code: def _update_marginals(
      pred_f_nat_mean: Tensor,
      pred_f_nat_cov: Tensor,
      omega_f_nat_mean: Tensor,
      omega_f_nat_cov: Tensor,
      N: int,
      P: int,
  ) -> Tuple[Tensor, Tensor]:
      r"""Computes the new marginal by summing up all the natural factors.
  
      Args:
          pred_f_nat_mean: A `batch_shape x M x (N + P)`-dim Tensor containing the
              natural predictive mean matrix for the objectives.
          pred_f_nat_cov: A `batch_shape x M x (N + P) x (N + P)`-dim Tensor containing
              the natural predictive covariance matrix for the objectives.
          omega_f_nat_mean: A `batch_shape x M x (N + P) x P x 2`-dim Tensor containing
              the omega natural mean factors for the objective matrix.
          omega_f_nat_cov: A `batch_shape x M x (N + P) x P x 2 x 2`-dim Tensor
              containing the omega natural covariance factors for the objective matrix.
          N: The number of design points.
          P: The number of Pareto optimal points.
  
      Returns:
          A two-element tuple containing
  
          - nat_mean_f: A `batch_shape x M x (N + P)`-dim Tensor containing the updated
              natural mean matrix for the objectives.
          - nat_cov_f: A `batch_shape x M x (N + P) x (N + P)`-dim Tensor containing
              the updated natural predictive covariance matrix for the objectives.
      """
  
      # `batch_shape x M x (N + P)`
      nat_mean_f = pred_f_nat_mean.clone()
      # `batch_shape x M x (N + P) x (N + P)
      nat_cov_f = pred_f_nat_cov.clone()
  
      ################################################################################
      # UPDATE THE OBJECTIVES
      ################################################################################
      # remove Pareto diagonal
      # zero out the diagonal
      omega_f_nat_mean[..., range(N, N + P), range(P), :] = 0
      omega_f_nat_cov[..., range(N, N + P), range(P), :, :] = 0
  
      # `batch_shape x M x (N + P)`
      # sum over the pareto dim
      nat_mean_f = nat_mean_f + omega_f_nat_mean[..., 0].sum(dim=-1)
      # `batch_shape x M x P`
      # sum over the data dim
      nat_mean_f[..., N:] = nat_mean_f[..., N:] + omega_f_nat_mean[
          ..., 1
      ].sum(dim=-2)
  
      # `batch_shape x M x (N + P)`
      nat_cov_f[..., range(N + P), range(N + P)] = nat_cov_f[
          ..., range(N + P), range(N + P)
      ] + omega_f_nat_cov[..., 0, 0].sum(dim=-1)
      # `batch_shape x M x P`
      nat_cov_f[..., range(N, N + P), range(N, N + P)] = nat_cov_f[
          ..., range(N, N + P), range(N, N + P)
      ] + omega_f_nat_cov[..., 1, 1].sum(dim=-2)
  
      for p in range(P):
          # `batch_shape x M x (N + P)`
          nat_cov_f[..., range(N + P), N + p] = (
              nat_cov_f[..., range(N + P), N + p]
              + omega_f_nat_cov[..., p, 0, 1]
          )
  
          # `batch_shape x M x (N + P)`
          nat_cov_f[..., N + p, range(N + P)] = (
              nat_cov_f[..., N + p, range(N + P)]
              + omega_f_nat_cov[..., p, 1, 0]
          )
  
      return nat_mean_f, nat_cov_f
  
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 7 for calling a global function with example usage: _update_marginals(pred_f_nat_mean, pred_f_nat_cov, omega_f_nat_mean, omega_f_nat_cov, N, P) and returns: nat_mean_f,  nat_cov_f