Interface for transformations of a Distribution
sample.
@abc.abstractmethod
oryx.bijectors.Bijector( graph_parents=None, is_constant_jacobian=False, validate_args=False, dtype=None, forward_min_event_ndims=UNSPECIFIED, inverse_min_event_ndims=UNSPECIFIED, parameters=None, name=None )
Bijectors can be used to represent any differentiable and injective
(one to one) function defined on an open subset of R^n
. Some noninjective
transformations are also supported (see 'Non Injective Transforms' below).
Mathematical Details
A Bijector
implements a smooth covering map, i.e., a local
diffeomorphism such that every point in the target has a neighborhood evenly
covered by a map (see also).
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
Useful for turning one random outcome into another random outcome from a different distribution.
Inverse
Useful for 'reversing' a transformation to compute one probability in terms of another.
log_det_jacobian(x)
'The log of the absolute value of the determinant of the matrix of all firstorder partial derivatives of the inverse function.'
Useful for inverting a transformation to compute one probability in terms of another. Geometrically, the Jacobian determinant is the volume of the transformation and is used to scale the probability.
We take the absolute value of the determinant before log to avoid NaN values. Geometrically, a negative determinant corresponds to an orientationreversing transformation. It is ok for us to discard the sign of the determinant because we only integrate everywherenonnegative functions (probability densities) and the correct orientation is always the one that produces a nonnegative integrand.
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 x != g(g(x)).
 Computing a loglikelihood:
def transformed_log_prob(bijector, log_prob, x):
return (bijector.inverse_log_det_jacobian(x, event_ndims=0) +
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, validate_args=False, name='exp'): super(Exp, self).__init__( validate_args=validate_args, forward_min_event_ndims=0, name=name) def _forward(self, x): return tf.exp(x) def _inverse(self, y): return tf.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): # Notice that we needn't do any reducing, even when`event_ndims > 0`. # The base Bijector class will handle reducing for us; it knows how # to do so because we called `super` `__init__` with # `forward_min_event_ndims = 0`. return x ```
'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) ```
Min_event_ndims and Naming
Bijectors are named for the dimensionality of data they act on (i.e. without
broadcasting). We can think of bijectors having an intrinsic min_event_ndims
, which is the minimum number of dimensions for the bijector act on. For
instance, a Cholesky decomposition requires a matrix, and hence
min_event_ndims=2
.
Some examples:
AffineScalar: min_event_ndims=0
Affine: min_event_ndims=1
Cholesky: min_event_ndims=2
Exp: min_event_ndims=0
Sigmoid: min_event_ndims=0
SoftmaxCentered: min_event_ndims=1
Note the difference between Affine
and AffineScalar
. AffineScalar
operates on scalar events, whereas Affine
operates on vectorvalued events.
More generally, there is a forward_min_event_ndims
and an
inverse_min_event_ndims
. In most cases, these will be the same.
However, for some shape changing bijectors, these will be different
(e.g. a bijector which pads an extra dimension at the end, might have
forward_min_event_ndims=0
and inverse_min_event_ndims=1
.
Additional Considerations for "Multi Tensor" Bijectors
Bijectors which operate on structures of Tensor
require structured
min_event_ndims
matching the structure of the inputs. In these cases,
min_event_ndims
describes both the minimum dimensionality and the
structure of arguments to forward
and inverse
. For example:
Split([sizes], axis):
forward_min_event_ndims=axis
inverse_min_event_ndims=[axis] * len(sizes)
Finally, some bijectors that operate on structures of inputs may not know
the minimum structured rank of their inputs without calltime shape information
(Composite bijectors, for example). In these cases, both min_event_ndims
properties will indicate the expected structure of inputs and outputs,
but the component values may be None
.
Jacobian Determinant
The Jacobian determinant is a reduction over event_ndims  min_event_ndims
(forward_min_event_ndims
for forward_log_det_jacobian
and
inverse_min_event_ndims
for inverse_log_det_jacobian
).
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
partitionedshape 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 determinant is a reduction over the event dimensions.
Another example is the Affine
Bijector
. Because min_event_ndims = 1
, the
Jacobian determinant reduction is over event_ndims  1
.
It is sometimes useful to implement the inverse Jacobian determinant as the negative forward Jacobian determinant. 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:(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 nonsingular 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
determinant. This should have superior numerical stability and will often
share subgraphs with the _inverse
implementation.
Is_constant_jacobian
Certain bijectors will have constant jacobian matrices. For instance, the
Affine
bijector encodes multiplication by a matrix plus a shift, with
jacobian matrix, the same aforementioned matrix.
is_constant_jacobian
encodes the fact that the jacobian matrix is constant.
The semantics of this argument are the following:
 Repeated calls to 'log_det_jacobian' functions with the same
event_ndims
(but not necessarily same input), will return the first computed jacobian (because the matrix is constant, and hence is input independent). log_det_jacobian
implementations are merely broadcastable to the truelog_det_jacobian
(because, again, the jacobian matrix is input independent). Specifically,log_det_jacobian
is implemented as the log jacobian determinant for a single input.class Identity(Bijector): def __init__(self, validate_args=False, name='identity'): super(Identity, self).__init__( is_constant_jacobian=True, validate_args=validate_args, forward_min_event_ndims=0, name=name) def _forward(self, x): return x def _inverse(self, y): return y def _inverse_log_det_jacobian(self, y): return self._forward_log_det_jacobian(self._inverse(y)) def _forward_log_det_jacobian(self, x): # The full log jacobian determinant would be tf.zero_like(x). # However, we circumvent materializing that, since the jacobian # calculation is input independent, and we specify it for one input. return tf.constant(0., x.dtype)
Subclass Requirements
Subclasses typically implement:
_forward
,_inverse
,_inverse_log_det_jacobian
,_forward_log_det_jacobian
(optional),_is_increasing
(scalar bijectors only)
The
_forward_log_det_jacobian
is called when the bijector is inverted via theInvert
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 eventshape is assumed unchanged from input.
Multipart bijectors, which operate on structures of tensors, may implement additional methods to propogate calltime dtype information over any changes to structure. These methods are:
 _forward_dtype
 _inverse_dtype
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:
 Sampling (e.g.,
sample
) only requires_forward
.  Probability functions (e.g.,
prob
,cdf
,survival
) only require_inverse
(and related).  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.) Sampling (e.g.,
Non Injective Transforms
Non injective maps g
are supported, provided their domain D
can be
partitioned into k
disjoint subsets, Union{D1, ..., Dk}
, such that,
ignoring sets of measure zero, the restriction of g
to each subset is a
differentiable bijection onto g(D)
. In particular, this implies that for
y in g(D)
, the set inverse, i.e. g^{1}(y) = {x in D : g(x) = y}
, always
contains exactly k
distinct points.
The property, _is_injective
is set to False
to indicate that the bijector
is not injective, yet satisfies the above condition.
The usual bijector API is modified in the case _is_injective is False
(see
method docstrings for specifics). Here we show by example the AbsoluteValue
bijector. In this case, the domain D = (inf, inf)
, can be partitioned
into D1 = (inf, 0)
, D2 = {0}
, and D3 = (0, inf)
. Let gi
be the
restriction of g
to Di
, then both g1
and g3
are bijections onto
(0, inf)
, with g1^{1}(y) = y
, and g3^{1}(y) = y
. We will use
g1
and g3
to define bijector methods over D1
and D3
. D2 = {0}
is
an oddball in that g2
is one to one, and the derivative is not well defined.
Fortunately, when considering transformations of probability densities
(e.g. in TransformedDistribution
), sets of measure zero have no effect in
theory, and only a small effect in 32 or 64 bit precision. For that reason,
we define inverse(0)
and inverse_log_det_jacobian(0)
both as [0, 0]
,
which is convenient and results in a leftsemicontinuous pdf.
abs = tfp.bijectors.AbsoluteValue()
abs.forward(1.)
==> 1.
abs.forward(1.)
==> 1.
abs.inverse(1.)
==> (1., 1.)
# The dX/dY is constant, == 1. So LogdX/dY == 0.
abs.inverse_log_det_jacobian(1., event_ndims=0)
==> (0., 0.)
# Special case handling of 0.
abs.inverse(0.)
==> (0., 0.)
abs.inverse_log_det_jacobian(0., event_ndims=0)
==> (0., 0.)
Args  

graph_parents

Python list of graph prerequisites of this Bijector .

is_constant_jacobian

Python bool indicating that the Jacobian matrix 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. For multipart bijectors, this value is expected to be the
same for all elements of the input and output structures.

forward_min_event_ndims

Python integer (structure) indicating the
minimum number of dimensions on which forward operates.

inverse_min_event_ndims

Python integer (structure) indicating the
minimum number of dimensions on which inverse operates. Will be set to
forward_min_event_ndims by default, if no value is provided.

parameters

Python dict of parameters used to instantiate this
Bijector . Bijector instances with identical types, names, and
parameters share an input/output cache. parameters dicts are
keyed by strings and are identical if their keys are identical and if
corresponding values have identical hashes (or object ids, for
unhashable objects).

name

The name to give Ops created by the initializer. 
Raises  

ValueError

If neither forward_min_event_ndims and
inverse_min_event_ndims are specified, or if either of them is
negative.

ValueError

If a member of graph_parents is not a Tensor .

Attributes  

dtype


forward_min_event_ndims

Returns the minimal number of dimensions bijector.forward operates on.
Multipart bijectors return structured 
graph_parents

Returns this Bijector 's graph_parents as a Python list.

has_static_min_event_ndims

Returns True if the bijector has staticallyknown min_event_ndims .

inverse_min_event_ndims

Returns the minimal number of dimensions bijector.inverse operates on.
Multipart bijectors return structured 
is_constant_jacobian

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

Returns the string name of this Bijector .

parameters

Dictionary of parameters used to instantiate this Bijector .

trainable_variables


validate_args

Returns True if Tensor arguments will be validated. 
variables

Methods
forward
forward(
x, name='forward', **kwargs
)
Returns the forward Bijector
evaluation, i.e., X = g(Y).
Args  

x

Tensor (structure). The input to the 'forward' evaluation.

name

The name to give this op. 
**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

Tensor (structure).

Raises  

TypeError

if self.dtype is specified and x.dtype is not
self.dtype .

NotImplementedError

if _forward is not implemented.

forward_dtype
forward_dtype(
dtype=UNSPECIFIED, name='forward_dtype', **kwargs
)
Returns the dtype returned by forward
for the provided input.
forward_event_ndims
forward_event_ndims(
event_ndims, **kwargs
)
Returns the number of event dimensions produced by forward
.
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 (structure) indicating eventportion shape
passed into forward function.

Returns  

forward_event_shape_tensor

TensorShape (structure) indicating
eventportion 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 (structure) indicating eventportion
shape passed into forward function.

name

name to give to the op 
Returns  

forward_event_shape_tensor

Tensor , int32 vector (structure)
indicating eventportion shape after applying forward .

forward_log_det_jacobian
forward_log_det_jacobian(
x, event_ndims, name='forward_log_det_jacobian', **kwargs
)
Returns both the forward_log_det_jacobian.
Args  

x

Tensor (structure). The input to the 'forward' Jacobian determinant
evaluation.

event_ndims

Number of dimensions in the probabilistic events being
transformed. Must be greater than or equal to
self.forward_min_event_ndims . The result is summed over the final
dimensions to produce a scalar Jacobian determinant for each event, i.e.
it has shape rank(x)  event_ndims dimensions.
Multipart bijectors require structured event_ndims, such that
rank(y[i])  rank(event_ndims[i]) is the same for all elements i of
the structured input. Furthermore, the first event_ndims[i] of each
x[i].shape must be the same for all i (broadcasting is not allowed).

name

The name to give this op. 
**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

Tensor (structure), if this bijector is injective.
If not injective this is not implemented.

Raises  

TypeError

if y 's dtype is incompatible with the expected output dtype.

NotImplementedError

if neither _forward_log_det_jacobian
nor {_inverse , _inverse_log_det_jacobian } are implemented, or
this is a noninjective bijector.

inverse
inverse(
y, name='inverse', **kwargs
)
Returns the inverse Bijector
evaluation, i.e., X = g^{1}(Y).
Args  

y

Tensor (structure). The input to the 'inverse' evaluation.

name

The name to give this op. 
**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

Tensor (structure), if this bijector is injective.
If not injective, returns the ktuple containing the unique
k points (x1, ..., xk) such that g(xi) = y .

Raises  

TypeError

if y 's structured dtype is incompatible with the expected
output dtype.

NotImplementedError

if _inverse is not implemented.

inverse_dtype
inverse_dtype(
dtype=UNSPECIFIED, name='inverse_dtype', **kwargs
)
Returns the dtype returned by inverse
for the provided input.
inverse_event_ndims
inverse_event_ndims(
event_ndims, **kwargs
)
Returns the number of event dimensions produced by inverse
.
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 (structure) indicating eventportion shape
passed into inverse function.

Returns  

inverse_event_shape_tensor

TensorShape (structure) indicating
eventportion 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 (structure) indicating
eventportion shape passed into inverse function.

name

name to give to the op 
Returns  

inverse_event_shape_tensor

Tensor , int32 vector (structure)
indicating eventportion shape after applying inverse .

inverse_log_det_jacobian
inverse_log_det_jacobian(
y, event_ndims, name='inverse_log_det_jacobian', **kwargs
)
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,
evaluated at g^{1}(y)
.
Args  

y

Tensor (structure). The input to the 'inverse' Jacobian determinant
evaluation.

event_ndims

Number of dimensions in the probabilistic events being
transformed. Must be greater than or equal to
self.inverse_min_event_ndims . The result is summed over the final
dimensions to produce a scalar Jacobian determinant for each event, i.e.
it has shape rank(y)  event_ndims dimensions.
Multipart bijectors require structured event_ndims, such that
rank(y[i])  rank(event_ndims[i]) is the same for all elements i of
the structured input. Furthermore, the first event_ndims[i] of each
x[i].shape must be the same for all i (broadcasting is not allowed).

name

The name to give this op. 
**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

ildj

Tensor , if this bijector is injective.
If not injective, returns the tuple of local log det
Jacobians, log(det(Dg_i^{1}(y))) , where g_i is the restriction
of g to the ith partition Di .

Raises  

TypeError

if x 's dtype is incompatible with the expected inversedtype.

NotImplementedError

if _inverse_log_det_jacobian is not implemented.

__call__
__call__(
value, name=None, **kwargs
)
Applies or composes the Bijector
, depending on input type.
This is a convenience function which applies the Bijector
instance in
three different ways, depending on the input:
 If the input is a
tfd.Distribution
instance, returntfd.TransformedDistribution(distribution=input, bijector=self)
.  If the input is a
tfb.Bijector
instance, returntfb.Chain([self, input])
.  Otherwise, return
self.forward(input)
Args  

value

A tfd.Distribution , tfb.Bijector , or a (structure of) Tensor .

name

Python str name given to ops created by this function.

**kwargs

Additional keyword arguments passed into the created
tfd.TransformedDistribution , tfb.Bijector , or self.forward .

Returns  

composition

A tfd.TransformedDistribution if the input was a
tfd.Distribution , a tfb.Chain if the input was a tfb.Bijector , or
a (structure of) Tensor computed by self.forward .

Examples
sigmoid = tfb.Reciprocal()(
tfb.Shift(shift=1.)(
tfb.Exp()(
tfb.Scale(scale=1.))))
# ==> `tfb.Chain([
# tfb.Reciprocal(),
# tfb.Shift(shift=1.),
# tfb.Exp(),
# tfb.Scale(scale=1.),
# ])` # ie, `tfb.Sigmoid()`
log_normal = tfb.Exp()(tfd.Normal(0, 1))
# ==> `tfd.TransformedDistribution(tfd.Normal(0, 1), tfb.Exp())`
tfb.Exp()([1., 0., 1.])
# ==> tf.exp([1., 0., 1.])
__eq__
__eq__(
other
)
Return self==value.