code: def _initialize_predictive_matrices(
      X: Tensor,
      model: Model,
      observation_noise: bool = True,
      jitter: float = 1e-4,
      natural: bool = True,
  ) -> Tuple[Tensor, Tensor, Tensor, Tensor]:
      r"""Initializes the natural predictive mean and covariance matrix. For a
      multivariate normal distribution with mean mu and covariance Sigma, the natural
      mean is Sigma^{-1} mu and the natural covariance is Sigma^{-1}.
  
      Args:
          X: A `batch_shape x R x d`-dim Tensor.
          model: The fitted model.
          observation_noise: If true, the posterior is computed with observation noise.
          jitter: The jitter added to the covariance matrix.
          natural: If true, we compute the natural statistics as well.
  
      Return:
          A four-element tuple containing
  
          - pred_nat_mean: A `batch_shape x num_outputs x R `-dim Tensor containing the
              predictive natural mean vectors.
          - pred_nat_cov: A `batch_shape x num_outputs x R x R`-dim Tensor containing
              the predictive natural covariance matrices.
          - pred_mean: A `batch_shape x num_outputs x R`-dim Tensor containing the
              predictive mean vectors.
          - pred_cov: A `batch_shape x num_outputs x R x R`-dim Tensor containing the
              predictive covariance matrices.
      """
      tkwargs = {"dtype": X.dtype, "device": X.device}
      # compute the predictive mean and covariances at X
      posterior = model.posterior(
          X, observation_noise=observation_noise
      )
  
      # `batch_shape x (R * num_outputs) x (R * num_outputs)`
      init_pred_cov = posterior.mvn.covariance_matrix
      num_outputs = model.num_outputs
      R = int(init_pred_cov.shape[-1] / num_outputs)
      pred_cov = [
          init_pred_cov[
              ..., (m * R) : ((m + 1) * R), (m * R) : ((m + 1) * R)
          ].unsqueeze(-1)
          for m in range(num_outputs)
      ]
      # `batch_shape x R x R x num_outputs` (before swap axes)
      # `batch_shape x num_outputs x R * R`
      pred_cov = (
          torch.cat(pred_cov, axis=-1).swapaxes(-2, -1).swapaxes(-3, -2)
      )
      identity = torch.diag_embed(
          torch.ones(pred_cov.shape[:-1], **tkwargs)
      )
      pred_cov = pred_cov + jitter * identity
  
      # `batch_shape x num_outputs x R`
      pred_mean = posterior.mean.swapaxes(-2, -1)
  
      #############################################################
      if natural:
          # natural parameters
          # `batch_shape x num_outputs x R x R`
          cholesky_pred_cov, _ = torch.linalg.cholesky_ex(pred_cov)
          pred_nat_cov = torch.cholesky_inverse(cholesky_pred_cov)
  
          # `batch_shape x num_outputs x R`
          pred_nat_mean = torch.einsum(
              "...ij,...j->...i", pred_nat_cov, pred_mean
          )
  
          return pred_nat_mean, pred_nat_cov, pred_mean, pred_cov
      else:
          return None, None, pred_mean, pred_cov
  
imports: from __future__ import annotations
  from typing import Any, Optional, Tuple
  import torch
  from botorch.acquisition.acquisition import AcquisitionFunction
  from botorch.exceptions import InputDataError
  from botorch.exceptions.errors import UnsupportedError
  from botorch.models.model import Model
  from botorch.models.model_list_gp_regression import ModelListGP
  from botorch.models.utils import check_no_nans
  from botorch.utils.transforms import concatenate_pending_points, t_batch_mode_transform
  from torch import Tensor
  from torch.distributions import Normal
label_desc: Use image node_id 2 for calling a global function with example usage: _initialize_predictive_matrices(X, model, observation_noise, jitter, natural) and returns: pred_nat_mean,  pred_nat_cov,  pred_mean,  pred_cov, None,  None,  pred_mean,  pred_cov