# ed.models.VectorDiffeomixture

## Class VectorDiffeomixture

Inherits From: RandomVariable

VectorDiffeomixture distribution.

The VectorDiffeomixture is an approximation to a compound distribution, i.e.,

p(x) = int_{X} q(x | v) p(v) dv
= lim_{Q->infty} sum{ prob[i] q(x | loc=sum_k^K lambda[k;i] loc[k],
scale=sum_k^K lambda[k;i] scale[k])
: i=0, ..., Q-1 }

where q(x | v) is a vector version of the distribution argument and p(v) is a SoftmaxNormal parameterized by mix_loc and mix_scale. The vector-ization of distribution entails an affine transformation of iid samples from distribution. The prob term is from quadrature and lambda[k] = sigmoid(mix_loc[k] + sqrt(2) mix_scale[k] grid[k]) where the grid points correspond to the probs.

In the non-approximation case, a draw from the mixture distribution (the “prior”) represents the convex weights for different affine transformations. I.e., draw a mixing vector v (from the K-1-simplex) and let the final sample be: y = (sum_k^K v[k] scale[k]) @ x + (sum_k^K v[k] loc[k]) where @ denotes matrix multiplication. However, the non-approximate distribution does not have an analytical probability density function (pdf). Therefore the VectorDiffeomixture class implements an approximation based on numerical quadrature (default: Gauss–Hermite quadrature). I.e., in Note: although the VectorDiffeomixture is approximately the SoftmaxNormal-Distribution compound distribution, it is itself a valid distribution. It possesses a sample, log_prob, mean, covariance which are all mutually consistent.

#### Intended Use

This distribution is noteworthy because it implements a mixture of Vector-ized distributions yet has samples differentiable in the distribution’s parameters (aka “reparameterized”). It has an analytical density function with O(dKQ) complexity. d is the vector dimensionality, K is the number of components, and Q is the number of quadrature points. These properties make it well-suited for Bayesian Variational Inference, i.e., as a surrogate family for the posterior.

For large values of mix_scale, the VectorDistribution behaves increasingly like a discrete mixture. (In most cases this limit is only achievable by also increasing the quadrature polynomial degree, Q.)

The term Vector is consistent with similar named Tensorflow Distributions. For more details, see the “About Vector distributions in Tensorflow.” section.

The term Diffeomixture is a portmanteau of diffeomorphism and compound mixture. For more details, see the “About Diffeomixtures and reparametrization.” section.

#### Mathematical Details

The VectorDiffeomixture approximates a SoftmaxNormal-mixed (“prior”) compound distribution. Using variable-substitution and numerical quadrature (default: Gauss–Hermite quadrature) we can redefine the distribution to be a parameter-less convex combination of K different affine combinations of a d iid samples from distribution.

That is, defined over R**d this distribution is parameterized by a (batch of) length-K mix_loc and mix_scale vectors, a length-K list of (a batch of) length-d loc vectors, and a length-K list of scale LinearOperators each operating on a (batch of) length-d vector space. Finally, a distribution parameter specifies the underlying base distribution which is “lifted” to become multivariate (“lifting” is the same concept as in TransformedDistribution).

The probability density function (pdf) is,

pdf(y; mix_loc, mix_scale, loc, scale, phi)
= sum{ prob[i] phi(f_inverse(x; i)) / abs(det(interp_scale[i]))
: i=0, ..., Q-1 }

where, phi is the base distribution pdf, and,

f_inverse(x; i) = inv(interp_scale[i]) @ (x - interp_loc[i])
interp_loc[i]   = sum{ lambda[k; i] loc[k]   : k=0, ..., K-1 }
interp_scale[i] = sum{ lambda[k; i] scale[k] : k=0, ..., K-1 }

and,

grid, weight = np.polynomial.hermite.hermgauss(quadrature_size)
prob[k]   = weight[k] / sqrt(pi)
lambda[k; i] = sigmoid(mix_loc[k] + sqrt(2) mix_scale[k] grid[i])

The distribution corresponding to phi must be a scalar-batch, scalar-event distribution. Typically it is reparameterized. If not, it must be a function of non-trainable parameters.

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!

#### 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, Y|v is a vector whose components are linear combinations of affine transformations, thus is itself an affine transformation. Therefore Y|v lives in the vector space generated by vectors of affine-transformed distributions.

Note: The marginals Y_1|v, ..., Y_d|v are not generally identical to some parameterization of distribution. This is due to the fact that the sum of draws from distribution are not generally itself the same distribution.

#### 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.

#### Examples

python tfd = tf.contrib.distributions

# k=1: loc=[2.]dims scale=LinOpDiag dims = 5 vdm = tfd.VectorDiffeomixture( mix_loc=[[0.], [1]], mix_scale=[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)

## 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.

### sample_shape

Sample shape of random variable.

### shape

Shape of random variable.

### validate_args

Python bool indicating possibly expensive checks are enabled.

## Methods

### init

__init__(
*args,
**kwargs
)

Constructs the VectorDiffeomixture on R**d.

#### Args:

• mix_loc: float-like Tensor with shape [b1, ..., bB, K-1]. Represents the location parameter of the SoftmaxNormal used for selecting one of the K affine transformations.
• mix_scale: float-like Tensor with shape [b1, ..., bB, K-1]. Represents the scale parameter of the SoftmaxNormal used for selecting one of the K affine transformations.
• 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.
• quadrature_fn: Python callable taking mix_loc, mix_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.

### abs

__abs__(
a,
*args
)

Computes the absolute value of a tensor.

Given a tensor x of complex numbers, this operation returns a tensor of type float32 or float64 that is the absolute value of each element in x. All elements in x must be complex numbers of the form $$a + bj$$. The absolute value is computed as . For example:

x = tf.constant([[-2.25 + 4.75j], [-3.25 + 5.75j]])
tf.abs(x)  # [5.25594902, 6.60492229]

#### Args:

• x: A Tensor or SparseTensor of type float32, float64, int32, int64, complex64 or complex128.
• name: A name for the operation (optional).

#### Returns:

A Tensor or SparseTensor the same size and type as x with absolute values. Note, for complex64 or complex128 input, the returned Tensor will be of type float32 or float64, respectively.

### add

__add__(
a,
*args
)

Returns x + y element-wise.

NOTE: Add supports broadcasting. AddN does not. More about broadcasting here

#### Args:

• x: A Tensor. Must be one of the following types: half, bfloat16, float32, float64, uint8, int8, int16, int32, int64, complex64, complex128, string.
• y: A Tensor. Must have the same type as x.
• name: A name for the operation (optional).

#### Returns:

A Tensor. Has the same type as x.

### and

__and__(
a,
*args
)

Returns the truth value of x AND y element-wise.

NOTE: LogicalAnd supports broadcasting. More about broadcasting here

#### Args:

• x: A Tensor of type bool.
• y: A Tensor of type bool.
• name: A name for the operation (optional).

#### Returns:

A Tensor of type bool.

### bool

__bool__()

### div

__div__(
a,
*args
)

Divide two values using Python 2 semantics. Used for Tensor.__div__.

#### Args:

• x: Tensor numerator of real numeric type.
• y: Tensor denominator of real numeric type.
• name: A name for the operation (optional).

#### Returns:

x / y returns the quotient of x and y.

### eq

__eq__(other)

### floordiv

__floordiv__(
a,
*args
)

Divides x / y elementwise, rounding toward the most negative integer.

The same as tf.div(x,y) for integers, but uses tf.floor(tf.div(x,y)) for floating point arguments so that the result is always an integer (though possibly an integer represented as floating point). This op is generated by x // y floor division in Python 3 and in Python 2.7 with from __future__ import division.

Note that for efficiency, floordiv uses C semantics for negative numbers (unlike Python and Numpy).

x and y must have the same type, and the result will have the same type as well.

#### Args:

• x: Tensor numerator of real numeric type.
• y: Tensor denominator of real numeric type.
• name: A name for the operation (optional).

#### Returns:

x / y rounded down (except possibly towards zero for negative integers).

#### Raises:

• TypeError: If the inputs are complex.

### ge

__ge__(
a,
*args
)

Returns the truth value of (x >= y) element-wise.

NOTE: GreaterEqual supports broadcasting. More about broadcasting here

#### Args:

• x: A Tensor. Must be one of the following types: float32, float64, int32, int64, uint8, int16, int8, uint16, half, uint32, uint64, bfloat16.
• y: A Tensor. Must have the same type as x.
• name: A name for the operation (optional).

#### Returns:

A Tensor of type bool.

### getitem

__getitem__(
a,
*args
)

This operation extracts the specified region from the tensor. The notation is similar to NumPy with the restriction that currently only support basic indexing. That means that using a tensor as input is not currently allowed

Some useful examples:

# strip leading and trailing 2 elements
foo = tf.constant([1,2,3,4,5,6])
print(foo[2:-2].eval())  # [3,4]

# skip every row and reverse every column
foo = tf.constant([[1,2,3], [4,5,6], [7,8,9]])
print(foo[::2,::-1].eval())  # [[3,2,1], [9,8,7]]

# Insert another dimension
foo = tf.constant([[1,2,3], [4,5,6], [7,8,9]])
print(foo[tf.newaxis, :, :].eval()) # => [[[1,2,3], [4,5,6], [7,8,9]]]
print(foo[:, tf.newaxis, :].eval()) # => [[[1,2,3]], [[4,5,6]], [[7,8,9]]]
print(foo[:, :, tf.newaxis].eval()) # => [[[1],[2],[3]], [[4],[5],[6]],
[[7],[8],[9]]]

# Ellipses (3 equivalent operations)
foo = tf.constant([[1,2,3], [4,5,6], [7,8,9]])
print(foo[tf.newaxis, :, :].eval())  # [[[1,2,3], [4,5,6], [7,8,9]]]
print(foo[tf.newaxis, ...].eval())  # [[[1,2,3], [4,5,6], [7,8,9]]]
print(foo[tf.newaxis].eval())  # [[[1,2,3], [4,5,6], [7,8,9]]]

Notes: - tf.newaxis is None as in NumPy. - An implicit ellipsis is placed at the end of the slice_spec - NumPy advanced indexing is currently not supported.

#### Args:

• tensor: An ops.Tensor object.
• slice_spec: The arguments to Tensor.__getitem__.
• var: In the case of variable slice assignment, the Variable object to slice (i.e. tensor is the read-only view of this variable).

#### Returns:

The appropriate slice of “tensor”, based on “slice_spec”.

#### Raises:

• ValueError: If a slice range is negative size.
• TypeError: If the slice indices aren’t int, slice, or Ellipsis.

### gt

__gt__(
a,
*args
)

Returns the truth value of (x > y) element-wise.

NOTE: Greater supports broadcasting. More about broadcasting here

#### Args:

• x: A Tensor. Must be one of the following types: float32, float64, int32, int64, uint8, int16, int8, uint16, half, uint32, uint64, bfloat16.
• y: A Tensor. Must have the same type as x.
• name: A name for the operation (optional).

#### Returns:

A Tensor of type bool.

### invert

__invert__(
a,
*args
)

Returns the truth value of NOT x element-wise.

#### Args:

• x: A Tensor of type bool.
• name: A name for the operation (optional).

#### Returns:

A Tensor of type bool.

### iter

__iter__()

### le

__le__(
a,
*args
)

Returns the truth value of (x <= y) element-wise.

NOTE: LessEqual supports broadcasting. More about broadcasting here

#### Args:

• x: A Tensor. Must be one of the following types: float32, float64, int32, int64, uint8, int16, int8, uint16, half, uint32, uint64, bfloat16.
• y: A Tensor. Must have the same type as x.
• name: A name for the operation (optional).

#### Returns:

A Tensor of type bool.

### lt

__lt__(
a,
*args
)

Returns the truth value of (x < y) element-wise.

NOTE: Less supports broadcasting. More about broadcasting here

#### Args:

• x: A Tensor. Must be one of the following types: float32, float64, int32, int64, uint8, int16, int8, uint16, half, uint32, uint64, bfloat16.
• y: A Tensor. Must have the same type as x.
• name: A name for the operation (optional).

#### Returns:

A Tensor of type bool.

### matmul

__matmul__(
a,
*args
)

Multiplies matrix a by matrix b, producing a * b.

The inputs must, following any transpositions, be tensors of rank >= 2 where the inner 2 dimensions specify valid matrix multiplication arguments, and any further outer dimensions match.

Both matrices must be of the same type. The supported types are: float16, float32, float64, int32, complex64, complex128.

Either matrix can be transposed or adjointed (conjugated and transposed) on the fly by setting one of the corresponding flag to True. These are False by default.

If one or both of the matrices contain a lot of zeros, a more efficient multiplication algorithm can be used by setting the corresponding a_is_sparse or b_is_sparse flag to True. These are False by default. This optimization is only available for plain matrices (rank-2 tensors) with datatypes bfloat16 or float32.

For example:

# 2-D tensor a
# [[1, 2, 3],
#  [4, 5, 6]]
a = tf.constant([1, 2, 3, 4, 5, 6], shape=[2, 3])

# 2-D tensor b
# [[ 7,  8],
#  [ 9, 10],
#  [11, 12]]
b = tf.constant([7, 8, 9, 10, 11, 12], shape=[3, 2])

# a * b
# [[ 58,  64],
#  [139, 154]]
c = tf.matmul(a, b)

# 3-D tensor a
# [[[ 1,  2,  3],
#   [ 4,  5,  6]],
#  [[ 7,  8,  9],
#   [10, 11, 12]]]
a = tf.constant(np.arange(1, 13, dtype=np.int32),
shape=[2, 2, 3])

# 3-D tensor b
# [[[13, 14],
#   [15, 16],
#   [17, 18]],
#  [[19, 20],
#   [21, 22],
#   [23, 24]]]
b = tf.constant(np.arange(13, 25, dtype=np.int32),
shape=[2, 3, 2])

# a * b
# [[[ 94, 100],
#   [229, 244]],
#  [[508, 532],
#   [697, 730]]]
c = tf.matmul(a, b)

# Since python >= 3.5 the @ operator is supported (see PEP 465).
# In TensorFlow, it simply calls the tf.matmul() function, so the
# following lines are equivalent:
d = a @ b @ [[10.], [11.]]
d = tf.matmul(tf.matmul(a, b), [[10.], [11.]])

#### Args:

• a: Tensor of type float16, float32, float64, int32, complex64, complex128 and rank > 1.
• b: Tensor with same type and rank as a.
• transpose_a: If True, a is transposed before multiplication.
• transpose_b: If True, b is transposed before multiplication.
• adjoint_a: If True, a is conjugated and transposed before multiplication.
• adjoint_b: If True, b is conjugated and transposed before multiplication.
• a_is_sparse: If True, a is treated as a sparse matrix.
• b_is_sparse: If True, b is treated as a sparse matrix.
• name: Name for the operation (optional).

#### Returns:

A Tensor of the same type as a and b where each inner-most matrix is the product of the corresponding matrices in a and b, e.g. if all transpose or adjoint attributes are False:

output[…, i, j] = sum_k (a[…, i, k] * b[…, k, j]), for all indices i, j.

• Note: This is matrix product, not element-wise product.

#### Raises:

• ValueError: If transpose_a and adjoint_a, or transpose_b and adjoint_b are both set to True.

### mod

__mod__(
a,
*args
)

Returns element-wise remainder of division. When x < 0 xor y < 0 is

true, this follows Python semantics in that the result here is consistent with a flooring divide. E.g. floor(x / y) * y + mod(x, y) = x.

NOTE: FloorMod supports broadcasting. More about broadcasting here

#### Args:

• x: A Tensor. Must be one of the following types: int32, int64, bfloat16, float32, float64.
• y: A Tensor. Must have the same type as x.
• name: A name for the operation (optional).

#### Returns:

A Tensor. Has the same type as x.

### mul

__mul__(
a,
*args
)

Dispatches cwise mul for “Dense*Dense" and “Dense*Sparse“.

### neg

__neg__(
a,
*args
)

Computes numerical negative value element-wise.

I.e., $$y = -x$$.

#### Args:

• x: A Tensor. Must be one of the following types: half, bfloat16, float32, float64, int32, int64, complex64, complex128.
• name: A name for the operation (optional).

#### Returns:

A Tensor. Has the same type as x.

### nonzero

__nonzero__()

### or

__or__(
a,
*args
)

Returns the truth value of x OR y element-wise.

NOTE: LogicalOr supports broadcasting. More about broadcasting here

#### Args:

• x: A Tensor of type bool.
• y: A Tensor of type bool.
• name: A name for the operation (optional).

#### Returns:

A Tensor of type bool.

### pow

__pow__(
a,
*args
)

Computes the power of one value to another.

Given a tensor x and a tensor y, this operation computes $$x^y$$ for corresponding elements in x and y. For example:

x = tf.constant([[2, 2], [3, 3]])
y = tf.constant([[8, 16], [2, 3]])
tf.pow(x, y)  # [[256, 65536], [9, 27]]

#### Args:

• x: A Tensor of type float32, float64, int32, int64, complex64, or complex128.
• y: A Tensor of type float32, float64, int32, int64, complex64, or complex128.
• name: A name for the operation (optional).

#### Returns:

A Tensor.

### radd

__radd__(
a,
*args
)

Returns x + y element-wise.

NOTE: Add supports broadcasting. AddN does not. More about broadcasting here

#### Args:

• x: A Tensor. Must be one of the following types: half, bfloat16, float32, float64, uint8, int8, int16, int32, int64, complex64, complex128, string.
• y: A Tensor. Must have the same type as x.
• name: A name for the operation (optional).

#### Returns:

A Tensor. Has the same type as x.

### rand

__rand__(
a,
*args
)

Returns the truth value of x AND y element-wise.

NOTE: LogicalAnd supports broadcasting. More about broadcasting here

#### Args:

• x: A Tensor of type bool.
• y: A Tensor of type bool.
• name: A name for the operation (optional).

#### Returns:

A Tensor of type bool.

### rdiv

__rdiv__(
a,
*args
)

Divide two values using Python 2 semantics. Used for Tensor.__div__.

#### Args:

• x: Tensor numerator of real numeric type.
• y: Tensor denominator of real numeric type.
• name: A name for the operation (optional).

#### Returns:

x / y returns the quotient of x and y.

### rfloordiv

__rfloordiv__(
a,
*args
)

Divides x / y elementwise, rounding toward the most negative integer.

The same as tf.div(x,y) for integers, but uses tf.floor(tf.div(x,y)) for floating point arguments so that the result is always an integer (though possibly an integer represented as floating point). This op is generated by x // y floor division in Python 3 and in Python 2.7 with from __future__ import division.

Note that for efficiency, floordiv uses C semantics for negative numbers (unlike Python and Numpy).

x and y must have the same type, and the result will have the same type as well.

#### Args:

• x: Tensor numerator of real numeric type.
• y: Tensor denominator of real numeric type.
• name: A name for the operation (optional).

#### Returns:

x / y rounded down (except possibly towards zero for negative integers).

#### Raises:

• TypeError: If the inputs are complex.

### rmatmul

__rmatmul__(
a,
*args
)

Multiplies matrix a by matrix b, producing a * b.

The inputs must, following any transpositions, be tensors of rank >= 2 where the inner 2 dimensions specify valid matrix multiplication arguments, and any further outer dimensions match.

Both matrices must be of the same type. The supported types are: float16, float32, float64, int32, complex64, complex128.

Either matrix can be transposed or adjointed (conjugated and transposed) on the fly by setting one of the corresponding flag to True. These are False by default.

If one or both of the matrices contain a lot of zeros, a more efficient multiplication algorithm can be used by setting the corresponding a_is_sparse or b_is_sparse flag to True. These are False by default. This optimization is only available for plain matrices (rank-2 tensors) with datatypes bfloat16 or float32.

For example:

# 2-D tensor a
# [[1, 2, 3],
#  [4, 5, 6]]
a = tf.constant([1, 2, 3, 4, 5, 6], shape=[2, 3])

# 2-D tensor b
# [[ 7,  8],
#  [ 9, 10],
#  [11, 12]]
b = tf.constant([7, 8, 9, 10, 11, 12], shape=[3, 2])

# a * b
# [[ 58,  64],
#  [139, 154]]
c = tf.matmul(a, b)

# 3-D tensor a
# [[[ 1,  2,  3],
#   [ 4,  5,  6]],
#  [[ 7,  8,  9],
#   [10, 11, 12]]]
a = tf.constant(np.arange(1, 13, dtype=np.int32),
shape=[2, 2, 3])

# 3-D tensor b
# [[[13, 14],
#   [15, 16],
#   [17, 18]],
#  [[19, 20],
#   [21, 22],
#   [23, 24]]]
b = tf.constant(np.arange(13, 25, dtype=np.int32),
shape=[2, 3, 2])

# a * b
# [[[ 94, 100],
#   [229, 244]],
#  [[508, 532],
#   [697, 730]]]
c = tf.matmul(a, b)

# Since python >= 3.5 the @ operator is supported (see PEP 465).
# In TensorFlow, it simply calls the tf.matmul() function, so the
# following lines are equivalent:
d = a @ b @ [[10.], [11.]]
d = tf.matmul(tf.matmul(a, b), [[10.], [11.]])

#### Args:

• a: Tensor of type float16, float32, float64, int32, complex64, complex128 and rank > 1.
• b: Tensor with same type and rank as a.
• transpose_a: If True, a is transposed before multiplication.
• transpose_b: If True, b is transposed before multiplication.
• adjoint_a: If True, a is conjugated and transposed before multiplication.
• adjoint_b: If True, b is conjugated and transposed before multiplication.
• a_is_sparse: If True, a is treated as a sparse matrix.
• b_is_sparse: If True, b is treated as a sparse matrix.
• name: Name for the operation (optional).

#### Returns:

A Tensor of the same type as a and b where each inner-most matrix is the product of the corresponding matrices in a and b, e.g. if all transpose or adjoint attributes are False:

output[…, i, j] = sum_k (a[…, i, k] * b[…, k, j]), for all indices i, j.

• Note: This is matrix product, not element-wise product.

#### Raises:

• ValueError: If transpose_a and adjoint_a, or transpose_b and adjoint_b are both set to True.

### rmod

__rmod__(
a,
*args
)

Returns element-wise remainder of division. When x < 0 xor y < 0 is

true, this follows Python semantics in that the result here is consistent with a flooring divide. E.g. floor(x / y) * y + mod(x, y) = x.

NOTE: FloorMod supports broadcasting. More about broadcasting here

#### Args:

• x: A Tensor. Must be one of the following types: int32, int64, bfloat16, float32, float64.
• y: A Tensor. Must have the same type as x.
• name: A name for the operation (optional).

#### Returns:

A Tensor. Has the same type as x.

### rmul

__rmul__(
a,
*args
)

Dispatches cwise mul for “Dense*Dense" and “Dense*Sparse“.

### ror

__ror__(
a,
*args
)

Returns the truth value of x OR y element-wise.

NOTE: LogicalOr supports broadcasting. More about broadcasting here

#### Args:

• x: A Tensor of type bool.
• y: A Tensor of type bool.
• name: A name for the operation (optional).

#### Returns:

A Tensor of type bool.

### rpow

__rpow__(
a,
*args
)

Computes the power of one value to another.

Given a tensor x and a tensor y, this operation computes $$x^y$$ for corresponding elements in x and y. For example:

x = tf.constant([[2, 2], [3, 3]])
y = tf.constant([[8, 16], [2, 3]])
tf.pow(x, y)  # [[256, 65536], [9, 27]]

#### Args:

• x: A Tensor of type float32, float64, int32, int64, complex64, or complex128.
• y: A Tensor of type float32, float64, int32, int64, complex64, or complex128.
• name: A name for the operation (optional).

#### Returns:

A Tensor.

### rsub

__rsub__(
a,
*args
)

Returns x - y element-wise.

NOTE: Subtract supports broadcasting. More about broadcasting here

#### Args:

• x: A Tensor. Must be one of the following types: half, bfloat16, float32, float64, uint8, int8, uint16, int16, int32, int64, complex64, complex128.
• y: A Tensor. Must have the same type as x.
• name: A name for the operation (optional).

#### Returns:

A Tensor. Has the same type as x.

### rtruediv

__rtruediv__(
a,
*args
)

### rxor

__rxor__(
a,
*args
)

x ^ y = (x | y) & ~(x & y).

### sub

__sub__(
a,
*args
)

Returns x - y element-wise.

NOTE: Subtract supports broadcasting. More about broadcasting here

#### Args:

• x: A Tensor. Must be one of the following types: half, bfloat16, float32, float64, uint8, int8, uint16, int16, int32, int64, complex64, complex128.
• y: A Tensor. Must have the same type as x.
• name: A name for the operation (optional).

#### Returns:

A Tensor. Has the same type as x.

### truediv

__truediv__(
a,
*args
)

### xor

__xor__(
a,
*args
)

x ^ y = (x | y) & ~(x & y).

### 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.

Note: the copy distribution may continue to depend on the original initialization arguments.

#### 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.

#### Args:

• other: tf.distributions.Distribution instance.
• name: Python str prepended to names of ops created by this function.

#### 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.

### eval

eval(
session=None,
feed_dict=None
)

In a session, computes and returns the value of this random variable.

This is not a graph construction method, it does not add ops to the graph.

This convenience method requires a session where the graph containing this variable has been launched. If no session is passed, the default session is used.

#### Args:

• session: tf.BaseSession. The tf.Session to use to evaluate this random variable. If none, the default session is used.
• feed_dict: dict. A dictionary that maps tf.Tensor objects to feed values. See tf.Session.run() for a description of the valid feed values.

#### Examples

x = Normal(0.0, 1.0)
with tf.Session() as sess:
# Usage passing the session explicitly.
print(x.eval(sess))
# Usage with the default session.  The 'with' block
# above makes 'sess' the default session.
print(x.eval())

### 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.

### get_ancestors

get_ancestors(collection=None)

Get ancestor random variables.

### get_blanket

get_blanket(collection=None)

Get the random variable’s Markov blanket.

### get_children

get_children(collection=None)

Get child random variables.

### get_descendants

get_descendants(collection=None)

Get descendant random variables.

### get_parents

get_parents(collection=None)

Get parent random variables.

### get_shape

get_shape()

Get shape of random variable.

### get_siblings

get_siblings(collection=None)

Get sibling random variables.

### get_variables

get_variables(collection=None)

Get TensorFlow variables that the random variable depends on.

### 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, (Shanon) cross entropy 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.

#### Args:

• other: tf.distributions.Distribution instance.
• name: Python str prepended to names of ops created by this function.

#### 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.

### value

value()

Get tensor that the random variable corresponds to.

### 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().