code: def test_fit_model(
      self,
      dtype: torch.dtype = torch.double,
      infer_noise: bool = False,
      task_rank: int = 1,
  ):
      tkwargs = {"device": self.device, "dtype": dtype}
      train_X, train_Y, train_Yvar, model = self._get_data_and_model(
          infer_noise=infer_noise, task_rank=task_rank, **tkwargs
      )
      n = train_X.shape[0]
      d = train_X.shape[1] - 1
  
      # Test init
      self.assertIsNone(model.mean_module)
      self.assertIsNone(model.covar_module)
      self.assertIsNone(model.likelihood)
      self.assertIsInstance(model.pyro_model, MultitaskSaasPyroModel)
      self.assertAllClose(train_X, model.pyro_model.train_X)
      self.assertAllClose(train_Y, model.pyro_model.train_Y)
      if infer_noise:
          self.assertIsNone(model.pyro_model.train_Yvar)
      else:
          self.assertAllClose(
              train_Yvar.clamp(MIN_INFERRED_NOISE_LEVEL),
              model.pyro_model.train_Yvar,
          )
  
      # Fit a model and check that the hyperparameters have the correct shape
      fit_fully_bayesian_model_nuts(
          model,
          warmup_steps=8,
          num_samples=5,
          thinning=2,
          disable_progbar=True,
      )
      self.assertEqual(model.batch_shape, torch.Size([3]))
      self.assertIsInstance(model.mean_module, ConstantMean)
      self.assertEqual(
          model.mean_module.raw_constant.shape, model.batch_shape
      )
      self.assertIsInstance(model.covar_module, ScaleKernel)
      self.assertEqual(
          model.covar_module.outputscale.shape, model.batch_shape
      )
      self.assertIsInstance(
          model.covar_module.base_kernel, MaternKernel
      )
      self.assertEqual(
          model.covar_module.base_kernel.lengthscale.shape,
          torch.Size([3, 1, d]),
      )
      if infer_noise:
          self.assertIsInstance(model.likelihood, GaussianLikelihood)
          self.assertEqual(
              model.likelihood.noise_covar.noise.shape,
              torch.Size([3, 1]),
          )
      else:
          self.assertIsInstance(
              model.likelihood, FixedNoiseGaussianLikelihood
          )
      self.assertIsInstance(model.task_covar_module, MaternKernel)
      self.assertEqual(
          model.task_covar_module.lengthscale.shape,
          torch.Size([3, 1, task_rank]),
      )
      self.assertEqual(
          model.latent_features.shape,
          torch.Size([3, self.num_tasks, task_rank]),
      )
  
      # Predict on some test points
      for batch_shape in [[5], [5, 2], [5, 2, 6]]:
          test_X = torch.rand(*batch_shape, d, **tkwargs)
          posterior = model.posterior(test_X)
          self.assertIsInstance(posterior, GaussianMixturePosterior)
          self.assertIsInstance(posterior, GaussianMixturePosterior)
  
          test_X = torch.rand(*batch_shape, d, **tkwargs)
          posterior = model.posterior(test_X)
          self.assertIsInstance(posterior, GaussianMixturePosterior)
          # Mean/variance
          expected_shape = (
              *batch_shape[: MCMC_DIM + 2],
              *model.batch_shape,
              *batch_shape[MCMC_DIM + 2 :],
              self.num_tasks,
          )
          expected_shape = torch.Size(expected_shape)
          mean, var = posterior.mean, posterior.variance
          self.assertEqual(mean.shape, expected_shape)
          self.assertEqual(var.shape, expected_shape)
  
          # Mixture mean/variance/median/quantiles
          mixture_mean = posterior.mixture_mean
          mixture_variance = posterior.mixture_variance
          quantile1 = posterior.quantile(value=torch.tensor(0.01))
          quantile2 = posterior.quantile(value=torch.tensor(0.99))
  
          # Marginalized mean/variance
          self.assertEqual(
              mixture_mean.shape,
              torch.Size(batch_shape + [self.num_tasks]),
          )
          self.assertEqual(
              mixture_variance.shape,
              torch.Size(batch_shape + [self.num_tasks]),
          )
          self.assertTrue(mixture_variance.min() > 0.0)
          self.assertEqual(
              quantile1.shape,
              torch.Size(batch_shape + [self.num_tasks]),
          )
          self.assertEqual(
              quantile2.shape,
              torch.Size(batch_shape + [self.num_tasks]),
          )
          self.assertTrue((quantile2 > quantile1).all())
  
          dist = torch.distributions.Normal(
              loc=posterior.mean, scale=posterior.variance.sqrt()
          )
          torch.allclose(
              dist.cdf(quantile1.unsqueeze(MCMC_DIM)).mean(
                  dim=MCMC_DIM
              ),
              0.05 * torch.ones(batch_shape + [1], **tkwargs),
          )
          torch.allclose(
              dist.cdf(quantile2.unsqueeze(MCMC_DIM)).mean(
                  dim=MCMC_DIM
              ),
              0.95 * torch.ones(batch_shape + [1], **tkwargs),
          )
          # Invalid quantile should raise
          for q in [-1.0, 0.0, 1.0, 1.3333]:
              with self.assertRaisesRegex(
                  ValueError, "value is expected to be in the range"
              ):
                  posterior.quantile(value=torch.tensor(q))
  
          # Test model lists with fully Bayesian models and mixed modeling
          deterministic = GenericDeterministicModel(
              f=lambda x: x[..., :1]
          )
          for ModelListClass, models, expected_outputs in zip(
              [ModelList, ModelListGP],
              [[deterministic, model], [model, model]],
              [3, 4],
          ):
              expected_shape = (
                  *batch_shape[: MCMC_DIM + 2],
                  *model.batch_shape,
                  *batch_shape[MCMC_DIM + 2 :],
                  expected_outputs,
              )
              expected_shape = torch.Size(expected_shape)
              model_list = ModelListClass(*models)
              posterior = model_list.posterior(test_X)
              mean, var = posterior.mean, posterior.variance
              self.assertEqual(mean.shape, expected_shape)
              self.assertEqual(var.shape, expected_shape)
  
      # Check properties
      median_lengthscale = model.median_lengthscale
      self.assertEqual(median_lengthscale.shape, torch.Size([d]))
      self.assertEqual(model.num_mcmc_samples, 3)
  
      # Check the keys in the state dict
      true_keys = EXPECTED_KEYS_NOISE if infer_noise else EXPECTED_KEYS
      self.assertEqual(set(model.state_dict().keys()), set(true_keys))
  
      # Check that we can load the state dict.
      state_dict = model.state_dict()
      _, _, _, m_new = self._get_data_and_model(
          infer_noise=infer_noise, task_rank=task_rank, **tkwargs
      )
      self.assertEqual(m_new.state_dict(), {})
      m_new.load_state_dict(state_dict)
      self.assertEqual(
          model.state_dict().keys(), m_new.state_dict().keys()
      )
      for k in model.state_dict().keys():
          self.assertTrue(
              (model.state_dict()[k] == m_new.state_dict()[k]).all()
          )
      preds1, preds2 = model.posterior(test_X), m_new.posterior(test_X)
      self.assertTrue(torch.equal(preds1.mean, preds2.mean))
      self.assertTrue(torch.equal(preds1.variance, preds2.variance))
  
      # Make sure the model shapes are set correctly
      self.assertEqual(
          model.pyro_model.train_X.shape, torch.Size([n, d + 1])
      )
      self.assertAllClose(model.pyro_model.train_X, train_X)
      model.train()  # Put the model in train mode
      self.assertAllClose(train_X, model.pyro_model.train_X)
      self.assertIsNone(model.mean_module)
      self.assertIsNone(model.covar_module)
      self.assertIsNone(model.likelihood)
      self.assertIsNone(model.task_covar_module)
  
imports: import itertools
  from typing import List, Optional
  import torch
  from botorch import fit_fully_bayesian_model_nuts
  from botorch.acquisition.analytic import ExpectedImprovement, PosteriorMean, ProbabilityOfImprovement, UpperConfidenceBound
  from botorch.acquisition.monte_carlo import qExpectedImprovement, qNoisyExpectedImprovement, qProbabilityOfImprovement, qSimpleRegret, qUpperConfidenceBound
  from botorch.acquisition.multi_objective import qExpectedHypervolumeImprovement, qNoisyExpectedHypervolumeImprovement
  from botorch.models import ModelList, ModelListGP
  from botorch.models.deterministic import GenericDeterministicModel
  from botorch.models.fully_bayesian import MCMC_DIM, MIN_INFERRED_NOISE_LEVEL
  from botorch.models.fully_bayesian_multitask import MultitaskSaasPyroModel, SaasFullyBayesianMultiTaskGP
  from botorch.models.transforms.input import Normalize
  from botorch.models.transforms.outcome import Standardize
  from botorch.posteriors import GaussianMixturePosterior
  from botorch.sampling.get_sampler import get_sampler
  from botorch.sampling.normal import IIDNormalSampler
  from botorch.utils.multi_objective.box_decompositions.non_dominated import NondominatedPartitioning
  from botorch.utils.test_helpers import gen_multi_task_dataset
  from botorch.utils.testing import BotorchTestCase
  from gpytorch.kernels import MaternKernel, ScaleKernel
  from gpytorch.likelihoods import FixedNoiseGaussianLikelihood
  from gpytorch.likelihoods.gaussian_likelihood import GaussianLikelihood
  from gpytorch.means import ConstantMean
label_desc: Use image node_id 5 for calling the TestFullyBayesianMultiTaskGP obj's underlying member method code with example usage: obj.test_fit_model(dtype, infer_noise, task_rank) without return types