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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) +
  • 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.


  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__(

    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):
      # 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)


  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.

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.


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__(

  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 array_ops.zero_like(x).
    # However, we circumvent materializing that, since the jacobian
    # calculation is input independent, and we specify it for one input.
    return constant_op.constant(0., x.dtype.base_dtype)

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.


    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 imples 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.distributions.bijectors.AbsoluteValue()

==> 1.

==> 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.
==> (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.
forward_min_event_ndims Python integer indicating the minimum number of dimensions forward operates on.
inverse_min_event_ndims Python integer indicating the minimum number of dimensions inverse operates on. Will be set to forward_min_event_ndims by default, if no value is provided.
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 dtype of Tensors transformable by this distribution.
forward_min_event_ndims Returns the minimal number of dimensions bijector.forward operates on.
graph_parents Returns this Bijector's graph_parents as a Python list.
inverse_min_event_ndims Returns the minimal number of dimensions bijector.inverse operates on.
is_constant_jacobian Returns true iff the Jacobian matrix is not a function of x.