tf.contrib.distributions.MultivariateNormalDiag

class tf.contrib.distributions.MultivariateNormalDiag

See the guide: Statistical Distributions (contrib) > Multivariate distributions

The multivariate normal distribution on R^k.

This distribution is defined by a 1-D mean mu and a 1-D diagonal diag_stdev, representing the standard deviations. This distribution assumes the random variables, (X_1,...,X_k) are independent, thus no non-diagonal terms of the covariance matrix are needed.

This allows for O(k) pdf evaluation, sampling, and storage.

Mathematical details

The PDF of this distribution is defined in terms of the diagonal covariance determined by diag_stdev: C_{ii} = diag_stdev[i]**2.

f(x) = (2 pi)^(-k/2) |det(C)|^(-1/2) exp(-1/2 (x - mu)^T C^{-1} (x - mu))

Examples

A single multi-variate Gaussian distribution is defined by a vector of means of length k, and the square roots of the (independent) random variables.

Extra leading dimensions, if provided, allow for batches.

# Initialize a single 3-variate Gaussian with diagonal standard deviation.
mu = [1, 2, 3.]
diag_stdev = [4, 5, 6.]
dist = tf.contrib.distributions.MultivariateNormalDiag(mu, diag_stdev)

# Evaluate this on an observation in R^3, returning a scalar.
dist.pdf([-1, 0, 1])

# Initialize a batch of two 3-variate Gaussians.
mu = [[1, 2, 3], [11, 22, 33]]  # shape 2 x 3
diag_stdev = ...  # shape 2 x 3, positive.
dist = tf.contrib.distributions.MultivariateNormalDiag(mu, diag_stdev)

# Evaluate this on a two observations, each in R^3, returning a length two
# tensor.
x = [[-1, 0, 1], [-11, 0, 11]]  # Shape 2 x 3.
dist.pdf(x)

Properties

allow_nan_stats

Python boolean 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 boolean.

dtype

The DType of Tensors handled by this Distribution.

is_continuous

is_reparameterized

mu

name

Name prepended to all ops created by this Distribution.

parameters

Dictionary of parameters used to instantiate this Distribution.

sigma

Dense (batch) covariance matrix, if available.

validate_args

Python boolean indicated possibly expensive checks are enabled.

Methods

__init__(mu, diag_stdev, validate_args=False, allow_nan_stats=True, name='MultivariateNormalDiag')

Multivariate Normal distributions on R^k.

User must provide means mu and standard deviations diag_stdev. Each batch member represents a random vector (X_1,...,X_k) of independent random normals. The mean of X_i is mu[i], and the standard deviation is diag_stdev[i].

Args:

  • mu: Rank N + 1 floating point tensor with shape [N1,...,Nb, k], b >= 0.
  • diag_stdev: Rank N + 1 Tensor with same dtype and shape as mu, representing the standard deviations. Must be positive.
  • validate_args: Boolean, default False. Whether to validate input with asserts. If validate_args is False, and the inputs are invalid, correct behavior is not guaranteed.
  • allow_nan_stats: Boolean, default True. If False, raise an exception if a statistic (e.g. mean/mode/etc...) is undefined for any batch member If True, batch members with valid parameters leading to undefined statistics will return NaN for this statistic.
  • name: The name to give Ops created by the initializer.

Raises:

  • TypeError: If mu and diag_stdev are different dtypes.

batch_shape(name='batch_shape')

Shape of a single sample from a single event index as a 1-D Tensor.

The product of the dimensions of the batch_shape is the number of independent distributions of this kind the instance represents.

Args:

  • name: name to give to the op

Returns:

  • batch_shape: Tensor.

cdf(value, name='cdf', **condition_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: The name to give this op. **condition_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(**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) intitialized from the union of self.parameters and override_parameters_kwargs, i.e., dict(self.parameters, **override_parameters_kwargs).

entropy(name='entropy')

Shannon entropy in nats.

event_shape(name='event_shape')

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.

get_batch_shape()

Shape of a single sample from a single event index as a TensorShape.

Same meaning as batch_shape. May be only partially defined.

Returns:

  • batch_shape: TensorShape, possibly unknown.

get_event_shape()

Shape of a single sample from a single batch as a TensorShape.

Same meaning as event_shape. May be only partially defined.

Returns:

  • event_shape: TensorShape, possibly unknown.

is_scalar_batch(name='is_scalar_batch')

Indicates that batch_shape == [].

Args:

  • name: The name to give this op.

Returns:

  • is_scalar_batch: Boolean scalar Tensor.

is_scalar_event(name='is_scalar_event')

Indicates that event_shape == [].

Args:

  • name: The name to give this op.

Returns:

  • is_scalar_event: Boolean scalar Tensor.

log_cdf(value, name='log_cdf', **condition_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: The name to give this op. **condition_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_pdf(value, name='log_pdf', **condition_kwargs)

Log probability density function.

Args:

  • value: float or double Tensor.
  • name: The name to give this op. **condition_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.

Raises:

  • TypeError: if not is_continuous.

log_pmf(value, name='log_pmf', **condition_kwargs)

Log probability mass function.

Args:

  • value: float or double Tensor.
  • name: The name to give this op. **condition_kwargs: Named arguments forwarded to subclass implementation.

Returns:

  • log_pmf: a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

Raises:

  • TypeError: if is_continuous.

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

Log probability density/mass function (depending on is_continuous).

Additional documentation from _MultivariateNormalOperatorPD:

x is a batch vector with compatible shape if x is a Tensor whose shape can be broadcast up to either:

self.batch_shape + self.event_shape

or

[M1,...,Mm] + self.batch_shape + self.event_shape

Args:

  • value: float or double Tensor.
  • name: The name to give this op. **condition_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_sigma_det(name='log_sigma_det')

Log of determinant of covariance matrix.

log_survival_function(value, name='log_survival_function', **condition_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: The name to give this op. **condition_kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

mean(name='mean')

Mean.

mode(name='mode')

Mode.

param_shapes(cls, sample_shape, name='DistributionParamShapes')

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

Subclasses should override static 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(cls, sample_shape)

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

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.

pdf(value, name='pdf', **condition_kwargs)

Probability density function.

Args:

  • value: float or double Tensor.
  • name: The name to give this op. **condition_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.

Raises:

  • TypeError: if not is_continuous.

pmf(value, name='pmf', **condition_kwargs)

Probability mass function.

Args:

  • value: float or double Tensor.
  • name: The name to give this op. **condition_kwargs: Named arguments forwarded to subclass implementation.

Returns:

  • pmf: a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

Raises:

  • TypeError: if is_continuous.

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

Probability density/mass function (depending on is_continuous).

Additional documentation from _MultivariateNormalOperatorPD:

x is a batch vector with compatible shape if x is a Tensor whose shape can be broadcast up to either:

self.batch_shape + self.event_shape

or

[M1,...,Mm] + self.batch_shape + self.event_shape

Args:

  • value: float or double Tensor.
  • name: The name to give this op. **condition_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.

sample(sample_shape=(), seed=None, name='sample', **condition_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. **condition_kwargs: Named arguments forwarded to subclass implementation.

Returns:

  • samples: a Tensor with prepended dimensions sample_shape.

sigma_det(name='sigma_det')

Determinant of covariance matrix.

std(name='std')

Standard deviation.

survival_function(value, name='survival_function', **condition_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: The name to give this op. **condition_kwargs: Named arguments forwarded to subclass implementation.

Returns:

Tensorof shapesample_shape(x) + self.batch_shapewith values of typeself.dtype`.

variance(name='variance')

Variance.

Defined in tensorflow/contrib/distributions/python/ops/mvn.py.