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# tfp.bijectors.IteratedSigmoidCentered

## Class `IteratedSigmoidCentered`

Bijector which applies a Stick Breaking procedure.

Inherits From: `Bijector`

Given a vector `x`, transform it in to a vector `y` such that `y[i] > 0, sum_i y[i] = 1.`. In other words, takes a vector in `R^{k-1}` (unconstrained space) and maps it to a vector in the unit simplex in `R^{k}`.

This transformation is centered in that it maps the zero vector `[0., 0., ... 0.]` to the center of the simplex `[1/k, ... 1/k]`.

This bijector arises from the stick-breaking procedure for constructing a Dirichlet distribution / Dirichlet process as defined in [Stan, 2018].

#### Example Use:

``````
bijector.IteratedSigmoidCentered().forward([0., 0., 0.])
# Result: [0.25, 0.25, 0.25, 0.25]
# Extra result: 0.25

bijector.IteratedSigmoidCentered().inverse([0.25, 0.25, 0.25, 0.25])
# Result: [0., 0., 0.]
# Extra coordinate removed.
``````

At first blush it may seem like the Invariance of domain theorem implies this implementation is not a bijection. However, the appended dimension makes the (forward) image non-open and the theorem does not directly apply.

: Stan Development Team. 2018. Stan Modeling Language Users Guide and Reference Manual, Version 2.18.0. http://mc-stan.org

## `__init__`

``````__init__(
validate_args=False,
name='iterated_sigmoid'
)
``````

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

### `__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:

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 `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 `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.])
``````

### `forward`

``````forward(
x,
name='forward',
**kwargs
)
``````

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.
• `**kwargs`: Named arguments forwarded to subclass implementation.

#### 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',
**kwargs
)
``````

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 `rank(x) - event_ndims` dimensions.
• `name`: The name to give this op.
• `**kwargs`: Named arguments forwarded to subclass implementation.

#### 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',
**kwargs
)
``````

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.
• `**kwargs`: Named arguments forwarded to subclass implementation.

#### 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',
**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`. 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.
• `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 `self.dtype` is specified and `y.dtype` is not `self.dtype`.
• `NotImplementedError`: if `_inverse_log_det_jacobian` is not implemented.