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tfp.substrates.numpy.distributions.MarkovChain

Distribution of a sequence generated by a memoryless process.

Inherits From: Distribution

A discrete-time Markov chain is a sequence of random variables in which the variable(s) at each step is independent of all previous variables, conditioned on the variable(s) at the immediate predecessor step. That is, there can be no (direct) long-term dependencies. This 'Markov property' is a simplifying assumption; for example, it enables efficient sampling. Many time-series models can be formulated as Markov chains.

Instances of tfd.MarkovChain represent fully-observed, discrete-time Markov chains, with one or more random variables at each step. These variables may take continuous or discrete values. Sampling is done sequentially, requiring time that scales with the length of the sequence; log_prob evaluation is vectorized over timesteps, and so requires only constant time given sufficient parallelism.

The discrete-valued Markov chains modeled by tfd.HiddenMarkovModel (using a trivial observation distribution) are a special case of those supported by this distribution, which enable exact inference over the values in an unobserved chain. Continuous-valued chains with linear Gaussian transitions are supported by tfd.LinearGaussianStateSpaceModel, which can similarly exploit the linear Gaussian structure for exact inference of hidden states. These distributions are limited to chains that have the respective (discrete or linear Gaussian) structure.

Autoregressive models that do not necessarily respect the Markov property are supported by tfd.Autoregressive, which is, in that sense, more general than this distribution. These models require a more involved specification, and sampling in general requires quadratic (rather than linear) time in the length of the sequence.

Exact inference for unobserved Markov chains is not possible in general; however, particle filtering exploits the Markov property to perform approximate inference, and is often a well-suited method for sequential inference tasks. Particle filtering is available in TFP using tfp.experimental.mcmc.particle_filter, and related methods.

Example: Gaussian random walk

One of the simplest continuous-valued Markov chains is a Gaussian random walk. This may also be viewed as a discretized Brownian motion.

tfd = tfp.distributions

gaussian_walk = tfd.MarkovChain(
  initial_state_prior=tfd.Normal(loc=0., scale=1.),
  transition_fn=lambda _, x: tfd.Normal(loc=x, scale=1.),
  num_steps=100)
# ==> `gaussian_walk.event_shape == [100]`
# ==> `gaussian_walk.batch_shape == []`

x = gaussian_walk.sample(5)  # Samples a matrix of 5 independent walks.
lp = gaussian_walk.log_prob(x)  # ==> `lp.shape == [5]`.

Example: batch of random walks

To spice things up, we'll now define a batch of random walks, each following a different distribution (in this case, different starting locations). We'll also demonstrate scales that differ across timesteps.

scales = tf.convert_to_tensor([0.5, 0.3, 0.2, 0.2, 0.3, 0.2, 0.7])
batch_gaussian_walk = tfd.MarkovChain(
  # The prior distribution determines the batch shape for the chain.
  # Transitions must respect this batch shape.
  initial_state_prior=tfd.Normal(loc=[-10., 0., 10.],
                                 scale=[1., 1., 1.]),
  transition_fn=lambda t, x: tfd.Normal(
    loc=x,
    # The `num_steps` dimension will always be leftmost in `x`, so we
    # pad the scale to the same rank as `x` to make their shapes line up.
    tf.reshape(tf.gather(scales, t),
               tf.concat([[-1],
                          tf.ones(tf.rank(x) - 1, dtype=tf.int32)], axis=0))),
  # Limit to eight steps since we only specified scales for seven transitions.
  num_steps=8)
# ==> `batch_gaussian_walk.event_shape == [8]`
# ==> `batch_gaussian_walk.batch_shape == [3]`

x = batch_gaussian_walk.sample(5)  # ==> `x.shape == [5, 3, 8]`.
lp = batch_gaussian_walk.log_prob(x)  # ==> `lp.shape == [5, 3]`.

Example: multivariate chain with longer-term dependence

We can also define multivariate Markov chains. In addition to the obvious use of modeling the joint evolution of multiple variables, multivariate chains can also help us work around the Markov limitation by the trick of folding state history into the current state as an auxiliary variable(s). The next example, a second-order autoregressive process with dynamic coefficients and scale, contains multiple time-dependent variables and also uses an auxiliary previous_level variable to enable the transition function to access the previous two steps of history:


def transition_fn(_, previous_state):
  return tfd.JointDistributionNamedAutoBatched(
      # The transition distribution must match the batch shape of the chain.
      # Since `log_scale` is a scalar quantity, its shape is the batch shape.
      batch_ndims=tf.rank(previous_state['log_scale']),
      model={
          # The autoregressive coefficients and the `log_scale` each follow
          # an independent slow-moving random walk.
          'coefs': tfd.Normal(loc=previous_state['coefs'], scale=0.01),
          'log_scale': tfd.Normal(loc=previous_state['log_scale'],
                                  scale=0.01),
          # The level is a linear combination of the previous *two* levels,
          # with additional noise of scale `exp(log_scale)`.
          'level': lambda coefs, log_scale: tfd.Normal(  
              loc=(coefs[..., 0] * previous_state['level'] +
                   coefs[..., 1] * previous_state['previous_level']),
              scale=tf.exp(log_scale)),
          # Store the previous level to access at the next step.
          'previous_level': tfd.Deterministic(previous_state['level'])})
process = tfd.MarkovChain(
    # For simplicity, define the prior as a 'transition' from fixed values.
    initial_state_prior=transition_fn(
        0, previous_state={
            'coefs': [0.7, -0.2],
            'log_scale': -1.,
            'level': 0.,
            'previous_level': 0.}),
    transition_fn=transition_fn,
    num_steps=100)
# ==> `process.event_shape == {'coefs': [100, 2], 'log_scale': [100],
#                              'level': [100], 'previous_level': [100]}`
# ==> `process.batch_shape == []`

x = process.sample(5)
# ==> `x['coefs'].shape == [5, 100, 2]`
# ==> `x['log_scale'].shape == [5, 100]`
# ==> `x['level'].shape == [5, 100]`
# ==> `x['previous_level'].shape == [5, 100]`
lp = process.log_prob(x)  # ==> `lp.shape == [5]`.

initial_state_prior tfd.Distribution instance describing a prior distribution on the state at step 0. This may be a joint distribution.
transition_fn Python callable with signature current_state_dist = transition_fn(previous_step, previous_state). The arguments are an integer previous_step, and previous_state, a (structure of) Tensor(s) like a sample from the initial_state_prior. The returned current_state_dist must have the same dtype, batch_shape, and event_shape as initial_state_prior.
num_steps Integer Tensor scalar number of steps in the chain.
experimental_use_kahan_sum If True, use Kahan summation to mitigate accumulation of floating-point error in log_prob calculation.
validate_args Python bool, default False. Whether to validate input with asserts. If validate_args is False, and the inputs are invalid, correct behavior is not guaranteed.
name The name to give ops created by this distribution.

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.

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

name Name prepended to all ops created by this Distribution.
num_steps

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_fn

validate_args Python bool indicating possibly expensive checks are enabled.
variables

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_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 Tensors, 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.

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.

mean

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Mean.

mode

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Mode.

param_shapes

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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

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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.

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.ParameterPropertiesdict mapping constructor argument names toParameterProperties` instances.

Raises
NotImplementedError if the distribution class does not implement _parameter_properties.

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

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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.

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

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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

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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__

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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.linalg.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__

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