code: def _prepare_base_samples(
      self, sample_shape: torch.Size, base_samples: Tensor = None
  ) -> Tuple[Tensor, Tensor]:
      covariance_matrix = self.joint_covariance_matrix
      joint_size = covariance_matrix.shape[-1]
      batch_shape = covariance_matrix.batch_shape
  
      # pre-allocated this as None
      test_noise_base_samples = None
      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."
                  f"provided base sample shape is {base_samples.shape} while"
                  f"the expected shape is {sample_shape}."
              )
  
          if base_samples.shape[-1] != 1:
              base_samples = base_samples.unsqueeze(-1)
          unsqueezed_dim = -2
  
          appended_shape = joint_size + self.train_train_covar.shape[-1]
          if self.observation_noise:
              appended_shape = (
                  appended_shape + self.test_noise.shape[-1]
              )
  
          if appended_shape != base_samples.shape[unsqueezed_dim]:
              # 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],
                      dtype=base_samples.dtype,
                      device=base_samples.device,
                  )
                  base_samples = torch.cat(
                      (base_samples, new_base_samples), dim=-1
                  )
                  base_samples = base_samples.unsqueeze(-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,
              1,
              device=covariance_matrix.device,
              dtype=covariance_matrix.dtype,
          )
  
          noise_base_samples = torch.randn(
              *sample_shape,
              *batch_shape,
              self.train_train_covar.shape[-1],
              1,
              device=covariance_matrix.device,
              dtype=covariance_matrix.dtype,
          )
          if self.observation_noise:
              test_noise_base_samples = torch.randn(
                  *sample_shape,
                  *self.test_noise.shape[:-1],
                  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, :]
          if self.observation_noise:
              test_noise_base_samples = noise_base_samples[
                  ..., -self.test_noise.shape[-1] :, :
              ]
              noise_base_samples = noise_base_samples[
                  ..., : -self.test_noise.shape[-1], :
              ]
  
      return base_samples, noise_base_samples, test_noise_base_samples
  
file_path: botorch/botorch/posteriors/multitask.py
imports: from typing import Optional, Tuple, Union
  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, to_linear_operator
  from torch import Tensor
label_desc: Use image node_id 4 for calling the MultitaskGPPosterior obj's underlying member method code with example usage: obj._prepare_base_samples(sample_shape, base_samples) and returns: base_samples,  noise_base_samples,  test_noise_base_samples