code: def _mean_cache(self, nan_policy: str) -> Tensor:
      mvn = self.likelihood(self.train_prior_dist, self.train_inputs)
      train_mean, train_train_covar = (
          mvn.loc,
          mvn.lazy_covariance_matrix,
      )
  
      train_labels_offset = (self.train_labels - train_mean).unsqueeze(
          -1
      )
  
      if nan_policy == "ignore":
          mean_cache = (
              train_train_covar.evaluate_kernel()
              .solve(train_labels_offset)
              .squeeze(-1)
          )
      elif nan_policy == "mask":
          # Mask all rows and columns in the kernel matrix corresponding to the missing observations.
          observed = settings.observation_nan_policy._get_observed(
              self.train_labels,
              torch.Size((self.train_labels.shape[-1],)),
          )
          mean_cache = torch.full_like(self.train_labels, torch.nan)
          kernel = MaskedLinearOperator(
              train_train_covar.evaluate_kernel(),
              observed.reshape(-1),
              observed.reshape(-1),
          )
          mean_cache[..., observed] = kernel.solve(
              train_labels_offset[..., observed, :]
          ).squeeze(-1)
      else:  # 'fill'
          # Fill all rows and columns in the kernel matrix corresponding to the missing observations with 0.
          # Don't touch the corresponding diagonal elements to ensure a unique solution.
          # This ensures that missing data is ignored during solving.
          warnings.warn(
              "Observation NaN policy 'fill' makes the kernel matrix dense during exact prediction.",
              RuntimeWarning,
          )
          kernel = train_train_covar.evaluate_kernel()
          missing = torch.isnan(self.train_labels)
          kernel_mask = (~missing).to(torch.float)
          kernel_mask = (
              kernel_mask[..., None] * kernel_mask[..., None, :]
          )
          torch.diagonal(kernel_mask, dim1=-2, dim2=-1)[...] = 1
          kernel = (
              kernel * kernel_mask
          )  # Unfortunately, this makes the kernel dense at the moment.
          train_labels_offset = (
              settings.observation_nan_policy._fill_tensor(
                  train_labels_offset
              )
          )
          mean_cache = kernel.solve(train_labels_offset).squeeze(-1)
          mean_cache[
              missing
          ] = (
              torch.nan
          )  # Ensure that nobody expects these values to be valid.
      if settings.detach_test_caches.on():
          mean_cache = mean_cache.detach()
  
      if mean_cache.grad_fn is not None:
          wrapper = functools.partial(clear_cache_hook, self)
          functools.update_wrapper(wrapper, clear_cache_hook)
          mean_cache.grad_fn.register_hook(wrapper)
  
      return mean_cache
  
imports: import functools
  import string
  import warnings
  import torch
  from linear_operator import to_dense, to_linear_operator
  from linear_operator.operators import AddedDiagLinearOperator, BatchRepeatLinearOperator, ConstantMulLinearOperator, DenseLinearOperator, InterpolatedLinearOperator, LinearOperator, LowRankRootAddedDiagLinearOperator, MaskedLinearOperator, MatmulLinearOperator, RootLinearOperator, ZeroLinearOperator
  from linear_operator.utils.cholesky import psd_safe_cholesky
  from linear_operator.utils.interpolation import left_interp, left_t_interp
  from torch import Tensor
  from ..None import settings
  from ..distributions import MultitaskMultivariateNormal
  from ..lazy import LazyEvaluatedKernelTensor
  from ..utils.memoize import add_to_cache, cached, clear_cache_hook, pop_from_cache
label_desc: Use image node_id 8 for calling the DefaultPredictionStrategy obj's underlying member method code with example usage: obj._mean_cache(nan_policy) and returns: mean_cache