def conditional(self, name, Xnew, **kwargs):
"""
Returns the conditional distribution evaluated over new input
locations `Xnew`.
`Xnew` will be split by columns and fed to the relevant
covariance functions based on their `input_dim`. For example, if
`cov_func1`, `cov_func2`, and `cov_func3` have `input_dim` of 2,
1, and 4, respectively, then `Xnew` must have 7 columns and a
covariance between the prediction points
.. code:: python
cov_func(Xnew) = cov_func1(Xnew[:, :2]) * cov_func1(Xnew[:, 2:3]) * cov_func1(Xnew[:, 3:])
The distribution returned by `conditional` does not have a
Kronecker structure regardless of whether the input points lie
on a full grid. Therefore, `Xnew` does not need to have grid
structure.
Parameters
----------
name : string
Name of the random variable
Xnew : array-like
Function input values. If one-dimensional, must be a column
vector with shape `(n, 1)`.
**kwargs
Extra keyword arguments that are passed to `MvNormal` distribution
constructor.
"""
mu, cov = self._build_conditional(Xnew)
shape = infer_shape(Xnew, kwargs.pop("shape", None))
return pm.MvNormal(name, mu=mu, cov=cov, shape=shape, **kwargs)
def conditional(self, name, Xnew, given=None, **kwargs):
R"""
Returns the conditional distribution evaluated over new input
locations `Xnew`.
Given a set of function values `f` that
the GP prior was over, the conditional distribution over a
set of new points, `f_*` is
.. math::
f_* \mid f, X, X_* \sim \mathcal{GP}\left(
K(X_*, X) K(X, X)^{-1} f \,,
K(X_*, X_*) - K(X_*, X) K(X, X)^{-1} K(X, X_*) \right)
Parameters
----------
name : string
Name of the random variable
Xnew : array-like
Function input values.
given : dict
Can optionally take as key value pairs: `X`, `y`, `noise`,
and `gp`. See the section in the documentation on additive GP
models in PyMC3 for more information.
**kwargs
Extra keyword arguments that are passed to `MvNormal` distribution
constructor.
"""
givens = self._get_given_vals(given)
mu, cov = self._build_conditional(Xnew, *givens)
chol = cholesky(stabilize(cov))
shape = infer_shape(Xnew, kwargs.pop("shape", None))
return pm.MvNormal(name, mu=mu, chol=chol, shape=shape, **kwargs)
def conditional(self, name, Xnew, **kwargs):
R"""
Returns the conditional distribution evaluated over new input
locations `Xnew`.
Given a set of function values `f` that
the TP prior was over, the conditional distribution over a
set of new points, `f_*` is
Parameters
----------
name : string
Name of the random variable
Xnew : array-like
Function input values.
**kwargs
Extra keyword arguments that are passed to `MvNormal` distribution
constructor.
"""
X = self.X
f = self.f
nu2, mu, covT = self._build_conditional(Xnew, X, f)
chol = cholesky(stabilize(covT))
shape = infer_shape(Xnew, kwargs.pop("shape", None))
return pm.MvStudentT(name, nu=nu2, mu=mu, chol=chol, shape=shape, **kwargs)
def conditional(self, name, Xnew, given=None, **kwargs):
R"""
Returns the conditional distribution evaluated over new input
locations `Xnew`.
Given a set of function values `f` that
the GP prior was over, the conditional distribution over a
set of new points, `f_*` is
.. math::
f_* \mid f, X, X_* \sim \mathcal{GP}\left(
K(X_*, X) K(X, X)^{-1} f \,,
K(X_*, X_*) - K(X_*, X) K(X, X)^{-1} K(X, X_*) \right)
Parameters
----------
name: string
Name of the random variable
Xnew: array-like
Function input values.
given: dict
Can optionally take as key value pairs: `X`, `y`, `noise`,
and `gp`. See the section in the documentation on additive GP
models in PyMC3 for more information.
**kwargs
Extra keyword arguments that are passed to `MvNormal` distribution
constructor.
"""
givens = self._get_given_vals(given)
mu, cov = self._build_conditional(Xnew, *givens)
shape = infer_shape(Xnew, kwargs.pop("shape", None))
return pm.MvNormal(name, mu=mu, cov=cov, size=shape, **kwargs)
def conditional(self, name, Xnew, **kwargs):
"""
Returns the conditional distribution evaluated over new input
locations `Xnew`.
`Xnew` will be split by columns and fed to the relevant
covariance functions based on their `input_dim`. For example, if
`cov_func1`, `cov_func2`, and `cov_func3` have `input_dim` of 2,
1, and 4, respectively, then `Xnew` must have 7 columns and a
covariance between the prediction points
.. code:: python
cov_func(Xnew) = cov_func1(Xnew[:, :2]) * cov_func1(Xnew[:, 2:3]) * cov_func1(Xnew[:, 3:])
The distribution returned by `conditional` does not have a
Kronecker structure regardless of whether the input points lie
on a full grid. Therefore, `Xnew` does not need to have grid
structure.
Parameters
----------
name: string
Name of the random variable
Xnew: array-like
Function input values. If one-dimensional, must be a column
vector with shape `(n, 1)`.
**kwargs
Extra keyword arguments that are passed to `MvNormal` distribution
constructor.
"""
mu, cov = self._build_conditional(Xnew)
shape = infer_shape(Xnew, kwargs.pop("shape", None))
return pm.MvNormal(name, mu=mu, cov=cov, shape=shape, **kwargs)
def conditional(self, name, Xnew, pred_noise=False, given=None, **kwargs):
R"""
Returns the approximate conditional distribution of the GP evaluated over
new input locations `Xnew`.
Parameters
----------
name: string
Name of the random variable
Xnew: array-like
Function input values. If one-dimensional, must be a column
vector with shape `(n, 1)`.
pred_noise: bool
Whether or not observation noise is included in the conditional.
Default is `False`.
given: dict
Can optionally take as key value pairs: `X`, `Xu`, `y`, `noise`,
and `gp`. See the section in the documentation on additive GP
models in PyMC3 for more information.
**kwargs
Extra keyword arguments that are passed to `MvNormal` distribution
constructor.
"""
givens = self._get_given_vals(given)
mu, cov = self._build_conditional(Xnew, pred_noise, False, *givens)
shape = infer_shape(Xnew, kwargs.pop("shape", None))
return pm.MvNormal(name, mu=mu, cov=cov, shape=shape, **kwargs)
def conditional(self, name, Xnew, **kwargs):
R"""
Returns the conditional distribution evaluated over new input
locations `Xnew`.
Given a set of function values `f` that
the TP prior was over, the conditional distribution over a
set of new points, `f_*` is
Parameters
----------
name: string
Name of the random variable
Xnew: array-like
Function input values.
**kwargs
Extra keyword arguments that are passed to `MvNormal` distribution
constructor.
"""
X = self.X
f = self.f
nu2, mu, cov = self._build_conditional(Xnew, X, f)
shape = infer_shape(Xnew, kwargs.pop("shape", None))
return pm.MvStudentT(name,
nu=nu2,
mu=mu,
cov=cov,
shape=shape,
**kwargs)
def conditional(self, name, Xnew, pred_noise=False, given=None, **kwargs):
R"""
Returns the approximate conditional distribution of the GP evaluated over
new input locations `Xnew`.
Parameters
----------
name : string
Name of the random variable
Xnew : array-like
Function input values. If one-dimensional, must be a column
vector with shape `(n, 1)`.
pred_noise : bool
Whether or not observation noise is included in the conditional.
Default is `False`.
given : dict
Can optionally take as key value pairs: `X`, `Xu`, `y`, `noise`,
and `gp`. See the section in the documentation on additive GP
models in PyMC3 for more information.
**kwargs
Extra keyword arguments that are passed to `MvNormal` distribution
constructor.
"""
givens = self._get_given_vals(given)
mu, cov = self._build_conditional(Xnew, pred_noise, False, *givens)
chol = cholesky(cov)
shape = infer_shape(Xnew, kwargs.pop("shape", None))
return pm.MvNormal(name, mu=mu, chol=chol, shape=shape, **kwargs)
def marginal_likelihood(self,
name,
X,
Xu,
y,
noise=None,
is_observed=True,
**kwargs):
R"""
Returns the approximate marginal likelihood distribution, given the input
locations `X`, inducing point locations `Xu`, data `y`, and white noise
standard deviations `sigma`.
Parameters
----------
name: string
Name of the random variable
X: array-like
Function input values. If one-dimensional, must be a column
vector with shape `(n, 1)`.
Xu: array-like
The inducing points. Must have the same number of columns as `X`.
y: array-like
Data that is the sum of the function with the GP prior and Gaussian
noise. Must have shape `(n, )`.
noise: scalar, Variable
Standard deviation of the Gaussian noise.
is_observed: bool
Whether to set `y` as an `observed` variable in the `model`.
Default is `True`.
**kwargs
Extra keyword arguments that are passed to `MvNormal` distribution
constructor.
"""
self.X = X
self.Xu = Xu
self.y = y
if noise is None:
sigma = kwargs.get("sigma")
if sigma is None:
raise ValueError("noise argument must be specified")
else:
self.sigma = sigma
warnings.warn(
"The 'sigma' argument has been deprecated. Use 'noise' instead.",
DeprecationWarning,
)
else:
self.sigma = noise
logp = functools.partial(self._build_marginal_likelihood_logp,
X=X,
Xu=Xu,
sigma=noise)
if is_observed:
return pm.DensityDist(name, logp, observed=y, **kwargs)
else:
shape = infer_shape(X, kwargs.pop("shape", None))
return pm.DensityDist(name, logp, shape=shape, **kwargs)
def _build_prior(self, name, X, reparameterize=True, **kwargs):
mu = self.mean_func(X)
chol = cholesky(stabilize(self.cov_func(X)))
shape = infer_shape(X, kwargs.pop("shape", None))
if reparameterize:
v = pm.Normal(name + "_rotated_", mu=0.0, sd=1.0, shape=shape, **kwargs)
f = pm.Deterministic(name, mu + tt.dot(chol, v))
else:
f = pm.MvNormal(name, mu=mu, chol=chol, shape=shape, **kwargs)
return f
def _build_prior(self, name, X, reparameterize=True, **kwargs):
mu = self.mean_func(X)
chol = cholesky(stabilize(self.cov_func(X)))
shape = infer_shape(X, kwargs.pop("shape", None))
if reparameterize:
v = pm.Normal(name + "_rotated_", mu=0.0, sd=1.0, shape=shape, **kwargs)
f = pm.Deterministic(name, mu + tt.dot(chol, v))
else:
f = pm.MvNormal(name, mu=mu, chol=chol, shape=shape, **kwargs)
return f
def _build_prior(self, name, X, reparameterize=True, **kwargs):
mu = self.mean_func(X)
chol = cholesky(stabilize(self.cov_func(X)))
shape = infer_shape(X, kwargs.pop("shape", None))
if reparameterize:
chi2 = pm.ChiSquared("chi2_", self.nu)
v = pm.Normal(name + "_rotated_", mu=0.0, sd=1.0, shape=shape, **kwargs)
f = pm.Deterministic(name, (tt.sqrt(self.nu) / chi2) * (mu + tt.dot(chol, v)))
else:
f = pm.MvStudentT(name, nu=self.nu, mu=mu, chol=chol, shape=shape, **kwargs)
return f
def _build_prior(self, name, X, reparameterize=True, **kwargs):
mu = self.mean_func(X)
chol = cholesky(stabilize(self.cov_func(X)))
shape = infer_shape(X, kwargs.pop("shape", None))
if reparameterize:
chi2 = pm.ChiSquared("chi2_", self.nu)
v = pm.Normal(name + "_rotated_", mu=0.0, sd=1.0, shape=shape, **kwargs)
f = pm.Deterministic(name, (tt.sqrt(self.nu) / chi2) * (mu + tt.dot(chol, v)))
else:
f = pm.MvStudentT(name, nu=self.nu, mu=mu, chol=chol, shape=shape, **kwargs)
return f
def marginal_likelihood(self,
name,
X,
y,
noise,
is_observed=True,
**kwargs):
R"""
Returns the marginal likelihood distribution, given the input
locations `X` and the data `y`.
This is integral over the product of the GP prior and a normal likelihood.
.. math::
y \mid X,\theta \sim \int p(y \mid f,\, X,\, \theta) \, p(f \mid X,\, \theta) \, df
Parameters
----------
name : string
Name of the random variable
X : array-like
Function input values. If one-dimensional, must be a column
vector with shape `(n, 1)`.
y : array-like
Data that is the sum of the function with the GP prior and Gaussian
noise. Must have shape `(n, )`.
noise : scalar, Variable, or Covariance
Standard deviation of the Gaussian noise. Can also be a Covariance for
non-white noise.
is_observed : bool
Whether to set `y` as an `observed` variable in the `model`.
Default is `True`.
**kwargs
Extra keyword arguments that are passed to `MvNormal` distribution
constructor.
"""
if not isinstance(noise, Covariance):
noise = pm.gp.cov.WhiteNoise(noise)
mu, cov = self._build_marginal_likelihood(X, noise)
chol = cholesky(stabilize(cov))
self.X = X
self.y = y
self.noise = noise
if is_observed:
return pm.MvNormal(name, mu=mu, chol=chol, observed=y, **kwargs)
else:
shape = infer_shape(X, kwargs.pop("shape", None))
return pm.MvNormal(name, mu=mu, chol=chol, shape=shape, **kwargs)
def marginal_likelihood(self, name, X, Xu, y, noise=None, is_observed=True, **kwargs):
R"""
Returns the approximate marginal likelihood distribution, given the input
locations `X`, inducing point locations `Xu`, data `y`, and white noise
standard deviations `sigma`.
Parameters
----------
name : string
Name of the random variable
X : array-like
Function input values. If one-dimensional, must be a column
vector with shape `(n, 1)`.
Xu: array-like
The inducing points. Must have the same number of columns as `X`.
y : array-like
Data that is the sum of the function with the GP prior and Gaussian
noise. Must have shape `(n, )`.
noise : scalar, Variable
Standard deviation of the Gaussian noise.
is_observed : bool
Whether to set `y` as an `observed` variable in the `model`.
Default is `True`.
**kwargs
Extra keyword arguments that are passed to `MvNormal` distribution
constructor.
"""
self.X = X
self.Xu = Xu
self.y = y
if noise is None:
sigma = kwargs.get('sigma')
if sigma is None:
raise ValueError('noise argument must be specified')
else:
self.sigma = sigma
warnings.warn(
"The 'sigma' argument has been deprecated. Use 'noise' instead.",
DeprecationWarning)
else:
self.sigma = noise
logp = functools.partial(self._build_marginal_likelihood_logp,
X=X, Xu=Xu, sigma=noise)
if is_observed:
return pm.DensityDist(name, logp, observed=y, **kwargs)
else:
shape = infer_shape(X, kwargs.pop("shape", None))
return pm.DensityDist(name, logp, shape=shape, **kwargs)
def _build_prior(self, name, X, reparameterize=True, **kwargs):
x = flatx
mu = self.mean_func(x)
cov = stabilize(self.cov_func(x))
shape = infer_shape(X, kwargs.pop("shape", None))
if reparameterize:
v = pm.Normal(name + "_rotated_",
mu=0.0,
sigma=1.0,
shape=shape,
**kwargs)
f = pm.Deterministic(name, mu + cholesky(cov).dot(v))
else:
f = pm.MvNormal(name, mu=mu, cov=cov, shape=shape, **kwargs)
return f
def marginal_likelihood(self,
name,
X,
Xu,
y,
sigma,
is_observed=True,
**kwargs):
R"""
Returns the approximate marginal likelihood distribution, given the input
locations `X`, inducing point locations `Xu`, data `y`, and white noise
standard deviations `sigma`.
Parameters
----------
name : string
Name of the random variable
X : array-like
Function input values. If one-dimensional, must be a column
vector with shape `(n, 1)`.
Xu: array-like
The inducing points. Must have the same number of columns as `X`.
y : array-like
Data that is the sum of the function with the GP prior and Gaussian
noise. Must have shape `(n, )`.
sigma : scalar, Variable
Standard deviation of the Gaussian noise.
is_observed : bool
Whether to set `y` as an `observed` variable in the `model`.
Default is `True`.
**kwargs
Extra keyword arguments that are passed to `MvNormal` distribution
constructor.
"""
self.X = X
self.Xu = Xu
self.y = y
self.sigma = sigma
logp = lambda y: self._build_marginal_likelihood_logp(X, Xu, y, sigma)
if is_observed:
return pm.DensityDist(name, logp, observed=y, **kwargs)
else:
shape = infer_shape(X, kwargs.pop("shape", None))
return pm.DensityDist(name, logp, shape=shape, **kwargs)
def marginal_likelihood(self, name, X, y, noise, is_observed=True, **kwargs):
R"""
Returns the marginal likelihood distribution, given the input
locations `X` and the data `y`.
This is integral over the product of the GP prior and a normal likelihood.
.. math::
y \mid X,\theta \sim \int p(y \mid f,\, X,\, \theta) \, p(f \mid X,\, \theta) \, df
Parameters
----------
name : string
Name of the random variable
X : array-like
Function input values. If one-dimensional, must be a column
vector with shape `(n, 1)`.
y : array-like
Data that is the sum of the function with the GP prior and Gaussian
noise. Must have shape `(n, )`.
noise : scalar, Variable, or Covariance
Standard deviation of the Gaussian noise. Can also be a Covariance for
non-white noise.
is_observed : bool
Whether to set `y` as an `observed` variable in the `model`.
Default is `True`.
**kwargs
Extra keyword arguments that are passed to `MvNormal` distribution
constructor.
"""
if not isinstance(noise, Covariance):
noise = pm.gp.cov.WhiteNoise(noise)
mu, cov = self._build_marginal_likelihood(X, noise)
chol = cholesky(stabilize(cov))
self.X = X
self.y = y
self.noise = noise
if is_observed:
return pm.MvNormal(name, mu=mu, chol=chol, observed=y, **kwargs)
else:
shape = infer_shape(X, kwargs.pop("shape", None))
return pm.MvNormal(name, mu=mu, chol=chol, shape=shape, **kwargs)
def conditional(self, name, Xnew, pred_noise=False, **kwargs):
"""
Returns the conditional distribution evaluated over new input
locations `Xnew`, just as in `Marginal`.
`Xnew` will be split
by columns and fed to the relevant covariance functions based on their
`input_dim`. For example, if `cov_func1`, `cov_func2`, and `cov_func3`
have `input_dim` of 2, 1, and 4, respectively, then `Xnew` must have
7 columns and a covariance between the prediction points
.. code:: python
cov_func(Xnew) = cov_func1(Xnew[:, :2]) * cov_func1(Xnew[:, 2:3]) * cov_func1(Xnew[:, 3:])
This `cov_func` does not have a Kronecker structure without a full
grid, but the conditional distribution does not have a Kronecker
structure regardless. Thus, the conditional method must fall back to
using `MvNormal` rather than `KroneckerNormal` in either case.
Parameters
----------
name : string
Name of the random variable
Xnew : array-like
Function input values. If one-dimensional, must be a column
vector with shape `(n, 1)`.
pred_noise : bool
Whether or not observation noise is included in the conditional.
Default is `False`.
**kwargs
Extra keyword arguments that are passed to `MvNormal` distribution
constructor.
"""
mu, cov = self._build_conditional(Xnew, pred_noise, False)
chol = cholesky(stabilize(cov))
shape = infer_shape(Xnew, kwargs.pop("shape", None))
return pm.MvNormal(name, mu=mu, chol=chol, shape=shape, **kwargs)
def conditional(self, name, Xnew, pred_noise=False, given=None, **kwargs):
R"""
Returns the conditional distribution evaluated over new input
locations `Xnew`.
Given a set of function values `f` that the GP prior was over, the
conditional distribution over a set of new points, `f_*` is:
.. math::
f_* \mid f, X, X_* \sim \mathcal{GP}\left(
K(X_*, X) [K(X, X) + K_{n}(X, X)]^{-1} f \,,
K(X_*, X_*) - K(X_*, X) [K(X, X) + K_{n}(X, X)]^{-1} K(X, X_*) \right)
Parameters
----------
name : string
Name of the random variable
Xnew : array-like
Function input values. If one-dimensional, must be a column
vector with shape `(n, 1)`.
pred_noise : bool
Whether or not observation noise is included in the conditional.
Default is `False`.
given : dict
Can optionally take as key value pairs: `X`, `y`, `noise`,
and `gp`. See the section in the documentation on additive GP
models in PyMC3 for more information.
**kwargs
Extra keyword arguments that are passed to `MvNormal` distribution
constructor.
"""
givens = self._get_given_vals(given)
mu, cov = self._build_conditional(Xnew, pred_noise, False, *givens)
chol = cholesky(cov)
shape = infer_shape(Xnew, kwargs.pop("shape", None))
return pm.MvNormal(name, mu=mu, chol=chol, shape=shape, **kwargs)
def marginal_likelihood(self, name, X, Xu, y, sigma, is_observed=True, **kwargs):
R"""
Returns the approximate marginal likelihood distribution, given the input
locations `X`, inducing point locations `Xu`, data `y`, and white noise
standard deviations `sigma`.
Parameters
----------
name : string
Name of the random variable
X : array-like
Function input values. If one-dimensional, must be a column
vector with shape `(n, 1)`.
Xu: array-like
The inducing points. Must have the same number of columns as `X`.
y : array-like
Data that is the sum of the function with the GP prior and Gaussian
noise. Must have shape `(n, )`.
sigma : scalar, Variable
Standard deviation of the Gaussian noise.
is_observed : bool
Whether to set `y` as an `observed` variable in the `model`.
Default is `True`.
**kwargs
Extra keyword arguments that are passed to `MvNormal` distribution
constructor.
"""
self.X = X
self.Xu = Xu
self.y = y
self.sigma = sigma
logp = lambda y: self._build_marginal_likelihood_logp(X, Xu, y, sigma)
if is_observed:
return pm.DensityDist(name, logp, observed=y, **kwargs)
else:
shape = infer_shape(X, kwargs.pop("shape", None))
return pm.DensityDist(name, logp, shape=shape, **kwargs)
def conditional(self, name, Xnew, pred_noise=False, **kwargs):
"""
Returns the conditional distribution evaluated over new input
locations `Xnew`, just as in `Marginal`.
`Xnew` will be split
by columns and fed to the relevant covariance functions based on their
`input_dim`. For example, if `cov_func1`, `cov_func2`, and `cov_func3`
have `input_dim` of 2, 1, and 4, respectively, then `Xnew` must have
7 columns and a covariance between the prediction points
.. code:: python
cov_func(Xnew) = cov_func1(Xnew[:, :2]) * cov_func1(Xnew[:, 2:3]) * cov_func1(Xnew[:, 3:])
This `cov_func` does not have a Kronecker structure without a full
grid, but the conditional distribution does not have a Kronecker
structure regardless. Thus, the conditional method must fall back to
using `MvNormal` rather than `KroneckerNormal` in either case.
Parameters
----------
name : string
Name of the random variable
Xnew : array-like
Function input values. If one-dimensional, must be a column
vector with shape `(n, 1)`.
pred_noise : bool
Whether or not observation noise is included in the conditional.
Default is `False`.
**kwargs
Extra keyword arguments that are passed to `MvNormal` distribution
constructor.
"""
mu, cov = self._build_conditional(Xnew, pred_noise, False)
chol = cholesky(stabilize(cov))
shape = infer_shape(Xnew, kwargs.pop("shape", None))
return pm.MvNormal(name, mu=mu, chol=chol, shape=shape, **kwargs)
def conditional(self, name, Xnew, pred_noise=False, given={}, **kwargs):
R"""
Returns the conditional distribution evaluated over new input
locations `Xnew`.
Given a set of function values `f` that the GP prior was over, the
conditional distribution over a set of new points, `f_*` is:
.. math::
f_* \mid f, X, X_* \sim \mathcal{GP}\left(
K(X_*, X) [K(X, X) + K_{n}(X, X)]^{-1} f \,,
K(X_*, X_*) - K(X_*, X) [K(X, X) + K_{n}(X, X)]^{-1} K(X, X_*) \right)
Parameters
----------
name : string
Name of the random variable
Xnew : array-like
Function input values. If one-dimensional, must be a column
vector with shape `(n, 1)`.
pred_noise : bool
Whether or not observation noise is included in the conditional.
Default is `False`.
given : dict
Can optionally take as key value pairs: `X`, `y`, `noise`,
and `gp`. See the section in the documentation on additive GP
models in PyMC3 for more information.
**kwargs
Extra keyword arguments that are passed to `MvNormal` distribution
constructor.
"""
givens = self._get_given_vals(given)
mu, cov = self._build_conditional(Xnew, pred_noise, False, *givens)
chol = cholesky(cov)
shape = infer_shape(Xnew, kwargs.pop("shape", None))
return pm.MvNormal(name, mu=mu, chol=chol, shape=shape, **kwargs)
