# tf.contrib.distributions.MultivariateNormalDiagPlusVDVT

### class tf.contrib.distributions.MultivariateNormalDiagPlusVDVT

The multivariate normal distribution on R^k.

Every batch member of this distribution is defined by a mean and a lightweight covariance matrix C.

#### Mathematical details

The PDF of this distribution in terms of the mean mu and covariance C is:

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


For every batch member, this distribution represents k random variables (X_1,...,X_k), with mean E[X_i] = mu[i], and covariance matrix C_{ij} := E[(X_i - mu[i])(X_j - mu[j])]

The user initializes this class by providing the mean mu, and a lightweight definition of C:

C = SS^T = SS = (M + V D V^T) (M + V D V^T)
M is diagonal (k x k)
V = is shape (k x r), typically r << k
D = is diagonal (r x r), optional (defaults to identity).


This allows for O(kr + r^3) pdf evaluation and determinant, and O(kr) sampling and storage (per batch member).

#### Examples

A single multi-variate Gaussian distribution is defined by a vector of means of length k, and square root of the covariance S = M + V D V^T. Extra leading dimensions, if provided, allow for batches.

# Initialize a single 3-variate Gaussian with covariance square root
# S = M + V D V^T, where V D V^T is a matrix-rank 2 update.
mu = [1, 2, 3.]
diag_large = [1.1, 2.2, 3.3]
v = ... # shape 3 x 2
diag_small = [4., 5.]
dist = tf.contrib.distributions.MultivariateNormalDiagPlusVDVT(
mu, diag_large, v, diag_small=diag_small)

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

# Initialize a batch of two 3-variate Gaussians.  This time, don't provide
# diag_small.  This means S = M + V V^T.
mu = [[1, 2, 3], [11, 22, 33]]  # shape 2 x 3
diag_large = ... # shape 2 x 3
v = ... # shape 2 x 3 x 1, a matrix-rank 1 update.
dist = tf.contrib.distributions.MultivariateNormalDiagPlusVDVT(
mu, diag_large, v)

# 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.

### 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_large, v, diag_small=None, validate_args=False, allow_nan_stats=True, name='MultivariateNormalDiagPlusVDVT')

Multivariate Normal distributions on R^k.

For every batch member, this distribution represents k random variables (X_1,...,X_k), with mean E[X_i] = mu[i], and covariance matrix C_{ij} := E[(X_i - mu[i])(X_j - mu[j])]

The user initializes this class by providing the mean mu, and a lightweight definition of C:

C = SS^T = SS = (M + V D V^T) (M + V D V^T)
M is diagonal (k x k)
V = is shape (k x r), typically r << k
D = is diagonal (r x r), optional (defaults to identity).


#### Args:

• mu: Rank n + 1 floating point tensor with shape [N1,...,Nn, k], n >= 0. The means.
• diag_large: Optional rank n + 1 floating point tensor, shape [N1,...,Nn, k] n >= 0. Defines the diagonal matrix M.
• v: Rank n + 1 floating point tensor, shape [N1,...,Nn, k, r] n >= 0. Defines the matrix V.
• diag_small: Rank n + 1 floating point tensor, shape [N1,...,Nn, k] n >= 0. Defines the diagonal matrix D. Default is None, which means D will be the identity matrix.
• 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.

### 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.

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.