Inherits From: `Bijector`

The affine autoregressive flow [(Papamakarios et al., 2016)][3] provides a relatively simple framework for user-specified (deep) architectures to learn a distribution over continuous events. Regarding terminology,

'Autoregressive models decompose the joint density as a product of conditionals, and model each conditional in turn. Normalizing flows transform a base density (e.g. a standard Gaussian) into the target density by an invertible transformation with tractable Jacobian.' [(Papamakarios et al., 2016)][3]

In other words, the 'autoregressive property' is equivalent to the decomposition, `p(x) = prod{ p(x[perm[i]] | x[perm[0:i]]) : i=0, ..., d }` where `perm` is some permutation of `{0, ..., d}`. In the simple case where the permutation is identity this reduces to: `p(x) = prod{ p(x[i] | x[0:i]) : i=0, ..., d }`.

In TensorFlow Probability, 'normalizing flows' are implemented as `tfp.bijectors.Bijector`s. The `forward` 'autoregression' is implemented using a `tf.while_loop` and a deep neural network (DNN) with masked weights such that the autoregressive property is automatically met in the `inverse`.

A `TransformedDistribution` using `MaskedAutoregressiveFlow(...)` uses the (expensive) forward-mode calculation to draw samples and the (cheap) reverse-mode calculation to compute log-probabilities. Conversely, a `TransformedDistribution` using `Invert(MaskedAutoregressiveFlow(...))` uses the (expensive) forward-mode calculation to compute log-probabilities and the (cheap) reverse-mode calculation to compute samples. See 'Example Use' [below] for more details.

Given a `shift_and_log_scale_fn`, the forward and inverse transformations are (a sequence of) affine transformations. A 'valid' `shift_and_log_scale_fn` must compute each `shift` (aka `loc` or 'mu' in [Germain et al. (2015)][1]) and `log(scale)` (aka 'alpha' in [Germain et al. (2015)][1]) such that each are broadcastable with the arguments to `forward` and `inverse`, i.e., such that the calculations in `forward`, `inverse` [below] are possible.

For convenience, `tfp.bijectors.AutoregressiveNetwork` is offered as a possible `shift_and_log_scale_fn` function. It implements the MADE architecture [(Germain et al., 2015)][1]. MADE is a feed-forward network that computes a `shift` and `log(scale)` using masked dense layers in a deep neural network. Weights are masked to ensure the autoregressive property. It is possible that this architecture is suboptimal for your task. To build alternative networks, either change the arguments to `tfp.bijectors.AutoregressiveNetwork` or use some other architecture, e.g., using `tf.keras.layers`.

Assuming `shift_and_log_scale_fn` has valid shape and autoregressive semantics, the forward transformation is

``````def forward(x):
y = zeros_like(x)
event_size = x.shape[-event_dims:].num_elements()
for _ in range(event_size):
shift, log_scale = shift_and_log_scale_fn(y)
y = x * tf.exp(log_scale) + shift
return y
``````

and the inverse transformation is

``````def inverse(y):
shift, log_scale = shift_and_log_scale_fn(y)
return (y - shift) / tf.exp(log_scale)
``````

Notice that the `inverse` does not need a for-loop. This is because in the forward pass each calculation of `shift` and `log_scale` is based on the `y` calculated so far (not `x`). In the `inverse`, the `y` is fully known, thus is equivalent to the scaling used in `forward` after `event_size` passes, i.e., the 'last' `y` used to compute `shift`, `log_scale`. (Roughly speaking, this also proves the transform is bijective.)

The `bijector_fn` argument allows specifying a more general coupling relation, such as the LSTM-inspired activation from [4], or Neural Spline Flow [5]. It must logically operate on each element of the input individually, and still obey the 'autoregressive property' described above. The forward transformation is

``````def forward(x):
y = zeros_like(x)
event_size = x.shape[-event_dims:].num_elements()
for _ in range(event_size):
bijector = bijector_fn(y)
y = bijector.forward(x)
return y
``````

and inverse transformation is

``````def inverse(y):
bijector = bijector_fn(y)
return bijector.inverse(y)
``````

#### Examples

``````tfd = tfp.distributions
tfb = tfp.bijectors

dims = 2

# A common choice for a normalizing flow is to use a Gaussian for the base
# distribution.  (However, any continuous distribution would work.) E.g.,
maf = tfd.TransformedDistribution(
distribution=tfd.Normal(loc=0., scale=1.),
shift_and_log_scale_fn=tfb.AutoregressiveNetwork(
params=2, hidden_units=[512, 512])),
event_shape=[dims])

x = maf.sample()  # Expensive; uses `tf.while_loop`, no Bijector caching.
maf.log_prob(x)   # Almost free; uses Bijector caching.
# Cheap; no `tf.while_loop` despite no Bijector caching.
maf.log_prob(tf.zeros(dims))

# [Papamakarios et al. (2016)][3] also describe an Inverse Autoregressive
# Flow [(Kingma et al., 2016)][2]:
iaf = tfd.TransformedDistribution(
distribution=tfd.Normal(loc=0., scale=1.),
shift_and_log_scale_fn=tfb.AutoregressiveNetwork(
params=2, hidden_units=[512, 512]))),
event_shape=[dims])

x = iaf.sample()  # Cheap; no `tf.while_loop` despite no Bijector caching.
iaf.log_prob(x)   # Almost free; uses Bijector caching.
# Expensive; uses `tf.while_loop`, no Bijector caching.
iaf.log_prob(tf.zeros(dims))

# In many (if not most) cases the default `shift_and_log_scale_fn` will be a
# poor choice.  Here's an example of using a 'shift only' version and with a
# different number/depth of hidden layers.
maf_no_scale_hidden2 = tfd.TransformedDistribution(
distribution=tfd.Normal(loc=0., scale=1.),
is_constant_jacobian=True),
event_shape=[dims])
# NOTE: The last line ensures that maf_no_scale_hidden2.trainable_variables
# will include all variables from `made`.
``````

#### Variable Tracking

A `tfb.MaskedAutoregressiveFlow` instance saves a reference to the values passed as `shift_and_log_scale_fn` and `bijector_fn` to its constructor. Thus, for most values passed as `shift_and_log_scale_fn` or `bijector_fn`, variables referenced by those values will be found and tracked by the `tfb.MaskedAutoregressiveFlow` instance. Please see the `tf.Module` documentation for further details.

However, if the value passed to `shift_and_log_scale_fn` or `bijector_fn` is a Python function, then `tfb.MaskedAutoregressiveFlow` cannot automatically track variables used inside `shift_and_log_scale_fn` or `bijector_fn`. To get `tfb.MaskedAutoregressiveFlow` to track such variables, either:

1. Replace the Python function with a `tf.Module`, `tf.keras.Layer`, or other callable object through which `tf.Module` can find variables.

2. Or, add a reference to the variables to the `tfb.MaskedAutoregressiveFlow` instance by setting an attribute -- for example:

``````made1 = tfb.AutoregressiveNetwork(params=1, hidden_units=[10, 10])
``````

#### References

[1]: Mathieu Germain, Karol Gregor, Iain Murray, and Hugo Larochelle. MADE: Masked Autoencoder for Distribution Estimation. In International Conference on Machine Learning, 2015. https://arxiv.org/abs/1502.03509

[2]: Diederik P. Kingma, Tim Salimans, Rafal Jozefowicz, Xi Chen, Ilya Sutskever, and Max Welling. Improving Variational Inference with Inverse Autoregressive Flow. In Neural Information Processing Systems, 2016. https://arxiv.org/abs/1606.04934

[3]: George Papamakarios, Theo Pavlakou, and Iain Murray. Masked Autoregressive Flow for Density Estimation. In Neural Information Processing Systems, 2017. https://arxiv.org/abs/1705.07057

[4]: Diederik P Kingma, Tim Salimans, Max Welling. Improving Variational Inference with Inverse Autoregressive Flow. In Neural Information Processing Systems, 2016. https://arxiv.org/abs/1606.04934

[5]: Conor Durkan, Artur Bekasov, Iain Murray, George Papamakarios. Neural Spline Flows, 2019. http://arxiv.org/abs/1906.04032

`shift_and_log_scale_fn` Python `callable` which computes `shift` and `log_scale` from the inverse domain (`y`). Calculation must respect the 'autoregressive property' (see class docstring). Suggested default `tfb.AutoregressiveNetwork(params=2, hidden_layers=...)`. Typically the function contains `tf.Variables`. Returning `None` for either (both) `shift`, `log_scale` is equivalent to (but more efficient than) returning zero. If `shift_and_log_scale_fn` returns a single `Tensor`, the returned value will be unstacked to get the `shift` and `log_scale`: `tf.unstack(shift_and_log_scale_fn(y), num=2, axis=-1)`.
`bijector_fn` Python `callable` which returns a `tfb.Bijector` which transforms event tensor with the signature `(input, **condition_kwargs) -> bijector`. The bijector must operate on scalar events and must not alter the rank of its input. The `bijector_fn` will be called with `Tensors` from the inverse domain (`y`). Calculation must respect the 'autoregressive property' (see class docstring).
`is_constant_jacobian` Python `bool`. Default: `False`. When `True` the implementation assumes `log_scale` does not depend on the forward domain (`x`) or inverse domain (`y`) values. (No validation is made; `is_constant_jacobian=False` is always safe but possibly computationally inefficient.)
`validate_args` Python `bool` indicating whether arguments should be checked for correctness.
`unroll_loop` Python `bool` indicating whether the `tf.while_loop` in `_forward` should be replaced with a static for loop. Requires that the final dimension of `x` be known at graph construction time. Defaults to `False`.
`event_ndims` Python `integer`, the intrinsic dimensionality of this bijector. 1 corresponds to a simple vector autoregressive bijector as implemented by the `tfp.bijectors.AutoregressiveNetwork`, 2 might be useful for a 2D convolutional `shift_and_log_scale_fn` and so on.
`name` Python `str`, name given to ops managed by this object.

`ValueError` If both or none of `shift_and_log_scale_fn` and `bijector_fn` are specified.

`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.

`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`. 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_dtype`

View source

Returns the dtype of the output of the forward transformation.

Args
`dtype` `tf.dtype`, or nested structure of `tf.dtype`s, of the input to `forward`.
`name` The name to give this op.
`**kwargs` Named arguments forwarded to subclass implementation.

Returns
`tf.dtype` or nested structure of `tf.dtype`s of the output of `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` 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`

View source

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`

View source

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`

View source

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_dtype`

View source

Returns the dtype of the output of the inverse transformation.

Args
`dtype` `tf.dtype`, or nested structure of `tf.dtype`s, of the input to `inverse`.
`name` The name to give this op.
`**kwargs` Named arguments forwarded to subclass implementation.

Returns
`tf.dtype` or nested structure of `tf.dtype`s of the output of `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` 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`

View source

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`

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`. 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.

### `__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 `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.])
``````