# tf.distributions.bijectors.Bijector

## Class Bijector

### Aliases:

• Class tf.contrib.distributions.bijectors.Bijector
• Class tf.distributions.bijectors.Bijector

See the guide: Random variable transformations (contrib) > Bijectors

Interface for invertible transformations of a Distribution sample.

#### Mathematical Details

A Bijector implements a diffeomorphism, i.e., a bijective, differentiable function. A Bijector is used by TransformedDistribution but can be generally used for transforming a Distribution generated Tensor. A Bijector is characterized by three operations:

1. Forward Evaluation

Useful for turning one random outcome into another random outcome from a different distribution.

1. Inverse Evaluation

Useful for "reversing" a transformation to compute one probability in terms of another.

1. (log o det o Jacobian o inverse)(x)

"The log of the determinant of the matrix of all first-order partial derivatives of the inverse function." Useful for inverting a transformation to compute one probability in terms of another. Geometrically, the det(Jacobian) is the volume of the transformation and is used to scale the probability.

By convention, transformations of random variables are named in terms of the forward transformation. The forward transformation creates samples, the inverse is useful for computing probabilities.

#### Example Uses

• Basic properties:
x = ...  # A tensor.
# Evaluate forward transformation.
fwd_x = my_bijector.forward(x)
x == my_bijector.inverse(fwd_x)
x != my_bijector.forward(fwd_x)  # Not equal because g(x) != g(g(x)).

• Computing a log-likelihood:
def transformed_log_prob(bijector, log_prob, x):
return (bijector.inverse_log_det_jacobian(x) +
log_prob(bijector.inverse(x)))

• Transforming a random outcome:
def transformed_sample(bijector, x):
return bijector.forward(x)


#### Example Bijectors

• "Exponential"
Y = g(X) = exp(X)
X ~ Normal(0, 1)  # Univariate.


Implies:

  g^{-1}(Y) = log(Y)
|Jacobian(g^{-1})(y)| = 1 / y
Y ~ LogNormal(0, 1), i.e.,
prob(Y=y) = |Jacobian(g^{-1})(y)| * prob(X=g^{-1}(y))
= (1 / y) Normal(log(y); 0, 1)


Here is an example of how one might implement the Exp bijector:

  class Exp(Bijector):

def __init__(self, event_ndims=0, validate_args=False, name="exp"):
super(Exp, self).__init__(
event_ndims=event_ndims, validate_args=validate_args, name=name)

def _forward(self, x):
return math_ops.exp(x)

def _inverse(self, y):
return math_ops.log(y)

def _inverse_log_det_jacobian(self, y):
return -self._forward_log_det_jacobian(self._inverse(y))

def _forward_log_det_jacobian(self, x):
if self.event_ndims is None:
raise ValueError("Jacobian requires known event_ndims.")
event_dims = array_ops.shape(x)[-self.event_ndims:]
return math_ops.reduce_sum(x, axis=event_dims)


"Affine"


Y = g(X) = sqrtSigma * X + mu X ~ MultivariateNormal(0, I_d)

Implies:

g^{-1}(Y) = inv(sqrtSigma) * (Y - mu)
|Jacobian(g^{-1})(y)| = det(inv(sqrtSigma))
Y ~ MultivariateNormal(mu, sqrtSigma) , i.e.,
prob(Y=y) = |Jacobian(g^{-1})(y)| * prob(X=g^{-1}(y))
= det(sqrtSigma)^(-d) *
MultivariateNormal(inv(sqrtSigma) * (y - mu); 0, I_d)




#### Jacobian

The Jacobian is a reduction over event dims. To see this, consider the Exp Bijector applied to a Tensor which has sample, batch, and event (S, B, E) shape semantics. Suppose the Tensor's partitioned-shape is (S=[4], B=[2], E=[3, 3]). The shape of the Tensor returned by forward and inverse is unchanged, i.e., [4, 2, 3, 3]. However the shape returned by inverse_log_det_jacobian is [4, 2] because the Jacobian is a reduction over the event dimensions.

It is sometimes useful to implement the inverse Jacobian as the negative forward Jacobian. For example,

def _inverse_log_det_jacobian(self, y):
return -self._forward_log_det_jac(self._inverse(y))  # Note negation.


The correctness of this approach can be seen from the following claim.

• Claim:

Assume Y = g(X) is a bijection whose derivative exists and is nonzero for its domain, i.e., dY/dX = d/dX g(X) != 0. Then:

none (log o det o jacobian o g^{-1})(Y) = -(log o det o jacobian o g)(X)

• Proof:

From the bijective, nonzero differentiability of g, the inverse function theorem implies g^{-1} is differentiable in the image of g. Applying the chain rule to y = g(x) = g(g^{-1}(y)) yields I = g'(g^{-1}(y))*g^{-1}'(y). The same theorem also implies g{-1}' is non-singular therefore: inv[ g'(g^{-1}(y)) ] = g^{-1}'(y). The claim follows from properties of determinant.

Generally its preferable to directly implement the inverse Jacobian. This should have superior numerical stability and will often share subgraphs with the _inverse implementation.

#### Subclass Requirements

• Subclasses typically implement:

• _forward,
• _inverse,
• _inverse_log_det_jacobian,
• _forward_log_det_jacobian (optional).

The _forward_log_det_jacobian is called when the bijector is inverted via the Invert bijector. If undefined, a slightly less efficiently calculation, -1 * _inverse_log_det_jacobian, is used.

If the bijector changes the shape of the input, you must also implement:

- _forward_event_shape_tensor,
- _forward_event_shape (optional),
- _inverse_event_shape_tensor,
- _inverse_event_shape (optional).


By default the event-shape is assumed unchanged from input.

• If the Bijector's use is limited to TransformedDistribution (or friends like QuantizedDistribution) then depending on your use, you may not need to implement all of _forward and _inverse functions.

Examples:

1. Sampling (e.g., sample) only requires _forward.
2. Probability functions (e.g., prob, cdf, survival) only require
_inverse (and related).
3. Only calling probability functions on the output of sample means
_inverse can be implemented as a cache lookup.


See "Example Uses" [above] which shows how these functions are used to transform a distribution. (Note: _forward could theoretically be implemented as a cache lookup but this would require controlling the underlying sample generation mechanism.)

## Properties

### dtype

dtype of Tensors transformable by this distribution.

### event_ndims

Returns then number of event dimensions this bijector operates on.

### graph_parents

Returns this Bijector's graph_parents as a Python list.

### is_constant_jacobian

Returns true iff the Jacobian is not a function of x.

#### Returns:

• is_constant_jacobian: Python bool.

### name

Returns the string name of this Bijector.

### validate_args

Returns True if Tensor arguments will be validated.

## Methods

### __init__

__init__(
event_ndims=None,
graph_parents=None,
is_constant_jacobian=False,
validate_args=False,
dtype=None,
name=None
)


Constructs Bijector.

A Bijector transforms random variables into new random variables.

Examples:

# Create the Y = g(X) = X transform which operates on vector events.
identity = Identity(event_ndims=1)

# Create the Y = g(X) = exp(X) transform which operates on matrices.
exp = Exp(event_ndims=2)


See Bijector subclass docstring for more details and specific examples.

#### Args:

• event_ndims: number of dimensions associated with event coordinates.
• graph_parents: Python list of graph prerequisites of this Bijector.
• is_constant_jacobian: Python bool indicating that the Jacobian is not a function of the input.
• validate_args: Python bool, default False. Whether to validate input with asserts. If validate_args is False, and the inputs are invalid, correct behavior is not guaranteed.
• dtype: tf.dtype supported by this Bijector. None means dtype is not enforced.
• name: The name to give Ops created by the initializer.

### forward

forward(
x,
name='forward'
)


Returns the forward Bijector evaluation, i.e., X = g(Y).

#### Args:

• x: Tensor. The input to the "forward" evaluation.
• name: The name to give this op.

#### Returns:

Tensor.

#### Raises:

• TypeError: if self.dtype is specified and x.dtype is not self.dtype.
• NotImplementedError: if _forward is not implemented.

### forward_event_shape

forward_event_shape(input_shape)


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

Same meaning as forward_event_shape_tensor. May be only partially defined.

#### Args:

• input_shape: TensorShape indicating event-portion shape passed into forward function.

#### Returns:

• forward_event_shape_tensor: TensorShape indicating event-portion shape after applying forward. Possibly unknown.

### forward_event_shape_tensor

forward_event_shape_tensor(
input_shape,
name='forward_event_shape_tensor'
)


Shape of a single sample from a single batch as an int32 1D Tensor.

#### Args:

• input_shape: Tensor, int32 vector indicating event-portion shape passed into forward function.
• name: name to give to the op

#### Returns:

• forward_event_shape_tensor: Tensor, int32 vector indicating event-portion shape after applying forward.

### forward_log_det_jacobian

forward_log_det_jacobian(
x,
name='forward_log_det_jacobian'
)


Returns both the forward_log_det_jacobian.

#### Args:

• x: Tensor. The input to the "forward" Jacobian evaluation.
• name: The name to give this op.

#### Returns:

Tensor.

#### Raises:

• TypeError: if self.dtype is specified and y.dtype is not self.dtype.
• NotImplementedError: if neither _forward_log_det_jacobian nor {_inverse, _inverse_log_det_jacobian} are implemented.

### inverse

inverse(
y,
name='inverse'
)


Returns the inverse Bijector evaluation, i.e., X = g^{-1}(Y).

#### Args:

• y: Tensor. The input to the "inverse" evaluation.
• name: The name to give this op.

#### Returns:

Tensor.

#### Raises:

• TypeError: if self.dtype is specified and y.dtype is not self.dtype.
• NotImplementedError: if _inverse is not implemented.

### inverse_event_shape

inverse_event_shape(output_shape)


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

Same meaning as inverse_event_shape_tensor. May be only partially defined.

#### Args:

• output_shape: TensorShape indicating event-portion shape passed into inverse function.

#### Returns:

• inverse_event_shape_tensor: TensorShape indicating event-portion shape after applying inverse. Possibly unknown.

### inverse_event_shape_tensor

inverse_event_shape_tensor(
output_shape,
name='inverse_event_shape_tensor'
)


Shape of a single sample from a single batch as an int32 1D Tensor.

#### Args:

• output_shape: Tensor, int32 vector indicating event-portion shape passed into inverse function.
• name: name to give to the op

#### Returns:

• inverse_event_shape_tensor: Tensor, int32 vector indicating event-portion shape after applying inverse.

### inverse_log_det_jacobian

inverse_log_det_jacobian(
y,
name='inverse_log_det_jacobian'
)


Returns the (log o det o Jacobian o inverse)(y).

Mathematically, returns: log(det(dX/dY))(Y). (Recall that: X=g^{-1}(Y).)

Note that forward_log_det_jacobian is the negative of this function.

#### Args:

• y: Tensor. The input to the "inverse" Jacobian evaluation.
• name: The name to give this op.

#### Returns:

Tensor.

#### Raises:

• TypeError: if self.dtype is specified and y.dtype is not self.dtype.
• NotImplementedError: if _inverse_log_det_jacobian is not implemented.