class_docstr: Posterior class for a Higher order Gaussian process model [Zhe2019hogp]_. Extends
  the standard GPyTorch posterior class by overwriting the rsample method.
  The posterior variance is handled internally by the HigherOrderGP model.
  HOGP is a tensorized GP model so the posterior covariance grows to be extremely
  large, but is highly structured, which means that we can exploit Kronecker
  identities to sample from the posterior using Matheron's rule as described in
  [Doucet2010sampl]_.
  
  In general, this posterior should ONLY be used for HOGP models
  that have highly structured covariances. It should also only be used internally when
  called from the HigherOrderGP.posterior(...) method. At this time, the posterior
  does not support gradients with respect to the training data.
code: def _prepare_base_samples(
      self, sample_shape: torch.Size, base_samples: Tensor = None
  ) -> Tensor:
      covariance_matrix = self.joint_covariance_matrix
      joint_size = covariance_matrix.shape[-1]
      batch_shape = covariance_matrix.batch_shape
  
      if base_samples is not None:
          if base_samples.shape[: len(sample_shape)] != sample_shape:
              raise RuntimeError(
                  "sample_shape disagrees with shape of base_samples."
              )
  
          appended_shape = joint_size + self.train_train_covar.shape[-1]
          if appended_shape != base_samples.shape[-1]:
              # get base_samples to the correct shape by expanding as sample shape,
              # batch shape, then rest of dimensions. We have to add first the sample
              # shape, then the batch shape of the model, and then finally the shape
              # of the test data points squeezed into a single dimension, accessed
              # from the test_train_covar.
              base_sample_shapes = (
                  sample_shape
                  + batch_shape
                  + self.test_train_covar.shape[-2:-1]
              )
              if base_samples.nelement() == base_sample_shapes.numel():
                  base_samples = base_samples.reshape(
                      base_sample_shapes
                  )
  
                  new_base_samples = torch.randn(
                      *sample_shape,
                      *batch_shape,
                      appended_shape - base_samples.shape[-1],
                      device=base_samples.device,
                      dtype=base_samples.dtype,
                  )
                  base_samples = torch.cat(
                      (base_samples, new_base_samples), dim=-1
                  )
              else:
                  raise BotorchTensorDimensionError(
                      "The base samples are not compatible with base sample shape. "
                      f"Received base samples of shape {base_samples.shape}, "
                      f"expected {base_sample_shapes}."
                  )
  
      if base_samples is None:
          # TODO: Allow qMC sampling
          base_samples = torch.randn(
              *sample_shape,
              *batch_shape,
              joint_size,
              device=covariance_matrix.device,
              dtype=covariance_matrix.dtype,
          )
  
          noise_base_samples = torch.randn(
              *sample_shape,
              *batch_shape,
              self.train_train_covar.shape[-1],
              device=covariance_matrix.device,
              dtype=covariance_matrix.dtype,
          )
      else:
          # finally split up the base samples
          noise_base_samples = base_samples[..., joint_size:]
          base_samples = base_samples[..., :joint_size]
  
      perm_list = [*range(1, base_samples.ndim), 0]
      return base_samples.permute(
          *perm_list
      ), noise_base_samples.permute(*perm_list)
  
file_path: botorch/botorch/posteriors/higher_order.py
imports: from typing import Optional, Tuple
  import torch
  from botorch.exceptions.errors import BotorchTensorDimensionError
  from botorch.posteriors.gpytorch import GPyTorchPosterior
  from gpytorch.distributions import MultivariateNormal
  from linear_operator.operators import LinearOperator
  from torch import Tensor
label_desc: Use image node_id 5 for calling the HigherOrderGPPosterior obj's underlying member method code with example usage: obj._prepare_base_samples(sample_shape, base_samples) and returns: base_samples,  noise_base_samples