code: def posterior_derivative(
      gp: Model, x: Tensor, kernel_type: str = "rbf"
  ) -> MultivariateNormal:
      r"""Computes the posterior of the derivative of the GP w.r.t. the given test
      points x.
      This follows the derivation used by GIBO in Sarah Muller, Alexander
      von Rohr, Sebastian Trimpe. "Local policy search with Bayesian optimization",
      Advances in Neural Information Processing Systems 34, NeurIPS 2021.
      Args:
          gp: Botorch model
          x: (n x D) Test points.
          kernel_type: Takes "rbf" or "matern"
      Returns:
          A Botorch Posterior.
      """
      if gp.prediction_strategy is None:
          gp.posterior(
              x
          )  # Call this to update prediction strategy of GPyTorch.
      if kernel_type not in ["rbf", "matern"]:
          raise ValueError("only matern and rbf kernels are supported")
      K_xX_dx = get_KxX_dx(gp, x, kernel_type=kernel_type)
      Kxx_dx2 = get_Kxx_dx2(gp, kernel_type=kernel_type)
      mean_d = (
          K_xX_dx
          @ get_KXX_inv(gp)
          @ (gp.train_targets - gp.mean_module(gp.train_inputs[0]))
      )
      variance_d = Kxx_dx2 - K_xX_dx @ get_KXX_inv(
          gp
      ) @ K_xX_dx.transpose(1, 2)
      variance_d = variance_d.clamp_min(1e-9)
      try:
          return MultivariateNormal(mean_d, variance_d)
      except RuntimeError:
          variance_d_diag = torch.diagonal(
              variance_d, offset=0, dim1=1, dim2=2
          )
          variance_d_new = torch.diag_embed(variance_d_diag)
          return MultivariateNormal(mean_d, variance_d_new)
  
file_path: Ax/ax/utils/sensitivity/derivative_gp.py
imports: import math
  import torch
  from botorch.models.model import Model
  from gpytorch.distributions import MultivariateNormal
  from torch import Tensor
label_desc: Use image node_id 4 for calling a global function with example usage: posterior_derivative(gp, x, kernel_type) and returns: MultivariateNormal, MultivariateNormal