tfp.substrates.numpy.bijectors.Bijector

Interface for transformations of a Distribution sample.

Bijectors can be used to represent any differentiable and injective (one to one) function defined on an open subset of R^n. Some non-injective 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:

  1. Forward

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

  2. Inverse

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

  3. log_det_jacobian(x)

    'The log of the absolute value 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 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 orientation-reversing transformation. It is ok for us to discard the sign of the determinant because we only integrate everywhere-nonnegative 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 log-likelihood:
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 vector-valued 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 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 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 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 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 true log_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 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.

    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 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.)

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 left-semicontinuous 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 Log|dX/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.)

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.

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.

dtype

forward_min_event_ndims Returns the minimal number of dimensions bijector.forward operates on.

Multipart bijectors return structured ndims, which indicates the expected structure of their inputs. Some multipart bijectors, notably Composites, may return structures of None.

graph_parents Returns this Bijector's graph_parents as a Python list.
has_static_min_event_ndims Returns True if the bijector has statically-known min_event_ndims.
inverse_min_event_ndims Returns the minimal number of dimensions bijector.inverse operates on.

Multipart bijectors return structured event_ndims, which indicates the expected structure of their outputs. Some multipart bijectors, notably Composites, may return structures of None.

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

View source

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

View source

Returns the dtype returned by forward for the provided input.

forward_event_ndims

View source

Returns the number of event dimensions produced by forward.

forward_event_shape

View source

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 event-portion shape passed into forward function.

Returns
forward_event_shape_tensor TensorShape (structure) indicating event-portion shape after applying forward. Possibly unknown.

forward_event_shape_tensor

View source

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

Args
input_shape Tensor, int32 vector (structure) indicating event-portion shape passed into forward function.
name name to give to the op

Returns
forward_event_shape_tensor Tensor, int32 vector (structure) indicating event-portion shape after applying forward.

forward_log_det_jacobian

View source

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 non-injective bijector.

inverse

View source

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 k-tuple 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

View source

Returns the dtype returned by inverse for the provided input.

inverse_event_ndims

View source

Returns the number of event dimensions produced by inverse.

inverse_event_shape

View source

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 event-portion shape passed into inverse function.

Returns
inverse_event_shape_tensor TensorShape (structure) indicating event-portion shape after applying inverse. Possibly unknown.

inverse_event_shape_tensor

View source

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

Args
output_shape Tensor, int32 vector (structure) indicating event-portion shape passed into inverse function.
name name to give to the op

Returns
inverse_event_shape_tensor Tensor, int32 vector (structure) indicating event-portion shape after applying inverse.

inverse_log_det_jacobian

View source

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 inverse-dtype.
NotImplementedError if _inverse_log_det_jacobian is not implemented.

__call__

View source

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:

  1. If the input is a tfd.Distribution instance, return tfd.TransformedDistribution(distribution=input, bijector=self).
  2. If the input is a tfb.Bijector instance, return tfb.Chain([self, input]).
  3. 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.AffineScalar(shift=1.)(
      tfb.Exp()(
        tfb.AffineScalar(scale=-1.))))
# ==> `tfb.Chain([
#         tfb.Reciprocal(),
#         tfb.AffineScalar(shift=1.),
#         tfb.Exp(),
#         tfb.AffineScalar(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__

View source

Return self==value.