tf.contrib.distributions.Autoregressive

Autoregressive distributions.

Inherits From: Distribution

tf.contrib.distributions.Autoregressive(
    distribution_fn, sample0=None, num_steps=None, validate_args=False,
    allow_nan_stats=True, name='Autoregressive'
)

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 tfp.distributions.Distribution-like instance, and x0 is a fixed initializing Tensor.

Examples

import tensorflow_probability as tfp
tfd = tfp.distributions

def normal_fn(self, event_size):
  n = event_size * (event_size + 1) / 2
  p = tf.Variable(tfd.Normal(loc=0., scale=1.).sample(n))
  affine = tfd.bijectors.Affine(
      scale_tril=tfd.fill_triangular(0.25 * p))
  def _fn(samples):
    scale = math_ops.exp(affine.forward(samples)).eval()
    return independent_lib.Independent(
        normal_lib.Normal(loc=0., scale=scale, validate_args=True),
        reinterpreted_batch_ndims=1)
  return _fn

batch_and_event_shape = [3, 2, 4]
sample0 = array_ops.zeros(batch_and_event_shape)
ar = autoregressive_lib.Autoregressive(
    self._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

Args
`distribution_fn`	Python `callable` which constructs a `tfp.distributions.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 `distribution_fn(sample0).event_shape.num_elements()` are both `None`.
`ValueError`	if `num_steps < 1`.

Attributes
`allow_nan_stats`	Python `bool` describing behavior when a stat is undefined. Stats return +/- infinity when it makes sense. E.g., the variance of a Cauchy distribution is infinity. However, sometimes the statistic is undefined, e.g., if a distribution's pdf does not achieve a maximum within the support of the distribution, the mode is undefined. If the mean is undefined, then by definition the variance is undefined. E.g. the mean for Student's T for df = 1 is undefined (no clear way to say it is either + or - infinity), so the variance = E[(X - mean)**2] is also undefined.
`batch_shape`	Shape of a single sample from a single event index as a `TensorShape`. May be partially defined or unknown. The batch dimensions are indexes into independent, non-identical parameterizations of this distribution.
`distribution0`
`distribution_fn`
`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.
`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 `distributions.FULLY_REPARAMETERIZED` or `distributions.NOT_REPARAMETERIZED`.
`sample0`
`validate_args`	Python `bool` indicating possibly expensive checks are enabled.

Args
`value`	`float` or `double` `Tensor`.
`name`	Python `str` prepended to names of ops created by this function.

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

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

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.

tf.contrib.distributions.Autoregressive

Mathematical Details

Examples

References

Methods

batch_shape_tensor

cdf

copy

covariance

cross_entropy

entropy

event_shape_tensor

is_scalar_batch

is_scalar_event

kl_divergence

log_cdf

log_prob

log_survival_function

mean

mode

param_shapes

param_static_shapes

prob

quantile

sample

stddev

survival_function

variance

`batch_shape_tensor`

`cdf`

`copy`

`covariance`

`cross_entropy`

`entropy`

`event_shape_tensor`

`is_scalar_batch`

`is_scalar_event`

`kl_divergence`

`log_cdf`

`log_prob`

`log_survival_function`

`mean`

`mode`

`param_shapes`

`param_static_shapes`

`prob`

`quantile`

`sample`

`stddev`

`survival_function`

`variance`