code: def test_KroneckerMultiTaskGP_default(self):
      bounds = torch.tensor([[-1.0, 0.0], [1.0, 1.0]])
  
      for batch_shape, dtype, use_intf, use_octf in itertools.product(
          (
              torch.Size(),
          ),  # torch.Size([3])), TODO: Fix and test batch mode
          (torch.float, torch.double),
          (False, True),
          (False, True),
      ):
          tkwargs = {"device": self.device, "dtype": dtype}
  
          octf = Standardize(m=2) if use_octf else None
  
          intf = (
              Normalize(
                  d=2,
                  bounds=bounds.to(**tkwargs),
                  transform_on_train=True,
              )
              if use_intf
              else None
          )
  
          # initialization with default settings
          model, train_X, _ = _gen_kronecker_model_and_data(
              model_kwargs={
                  "outcome_transform": octf,
                  "input_transform": intf,
              },
              batch_shape=batch_shape,
              **tkwargs,
          )
          self.assertIsInstance(model, KroneckerMultiTaskGP)
          self.assertEqual(model.num_outputs, 2)
          self.assertIsInstance(
              model.likelihood, MultitaskGaussianLikelihood
          )
          self.assertEqual(model.likelihood.rank, 0)
          self.assertIsInstance(model.mean_module, MultitaskMean)
          self.assertIsInstance(model.covar_module, MultitaskKernel)
          base_kernel = model.covar_module
          self.assertIsInstance(
              base_kernel.data_covar_module, MaternKernel
          )
          self.assertIsInstance(
              base_kernel.task_covar_module, IndexKernel
          )
          task_covar_prior = (
              base_kernel.task_covar_module.IndexKernelPrior
          )
          self.assertIsInstance(task_covar_prior, LKJCovariancePrior)
          self.assertEqual(task_covar_prior.correlation_prior.eta, 1.5)
          self.assertIsInstance(
              task_covar_prior.sd_prior, SmoothedBoxPrior
          )
          lengthscale_prior = (
              base_kernel.data_covar_module.lengthscale_prior
          )
          self.assertIsInstance(lengthscale_prior, GammaPrior)
          self.assertEqual(lengthscale_prior.concentration, 3.0)
          self.assertEqual(lengthscale_prior.rate, 6.0)
          self.assertEqual(
              base_kernel.task_covar_module.covar_factor.shape[-1], 2
          )
  
          # test model fitting
          mll = ExactMarginalLogLikelihood(model.likelihood, model)
          with warnings.catch_warnings():
              warnings.filterwarnings(
                  "ignore", category=OptimizationWarning
              )
              mll = fit_gpytorch_mll(
                  mll,
                  optimizer_kwargs={"options": {"maxiter": 1}},
                  max_attempts=1,
              )
  
          # test posterior
          test_x = torch.rand(2, 2, **tkwargs)
          posterior_f = model.posterior(test_x)
          if not use_octf:
              self.assertIsInstance(posterior_f, GPyTorchPosterior)
              self.assertIsInstance(
                  posterior_f.distribution, MultitaskMultivariateNormal
              )
          else:
              self.assertIsInstance(posterior_f, TransformedPosterior)
              self.assertIsInstance(
                  posterior_f._posterior.distribution,
                  MultitaskMultivariateNormal,
              )
  
          self.assertEqual(posterior_f.mean.shape, torch.Size([2, 2]))
          self.assertEqual(
              posterior_f.variance.shape, torch.Size([2, 2])
          )
  
          if use_octf:
              # ensure un-transformation is applied
              tmp_tf = model.outcome_transform
              del model.outcome_transform
              p_tf = model.posterior(test_x)
              model.outcome_transform = tmp_tf
              expected_var = tmp_tf.untransform_posterior(p_tf).variance
              self.assertAllClose(posterior_f.variance, expected_var)
          else:
              # test observation noise
              # TODO: outcome transform + likelihood noise?
              posterior_noisy = model.posterior(
                  test_x, observation_noise=True
              )
              self.assertTrue(
                  torch.allclose(
                      posterior_noisy.variance,
                      model.likelihood(
                          posterior_f.distribution
                      ).variance,
                  )
              )
  
          # test posterior (batch eval)
          test_x = torch.rand(3, 2, 2, **tkwargs)
          posterior_f = model.posterior(test_x)
          if not use_octf:
              self.assertIsInstance(posterior_f, GPyTorchPosterior)
              self.assertIsInstance(
                  posterior_f.distribution, MultitaskMultivariateNormal
              )
          else:
              self.assertIsInstance(posterior_f, TransformedPosterior)
              self.assertIsInstance(
                  posterior_f._posterior.distribution,
                  MultitaskMultivariateNormal,
              )
          self.assertEqual(
              posterior_f.mean.shape, torch.Size([3, 2, 2])
          )
          self.assertEqual(
              posterior_f.variance.shape, torch.Size([3, 2, 2])
          )
  
          # test that using a posterior transform throws error
          post_tf = ScalarizedPosteriorTransform(
              weights=torch.ones(2, **tkwargs)
          )
          with self.assertRaises(NotImplementedError):
              model.posterior(test_x, posterior_transform=post_tf)
  
imports: import itertools
  import math
  import warnings
  from typing import List, Optional
  import torch
  from botorch.acquisition.objective import ScalarizedPosteriorTransform
  from botorch.exceptions import OptimizationWarning
  from botorch.fit import fit_gpytorch_mll
  from botorch.models.multitask import FixedNoiseMultiTaskGP, KroneckerMultiTaskGP, MultiTaskGP
  from botorch.models.transforms.input import Normalize
  from botorch.models.transforms.outcome import Standardize
  from botorch.posteriors import GPyTorchPosterior
  from botorch.posteriors.transformed import TransformedPosterior
  from botorch.utils.test_helpers import gen_multi_task_dataset
  from botorch.utils.testing import BotorchTestCase
  from gpytorch.distributions import MultitaskMultivariateNormal, MultivariateNormal
  from gpytorch.kernels import IndexKernel, MaternKernel, MultitaskKernel, RBFKernel, ScaleKernel
  from gpytorch.likelihoods import FixedNoiseGaussianLikelihood, GaussianLikelihood, MultitaskGaussianLikelihood
  from gpytorch.means import ConstantMean, MultitaskMean
  from gpytorch.means.linear_mean import LinearMean
  from gpytorch.mlls.exact_marginal_log_likelihood import ExactMarginalLogLikelihood
  from gpytorch.priors import GammaPrior, LogNormalPrior, SmoothedBoxPrior
  from gpytorch.priors.lkj_prior import LKJCovariancePrior
  from gpytorch.settings import max_cholesky_size, max_root_decomposition_size
label_desc: Use image node_id 1 for calling the TestKroneckerMultiTaskGP obj's underlying member method code with example usage: obj.test_KroneckerMultiTaskGP_default() without return types