# tfp.distributions.HiddenMarkovModel

Hidden Markov model distribution.

Inherits From: `Distribution`

### Used in the notebooks

Used in the tutorials

The `HiddenMarkovModel` distribution implements a (batch of) discrete hidden Markov models where the initial states, transition probabilities and observed states are all given by user-provided distributions.

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 `x[i]` given `z[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

`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_transition_distribution` Python `bool`, default `False`. When `True`, the transition_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 -1` and is the second-to-last batch axis in the batch dimensions (just to the left of the dimension for the number of states). Because transitions only happen between steps, the number of transitions is one less than num_steps.
`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).
`mask` optional bool-type `tensor` with rightmost dimension matching `num_steps`, indicating which observations should be ignored (not conditioned on) for posterior inference and `log_prob` evaluation. This may be overridden by passing the `mask` arg to individual methods. When the mask has value `True`, the corresponding observations aren't used. If `mask` is `None` then all of the observations are used. Default value: `None`.
`name` Python `str` name prefixed to Ops created by this class. Default value: "HiddenMarkovModel".

`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.

`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 `Tensor`s handled by this `Distribution`.
`event_shape` Shape of a single sample from a single batch as a `TensorShape`.

May be partially defined or unknown.

`experimental_shard_axis_names` The list or structure of lists of active shard axis names.
`initial_distribution`

`mask`

`name` Name prepended to all ops created by this `Distribution`.
`name_scope` Returns a `tf.name_scope` instance for this class.
`non_trainable_variables` Sequence of non-trainable variables owned by this module and its submodules.

`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`.

`submodules` Sequence of all sub-modules.

Submodules are modules which are properties of this module, or found as properties of modules which are properties of this module (and so on).

````a = tf.Module()`
`b = tf.Module()`
`c = tf.Module()`
`a.b = b`
`b.c = c`
`list(a.submodules) == [b, c]`
`True`
`list(b.submodules) == [c]`
`True`
`list(c.submodules) == []`
`True`
```

`trainable_variables` Sequence of trainable variables owned by this module and its submodules.

`transition_distribution`

`validate_args` Python `bool` indicating possibly expensive checks are enabled.
`variables` Sequence of variables owned by this module and its submodules.

## Methods

### `batch_shape_tensor`

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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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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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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.

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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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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Shannon entropy in nats.

### `event_shape_tensor`

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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`.

### `experimental_default_event_space_bijector`

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Bijector mapping the reals (R**n) to the event space of the distribution.

Distributions with continuous support may implement `_default_event_space_bijector` which returns a subclass of `tfp.bijectors.Bijector` that maps R**n to the distribution's event space. For example, the default bijector for the `Beta` distribution is `tfp.bijectors.Sigmoid()`, which maps the real line to `[0, 1]`, the support of the `Beta` distribution. The default bijector for the `CholeskyLKJ` distribution is `tfp.bijectors.CorrelationCholesky`, which maps R^(k * (k-1) // 2) to the submanifold of k x k lower triangular matrices with ones along the diagonal.

The purpose of `experimental_default_event_space_bijector` is to enable gradient descent in an unconstrained space for Variational Inference and Hamiltonian Monte Carlo methods. Some effort has been made to choose bijectors such that the tails of the distribution in the unconstrained space are between Gaussian and Exponential.

For distributions with discrete event space, or for which TFP currently lacks a suitable bijector, this function returns `None`.

Args
`*args` Passed to implementation `_default_event_space_bijector`.
`**kwargs` Passed to implementation `_default_event_space_bijector`.

Returns
`event_space_bijector` `Bijector` instance or `None`.

### `experimental_fit`

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Instantiates a distribution that maximizes the likelihood of `x`.

Args
`value` a `Tensor` valid sample from this distribution family.
`sample_ndims` Positive `int` Tensor number of leftmost dimensions of `value` that index i.i.d. samples. Default value: `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`.
`**init_kwargs` Additional keyword arguments passed through to `cls.__init__`. These take precedence in case of collision with the fitted parameters; for example, `tfd.Normal.experimental_fit([1., 1.], scale=20.)` returns a Normal distribution with `scale=20.` rather than the maximum likelihood parameter `scale=0.`.

Returns
`maximum_likelihood_instance` instance of `cls` with parameters that maximize the likelihood of `value`.

### `experimental_local_measure`

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Returns a log probability density together with a `TangentSpace`.

A `TangentSpace` allows us to calculate the correct push-forward density when we apply a transformation to a `Distribution` on a strict submanifold of R^n (typically via a `Bijector` in the `TransformedDistribution` subclass). The density correction uses the basis of the tangent space.

Args
`value` `float` or `double` `Tensor`.
`backward_compat` `bool` specifying whether to fall back to returning `FullSpace` as the tangent space, and representing R^n with the standard basis.
`**kwargs` Named arguments forwarded to subclass implementation.

Returns
`log_prob` a `Tensor` representing the log probability density, of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`.
`tangent_space` a `TangentSpace` object (by default `FullSpace`) representing the tangent space to the manifold at `value`.

Raises
UnspecifiedTangentSpaceError if `backward_compat` is False and the `_experimental_tangent_space` attribute has not been defined.

### `experimental_sample_and_log_prob`

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Samples from this distribution and returns the log density of the sample.

The default implementation simply calls `sample` and `log_prob`:

``````def _sample_and_log_prob(self, sample_shape, seed, **kwargs):
x = self.sample(sample_shape=sample_shape, seed=seed, **kwargs)
return x, self.log_prob(x, **kwargs)
``````

However, some subclasses may provide more efficient and/or numerically stable implementations.

Args
`sample_shape` integer `Tensor` desired shape of samples to draw. Default value: `()`.
`seed` PRNG seed; see `tfp.random.sanitize_seed` for details. Default value: `None`.
`name` name to give to the op. Default value: `'sample_and_log_prob'`.
`**kwargs` Named arguments forwarded to subclass implementation.

Returns
`samples` a `Tensor`, or structure of `Tensor`s, with prepended dimensions `sample_shape`.
`log_prob` a `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`.

### `is_scalar_batch`

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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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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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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 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 probability density/mass function.

Additional documentation from `HiddenMarkovModel`:

##### `kwargs`:
• `mask`: optional bool-type `tensor` with rightmost dimension matching `num_steps`, indicating which observations should be ignored (not conditioned on). When the mask has value `True`, the corresponding observations aren't used. If no mask is specified (`mask` and `self.mask` are both `None`) then all of the observations are used. The leftmost dimensions `shape(mask)[:-1]` must broadcast with `self.batch_shape`. shape of `x`. Default value: `None` (falls back to `self.mask`).

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.

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`.

View source

Mean.

View source

Mode.

### `num_states_tensor`

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The number of hidden states in the hidden Markov model.

### `param_shapes`

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

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`

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

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.

### `parameter_properties`

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Returns a dict mapping constructor arg names to property annotations.

This dict should include an entry for each of the distribution's `Tensor`-valued constructor arguments.

Distribution subclasses are not required to implement `_parameter_properties`, so this method may raise `NotImplementedError`. Providing a `_parameter_properties` implementation enables several advanced features, including:

• Distribution batch slicing (`sliced_distribution = distribution[i:j]`).
• Automatic inference of `_batch_shape` and `_batch_shape_tensor`, which must otherwise be computed explicitly.
• Automatic instantiation of the distribution within TFP's internal property tests.
• Automatic construction of 'trainable' instances of the distribution using appropriate bijectors to avoid violating parameter constraints. This enables the distribution family to be used easily as a surrogate posterior in variational inference.

In the future, parameter property annotations may enable additional functionality; for example, returning Distribution instances from `tf.vectorized_map`.

Args
`dtype` Optional float `dtype` to assume for continuous-valued parameters. Some constraining bijectors require advance knowledge of the dtype because certain constants (e.g., `tfb.Softplus.low`) must be instantiated with the same dtype as the values to be transformed.
`num_classes` Optional `int` `Tensor` number of classes to assume when inferring the shape of parameters for categorical-like distributions. Otherwise ignored.

Returns
`parameter_properties` A `str ->`tfp.python.internal.parameter_properties.ParameterProperties`dict mapping constructor argument names to`ParameterProperties` instances.

Raises
`NotImplementedError` if the distribution class does not implement `_parameter_properties`.

### `posterior_marginals`

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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 dimensions of this tensor correspond to the dimensions of the observation distributions of the underlying Markov chain, if the observations are non-scalar. 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 should be ignored (not conditioned on). When the mask has value `True`, the corresponding observations aren't used. If no mask is specified (`mask` and `self.mask` are both `None`) then all of the observations are used. The leftmost dimensions `shape(mask)[:-1]` must broadcast with `self.batch_shape`. Default value: `None` (falls back to `self.mask`).
`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`

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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, if the observations are non-scalar. 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 should be ignored (not conditioned on). When the mask has value `True`, the corresponding observations aren't used. If no mask is specified (`mask` and `self.mask` are both `None`) then all of the observations are used. The leftmost dimensions `shape(mask)[:-1]` must broadcast with `self.batch_shape`.
`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.
``````

### `prior_marginals`

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Compute prior marginal distribution for each state.

This function computes, for each time step, the prior probability that the hidden Markov model is at a given state. In other words this function computes: `P(z[i])` for all `i` from `0` to `num_steps - 1`.

Args
`name` Python `str` name prefixed to Ops created by this class. Default value: "priors".

Returns
`prior` A `Categorical` distribution object representing the prior 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 prior distribution for each step.

### `prob`

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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 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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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` PRNG seed; see `tfp.random.sanitize_seed` for details.
`name` name to give to the op.
`**kwargs` Named arguments forwarded to subclass implementation.

Returns
`samples` a `Tensor` with prepended dimensions `sample_shape`.

### `stddev`

View source

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`

View source

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`.

### `unnormalized_log_prob`

View source

Potentially unnormalized log probability density/mass function.

This function is similar to `log_prob`, but does not require that the return value be normalized. (Normalization here refers to the total integral of probability being one, as it should be by definition for any probability distribution.) This is useful, for example, for distributions where the normalization constant is difficult or expensive to compute. By default, this simply calls `log_prob`.

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

### `variance`

View source

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()`.

### `with_name_scope`

Decorator to automatically enter the module name scope.

````class MyModule(tf.Module):`
`  @tf.Module.with_name_scope`
`  def __call__(self, x):`
`    if not hasattr(self, 'w'):`
`      self.w = tf.Variable(tf.random.normal([x.shape[1], 3]))`
`    return tf.matmul(x, self.w)`
```

Using the above module would produce `tf.Variable`s and `tf.Tensor`s whose names included the module name:

````mod = MyModule()`
`mod(tf.ones([1, 2]))`
`<tf.Tensor: shape=(1, 3), dtype=float32, numpy=..., dtype=float32)>`
`mod.w`
`<tf.Variable 'my_module/Variable:0' shape=(2, 3) dtype=float32,`
`numpy=..., dtype=float32)>`
```

Args
`method` The method to wrap.

Returns
The original method wrapped such that it enters the module's name scope.

### `__getitem__`

View source

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.normal([5, 3, 2, 2])
cov = tf.matmul(x, x, transpose_b=True)
chol = tf.linalg.cholesky(cov)
loc = tf.random.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.

View source

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