code: def forward(self, X: Tensor) -> Tensor:
      posterior = self.model.posterior(
          X, posterior_transform=self.posterior_transform
      )
      mu = posterior.mean
      sigma = posterior.variance.clamp_min(1e-9).sqrt()
      # clamp here, since upper_bounds will contain `inf`s, which
      # are not differentiable
      cell_upper_bounds = self.cell_upper_bounds.clamp_max(
          1e10 if X.dtype == torch.double else 1e8
      )
      # Compute psi(lower_i, upper_i, mu_i, sigma_i) for i=0, ... m-2
      psi_lu = self.psi(
          lower=self.cell_lower_bounds,
          upper=cell_upper_bounds,
          mu=mu,
          sigma=sigma,
      )
      # Compute psi(lower_m, lower_m, mu_m, sigma_m)
      psi_ll = self.psi(
          lower=self.cell_lower_bounds,
          upper=self.cell_lower_bounds,
          mu=mu,
          sigma=sigma,
      )
      # Compute nu(lower_m, upper_m, mu_m, sigma_m)
      nu = self.nu(
          lower=self.cell_lower_bounds,
          upper=cell_upper_bounds,
          mu=mu,
          sigma=sigma,
      )
      # compute the difference psi_ll - psi_lu
      psi_diff = psi_ll - psi_lu
  
      # this is batch_shape x num_cells x 2 x (m-1)
      stacked_factors = torch.stack([psi_diff, nu], dim=-2)
  
      # Take the cross product of psi_diff and nu across all outcomes
      # e.g. for m = 2
      # for each batch and cell, compute
      # [psi_diff_0, psi_diff_1]
      # [nu_0, psi_diff_1]
      # [psi_diff_0, nu_1]
      # [nu_0, nu_1]
      # this tensor has shape: `batch_shape x num_cells x 2^m x m`
      all_factors_up_to_last = stacked_factors.gather(
          dim=-2,
          index=self._cross_product_indices.expand(
              stacked_factors.shape[:-2]
              + self._cross_product_indices.shape
          ),
      )
      # compute product for all 2^m terms,
      # sum across all terms and hypercells
      return all_factors_up_to_last.prod(dim=-1).sum(dim=-1).sum(dim=-1)
  
imports: from __future__ import annotations
  from abc import abstractmethod
  from itertools import product
  from typing import List, Optional
  import torch
  from botorch.acquisition.acquisition import AcquisitionFunction
  from botorch.acquisition.objective import PosteriorTransform
  from botorch.exceptions.errors import UnsupportedError
  from botorch.models.model import Model
  from botorch.utils.multi_objective.box_decompositions.non_dominated import NondominatedPartitioning
  from botorch.utils.transforms import t_batch_mode_transform
  from torch import Tensor
  from torch.distributions import Normal
label_desc: Use image node_id 4 for calling the ExpectedHypervolumeImprovement obj's underlying member method code with example usage: obj.forward(X) and returns: all_factors_up_to_last