text: import tensorflow as tf
  
  from .. import kernels
  from .. import mean_functions as mfn
  from ..inducing_variables import InducingPoints
  from ..probability_distributions import DiagonalGaussian, Gaussian, MarkovGaussian
  from . import dispatch
  from .expectations import expectation
  
  
  NoneType = type(None)
  
  
  @dispatch.expectation.register(Gaussian, kernels.Linear, NoneType, NoneType, NoneType)
  def _E(p, kernel, _, __, ___, nghp=None):
      """
      Compute the expectation:
      <diag(K_{X, X})>_p(X)
          - K_{.,.} :: Linear kernel
  
      :return: N
      """
      # use only active dimensions
      Xmu, _ = kernel.slice(p.mu, None)
      Xcov = kernel.slice_cov(p.cov)
  
      return tf.reduce_sum(kernel.variance * (tf.linalg.diag_part(Xcov) + Xmu ** 2), 1)
  
  
  @dispatch.expectation.register(Gaussian, kernels.Linear, InducingPoints, NoneType, NoneType)
  def _E(p, kernel, inducing_variable, _, __, nghp=None):
      """
      Compute the expectation:
      <K_{X, Z}>_p(X)
          - K_{.,.} :: Linear kernel
  
      :return: NxM
      """
      # use only active dimensions
      Z, Xmu = kernel.slice(inducing_variable.Z, p.mu)
  
      return tf.linalg.matmul(Xmu, Z * kernel.variance, transpose_b=True)
  
  
  @dispatch.expectation.register(Gaussian, kernels.Linear, InducingPoints, mfn.Identity, NoneType)
  def _E(p, kernel, inducing_variable, mean, _, nghp=None):
      """
      Compute the expectation:
      expectation[n] = <K_{Z, x_n} x_n^T>_p(x_n)
          - K_{.,.} :: Linear kernel
  
      :return: NxMxD
      """
      Xmu, Xcov = p.mu, p.cov
  
      N = tf.shape(Xmu)[0]
      var_Z = kernel.variance * inducing_variable.Z  # MxD
      tiled_Z = tf.tile(tf.expand_dims(var_Z, 0), (N, 1, 1))  # NxMxD
      return tf.linalg.matmul(tiled_Z, Xcov + (Xmu[..., None] * Xmu[:, None, :]))
  
  
  @dispatch.expectation.register(
      MarkovGaussian, kernels.Linear, InducingPoints, mfn.Identity, NoneType
  )
  def _E(p, kernel, inducing_variable, mean, _, nghp=None):
      """
      Compute the expectation:
      expectation[n] = <K_{Z, x_n} x_{n+1}^T>_p(x_{n:n+1})
          - K_{.,.} :: Linear kernel
          - p       :: MarkovGaussian distribution (p.cov 2x(N+1)xDxD)
  
      :return: NxMxD
      """
      Xmu, Xcov = p.mu, p.cov
  
      N = tf.shape(Xmu)[0] - 1
      var_Z = kernel.variance * inducing_variable.Z  # MxD
      tiled_Z = tf.tile(tf.expand_dims(var_Z, 0), (N, 1, 1))  # NxMxD
      eXX = Xcov[1, :-1] + (Xmu[:-1][..., None] * Xmu[1:][:, None, :])  # NxDxD
      return tf.linalg.matmul(tiled_Z, eXX)
  
  
  @dispatch.expectation.register(
      (Gaussian, DiagonalGaussian), kernels.Linear, InducingPoints, kernels.Linear, InducingPoints
  )
  def _E(p, kern1, feat1, kern2, feat2, nghp=None):
      """
      Compute the expectation:
      expectation[n] = <Ka_{Z1, x_n} Kb_{x_n, Z2}>_p(x_n)
          - Ka_{.,.}, Kb_{.,.} :: Linear kernels
      Ka and Kb as well as Z1 and Z2 can differ from each other, but this is supported
      only if the Gaussian p is Diagonal (p.cov NxD) and Ka, Kb have disjoint active_dims
      in which case the joint expectations simplify into a product of expectations
  
      :return: NxMxM
      """
      if kern1.on_separate_dims(kern2) and isinstance(
          p, DiagonalGaussian
      ):  # no joint expectations required
          eKxz1 = expectation(p, (kern1, feat1))
          eKxz2 = expectation(p, (kern2, feat2))
          return eKxz1[:, :, None] * eKxz2[:, None, :]
  
      if kern1 != kern2 or feat1 != feat2:
          raise NotImplementedError(
              "The expectation over two kernels has only an "
              "analytical implementation if both kernels are equal."
          )
  
      kernel = kern1
      inducing_variable = feat1
  
      # use only active dimensions
      Xcov = kernel.slice_cov(tf.linalg.diag(p.cov) if isinstance(p, DiagonalGaussian) else p.cov)
      Z, Xmu = kernel.slice(inducing_variable.Z, p.mu)
  
      N = tf.shape(Xmu)[0]
      var_Z = kernel.variance * Z
      tiled_Z = tf.tile(tf.expand_dims(var_Z, 0), (N, 1, 1))  # NxMxD
      XX = Xcov + tf.expand_dims(Xmu, 1) * tf.expand_dims(Xmu, 2)  # NxDxD
      return tf.linalg.matmul(tf.linalg.matmul(tiled_Z, XX), tiled_Z, transpose_b=True)
  