution.
transition_distribution	A Categorical-like instance. The rightmost batch dimension indexes the probability distribution of each hidden state conditioned on the previous hidden state.
observation_distribution	A tfp.distributions.Distribution-like instance. The rightmost batch dimension indexes the distribution of each observation conditioned on the corresponding hidden state.
num_steps	The number of steps taken in Markov chain. An integer valued tensor. The number of transitions is num_steps - 1.
validate_args	Python bool, default False. When True distribution parameters are checked for validity despite possibly degrading runtime performance. When False invalid inputs may silently render incorrect outputs. Default value: False.
allow_nan_stats	Python bool, default True. When True, statistics (e.g., mean, mode, variance) use the value "NaN" to indicate the result is undefined. When False, an exception is raised if one or more of the statistic's batch members are undefined. Default value: True.
time_varying_observation_distribution	Python bool, default False. When True, the observation_distribution has an additional batch dimension that indexes the distribution of each observation conditioned on the corresponding timestep. This dimension size should always match num_steps and is the second-to-last batch axis in the batch dimensions (just to the left of the dimension for the number of states).
name	Python str name prefixed to Ops created by this class. Default value: "HiddenMarkovModel".

Raises
ValueError	if num_steps is not at least 1.
ValueError	if initial_distribution does not have scalar event_shape.
ValueError	if transition_distribution does not have scalar event_shape.
ValueError	if transition_distribution and observation_distribution are fully defined but don't have matching rightmost dimension.

Attributes
allow_nan_stats	Python bool describing behavior when a stat is undefined. Stats return +/- infinity when it makes sense. E.g., the variance of a Cauchy distribution is infinity. However, sometimes the statistic is undefined, e.g., if a distribution's pdf does not achieve a maximum within the support of the distribution, the mode is undefined. If the mean is undefined, then by definition the variance is undefined. E.g. the mean for Student's T for df = 1 is undefined (no clear way to say it is either + or - infinity), so the variance = E[(X - mean)**2] is also undefined.
batch_shape	Shape of a single sample from a single event index as a TensorShape. May be partially defined or unknown. The batch dimensions are indexes into independent, non-identical parameterizations of this distribution.
dtype	The DType of Tensors handled by this Distribution.
event_shape	Shape of a single sample from a single batch as a TensorShape. May be partially defined or unknown.
initial_distribution
name	Name prepended to all ops created by this Distribution.
num_states
num_states_static	The number of hidden states in the hidden Markov model.
num_steps
observation_distribution
parameters	Dictionary of parameters used to instantiate this Distribution.
reparameterization_type	Describes how samples from the distribution are reparameterized. Currently this is one of the static instances tfd.FULLY_REPARAMETERIZED or tfd.NOT_REPARAMETERIZED.
trainable_variables
transition_distribution
validate_args	Python bool indicating possibly expensive checks are enabled.
variables

Methods

batch_shape_tensor

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batch_shape_tensor(
    name='batch_shape_tensor'
)

Shape of a single sample from a single event index as a 1-D Tensor.

The batch dimensions are indexes into independent, non-identical parameterizations of this distribution.

Args
name	name to give to the op

Returns
batch_shape	Tensor.

cdf

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cdf(
    value, name='cdf', **kwargs
)

Cumulative distribution function.

Given random variable X, the cumulative distribution function cdf is:

cdf(x) := P[X <= x]

Args
value	float or double Tensor.
name	Python str prepended to names of ops created by this function.
**kwargs	Named arguments forwarded to subclass implementation.

Returns
cdf	a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

copy

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copy(
    **override_parameters_kwargs
)

Creates a deep copy of the distribution.

Args
**override_parameters_kwargs	String/value dictionary of initialization arguments to override with new values.

Returns
distribution	A new instance of type(self) initialized from the union of self.parameters and override_parameters_kwargs, i.e., dict(self.parameters, **override_parameters_kwargs).

covariance

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covariance(
    name='covariance', **kwargs
)

Covariance.

Covariance is (possibly) defined only for non-scalar-event distributions.

For example, for a length-k, vector-valued distribution, it is calculated as,

Cov[i, j] = Covariance(X_i, X_j) = E[(X_i - E[X_i]) (X_j - E[X_j])]

where Cov is a (batch of) k x k matrix, 0 <= (i, j) < k, and E denotes expectation.

Alternatively, for non-vector, multivariate distributions (e.g., matrix-valued, Wishart), Covariance shall return a (batch of) matrices under some vectorization of the events, i.e.,

Cov[i, j] = Covariance(Vec(X)_i, Vec(X)_j) = [as above]

where Cov is a (batch of) k' x k' matrices, 0 <= (i, j) < k' = reduce_prod(event_shape), and Vec is some function mapping indices of this distribution's event dimensions to indices of a length-k' vector.

Args
name	Python str prepended to names of ops created by this function.
**kwargs	Named arguments forwarded to subclass implementation.

Returns
covariance	Floating-point Tensor with shape [B1, ..., Bn, k', k'] where the first n dimensions are batch coordinates and k' = reduce_prod(self.event_shape).

cross_entropy

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cross_entropy(
    other, name='cross_entropy'
)

Computes the (Shannon) cross entropy.

Denote this distribution (self) by P and the other distribution by Q. Assuming P, Q are absolutely continuous with respect to one another and permit densities p(x) dr(x) and q(x) dr(x), (Shannon) cross entropy is defined as:

H[P, Q] = E_p[-log q(X)] = -int_F p(x) log q(x) dr(x)

where F denotes the support of the random variable X ~ P.

Args
other	tfp.distributions.Distribution instance.
name	Python str prepended to names of ops created by this function.

Returns
cross_entropy	self.dtype Tensor with shape [B1, ..., Bn] representing n different calculations of (Shannon) cross entropy.

entropy

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entropy(
    name='entropy', **kwargs
)

Shannon entropy in nats.

event_shape_tensor

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event_shape_tensor(
    name='event_shape_tensor'
)

Shape of a single sample from a single batch as a 1-D int32 Tensor.

Args
name	name to give to the op

Returns
event_shape	Tensor.

is_scalar_batch

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is_scalar_batch(
    name='is_scalar_batch'
)

Indicates that batch_shape == [].

Args
name	Python str prepended to names of ops created by this function.

Returns
is_scalar_batch	bool scalar Tensor.

is_scalar_event

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is_scalar_event(
    name='is_scalar_event'
)

Indicates that event_shape == [].

Args
name	Python str prepended to names of ops created by this function.

Returns
is_scalar_event	bool scalar Tensor.

kl_divergence

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kl_divergence(
    other, name='kl_divergence'
)

Computes the Kullback--Leibler divergence.

Denote this distribution (self) by p and the other distribution by q. Assuming p, q are absolutely continuous with respect to reference measure r, the KL divergence is defined as:

KL[p, q] = E_p[log(p(X)/q(X))]
         = -int_F p(x) log q(x) dr(x) + int_F p(x) log p(x) dr(x)
         = H[p, q] - H[p]

where F denotes the support of the random variable X ~ p, H[., .] denotes (Shannon) cross entropy, and H[.] denotes (Shannon) entropy.

Args
other	tfp.distributions.Distribution instance.
name	Python str prepended to names of ops created by this function.

Returns
kl_divergence	self.dtype Tensor with shape [B1, ..., Bn] representing n different calculations of the Kullback-Leibler divergence.

log_cdf

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log_cdf(
    value, name='log_cdf', **kwargs
)

Log cumulative distribution function.

Given random variable X, the cumulative distribution function cdf is:

log_cdf(x) := Log[ P[X <= x] ]

Often, a numerical approximation can be used for log_cdf(x) that yields a more accurate answer than simply taking the logarithm of the cdf when x << -1.

Args
value	float or double Tensor.
name	Python str prepended to names of ops created by this function.
**kwargs	Named arguments forwarded to subclass implementation.

Returns
logcdf	a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

log_prob

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log_prob(
    value, name='log_prob', **kwargs
)

Log probability density/mass function.

Args
value	float or double Tensor.
name	Python str prepended to names of ops created by this function.
**kwargs	Named arguments forwarded to subclass implementation.

Returns
log_prob	a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

log_survival_function

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log_survival_function(
    value, name='log_survival_function', **kwargs
)

Log survival function.

Given random variable X, the survival function is defined:

log_survival_function(x) = Log[ P[X > x] ]
                         = Log[ 1 - P[X <= x] ]
                         = Log[ 1 - cdf(x) ]

Typically, different numerical approximations can be used for the log survival function, which are more accurate than 1 - cdf(x) when x >> 1.

Args
value	float or double Tensor.
name	Python str prepended to names of ops created by this function.
**kwargs	Named arguments forwarded to subclass implementation.

Returns
Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

mean

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mean(
    name='mean', **kwargs
)

Mean.

mode

View source

mode(
    name='mode', **kwargs
)

Mode.

num_states_tensor

View source

num_states_tensor()

The number of hidden states in the hidden Markov model.

param_shapes

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@classmethod
param_shapes(
    sample_shape, name='DistributionParamShapes'
)

Shapes of parameters given the desired shape of a call to sample().

This is a class method that describes what key/value arguments are required to instantiate the given Distribution so that a particular shape is returned for that instance's call to sample().

Subclasses should override class method _param_shapes.

Args
sample_shape	Tensor or python list/tuple. Desired shape of a call to sample().
name	name to prepend ops with.

Returns
dict of parameter name to Tensor shapes.

param_static_shapes

View source

@classmethod
param_static_shapes(
    sample_shape
)

param_shapes with static (i.e. TensorShape) shapes.

This is a class method that describes what key/value arguments are required to instantiate the given Distribution so that a particular shape is returned for that instance's call to sample(). Assumes that the sample's shape is known statically.

Subclasses should override class method _param_shapes to return constant-valued tensors when constant values are fed.

Args
sample_shape	TensorShape or python list/tuple. Desired shape of a call to sample().

Returns
dict of parameter name to TensorShape.

Raises
ValueError	if sample_shape is a TensorShape and is not fully defined.

posterior_marginals

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posterior_marginals(
    observations, mask=None, name='posterior_marginals'
)

Compute marginal posterior distribution for each state.

This function computes, for each time step, the marginal conditional probability that the hidden Markov model was in each possible state given the observations that were made at each time step. So if the hidden states are z[0],...,z[num_steps - 1] and the observations are x[0], ..., x[num_steps - 1], then this function computes P(z[i] | x[0], ..., x[num_steps - 1]) for all i from 0 to num_steps - 1.

This operation is sometimes called smoothing. It uses a form of the forward-backward algorithm.

Args
observations	A tensor representing a batch of observations made on the hidden Markov model. The rightmost dimension of this tensor gives the steps in a sequence of observations from a single sample from the hidden Markov model. The size of this dimension should match the num_steps parameter of the hidden Markov model object. The other dimensions are the dimensions of the batch and these are broadcast with the hidden Markov model's parameters.
mask	optional bool-type tensor with rightmost dimension matching num_steps indicating which observations the result of this function should be conditioned on. When the mask has value True the corresponding observations aren't used. if mask is None then all of the observations are used. the mask dimensions left of the last are broadcast with the hmm batch as well as with the observations.
name	Python str name prefixed to Ops created by this class. Default value: "HiddenMarkovModel".

Returns
posterior_marginal	A Categorical distribution object representing the marginal probability of the hidden Markov model being in each state at each step. The rightmost dimension of the Categorical distributions batch will equal the num_steps parameter providing one marginal distribution for each step. The other dimensions are the dimensions corresponding to the batch of observations.

Raises
ValueError	if rightmost dimension of observations does not have size num_steps.

posterior_mode

View source

posterior_mode(
    observations, mask=None, name='posterior_mode'
)

Compute maximum likelihood sequence of hidden states.

When this function is provided with a sequence of observations x[0], ..., x[num_steps - 1], it returns the sequence of hidden states z[0], ..., z[num_steps - 1], drawn from the underlying Markov chain, that is most likely to yield those observations.

It uses the Viterbi algorithm.

Args
observations	A tensor representing a batch of observations made on the hidden Markov model. The rightmost dimensions of this tensor correspond to the dimensions of the observation distributions of the underlying Markov chain. The next dimension from the right indexes the steps in a sequence of observations from a single sample from the hidden Markov model. The size of this dimension should match the num_steps parameter of the hidden Markov model object. The other dimensions are the dimensions of the batch and these are broadcast with the hidden Markov model's parameters.
mask	optional bool-type tensor with rightmost dimension matching num_steps indicating which observations the result of this function should be conditioned on. When the mask has value True the corresponding observations aren't used. if mask is None then all of the observations are used. the mask dimensions left of the last are broadcast with the hmm batch as well as with the observations.
name	Python str name prefixed to Ops created by this class. Default value: "HiddenMarkovModel".

Returns
posterior_mode	A Tensor representing the most likely sequence of hidden states. The rightmost dimension of this tensor will equal the num_steps parameter providing one hidden state for each step. The other dimensions are those of the batch.

Raises
ValueError	if the observations tensor does not consist of sequences of num_steps observations.

Examples

tfd = tfp.distributions

# A simple weather model.

# Represent a cold day with 0 and a hot day with 1.
# Suppose the first day of a sequence has a 0.8 chance of being cold.

initial_distribution = tfd.Categorical(probs=[0.8, 0.2])

# Suppose a cold day has a 30% chance of being followed by a hot day
# and a hot day has a 20% chance of being followed by a cold day.

transition_distribution = tfd.Categorical(probs=[[0.7, 0.3],
                                                 [0.2, 0.8]])

# Suppose additionally that on each day the temperature is
# normally distributed with mean and standard deviation 0 and 5 on
# a cold day and mean and standard deviation 15 and 10 on a hot day.

observation_distribution = tfd.Normal(loc=[0., 15.], scale=[5., 10.])

# This gives the hidden Markov model:

model = tfd.HiddenMarkovModel(
    initial_distribution=initial_distribution,
    transition_distribution=transition_distribution,
    observation_distribution=observation_distribution,
    num_steps=7)

# Suppose we observe gradually rising temperatures over a week:
temps = [-2., 0., 2., 4., 6., 8., 10.]

# We can now compute the most probable sequence of hidden states:

model.posterior_mode(temps)

# The result is [0 0 0 0 0 1 1] telling us that the transition
# from "cold" to "hot" most likely happened between the
# 5th and 6th days.

prob

View source

prob(
    value, name='prob', **kwargs
)

Probability density/mass function.

Args
value	float or double Tensor.
name	Python str prepended to names of ops created by this function.
**kwargs	Named arguments forwarded to subclass implementation.

Returns
prob	a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

quantile

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quantile(
    value, name='quantile', **kwargs
)

Quantile function. Aka 'inverse cdf' or 'percent point function'.

Given random variable X and p in [0, 1], the quantile is:

quantile(p) := x such that P[X <= x] == p

Args
value	float or double Tensor.
name	Python str prepended to names of ops created by this function.
**kwargs	Named arguments forwarded to subclass implementation.

Returns
quantile	a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

sample

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sample(
    sample_shape=(), seed=None, name='sample', **kwargs
)

Generate samples of the specified shape.

Note that a call to sample() without arguments will generate a single sample.

Args
sample_shape	0D or 1D int32 Tensor. Shape of the generated samples.
seed	Python integer or tfp.util.SeedStream instance, for seeding PRNG.
name	name to give to the op.
**kwargs	Named arguments forwarded to subclass implementation.

Returns
samples	a Tensor with prepended dimensions sample_shape.

stddev

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stddev(
    name='stddev', **kwargs
)

Standard deviation.

Standard deviation is defined as,

stddev = E[(X - E[X])**2]**0.5

where X is the random variable associated with this distribution, E denotes expectation, and stddev.shape = batch_shape + event_shape.

Args
name	Python str prepended to names of ops created by this function.
**kwargs	Named arguments forwarded to subclass implementation.

Returns
stddev	Floating-point Tensor with shape identical to batch_shape + event_shape, i.e., the same shape as self.mean().

survival_function

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survival_function(
    value, name='survival_function', **kwargs
)

Survival function.

Given random variable X, the survival function is defined:

survival_function(x) = P[X > x]
                     = 1 - P[X <= x]
                     = 1 - cdf(x).

Args
value	float or double Tensor.
name	Python str prepended to names of ops created by this function.
**kwargs	Named arguments forwarded to subclass implementation.

Returns
Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

variance

View source

variance(
    name='variance', **kwargs
)

Variance.

Variance is defined as,

Var = E[(X - E[X])**2]

where X is the random variable associated with this distribution, E denotes expectation, and Var.shape = batch_shape + event_shape.

Args
name	Python str prepended to names of ops created by this function.
**kwargs	Named arguments forwarded to subclass implementation.

Returns
variance	Floating-point Tensor with shape identical to batch_shape + event_shape, i.e., the same shape as self.mean().

__getitem__

View source

__getitem__(
    slices
)

Slices the batch axes of this distribution, returning a new instance.

b = tfd.Bernoulli(logits=tf.zeros([3, 5, 7, 9]))
b.batch_shape  # => [3, 5, 7, 9]
b2 = b[:, tf.newaxis, ..., -2:, 1::2]
b2.batch_shape  # => [3, 1, 5, 2, 4]

x = tf.random.stateless_normal([5, 3, 2, 2])
cov = tf.matmul(x, x, transpose_b=True)
chol = tf.cholesky(cov)
loc = tf.random.stateless_normal([4, 1, 3, 1])
mvn = tfd.MultivariateNormalTriL(loc, chol)
mvn.batch_shape  # => [4, 5, 3]
mvn.event_shape  # => [2]
mvn2 = mvn[:, 3:, ..., ::-1, tf.newaxis]
mvn2.batch_shape  # => [4, 2, 3, 1]
mvn2.event_shape  # => [2]

Args
slices	slices from the [] operator

Returns
dist	A new tfd.Distribution instance with sliced parameters.

__iter__

View source

__iter__()