code: def test_FixedNoiseMultiTaskGP(self):
      bounds = torch.tensor([[-1.0, 0.0], [1.0, 1.0]])
      for dtype, use_intf, use_octf in itertools.product(
          (torch.float, torch.double), (False, True), (False, True)
      ):
          tkwargs = {"device": self.device, "dtype": dtype}
          octf = Standardize(m=1) if use_octf else None
          intf = (
              Normalize(
                  d=2,
                  bounds=bounds.to(**tkwargs),
                  transform_on_train=True,
              )
              if use_intf
              else None
          )
          model, _, (train_X, _, _) = _gen_fixed_noise_model_and_data(
              input_transform=intf, outcome_transform=octf, **tkwargs
          )
          self.assertIsInstance(model, MultiTaskGP)
          self.assertEqual(model.num_outputs, 2)
          self.assertIsInstance(
              model.likelihood, FixedNoiseGaussianLikelihood
          )
          self.assertIsInstance(model.mean_module, ConstantMean)
          self.assertIsInstance(model.covar_module, ScaleKernel)
          matern_kernel = model.covar_module.base_kernel
          self.assertIsInstance(matern_kernel, MaternKernel)
          self.assertIsInstance(
              matern_kernel.lengthscale_prior, GammaPrior
          )
          self.assertIsInstance(model.task_covar_module, IndexKernel)
          self.assertEqual(model._rank, 2)
          self.assertEqual(
              model.task_covar_module.covar_factor.shape[-1],
              model._rank,
          )
          if use_octf:
              self.assertIsInstance(
                  model.outcome_transform, Standardize
              )
          if use_intf:
              self.assertIsInstance(model.input_transform, Normalize)
  
          # 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,
              )
  
          # check that training data has input transform applied
          # check that the train inputs have been transformed and set on the model
          if use_intf:
              self.assertTrue(
                  torch.equal(
                      model.train_inputs[0],
                      model.input_transform(train_X),
                  )
              )
  
          # test posterior
          test_x = torch.rand(2, 1, **tkwargs)
          posterior_f = model.posterior(test_x)
          self.assertIsInstance(posterior_f, GPyTorchPosterior)
          self.assertIsInstance(
              posterior_f.distribution, MultitaskMultivariateNormal
          )
          self.assertEqual(posterior_f.mean.shape, torch.Size([2, 2]))
          self.assertEqual(
              posterior_f.variance.shape, torch.Size([2, 2])
          )
  
          # check posterior transform is applied
          if use_octf:
              posterior_pred = model.posterior(test_x)
              tmp_tf = model.outcome_transform
              del model.outcome_transform
              pp_tf = model.posterior(test_x)
              model.outcome_transform = tmp_tf
              expected_var = tmp_tf.untransform_posterior(
                  pp_tf
              ).variance
              self.assertAllClose(posterior_pred.variance, expected_var)
  
          # test that posterior w/ observation noise raises appropriate error
          with self.assertRaises(NotImplementedError):
              model.posterior(test_x, observation_noise=True)
          with self.assertRaises(NotImplementedError):
              model.posterior(
                  test_x, observation_noise=torch.rand(2, **tkwargs)
              )
  
          # test posterior w/ single output index
          posterior_f = model.posterior(test_x, output_indices=[0])
          self.assertIsInstance(posterior_f, GPyTorchPosterior)
          self.assertIsInstance(
              posterior_f.distribution, MultivariateNormal
          )
          self.assertEqual(posterior_f.mean.shape, torch.Size([2, 1]))
          self.assertEqual(
              posterior_f.variance.shape, torch.Size([2, 1])
          )
  
          # test posterior w/ bad output index
          with self.assertRaises(ValueError):
              model.posterior(test_x, output_indices=[2])
  
          # test posterior (batch eval)
          test_x = torch.rand(3, 2, 1, **tkwargs)
          posterior_f = model.posterior(test_x)
          self.assertIsInstance(posterior_f, GPyTorchPosterior)
          self.assertIsInstance(
              posterior_f.distribution, MultitaskMultivariateNormal
          )
  
          # test that unsupported batch shape MTGPs throw correct error
          with self.assertRaises(ValueError):
              FixedNoiseMultiTaskGP(
                  torch.rand(2, 2, 2),
                  torch.rand(2, 2, 1),
                  torch.rand(2, 2, 1),
                  0,
              )
  
          # test that bad feature index throws correct error
          _, (train_X, train_Y, _) = gen_multi_task_dataset(**tkwargs)
          train_Yvar = torch.full_like(train_Y, 0.05)
          with self.assertRaises(ValueError):
              FixedNoiseMultiTaskGP(train_X, train_Y, train_Yvar, 2)
  
          # test that bad output task throws correct error
          with self.assertRaises(RuntimeError):
              FixedNoiseMultiTaskGP(
                  train_X, train_Y, train_Yvar, 0, output_tasks=[2]
              )
  
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 2 for calling the TestFixedNoiseMultiTaskGP obj's underlying member method code with example usage: obj.test_FixedNoiseMultiTaskGP() without return types