misc.py
``````import tensorflow as tf

from .. import kernels
from .. import mean_functions as mfn
from ..features import InducingFeature, InducingPoints
from ..probability_distributions import DiagonalGaussian, Gaussian, MarkovGaussian
from . import dispatch
from .expectations import expectation

NoneType = type(None)

# ================ exKxz transpose and mean function handling =================

@dispatch.expectation.register((Gaussian, MarkovGaussian), mfn.Identity, NoneType, kernels.Linear, InducingPoints)
def _E(p, mean, _, kernel, feature, nghp=None):
"""
Compute the expectation:
expectation[n] = <x_n K_{x_n, Z}>_p(x_n)
- K_{.,} :: Linear kernel
or the equivalent for MarkovGaussian

:return: NxDxM
"""

@dispatch.expectation.register((Gaussian, MarkovGaussian), kernels.Kernel, InducingFeature, mfn.MeanFunction, NoneType)
def _E(p, kernel, feature, mean, _, nghp=None):
"""
Compute the expectation:
expectation[n] = <K_{Z, x_n} m(x_n)>_p(x_n)
or the equivalent for MarkovGaussian

:return: NxMxQ
"""
return tf.linalg.adjoint(expectation(p, mean, (kernel, feature), nghp=nghp))

@dispatch.expectation.register(Gaussian, mfn.Constant, NoneType, kernels.Kernel, InducingPoints)
def _E(p, constant_mean, _, kernel, feature, nghp=None):
"""
Compute the expectation:
expectation[n] = <m(x_n)^T K_{x_n, Z}>_p(x_n)
- m(x_i) = c :: Constant function
- K_{.,.}    :: Kernel function

:return: NxQxM
"""
c = constant_mean(p.mu)  # NxQ
eKxz = expectation(p, (kernel, feature), nghp=nghp)  # NxM

return c[..., None] * eKxz[:, None, :]

@dispatch.expectation.register(Gaussian, mfn.Linear, NoneType, kernels.Kernel, InducingPoints)
def _E(p, linear_mean, _, kernel, feature, nghp=None):
"""
Compute the expectation:
expectation[n] = <m(x_n)^T K_{x_n, Z}>_p(x_n)
- m(x_i) = A x_i + b :: Linear mean function
- K_{.,.}            :: Kernel function

:return: NxQxM
"""
N = p.mu.shape[0]
D = p.mu.shape[1]
exKxz = expectation(p, mfn.Identity(D), (kernel, feature), nghp=nghp)
eKxz = expectation(p, (kernel, feature), nghp=nghp)
eAxKxz = tf.linalg.matmul(tf.tile(linear_mean.A[None, :, :], (N, 1, 1)), exKxz, transpose_a=True)
ebKxz = linear_mean.b[None, :, None] * eKxz[:, None, :]
return eAxKxz + ebKxz

@dispatch.expectation.register(Gaussian, mfn.Identity, NoneType, kernels.Kernel, InducingPoints)
def _E(p, identity_mean, _, kernel, feature, nghp=None):
"""
This prevents infinite recursion for kernels that don't have specific
implementations of _expectation(p, identity_mean, None, kernel, feature).
Recursion can arise because Identity is a subclass of Linear mean function
so _expectation(p, linear_mean, none, kernel, feature) would call itself.
More specific signatures (e.g. (p, identity_mean, None, RBF, feature)) will
be found and used whenever available
"""
raise NotImplementedError

# ============== Conversion to Gaussian from Diagonal or Markov ===============
# Catching missing DiagonalGaussian implementations by converting to full Gaussian:

@dispatch.expectation.register(DiagonalGaussian, object, (InducingFeature, NoneType), object,
(InducingFeature, NoneType))
def _E(p, obj1, feat1, obj2, feat2, nghp=None):
gaussian = Gaussian(p.mu, tf.linalg.diag(p.cov))
return expectation(gaussian, (obj1, feat1), (obj2, feat2), nghp=nghp)

# Catching missing MarkovGaussian implementations by converting to Gaussian (when indifferent):

@dispatch.expectation.register(MarkovGaussian, object, (InducingFeature, NoneType), object,
(InducingFeature, NoneType))
def _E(p, obj1, feat1, obj2, feat2, nghp=None):
"""
Nota Bene: if only one object is passed, obj1 is
associated with x_n, whereas obj2 with x_{n+1}

"""
if obj2 is None:
gaussian = Gaussian(p.mu[:-1], p.cov[0, :-1])
return expectation(gaussian, (obj1, feat1), nghp=nghp)
elif obj1 is None:
gaussian = Gaussian(p.mu[1:], p.cov[0, 1:])
return expectation(gaussian, (obj2, feat2), nghp=nghp)
else:
return expectation(p, (obj1, feat1), (obj2, feat2), nghp=nghp)
``````