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VectorDiffeomixture distribution.
Inherits From: Distribution
tf.contrib.distributions.VectorDiffeomixture(
mix_loc, temperature, distribution, loc=None, scale=None, quadrature_size=8, qua
drature_fn=tf.contrib.distributions.quadrature_scheme_softmaxnormal_quantiles,
validate_args=False, allow_nan_stats=True, name='VectorDiffeomixture'
)
A vector diffeomixture (VDM) is a distribution parameterized by a convex
combination of K
component loc
vectors, loc[k], k = 0,...,K-1
, and K
scale
matrices scale[k], k = 0,..., K-1
. It approximates the following
compound distribution
p(x) = int p(x | z) p(z) dz,
where z is in the K-simplex, and
p(x | z) := p(x | loc=sum_k z[k] loc[k], scale=sum_k z[k] scale[k])
The integral int p(x | z) p(z) dz
is approximated with a quadrature scheme
adapted to the mixture density p(z)
. The N
quadrature points z_{N, n}
and weights w_{N, n}
(which are non-negative and sum to 1) are chosen
such that
as N --> infinity
.
Since q_N(x)
is in fact a mixture (of N
points), we may sample from
q_N
exactly. It is important to note that the VDM is defined as q_N
above, and not p(x)
. Therefore, sampling and pdf may be implemented as
exact (up to floating point error) methods.
A common choice for the conditional p(x | z)
is a multivariate Normal.
The implemented marginal p(z)
is the SoftmaxNormal
, which is a
K-1
dimensional Normal transformed by a SoftmaxCentered
bijector, making
it a density on the K
-simplex. That is,
Z = SoftmaxCentered(X),
X = Normal(mix_loc / temperature, 1 / temperature)
The default quadrature scheme chooses z_{N, n}
as N
midpoints of
the quantiles of p(z)
(generalized quantiles if K > 2
).
See [Dillon and Langmore (2018)][1] for more details.
About Vector
distributions in TensorFlow.
The VectorDiffeomixture
is a non-standard distribution that has properties
particularly useful in variational Bayesian
methods.
Conditioned on a draw from the SoftmaxNormal, X|z
is a vector whose
components are linear combinations of affine transformations, thus is itself
an affine transformation.
About Diffeomixture
s and reparameterization.
The VectorDiffeomixture
is designed to be reparameterized, i.e., its
parameters are only used to transform samples from a distribution which has no
trainable parameters. This property is important because backprop stops at
sources of stochasticity. That is, as long as the parameters are used after
the underlying source of stochasticity, the computed gradient is accurate.
Reparametrization means that we can use gradient-descent (via backprop) to optimize Monte-Carlo objectives. Such objectives are a finite-sample approximation of an expectation and arise throughout scientific computing.
Examples
import tensorflow_probability as tfp
tfd = tfp.distributions
# Create two batches of VectorDiffeomixtures, one with mix_loc=[0.],
# another with mix_loc=[1]. In both cases, `K=2` and the affine
# transformations involve:
# k=0: loc=zeros(dims) scale=LinearOperatorScaledIdentity
# k=1: loc=[2.]*dims scale=LinOpDiag
dims = 5
vdm = tfd.VectorDiffeomixture(
mix_loc=[[0.], [1]],
temperature=[1.],
distribution=tfd.Normal(loc=0., scale=1.),
loc=[
None, # Equivalent to `np.zeros(dims, dtype=np.float32)`.
np.float32([2.]*dims),
],
scale=[
tf.linalg.LinearOperatorScaledIdentity(
num_rows=dims,
multiplier=np.float32(1.1),
is_positive_definite=True),
tf.linalg.LinearOperatorDiag(
diag=np.linspace(2.5, 3.5, dims, dtype=np.float32),
is_positive_definite=True),
],
validate_args=True)
References
[1]: Joshua Dillon and Ian Langmore. Quadrature Compound: An approximating family of distributions. arXiv preprint arXiv:1801.03080, 2018. https://arxiv.org/abs/1801.03080
Args | |
---|---|
mix_loc
|
float -like Tensor with shape [b1, ..., bB, K-1] .
In terms of samples, larger mix_loc[..., k] ==>
Z is more likely to put more weight on its kth component.
|
temperature
|
float -like Tensor . Broadcastable with mix_loc .
In terms of samples, smaller temperature means one component is more
likely to dominate. I.e., smaller temperature makes the VDM look more
like a standard mixture of K components.
|
distribution
|
tf.Distribution -like instance. Distribution from which d
iid samples are used as input to the selected affine transformation.
Must be a scalar-batch, scalar-event distribution. Typically
distribution.reparameterization_type = FULLY_REPARAMETERIZED or it is
a function of non-trainable parameters. WARNING: If you backprop through
a VectorDiffeomixture sample and the distribution is not
FULLY_REPARAMETERIZED yet is a function of trainable variables, then
the gradient will be incorrect!
|
loc
|
Length-K list of float -type Tensor s. The k -th element
represents the shift used for the k -th affine transformation. If
the k -th item is None , loc is implicitly 0 . When specified,
must have shape [B1, ..., Bb, d] where b >= 0 and d is the event
size.
|
scale
|
Length-K list of LinearOperator s. Each should be
positive-definite and operate on a d -dimensional vector space. The
k -th element represents the scale used for the k -th affine
transformation. LinearOperator s must have shape [B1, ..., Bb, d, d] ,
b >= 0 , i.e., characterizes b -batches of d x d matrices
|
quadrature_size
|
Python int scalar representing number of
quadrature points. Larger quadrature_size means q_N(x) better
approximates p(x) .
|
quadrature_fn
|
Python callable taking normal_loc , normal_scale ,
quadrature_size , validate_args and returning tuple(grid, probs)
representing the SoftmaxNormal grid and corresponding normalized weight.
normalized) weight.
Default value: quadrature_scheme_softmaxnormal_quantiles .
|
validate_args
|
Python bool , default False . When True distribution
parameters are checked for validity despite possibly degrading runtime
performance. When False invalid inputs may silently render incorrect
outputs.
|
allow_nan_stats
|
Python bool , default True . When True ,
statistics (e.g., mean, mode, variance) use the value "NaN " to
indicate the result is undefined. When False , an exception is raised
if one or more of the statistic's batch members are undefined.
|
name
|
Python str name prefixed to Ops created by this class.
|
Raises | |
---|---|
ValueError
|
if not scale or len(scale) < 2 .
|
ValueError
|
if len(loc) != len(scale)
|
ValueError
|
if quadrature_grid_and_probs is not None and
len(quadrature_grid_and_probs[0]) != len(quadrature_grid_and_probs[1])
|
ValueError
|
if validate_args and any not scale.is_positive_definite.
|
TypeError
|
if any scale.dtype != scale[0].dtype. |
TypeError
|
if any loc.dtype != scale[0].dtype. |
NotImplementedError
|
if len(scale) != 2 .
|
ValueError
|
if not distribution.is_scalar_batch .
|
ValueError
|
if not distribution.is_scalar_event .
|
Attributes | |
---|---|
allow_nan_stats
|
Python bool describing behavior when a stat is undefined.
Stats return +/- infinity when it makes sense. E.g., the variance of a Cauchy distribution is infinity. However, sometimes the statistic is undefined, e.g., if a distribution's pdf does not achieve a maximum within the support of the distribution, the mode is undefined. If the mean is undefined, then by definition the variance is undefined. E.g. the mean for Student's T for df = 1 is undefined (no clear way to say it is either + or - infinity), so the variance = E[(X - mean)**2] is also undefined. |
batch_shape
|
Shape of a single sample from a single event index as a TensorShape .
May be partially defined or unknown. The batch dimensions are indexes into independent, non-identical parameterizations of this distribution. |
distribution
|
Base scalar-event, scalar-batch distribution. |
dtype
|
The DType of Tensor s handled by this Distribution .
|
endpoint_affine
|
Affine transformation for each of K components.
|
event_shape
|
Shape of a single sample from a single batch as a TensorShape .
May be partially defined or unknown. |
grid
|
Grid of mixing probabilities, one for each grid point. |
interpolated_affine
|
Affine transformation for each convex combination of K components.
|
mixture_distribution
|
Distribution used to select a convex combination of affine transforms. |
name
|
Name prepended to all ops created by this Distribution .
|
parameters
|
Dictionary of parameters used to instantiate this Distribution .
|
reparameterization_type
|
Describes how samples from the distribution are reparameterized.
Currently this is one of the static instances
|
validate_args
|
Python bool indicating possibly expensive checks are enabled.
|
Methods
batch_shape_tensor
batch_shape_tensor(
name='batch_shape_tensor'
)
Shape of a single sample from a single event index as a 1-D Tensor
.
The batch dimensions are indexes into independent, non-identical parameterizations of this distribution.
Args | |
---|---|
name
|
name to give to the op |
Returns | |
---|---|
batch_shape
|
Tensor .
|
cdf
cdf(
value, name='cdf'
)
Cumulative distribution function.
Given random variable X
, the cumulative distribution function cdf
is:
cdf(x) := P[X <= x]
Args | |
---|---|
value
|
float or double Tensor .
|
name
|
Python str prepended to names of ops created by this function.
|
Returns | |
---|---|
cdf
|
a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .
|
copy
copy(
**override_parameters_kwargs
)
Creates a deep copy of the distribution.
Args | |
---|---|
**override_parameters_kwargs
|
String/value dictionary of initialization arguments to override with new values. |
Returns | |
---|---|
distribution
|
A new instance of type(self) initialized from the union
of self.parameters and override_parameters_kwargs, i.e.,
dict(self.parameters, **override_parameters_kwargs) .
|
covariance
covariance(
name='covariance'
)
Covariance.
Covariance is (possibly) defined only for non-scalar-event distributions.
For example, for a length-k
, vector-valued distribution, it is calculated
as,
Cov[i, j] = Covariance(X_i, X_j) = E[(X_i - E[X_i]) (X_j - E[X_j])]
where Cov
is a (batch of) k x k
matrix, 0 <= (i, j) < k
, and E
denotes expectation.
Alternatively, for non-vector, multivariate distributions (e.g.,
matrix-valued, Wishart), Covariance
shall return a (batch of) matrices
under some vectorization of the events, i.e.,
Cov[i, j] = Covariance(Vec(X)_i, Vec(X)_j) = [as above]
where Cov
is a (batch of) k' x k'
matrices,
0 <= (i, j) < k' = reduce_prod(event_shape)
, and Vec
is some function
mapping indices of this distribution's event dimensions to indices of a
length-k'
vector.
Args | |
---|---|
name
|
Python str prepended to names of ops created by this function.
|
Returns | |
---|---|
covariance
|
Floating-point Tensor with shape [B1, ..., Bn, k', k']
where the first n dimensions are batch coordinates and
k' = reduce_prod(self.event_shape) .
|
cross_entropy
cross_entropy(
other, name='cross_entropy'
)
Computes the (Shannon) cross entropy.
Denote this distribution (self
) by P
and the other
distribution by
Q
. Assuming P, Q
are absolutely continuous with respect to
one another and permit densities p(x) dr(x)
and q(x) dr(x)
, (Shanon)
cross entropy is defined as:
H[P, Q] = E_p[-log q(X)] = -int_F p(x) log q(x) dr(x)
where F
denotes the support of the random variable X ~ P
.
Args | |
---|---|
other
|
tfp.distributions.Distribution instance.
|
name
|
Python str prepended to names of ops created by this function.
|
Returns | |
---|---|
cross_entropy
|
self.dtype Tensor with shape [B1, ..., Bn]
representing n different calculations of (Shanon) cross entropy.
|
entropy
entropy(
name='entropy'
)
Shannon entropy in nats.
event_shape_tensor
event_shape_tensor(
name='event_shape_tensor'
)
Shape of a single sample from a single batch as a 1-D int32 Tensor
.
Args | |
---|---|
name
|
name to give to the op |
Returns | |
---|---|
event_shape
|
Tensor .
|
is_scalar_batch
is_scalar_batch(
name='is_scalar_batch'
)
Indicates that batch_shape == []
.
Args | |
---|---|
name
|
Python str prepended to names of ops created by this function.
|
Returns | |
---|---|
is_scalar_batch
|
bool scalar Tensor .
|
is_scalar_event
is_scalar_event(
name='is_scalar_event'
)
Indicates that event_shape == []
.
Args | |
---|---|
name
|
Python str prepended to names of ops created by this function.
|
Returns | |
---|---|
is_scalar_event
|
bool scalar Tensor .
|
kl_divergence
kl_divergence(
other, name='kl_divergence'
)
Computes the Kullback--Leibler divergence.
Denote this distribution (self
) by p
and the other
distribution by
q
. Assuming p, q
are absolutely continuous with respect to reference
measure r
, the KL divergence is defined as:
KL[p, q] = E_p[log(p(X)/q(X))]
= -int_F p(x) log q(x) dr(x) + int_F p(x) log p(x) dr(x)
= H[p, q] - H[p]
where F
denotes the support of the random variable X ~ p
, H[., .]
denotes (Shanon) cross entropy, and H[.]
denotes (Shanon) entropy.
Args | |
---|---|
other
|
tfp.distributions.Distribution instance.
|
name
|
Python str prepended to names of ops created by this function.
|
Returns | |
---|---|
kl_divergence
|
self.dtype Tensor with shape [B1, ..., Bn]
representing n different calculations of the Kullback-Leibler
divergence.
|
log_cdf
log_cdf(
value, name='log_cdf'
)
Log cumulative distribution function.
Given random variable X
, the cumulative distribution function cdf
is:
log_cdf(x) := Log[ P[X <= x] ]
Often, a numerical approximation can be used for log_cdf(x)
that yields
a more accurate answer than simply taking the logarithm of the cdf
when
x << -1
.
Args | |
---|---|
value
|
float or double Tensor .
|
name
|
Python str prepended to names of ops created by this function.
|
Returns | |
---|---|
logcdf
|
a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .
|
log_prob
log_prob(
value, name='log_prob'
)
Log probability density/mass function.
Args | |
---|---|
value
|
float or double Tensor .
|
name
|
Python str prepended to names of ops created by this function.
|
Returns | |
---|---|
log_prob
|
a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .
|
log_survival_function
log_survival_function(
value, name='log_survival_function'
)
Log survival function.
Given random variable X
, the survival function is defined:
log_survival_function(x) = Log[ P[X > x] ]
= Log[ 1 - P[X <= x] ]
= Log[ 1 - cdf(x) ]
Typically, different numerical approximations can be used for the log
survival function, which are more accurate than 1 - cdf(x)
when x >> 1
.
Args | |
---|---|
value
|
float or double Tensor .
|
name
|
Python str prepended to names of ops created by this function.
|
Returns | |
---|---|
Tensor of shape sample_shape(x) + self.batch_shape with values of type
self.dtype .
|
mean
mean(
name='mean'
)
Mean.
mode
mode(
name='mode'
)
Mode.
param_shapes
@classmethod
param_shapes( sample_shape, name='DistributionParamShapes' )
Shapes of parameters given the desired shape of a call to sample()
.
This is a class method that describes what key/value arguments are required
to instantiate the given Distribution
so that a particular shape is
returned for that instance's call to sample()
.
Subclasses should override class method _param_shapes
.
Args | |
---|---|
sample_shape
|
Tensor or python list/tuple. Desired shape of a call to
sample() .
|
name
|
name to prepend ops with. |
Returns | |
---|---|
dict of parameter name to Tensor shapes.
|
param_static_shapes
@classmethod
param_static_shapes( sample_shape )
param_shapes with static (i.e. TensorShape
) shapes.
This is a class method that describes what key/value arguments are required
to instantiate the given Distribution
so that a particular shape is
returned for that instance's call to sample()
. Assumes that the sample's
shape is known statically.
Subclasses should override class method _param_shapes
to return
constant-valued tensors when constant values are fed.
Args | |
---|---|
sample_shape
|
TensorShape or python list/tuple. Desired shape of a call
to sample() .
|
Returns | |
---|---|
dict of parameter name to TensorShape .
|
Raises | |
---|---|
ValueError
|
if sample_shape is a TensorShape and is not fully defined.
|
prob
prob(
value, name='prob'
)
Probability density/mass function.
Args | |
---|---|
value
|
float or double Tensor .
|
name
|
Python str prepended to names of ops created by this function.
|
Returns | |
---|---|
prob
|
a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .
|
quantile
quantile(
value, name='quantile'
)
Quantile function. Aka "inverse cdf" or "percent point function".
Given random variable X
and p in [0, 1]
, the quantile
is:
quantile(p) := x such that P[X <= x] == p
Args | |
---|---|
value
|
float or double Tensor .
|
name
|
Python str prepended to names of ops created by this function.
|
Returns | |
---|---|
quantile
|
a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .
|
sample
sample(
sample_shape=(), seed=None, name='sample'
)
Generate samples of the specified shape.
Note that a call to sample()
without arguments will generate a single
sample.
Args | |
---|---|
sample_shape
|
0D or 1D int32 Tensor . Shape of the generated samples.
|
seed
|
Python integer seed for RNG |
name
|
name to give to the op. |
Returns | |
---|---|
samples
|
a Tensor with prepended dimensions sample_shape .
|
stddev
stddev(
name='stddev'
)
Standard deviation.
Standard deviation is defined as,
stddev = E[(X - E[X])**2]**0.5
where X
is the random variable associated with this distribution, E
denotes expectation, and stddev.shape = batch_shape + event_shape
.
Args | |
---|---|
name
|
Python str prepended to names of ops created by this function.
|
Returns | |
---|---|
stddev
|
Floating-point Tensor with shape identical to
batch_shape + event_shape , i.e., the same shape as self.mean() .
|
survival_function
survival_function(
value, name='survival_function'
)
Survival function.
Given random variable X
, the survival function is defined:
survival_function(x) = P[X > x]
= 1 - P[X <= x]
= 1 - cdf(x).
Args | |
---|---|
value
|
float or double Tensor .
|
name
|
Python str prepended to names of ops created by this function.
|
Returns | |
---|---|
Tensor of shape sample_shape(x) + self.batch_shape with values of type
self.dtype .
|
variance
variance(
name='variance'
)
Variance.
Variance is defined as,
Var = E[(X - E[X])**2]
where X
is the random variable associated with this distribution, E
denotes expectation, and Var.shape = batch_shape + event_shape
.
Args | |
---|---|
name
|
Python str prepended to names of ops created by this function.
|
Returns | |
---|---|
variance
|
Floating-point Tensor with shape identical to
batch_shape + event_shape , i.e., the same shape as self.mean() .
|