tfp.substrates.jax.distributions.MarkovChain

View source on GitHub

Distribution of a sequence generated by a memoryless process.

Inherits From: Distribution

tfp.substrates.jax.distributions.MarkovChain(
    initial_state_prior,
    transition_fn,
    num_steps,
    experimental_use_kahan_sum=False,
    validate_args=False,
    name='MarkovChain'
)

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.

Related distributions

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]`.

Methods

batch_shape_tensor

View source

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.

cdf

View source

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

Cumulative distribution function.

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

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

copy

View source

copy(
    **override_parameters_kwargs
)

Creates a deep copy of the distribution.

covariance

View source

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.

cross_entropy

View source

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.

entropy

View source

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

Shannon entropy in nats.

event_shape_tensor

View source

event_shape_tensor(
    name='event_shape_tensor'
)

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

experimental_default_event_space_bijector

View source

experimental_default_event_space_bijector(
    *args, **kwargs
)

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.

experimental_fit

View source

@classmethod
experimental_fit(
    value, sample_ndims=1, validate_args=False, **init_kwargs
)

Instantiates a distribution that maximizes the likelihood of x.

experimental_local_measure

View source

experimental_local_measure(
    value, backward_compat=False, **kwargs
)

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.

experimental_sample_and_log_prob

View source

experimental_sample_and_log_prob(
    sample_shape=(), seed=None, name='sample_and_log_prob', **kwargs
)

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.

is_scalar_batch

View source

is_scalar_batch(
    name='is_scalar_batch'
)

Indicates that batch_shape == [].

is_scalar_event

View source

is_scalar_event(
    name='is_scalar_event'
)

Indicates that event_shape == [].

kl_divergence

View source

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.

log_cdf

View source

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.

log_prob

View source

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

Log probability density/mass function.

log_survival_function

View source

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.

mean

View source

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

Mean.

mode

View source

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

Mode.

param_shapes

View source

@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.

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.

parameter_properties

View source

@classmethod
parameter_properties(
    dtype=tf.float32, num_classes=None
)

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.

prob

View source

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

Probability density/mass function.

quantile

View source

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

sample

View source

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.

stddev

View source

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.

survival_function

View source

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).

unnormalized_log_prob

View source

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

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.

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.

__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.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]

__iter__

View source

__iter__()