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ed.models.VectorDiffeomixture

Class `VectorDiffeomixture`

Inherits From: RandomVariable

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] (https://en.wikipedia.org/wiki/Compound_probability_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 [1] for more details.

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

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.

Note: The marginals X_1|v, ..., X_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 `Diffeomixture`s 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

```python 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)

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.

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.

`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, uint8, int16, int8, int64, bfloat16, uint16, half, uint32, uint64.
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
)

Overload for Tensor.__getitem__.

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 non-scalar 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]]

# Use scalar tensors as indices on both dimensions
print(foo[tf.constant(0), tf.constant(2)].eval())  # => 3

# 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, uint8, int16, int8, int64, bfloat16, uint16, half, uint32, uint64.
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, uint8, int16, int8, int64, bfloat16, uint16, half, uint32, uint64.
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, uint8, int16, int8, int64, bfloat16, uint16, half, uint32, uint64.
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.

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.

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, 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.

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().

Edward

ed.models.VectorDiffeomixture

Class VectorDiffeomixture

About Vector distributions in TensorFlow.

About Diffeomixtures and reparameterization.

Examples

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

Properties

allow_nan_stats

Returns:

batch_shape

Returns:

distribution

dtype

endpoint_affine

event_shape

Returns:

grid

interpolated_affine

mixture_distribution

name

parameters

reparameterization_type

Returns:

sample_shape

shape

validate_args

Methods

init

Args:

Raises:

abs

Args:

Returns:

add

Args:

Returns:

and

Args:

Returns:

bool

div

Args:

Returns:

eq

floordiv

Args:

Returns:

Raises:

ge

Args:

Returns:

getitem

Args:

Returns:

Raises:

gt

Args:

Returns:

invert

Args:

Returns:

iter

le

Args:

Returns:

lt

Args:

Returns:

matmul

Args:

Returns:

Raises:

mod

Args:

Returns:

mul

Class `VectorDiffeomixture`

About `Vector` distributions in TensorFlow.

About `Diffeomixture`s and reparameterization.

another with mix_loc=[1]. In both cases, `K=2` and the affine

`allow_nan_stats`

`batch_shape`

`distribution`

`dtype`

`endpoint_affine`

`event_shape`

`grid`

`interpolated_affine`

`mixture_distribution`

`name`

`parameters`

`reparameterization_type`

`sample_shape`

`shape`

`validate_args`

`init`

`abs`

`add`

`and`

`bool`

`div`

`eq`

`floordiv`

`ge`

`getitem`

`gt`

`invert`

`iter`

`le`

`lt`

`matmul`

`mod`

`mul`

`neg`

`nonzero`

`or`

`pow`

`radd`

`rand`

`rdiv`

`rfloordiv`

`rmatmul`

`rmod`

`rmul`

`ror`

`rpow`

`rsub`

`rtruediv`

`rxor`

`sub`

`truediv`

`xor`

`batch_shape_tensor`

`cdf`

`copy`

`covariance`

`cross_entropy`

`entropy`

`eval`

`event_shape_tensor`