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tfp.experimental.substrates.jax.distributions.Autoregressive

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Class Autoregressive

Autoregressive distributions.

Inherits From: Distribution

The Autoregressive distribution enables learning (often) richer multivariate distributions by repeatedly applying a diffeomorphic transformation (such as implemented by Bijectors). Regarding terminology,

"Autoregressive models decompose the joint density as a product of conditionals, and model each conditional in turn. Normalizing flows transform a base density (e.g. a standard Gaussian) into the target density by an invertible transformation with tractable Jacobian." [(Papamakarios et al., 2016)][1]

In other words, the "autoregressive property" is equivalent to the decomposition, p(x) = prod{ p(x[i] | x[0:i]) : i=0, ..., d }. The provided shift_and_log_scale_fn, masked_autoregressive_default_template, achieves this property by zeroing out weights in its masked_dense layers.

Practically speaking the autoregressive property means that there exists a permutation of the event coordinates such that each coordinate is a diffeomorphic function of only preceding coordinates [(van den Oord et al., 2016)][2].

Mathematical Details

The probability function is

prob(x; fn, n) = fn(x).prob(x)

And a sample is generated by

x = fn(...fn(fn(x0).sample()).sample()).sample()

where the ellipses (...) represent n-2 composed calls to fn, fn constructs a tfd.Distribution-like instance, and x0 is a fixed initializing Tensor.

Examples

tfd = tfp.distributions
tfb = tfp.bijectors

def _normal_fn(event_size):
  n = event_size * (event_size + 1) // 2
  p = tf.Variable(tfd.Normal(loc=0., scale=1.).sample(n))
  affine = tfb.Affine(
      scale_tril=tfp.math.fill_triangular(0.25 * p))
  def _fn(samples):
    scale = tf.exp(affine.forward(samples))
    return tfd.Independent(
        tfd.Normal(loc=0., scale=scale, validate_args=True),
        reinterpreted_batch_ndims=1)
  return _fn

batch_and_event_shape = [3, 2, 4]
sample0 = tf.zeros(batch_and_event_shape)
ar = tfd.Autoregressive(
    _normal_fn(batch_and_event_shape[-1]), sample0)
x = ar.sample([6, 5])
# ==> x.shape = [6, 5, 3, 2, 4]
prob_x = ar.prob(x)
# ==> x.shape = [6, 5, 3, 2]

References

[1]: George Papamakarios, Theo Pavlakou, and Iain Murray. Masked Autoregressive Flow for Density Estimation. In Neural Information Processing Systems, 2017. https://arxiv.org/abs/1705.07057

[2]: Aaron van den Oord, Nal Kalchbrenner, Oriol Vinyals, Lasse Espeholt, Alex Graves, and Koray Kavukcuoglu. Conditional Image Generation with PixelCNN Decoders. In Neural Information Processing Systems, 2016. https://arxiv.org/abs/1606.05328

__init__

__init__(
    distribution_fn,
    sample0=None,
    num_steps=None,
    validate_args=False,
    allow_nan_stats=True,
    name='Autoregressive'
)

Construct an Autoregressive distribution.

Args:

  • distribution_fn: Python callable which constructs a tfd.Distribution-like instance from a Tensor (e.g., sample0). The function must respect the "autoregressive property", i.e., there exists a permutation of event such that each coordinate is a diffeomorphic function of on preceding coordinates.
  • sample0: Initial input to distribution_fn; used to build the distribution in __init__ which in turn specifies this distribution's properties, e.g., event_shape, batch_shape, dtype. If unspecified, then distribution_fn should be default constructable.
  • num_steps: Number of times distribution_fn is composed from samples, e.g., num_steps=2 implies distribution_fn(distribution_fn(sample0).sample(n)).sample().
  • validate_args: Python bool. Whether to validate input with asserts. If validate_args is False, and the inputs are invalid, correct behavior is not guaranteed.
  • 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.
  • name: Python str name prefixed to Ops created by this class. Default value: "Autoregressive".

Raises:

  • ValueError: if num_steps and num_elements(distribution_fn(sample0).event_shape) are both None.
  • ValueError: if num_steps < 1.

Properties

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.

Returns:

  • allow_nan_stats: Python bool.

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.

Returns:

  • batch_shape: TensorShape, possibly unknown.

distribution0

distribution_fn

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.

Returns:

  • event_shape: TensorShape, possibly unknown.

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.

Returns:

An instance of ReparameterizationType.

sample0

trainable_variables

validate_args

Python bool indicating possibly expensive checks are enabled.

variables

Methods

__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.normal([5, 3, 2, 2])
cov = tf.matmul(x, x, transpose_b=True)
chol = tf.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.

__iter__

View source

__iter__()

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.

Args:

  • name: name to give to the op

Returns:

  • batch_shape: Tensor.

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]

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

View source

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

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.

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

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.

Args:

Returns:

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

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.

Args:

  • name: name to give to the op

Returns:

  • event_shape: Tensor.

is_scalar_batch

View source

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

View source

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

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.

Args:

Returns:

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

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.

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

View source

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

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.

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

View source

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

Mean.

mode

View source

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

Mode.

param_shapes

View source

param_shapes(
    cls,
    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

param_static_shapes(
    cls,
    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.

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

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

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

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.

Args:

  • sample_shape: 0D or 1D int32 Tensor. Shape of the generated samples.
  • seed: Python integer seed for RNG
  • 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

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

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).

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().