View source on GitHub 
Affine MaskedAutoregressiveFlow bijector.
Inherits From: Bijector
tfp.experimental.substrates.numpy.bijectors.MaskedAutoregressiveFlow(
shift_and_log_scale_fn=None, bijector_fn=None, is_constant_jacobian=False,
validate_args=False, unroll_loop=False, event_ndims=1, name=None
)
The affine autoregressive flow [(Papamakarios et al., 2016)][3] provides a relatively simple framework for userspecified (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) forwardmode calculation to draw samples and the (cheap)
reversemode calculation to compute logprobabilities. Conversely, a
TransformedDistribution
using Invert(MaskedAutoregressiveFlow(...))
uses
the (expensive) forwardmode calculation to compute logprobabilities and the
(cheap) reversemode 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 feedforward 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 forloop. 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 LSTMinspired 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.),
bijector=tfb.MaskedAutoregressiveFlow(
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.),
bijector=tfb.Invert(tfb.MaskedAutoregressiveFlow(
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.
made = tfb.AutoregressiveNetwork(params=1, hidden_units=[32])
maf_no_scale_hidden2 = tfd.TransformedDistribution(
distribution=tfd.Normal(loc=0., scale=1.),
bijector=tfb.MaskedAutoregressiveFlow(
lambda y: (made(y)[..., 0], None),
is_constant_jacobian=True),
event_shape=[dims])
maf_no_scale_hidden2._made = made # Ensure maf_no_scale_hidden2.trainable
# 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:
Replace the Python function with a
tf.Module
,tf.keras.Layer
, or other callable object through whichtf.Module
can find variables.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]) made2 = tfb.AutoregressiveNetwork(params=1, hidden_units=[10, 10]) maf = tfb.MaskedAutoregressiveFlow(lambda y: (made1(y), made2(y) + 1.)) maf._made_variables = made1.variables + made2.variables
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
Args  

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.

Raises  

ValueError

If both or none of shift_and_log_scale_fn and bijector_fn
are specified.

Attributes  

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
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_dtype
forward_dtype(
dtype, name='forward_dtype', **kwargs
)
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
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 eventportion shape passed into
forward function.

Returns  

forward_event_shape_tensor

TensorShape indicating eventportion 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 eventportion shape
passed into forward function.

name

name to give to the op 
Returns  

forward_event_shape_tensor

Tensor , int32 vector indicating
eventportion 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 noninjective 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 ktuple 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
inverse_dtype(
dtype, name='inverse_dtype', **kwargs
)
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
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 eventportion shape passed into
inverse function.

Returns  

inverse_event_shape_tensor

TensorShape indicating eventportion 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 eventportion shape
passed into inverse function.

name

name to give to the op 
Returns  

inverse_event_shape_tensor

Tensor , int32 vector indicating
eventportion 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.

__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:
 If the input is a
tfd.Distribution
instance, returntfd.TransformedDistribution(distribution=input, bijector=self)
.  If the input is a
tfb.Bijector
instance, returntfb.Chain([self, input])
.  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.])