code: def test_posterior(self):
      for dtype in [torch.float, torch.double]:
          for mcs in [800, 10]:
              torch.random.manual_seed(0)
              with max_cholesky_size(mcs):
                  test_x = torch.rand(2, 12, 1).to(
                      device=self.device, dtype=dtype
                  )
  
                  self.model.to(dtype)
                  # clear caches
                  self.model.train()
                  self.model.eval()
                  # test the posterior works
                  posterior = self.model.posterior(test_x)
                  self.assertIsInstance(posterior, GPyTorchPosterior)
  
                  # test that a posterior transform raises an error
                  with self.assertRaises(NotImplementedError):
                      self.model.posterior(
                          test_x,
                          posterior_transform=DummyPosteriorTransform(),
                      )
  
                  # test the posterior works with observation noise
                  posterior = self.model.posterior(
                      test_x, observation_noise=True
                  )
                  self.assertIsInstance(posterior, GPyTorchPosterior)
  
                  # test the posterior works with no variances
                  # some funkiness in MVNs registration so the variance is non-zero.
                  with skip_posterior_variances():
                      posterior = self.model.posterior(test_x)
                      self.assertIsInstance(
                          posterior, GPyTorchPosterior
                      )
                      self.assertLessEqual(
                          posterior.variance.max(), 1e-6
                      )
  
imports: import itertools
  from unittest import mock
  import torch
  from botorch.acquisition.objective import PosteriorTransform
  from botorch.models import HigherOrderGP
  from botorch.models.higher_order_gp import FlattenedStandardize
  from botorch.models.transforms.input import Normalize
  from botorch.models.transforms.outcome import Standardize
  from botorch.optim.fit import fit_gpytorch_mll_torch
  from botorch.posteriors import GPyTorchPosterior, TransformedPosterior
  from botorch.sampling import IIDNormalSampler
  from botorch.utils.testing import BotorchTestCase
  from gpytorch.kernels import RBFKernel
  from gpytorch.likelihoods import GaussianLikelihood
  from gpytorch.mlls import ExactMarginalLogLikelihood
  from gpytorch.settings import max_cholesky_size, skip_posterior_variances
label_desc: Use image node_id 3 for calling the TestHigherOrderGP obj's underlying member method code with example usage: obj.test_posterior() without return types