Class GaussianProcessRegressionModel
Inherits From: MultivariateNormalLinearOperator
Posterior predictive distribution in a conjugate GP regression model.
This class represents the distribution over function values at a set of points
in some index set, conditioned on noisy observations at some other set of
points. More specifically, we assume a Gaussian process prior, f ~ GP(m, k)
with IID normal noise on observations of function values. In this model
posterior inference can be done analytically. This Distribution
is
parameterized by
- the mean and covariance functions of the GP prior,
- the set of (noisy) observations and index points to which they correspond,
- the set of index points at which the resulting posterior predictive distribution over function values is defined,
- the observation noise variance,
- jitter, to compensate for numerical instability of Cholesky decomposition,
in addition to the usual params like validate_args
and allow_nan_stats
.
Mathematical Details
Gaussian process regression (GPR) assumes a Gaussian process (GP) prior and a
normal likelihood as a generative model for data. Given GP mean function m
,
covariance kernel k
, and observation noise variance v
, we have
f ~ GP(m, k)
iid
(y[i] | f, x[i]) ~ Normal(f(x[i]), v), i = 1, ... , N
where y[i]
are the noisy observations of function values at points x[i]
.
In practice, f
is an infinite object (eg, a function over R^n
) which can't
be realized on a finite machine, but fortunately the marginal distribution
over function values at a finite set of points is just a multivariate normal
with mean and covariance given by the mean and covariance functions applied at
our finite set of points (see [Rasmussen and Williams, 2006][1] for a more
extensive discussion of these facts).
We spell out the generative model in detail below, but first, a digression on
notation. In what follows we drop the indices on vectorial objects such as
x[i]
, it being implied that we are generally considering finite collections
of index points and corresponding function values and noisy observations
thereof. Thus x
should be considered to stand for a collection of index
points (indeed, themselves often vectorial). Furthermore:
f(x)
refers to the collection of function values at the index points in the collectionx
",m(t)
refers to the collection of values of the mean function at the index points in the collectiont
, andk(x, t)
refers to the matrix whose entries are values of the kernel functionk
at all pairs of index points fromx
andt
.
With these conventions in place, we may write
(f(x) | x) ~ MVN(m(x), k(x, x))
(y | f(x), x) ~ Normal(f(x), v)
When we condition on observed data y
at the points x
, we can derive the
posterior distribution over function values f(x)
at those points. We can
then compute the posterior predictive distribution over function values f(t)
at a new set of points t
, conditional on those observed data.
(f(t) | t, x, f(x)) ~ MVN(loc, cov)
where
loc = k(t, x) @ inv(k(x, x) + v * I) @ (y - loc)
cov = k(t, t) - k(t, x) @ inv(k(x, x) + v * I) @ k(x, t)
where I
is the identity matrix of appropriate dimension. Finally, the
distribution over noisy observations at the new set of points t
is obtained
by adding IID noise from Normal(0., observation_noise_variance)
.
Examples
Draw joint samples from the posterior predictive distribution in a GP
regression model
import numpy as np
import tensorflow as tf
import tensorflow_probability as tfp
tfd = tfp.distributions
psd_kernels = tfp.positive_semidefinite_kernels
# Generate noisy observations from a known function at some random points.
observation_noise_variance = .5
f = lambda x: np.sin(10*x[..., 0]) * np.exp(-x[..., 0]**2)
observation_index_points = np.random.uniform(-1., 1., 50)[..., np.newaxis]
observations = (f(observation_index_points) +
np.random.normal(0., np.sqrt(observation_noise_variance))
index_points = np.linspace(-1., 1., 100)[..., np.newaxis]
kernel = psd_kernels.MaternFiveHalves()
gprm = tfd.GaussianProcessRegressionModel(
kernel=kernel,
index_points=index_points,
observation_index_points=observation_index_points,
observations=observations,
observation_noise_variance=observation_noise_variance)
samples = gprm.sample(10)
# ==> 10 independently drawn, joint samples at `index_points`.
Above, we have used the kernel with default parameters, which are unlikely to be good. Instead, we can train the kernel hyperparameters on the data, as in the next example.
Optimize model parameters via maximum marginal likelihood
Here we learn the kernel parameters as well as the observation noise variance using gradient descent on the maximum marginal likelihood.
# Suppose we have some data from a known function. Note the index points in
# general have shape `[b1, ..., bB, f1, ..., fF]` (here we assume `F == 1`),
# so we need to explicitly consume the feature dimensions (just the last one
# here).
f = lambda x: np.sin(10*x[..., 0]) * np.exp(-x[..., 0]**2)
observation_index_points = np.random.uniform(-1., 1., 50)[..., np.newaxis]
observations = f(observation_index_points) + np.random.normal(0., .05, 50)
# Define a kernel with trainable parameters. Note we transform the trainable
# variables to apply a positivity constraint.
amplitude = tf.exp(tf.Variable(np.float64(0)), name='amplitude')
length_scale = tf.exp(tf.Variable(np.float64(0)), name='length_scale')
kernel = psd_kernels.ExponentiatedQuadratic(amplitude, length_scale)
observation_noise_variance = tf.exp(
tf.Variable(np.float64(-5)), name='observation_noise_variance')
# We'll use an unconditioned GP to train the kernel parameters.
gp = tfd.GaussianProcess(
kernel=kernel,
index_points=observation_index_points,
observation_noise_variance=observation_noise_variance)
neg_log_likelihood = -gp.log_prob(observations)
optimizer = tf.train.AdamOptimizer(learning_rate=.05, beta1=.5, beta2=.99)
optimize = optimizer.minimize(neg_log_likelihood)
# We can construct the posterior at a new set of `index_points` using the same
# kernel (with the same parameters, which we'll optimize below).
index_points = np.linspace(-1., 1., 100)[..., np.newaxis]
gprm = tfd.GaussianProcessRegressionModel(
kernel=kernel,
index_points=index_points,
observation_index_points=observation_index_points,
observations=observations,
observation_noise_variance=observation_noise_variance)
samples = gprm.sample(10)
# ==> 10 independently drawn, joint samples at `index_points`.
# Now execute the above ops in a Session, first training the model
# parameters, then drawing and plotting posterior samples.
with tf.Session() as sess:
sess.run(tf.global_variables_initializer())
for i in range(1000):
_, neg_log_likelihood_ = sess.run([optimize, neg_log_likelihood])
if i % 100 == 0:
print("Step {}: NLL = {}".format(i, neg_log_likelihood_))
print("Final NLL = {}".format(neg_log_likelihood_))
samples_ = sess.run(samples)
plt.scatter(np.squeeze(observation_index_points), observations)
plt.plot(np.stack([index_points[:, 0]]*10).T, samples_.T, c='r', alpha=.2)
Marginalization of model hyperparameters
Here we use TensorFlow Probability's MCMC functionality to perform marginalization of the model hyperparameters: kernel params as well as observation noise variance.
f = lambda x: np.sin(10*x[..., 0]) * np.exp(-x[..., 0]**2)
observation_index_points = np.random.uniform(-1., 1., 25)[..., np.newaxis]
observations = np.random.normal(f(observation_index_points), .05)
def joint_log_prob(
index_points, observations, amplitude, length_scale, noise_variance):
# Hyperparameter Distributions.
rv_amplitude = tfd.LogNormal(np.float64(0.), np.float64(1))
rv_length_scale = tfd.LogNormal(np.float64(0.), np.float64(1))
rv_noise_variance = tfd.LogNormal(np.float64(0.), np.float64(1))
gp = tfd.GaussianProcess(
kernel=psd_kernels.ExponentiatedQuadratic(amplitude, length_scale),
index_points=index_points,
observation_noise_variance=noise_variance)
return (
rv_amplitude.log_prob(amplitude) +
rv_length_scale.log_prob(length_scale) +
rv_noise_variance.log_prob(noise_variance) +
gp.log_prob(observations)
)
initial_chain_states = [
1e-1 * tf.ones([], dtype=np.float64, name='init_amplitude'),
1e-1 * tf.ones([], dtype=np.float64, name='init_length_scale'),
1e-1 * tf.ones([], dtype=np.float64, name='init_obs_noise_variance')
]
# Since HMC operates over unconstrained space, we need to transform the
# samples so they live in real-space.
unconstraining_bijectors = [
tfp.bijectors.Softplus(),
tfp.bijectors.Softplus(),
tfp.bijectors.Softplus(),
]
def unnormalized_log_posterior(amplitude, length_scale, noise_variance):
return joint_log_prob(
observation_index_points, observations, amplitude, length_scale,
noise_variance)
num_results = 200
[
amplitudes,
length_scales,
observation_noise_variances
], kernel_results = tfp.mcmc.sample_chain(
num_results=num_results,
num_burnin_steps=500,
num_steps_between_results=3,
current_state=initial_chain_states,
kernel=tfp.mcmc.TransformedTransitionKernel(
inner_kernel = tfp.mcmc.HamiltonianMonteCarlo(
target_log_prob_fn=unnormalized_log_posterior,
step_size=[np.float64(.15)],
num_leapfrog_steps=3),
bijector=unconstraining_bijectors))
# Now we can sample from the posterior predictive distribution at a new set
# of index points.
gprm = tfd.GaussianProcessRegressionModel(
# Batch of `num_results` kernels parameterized by the MCMC samples.
kernel=psd_kernels.ExponentiatedQuadratic(amplitudes, length_scales),
index_points=np.linspace(-2., 2., 200)[..., np.newaxis],
observation_index_points=observation_index_points,
observations=observations,
# We reshape this to align batch dimensions.
observation_noise_variance=observation_noise_variances[..., np.newaxis])
samples = gprm.sample()
with tf.Session() as sess:
kernel_results_, samples_ = sess.run([kernel_results, samples])
print("Acceptance rate: {}".format(
np.mean(kernel_results_.inner_results.is_accepted)))
# Plot posterior samples and their mean, target function, and observations.
plt.plot(np.stack([index_points[:, 0]]*num_results).T,
samples_.T,
c='r',
alpha=.01)
plt.plot(index_points[:, 0], np.mean(samples_, axis=0), c='k')
plt.plot(index_points[:, 0], f(index_points))
plt.scatter(observation_index_points[:, 0], observations)
References
[1]: Carl Rasmussen, Chris Williams. Gaussian Processes For Machine Learning, 2006.
__init__
__init__(
kernel,
index_points,
observation_index_points=None,
observations=None,
observation_noise_variance=0.0,
predictive_noise_variance=None,
mean_fn=None,
jitter=1e-06,
validate_args=False,
allow_nan_stats=False,
name='GaussianProcessRegressionModel'
)
Construct a GaussianProcessRegressionModel instance.
Args:
kernel
:PositiveSemidefiniteKernel
-like instance representing the GP's covariance function.index_points
:float
Tensor
representing finite collection, or batch of collections, of points in the index set over which the GP is defined. Shape has the form[b1, ..., bB, e, f1, ..., fF]
whereF
is the number of feature dimensions and must equalkernel.feature_ndims
ande
is the number (size) of index points in each batch. Ultimately this distribution corresponds to ane
-dimensional multivariate normal. The batch shape must be broadcastable withkernel.batch_shape
and any batch dims yielded bymean_fn
.observation_index_points
:float
Tensor
representing finite collection, or batch of collections, of points in the index set for which some data has been observed. Shape has the form[b1, ..., bB, e, f1, ..., fF]
whereF
is the number of feature dimensions and must equalkernel.feature_ndims
, ande
is the number (size) of index points in each batch.[b1, ..., bB, e]
must be broadcastable with the shape ofobservations
, and[b1, ..., bB]
must be broadcastable with the shapes of all other batched parameters (kernel.batch_shape
,index_points
, etc). The default value isNone
, which corresponds to the empty set of observations, and simply results in the prior predictive model (a GP with noise of variancepredictive_noise_variance
).observations
:float
Tensor
representing collection, or batch of collections, of observations corresponding toobservation_index_points
. Shape has the form[b1, ..., bB, e]
, which must be brodcastable with the batch and example shapes ofobservation_index_points
. The batch shape[b1, ..., bB]
must be broadcastable with the shapes of all other batched parameters (kernel.batch_shape
,index_points
, etc.). The default value isNone
, which corresponds to the empty set of observations, and simply results in the prior predictive model (a GP with noise of variancepredictive_noise_variance
).observation_noise_variance
:float
Tensor
representing the variance of the noise in the Normal likelihood distribution of the model. May be batched, in which case the batch shape must be broadcastable with the shapes of all other batched parameters (kernel.batch_shape
,index_points
, etc.). Default value:0.
predictive_noise_variance
:float
Tensor
representing the variance in the posterior predictive model. IfNone
, we simply re-useobservation_noise_variance
for the posterior predictive noise. If set explicitly, however, we use this value. This allows us, for example, to omit predictive noise variance (by setting this to zero) to obtain noiseless posterior predictions of function values, conditioned on noisy observations.mean_fn
: Pythoncallable
that acts onindex_points
to produce a collection, or batch of collections, of mean values atindex_points
. Takes aTensor
of shape[b1, ..., bB, f1, ..., fF]
and returns aTensor
whose shape is broadcastable with[b1, ..., bB]
. Default value:None
implies the constant zero function.jitter
:float
scalarTensor
added to the diagonal of the covariance matrix to ensure positive definiteness of the covariance matrix. Default value:1e-6
.validate_args
: Pythonbool
, defaultFalse
. WhenTrue
distribution parameters are checked for validity despite possibly degrading runtime performance. WhenFalse
invalid inputs may silently render incorrect outputs. Default value:False
.allow_nan_stats
: Pythonbool
, defaultTrue
. WhenTrue
, statistics (e.g., mean, mode, variance) use the valueNaN
to indicate the result is undefined. WhenFalse
, an exception is raised if one or more of the statistic's batch members are undefined. Default value:False
.name
: Pythonstr
name prefixed to Ops created by this class. Default value: 'GaussianProcessRegressionModel'.
Raises:
ValueError
: if either- only one of
observations
andobservation_index_points
is given, or mean_fn
is notNone
and not callable.
- only one of
Properties
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.
Returns:
allow_nan_stats
: Pythonbool
.
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.
Returns:
batch_shape
:TensorShape
, possibly unknown.
bijector
Function transforming x => y.
distribution
Base distribution, p(x).
dtype
The DType
of Tensor
s handled by this Distribution
.
event_shape
Shape of a single sample from a single batch as a TensorShape
.
May be partially defined or unknown.
Returns:
event_shape
:TensorShape
, possibly unknown.
index_points
jitter
kernel
loc
The loc
Tensor
in Y = scale @ X + loc
.
mean_fn
name
Name prepended to all ops created by this Distribution
.
observation_index_points
observation_noise_variance
observations
parameters
Dictionary of parameters used to instantiate this Distribution
.
predictive_noise_variance
reparameterization_type
Describes how samples from the distribution are reparameterized.
Currently this is one of the static instances
tfd.FULLY_REPARAMETERIZED
or tfd.NOT_REPARAMETERIZED
.
Returns:
An instance of ReparameterizationType
.
scale
The scale
LinearOperator
in Y = scale @ X + loc
.
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
ordouble
Tensor
.name
: Pythonstr
prepended to names of ops created by this function.
Returns:
cdf
: aTensor
of shapesample_shape(x) + self.batch_shape
with values of typeself.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 oftype(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
: Pythonstr
prepended to names of ops created by this function.
Returns:
covariance
: Floating-pointTensor
with shape[B1, ..., Bn, k', k']
where the firstn
dimensions are batch coordinates andk' = 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
: Pythonstr
prepended to names of ops created by this function.
Returns:
cross_entropy
:self.dtype
Tensor
with shape[B1, ..., Bn]
representingn
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
: Pythonstr
prepended to names of ops created by this function.
Returns:
is_scalar_batch
:bool
scalarTensor
.
is_scalar_event
is_scalar_event(name='is_scalar_event')
Indicates that event_shape == []
.
Args:
name
: Pythonstr
prepended to names of ops created by this function.
Returns:
is_scalar_event
:bool
scalarTensor
.
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
: Pythonstr
prepended to names of ops created by this function.
Returns:
kl_divergence
:self.dtype
Tensor
with shape[B1, ..., Bn]
representingn
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
ordouble
Tensor
.name
: Pythonstr
prepended to names of ops created by this function.
Returns:
logcdf
: aTensor
of shapesample_shape(x) + self.batch_shape
with values of typeself.dtype
.
log_prob
log_prob(
value,
name='log_prob'
)
Log probability density/mass function.
Additional documentation from MultivariateNormalLinearOperator
:
value
is a batch vector with compatible shape if value
is a Tensor
whose
shape can be broadcast up to either:
self.batch_shape + self.event_shape
or
[M1, ..., Mm] + self.batch_shape + self.event_shape
Args:
value
:float
ordouble
Tensor
.name
: Pythonstr
prepended to names of ops created by this function.
Returns:
log_prob
: aTensor
of shapesample_shape(x) + self.batch_shape
with values of typeself.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
ordouble
Tensor
.name
: Pythonstr
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
param_shapes(
cls,
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 tosample()
.name
: name to prepend ops with.
Returns:
dict
of parameter name to Tensor
shapes.
param_static_shapes
param_static_shapes(
cls,
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 tosample()
.
Returns:
dict
of parameter name to TensorShape
.
Raises:
ValueError
: ifsample_shape
is aTensorShape
and is not fully defined.
prob
prob(
value,
name='prob'
)
Probability density/mass function.
Additional documentation from MultivariateNormalLinearOperator
:
value
is a batch vector with compatible shape if value
is a Tensor
whose
shape can be broadcast up to either:
self.batch_shape + self.event_shape
or
[M1, ..., Mm] + self.batch_shape + self.event_shape
Args:
value
:float
ordouble
Tensor
.name
: Pythonstr
prepended to names of ops created by this function.
Returns:
prob
: aTensor
of shapesample_shape(x) + self.batch_shape
with values of typeself.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
ordouble
Tensor
.name
: Pythonstr
prepended to names of ops created by this function.
Returns:
quantile
: aTensor
of shapesample_shape(x) + self.batch_shape
with values of typeself.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 1Dint32
Tensor
. Shape of the generated samples.seed
: Python integer seed for RNGname
: name to give to the op.
Returns:
samples
: aTensor
with prepended dimensionssample_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
: Pythonstr
prepended to names of ops created by this function.
Returns:
stddev
: Floating-pointTensor
with shape identical tobatch_shape + event_shape
, i.e., the same shape asself.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
ordouble
Tensor
.name
: Pythonstr
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
: Pythonstr
prepended to names of ops created by this function.
Returns:
variance
: Floating-pointTensor
with shape identical tobatch_shape + event_shape
, i.e., the same shape asself.mean()
.