# tf.contrib.distributions.VectorDiffeomixture

## Class VectorDiffeomixture

Inherits From: Distribution

VectorDiffeomixture distribution.

A vector diffeomixture (VDM) is a distribution parameterized by a convex combination of K component loc vectors, loc[k], k = 0,...,K-1, and K scale matrices scale[k], k = 0,..., K-1. It approximates the following compound distribution

p(x) = int p(x | z) p(z) dz,
where z is in the K-simplex, and
p(x | z) := p(x | loc=sum_k z[k] loc[k], scale=sum_k z[k] scale[k])


The integral int p(x | z) p(z) dz is approximated with a quadrature scheme adapted to the mixture density p(z). The N quadrature points z_{N, n} and weights w_{N, n} (which are non-negative and sum to 1) are chosen such that

q_N(x) := sum_{n=1}^N w_{n, N} p(x | z_{N, n}) --> p(x)

as N --> infinity.

Since q_N(x) is in fact a mixture (of N points), we may sample from q_N exactly. It is important to note that the VDM is defined as q_N above, and not p(x). Therefore, sampling and pdf may be implemented as exact (up to floating point error) methods.

A common choice for the conditional p(x | z) is a multivariate Normal.

The implemented marginal p(z) is the SoftmaxNormal, which is a K-1 dimensional Normal transformed by a SoftmaxCentered bijector, making it a density on the K-simplex. That is,

Z = SoftmaxCentered(X),
X = Normal(mix_loc / temperature, 1 / temperature)


The default quadrature scheme chooses z_{N, n} as N midpoints of the quantiles of p(z) (generalized quantiles if K > 2).

See [Dillon and Langmore (2018)][1] for more details.

#### About Vector distributions in TensorFlow.

The VectorDiffeomixture is a non-standard distribution that has properties particularly useful in variational Bayesian methods.

Conditioned on a draw from the SoftmaxNormal, X|z is a vector whose components are linear combinations of affine transformations, thus is itself an affine transformation.

#### About Diffeomixtures and reparameterization.

The VectorDiffeomixture is designed to be reparameterized, i.e., its parameters are only used to transform samples from a distribution which has no trainable parameters. This property is important because backprop stops at sources of stochasticity. That is, as long as the parameters are used after the underlying source of stochasticity, the computed gradient is accurate.

Reparametrization means that we can use gradient-descent (via backprop) to optimize Monte-Carlo objectives. Such objectives are a finite-sample approximation of an expectation and arise throughout scientific computing.

WARNING: If you backprop through a VectorDiffeomixture sample and the "base" distribution is both: not FULLY_REPARAMETERIZED and a function of trainable variables, then the gradient is not guaranteed correct!

#### Examples

tfd = tf.contrib.distributions

# Create two batches of VectorDiffeomixtures, one with mix_loc=[0.],
# another with mix_loc=[1]. In both cases, K=2 and the affine
# transformations involve:
# k=0: loc=zeros(dims)  scale=LinearOperatorScaledIdentity
# k=1: loc=[2.]*dims    scale=LinOpDiag
dims = 5
vdm = tfd.VectorDiffeomixture(
mix_loc=[[0.], [1]],
temperature=[1.],
distribution=tfd.Normal(loc=0., scale=1.),
loc=[
None,  # Equivalent to np.zeros(dims, dtype=np.float32).
np.float32([2.]*dims),
],
scale=[
tf.linalg.LinearOperatorScaledIdentity(
num_rows=dims,
multiplier=np.float32(1.1),
is_positive_definite=True),
tf.linalg.LinearOperatorDiag(
diag=np.linspace(2.5, 3.5, dims, dtype=np.float32),
is_positive_definite=True),
],
validate_args=True)


#### References

[1]: Joshua Dillon and Ian Langmore. Quadrature Compound: An approximating family of distributions. arXiv preprint arXiv:1801.03080, 2018. https://arxiv.org/abs/1801.03080

## 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.

### distribution

Base scalar-event, scalar-batch distribution.

### dtype

The DType of Tensors handled by this Distribution.

### endpoint_affine

Affine transformation for each of K components.

### 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.

### grid

Grid of mixing probabilities, one for each grid point.

### interpolated_affine

Affine transformation for each convex combination of K components.

### mixture_distribution

Distribution used to select a convex combination of affine transforms.

### name

Name prepended to all ops created by this Distribution.

### 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.

#### Returns:

An instance of ReparameterizationType.

### validate_args

Python bool indicating possibly expensive checks are enabled.

## Methods

### __init__

__init__(
mix_loc,
temperature,
distribution,
loc=None,
scale=None,
validate_args=False,
allow_nan_stats=True,
name='VectorDiffeomixture'
)


Constructs the VectorDiffeomixture on R^d.

The vector diffeomixture (VDM) approximates the compound distribution

p(x) = int p(x | z) p(z) dz,
where z is in the K-simplex, and
p(x | z) := p(x | loc=sum_k z[k] loc[k], scale=sum_k z[k] scale[k])


#### Args:

• mix_loc: float-like Tensor with shape [b1, ..., bB, K-1]. In terms of samples, larger mix_loc[..., k] ==> Z is more likely to put more weight on its kth component.
• temperature: float-like Tensor. Broadcastable with mix_loc. In terms of samples, smaller temperature means one component is more likely to dominate. I.e., smaller temperature makes the VDM look more like a standard mixture of K components.
• distribution: tf.Distribution-like instance. Distribution from which d iid samples are used as input to the selected affine transformation. Must be a scalar-batch, scalar-event distribution. Typically distribution.reparameterization_type = FULLY_REPARAMETERIZED or it is a function of non-trainable parameters. WARNING: If you backprop through a VectorDiffeomixture sample and the distribution is not FULLY_REPARAMETERIZED yet is a function of trainable variables, then the gradient will be incorrect!
• loc: Length-K list of float-type Tensors. The k-th element represents the shift used for the k-th affine transformation. If the k-th item is None, loc is implicitly 0. When specified, must have shape [B1, ..., Bb, d] where b >= 0 and d is the event size.
• scale: Length-K list of LinearOperators. Each should be positive-definite and operate on a d-dimensional vector space. The k-th element represents the scale used for the k-th affine transformation. LinearOperators must have shape [B1, ..., Bb, d, d], b >= 0, i.e., characterizes b-batches of d x d matrices
• quadrature_size: Python int scalar representing number of quadrature points. Larger quadrature_size means q_N(x) better approximates p(x).
• quadrature_fn: Python callable taking normal_loc, normal_scale, quadrature_size, validate_args and returning tuple(grid, probs) representing the SoftmaxNormal grid and corresponding normalized weight. normalized) weight. Default value: quadrature_scheme_softmaxnormal_quantiles.
• validate_args: Python bool, default False. When True distribution parameters are checked for validity despite possibly degrading runtime performance. When False invalid inputs may silently render incorrect outputs.
• 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.

#### Raises:

• ValueError: if not scale or len(scale) < 2.
• ValueError: if len(loc) != len(scale)
• ValueError: if quadrature_grid_and_probs is not None and len(quadrature_grid_and_probs[0]) != len(quadrature_grid_and_probs[1])
• ValueError: if validate_args and any not scale.is_positive_definite.
• TypeError: if any scale.dtype != scale[0].dtype.
• TypeError: if any loc.dtype != scale[0].dtype.
• NotImplementedError: if len(scale) != 2.
• ValueError: if not distribution.is_scalar_batch.
• ValueError: if not distribution.is_scalar_event.

### batch_shape_tensor

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

cdf(
value,
name='cdf'
)


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.

#### Returns:

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

### copy

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

covariance(name='covariance')


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.

#### 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

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), (Shanon) 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.

#### Returns:

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

### entropy

entropy(name='entropy')


Shannon entropy in nats.

### event_shape_tensor

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

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

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

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 (Shanon) cross entropy, and H[.] denotes (Shanon) entropy.

#### Returns:

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

### log_cdf

log_cdf(
value,
name='log_cdf'
)


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.

#### Returns:

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

### log_prob

log_prob(
value,
name='log_prob'
)


Log probability density/mass function.

#### Args:

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

#### Returns:

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

### log_survival_function

log_survival_function(
value,
name='log_survival_function'
)


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.

#### Returns:

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

### mean

mean(name='mean')


Mean.

### mode

mode(name='mode')


Mode.

### param_shapes

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

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

prob(
value,
name='prob'
)


Probability density/mass function.

#### Args:

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

#### Returns:

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

### quantile

quantile(
value,
name='quantile'
)


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.

#### Returns:

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

### sample

sample(
sample_shape=(),
seed=None,
name='sample'
)


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.

#### Returns:

• samples: a Tensor with prepended dimensions sample_shape.

### stddev

stddev(name='stddev')


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.

#### Returns:

• stddev: Floating-point Tensor with shape identical to batch_shape + event_shape, i.e., the same shape as self.mean().

### survival_function

survival_function(
value,
name='survival_function'
)


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.

#### Returns:

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

### variance

variance(name='variance')


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.

#### Returns:

• variance: Floating-point Tensor with shape identical to batch_shape + event_shape, i.e., the same shape as self.mean().