# tf.contrib.distributions.bijectors.Bijector

## Class `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 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)

``````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`.

#### 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 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.

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

WARNING Handing of non-injective transforms is subject to change.

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 = tf.contrib.distributions.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.)
``````

## `__init__`

``````__init__(
graph_parents=None,
is_constant_jacobian=False,
validate_args=False,
dtype=None,
forward_min_event_ndims=None,
inverse_min_event_ndims=None,
name=None
)
``````

Constructs Bijector.

A `Bijector` transforms random variables into new random variables.

Examples:

``````# Create the Y = g(X) = X transform.
identity = Identity()

# Create the Y = g(X) = exp(X) transform.
exp = Exp()
``````

See `Bijector` subclass docstring for more details and specific examples.

#### 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.
• `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.

#### 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`.

## Properties

### `dtype`

dtype of `Tensor`s 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.

#### 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

### `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,
event_ndims,
name='forward_log_det_jacobian'
)
``````

Returns both the forward_log_det_jacobian.

#### Args:

• `x`: `Tensor`. 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 `x.shape.ndims - event_ndims` dimensions.
• `name`: The name to give this op.

#### Returns:

`Tensor`, if this bijector is injective. If not injective this is not implemented.

#### 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, or this is a non-injective bijector.

### `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`, 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 `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,
event_ndims,
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, evaluated at `g^{-1}(y)`.

#### Args:

• `y`: `Tensor`. 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 `y.shape.ndims - event_ndims` dimensions.
• `name`: The name to give this op.

#### Returns:

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