contrib.distributions.bijectors.Bijector
tf.contrib.distributions.bijectors.Bijector
class tf.contrib.distributions.bijectors.Bijector
Defined in tensorflow/python/ops/distributions/bijector_impl.py
.
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:
- Forward Evaluation
Useful for turning one random outcome into another random outcome from a different distribution.
- Inverse Evaluation
Useful for "reversing" a transformation to compute one probability in terms of another.
- (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 impliesg^{-1}
is differentiable in the image ofg
. Applying the chain rule toy = g(x) = g(g^{-1}(y))
yieldsI = g'(g^{-1}(y))*g^{-1}'(y)
. The same theorem also impliesg{-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 toTransformedDistribution
(or friends likeQuantizedDistribution
) 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 Tensor
s 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.
Note: Jacobian is either constant for both forward and inverse or neither.
Returns:
-
is_constant_jacobian
: Pythonbool
.
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 thisBijector
. -
is_constant_jacobian
: Pythonbool
indicating that the Jacobian is not a function of the input. -
validate_args
: Pythonbool
, defaultFalse
. Whether to validate input with asserts. Ifvalidate_args
isFalse
, and the inputs are invalid, correct behavior is not guaranteed. -
dtype
:tf.dtype
supported by thisBijector
.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
: ifself.dtype
is specified andx.dtype
is notself.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 intoforward
function.
Returns:
-
forward_event_shape_tensor
:TensorShape
indicating event-portion shape after applyingforward
. 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 intoforward
function. -
name
: name to give to the op
Returns:
-
forward_event_shape_tensor
:Tensor
,int32
vector indicating event-portion shape after applyingforward
.
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
: ifself.dtype
is specified andy.dtype
is notself.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
: ifself.dtype
is specified andy.dtype
is notself.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 intoinverse
function.
Returns:
-
inverse_event_shape_tensor
:TensorShape
indicating event-portion shape after applyinginverse
. 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 intoinverse
function. -
name
: name to give to the op
Returns:
-
inverse_event_shape_tensor
:Tensor
,int32
vector indicating event-portion shape after applyinginverse
.
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
: ifself.dtype
is specified andy.dtype
is notself.dtype
. -
NotImplementedError
: if_inverse_log_det_jacobian
is not implemented.
© 2017 The TensorFlow Authors. All rights reserved.
Licensed under the Creative Commons Attribution License 3.0.
Code samples licensed under the Apache 2.0 License.
https://www.tensorflow.org/api_docs/python/tf/contrib/distributions/bijectors/Bijector