class_docstr: This method approximates the Gaussian process posterior using a multivariate Gaussian.
  
  The key reference is :cite:t:`Opper:2009`.
  
  The idea is that the posterior over the function-value vector F is
  approximated by a Gaussian, and the KL divergence is minimised between
  the approximation and the posterior. It turns out that the optimal
  posterior precision shares off-diagonal elements with the prior, so
  only the diagonal elements of the precision need be adjusted.
  The posterior approximation is
  
  .. math::
  
     q(\mathbf f) = N(\mathbf f \,|\, \mathbf K \boldsymbol \alpha,
                       [\mathbf K^{-1} + \textrm{diag}(\boldsymbol \lambda))^2]^{-1})
  
  This approach has only 2ND parameters, rather than the N + N^2 of vgp,
  but the optimization is non-convex and in practice may cause difficulty.
docstr: The posterior variance of F is given by
  
  .. math::
  
     q(f) = N(f |
         K \alpha + \textrm{mean}, [K^-1 + \textrm{diag}(\lambda**2)]^-1)
  
  Here we project this to F*, the values of the GP at Xnew which is given
  by
  
  .. math::
  
     q(F*) = N ( F* | K_{*F} \alpha + \textrm{mean}, K_{**} - K_{*f}[K_{ff} +
                                     \textrm{diag}(\lambda**-2)]^-1 K_{f*} )
  
  Note: This model currently does not allow full output covariances
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code: def predict_f(
      self,
      Xnew: InputData,
      full_cov: bool = False,
      full_output_cov: bool = False,
  ) -> MeanAndVariance:
      r"""
      The posterior variance of F is given by
  
      .. math::
  
         q(f) = N(f |
             K \alpha + \textrm{mean}, [K^-1 + \textrm{diag}(\lambda**2)]^-1)
  
      Here we project this to F*, the values of the GP at Xnew which is given
      by
  
      .. math::
  
         q(F*) = N ( F* | K_{*F} \alpha + \textrm{mean}, K_{**} - K_{*f}[K_{ff} +
                                         \textrm{diag}(\lambda**-2)]^-1 K_{f*} )
  
      Note: This model currently does not allow full output covariances
      """
      assert_params_false(
          self.predict_f, full_output_cov=full_output_cov
      )
  
      X_data, _ = self.data
      # compute kernel things
      Kx = self.kernel(X_data, Xnew)
      K = self.kernel(X_data)
  
      # predictive mean
      f_mean = tf.linalg.matmul(
          Kx, self.q_alpha, transpose_a=True
      ) + self.mean_function(Xnew)
  
      # predictive var
      A = K + tf.linalg.diag(
          tf.transpose(1.0 / tf.square(self.q_lambda))
      )
      L = tf.linalg.cholesky(A)
      Kx_tiled = tf.tile(Kx[None, ...], [self.num_latent_gps, 1, 1])
      LiKx = tf.linalg.triangular_solve(L, Kx_tiled)
      if full_cov:
          f_var = self.kernel(Xnew) - tf.linalg.matmul(
              LiKx, LiKx, transpose_a=True
          )
      else:
          f_var = self.kernel(Xnew, full_cov=False) - tf.reduce_sum(
              tf.square(LiKx), axis=1
          )
      return f_mean, tf.transpose(f_var)
  
imports: from typing import Optional
  import numpy
  import tensorflow
  from check_shapes import check_shapes, inherit_check_shapes
  import gpflow
  from ..None import posteriors
  from ..base import InputData, MeanAndVariance, Parameter, RegressionData
  from ..conditionals import conditional
  from ..config import default_float, default_jitter
  from ..kernels import Kernel
  from ..kullback_leiblers import gauss_kl
  from ..likelihoods import Likelihood
  from ..mean_functions import MeanFunction
  from ..utilities import assert_params_false, is_variable, triangular, triangular_size
  from .model import GPModel
  from .training_mixins import InternalDataTrainingLossMixin
  from .util import data_input_to_tensor
label_desc: Use image node_id 4 for calling the VGPOpperArchambeau obj's underlying member method code with example usage: obj.predict_f(Xnew, full_cov, full_output_cov) and returns: f_mean,  tf