class_docstr: A fully Bayesian multi-task GP model with the SAAS prior.
  
  This model assumes that the inputs have been normalized to [0, 1]^d and that the
  output has been stratified standardized to have zero mean and unit variance for
  each task.The SAAS model [Eriksson2021saasbo]_ with a Matern-5/2 is used as data
  kernel by default.
  
  You are expected to use `fit_fully_bayesian_model_nuts` to fit this model as it
  isn't compatible with `fit_gpytorch_model`.
  
  Example:
      >>> X1, X2 = torch.rand(10, 2), torch.rand(20, 2)
      >>> i1, i2 = torch.zeros(10, 1), torch.ones(20, 1)
      >>> train_X = torch.cat([
      >>>     torch.cat([X1, i1], -1), torch.cat([X2, i2], -1),
      >>> ])
      >>> train_Y = torch.cat(f1(X1), f2(X2)).unsqueeze(-1)
      >>> train_Yvar = 0.01 * torch.ones_like(train_Y)
      >>> mtsaas_gp = SaasFullyBayesianMultiTaskGP(
      >>>     train_X, train_Y, train_Yvar, task_feature=-1,
      >>> )
      >>> fit_fully_bayesian_model_nuts(mtsaas_gp)
      >>> posterior = mtsaas_gp.posterior(test_X)
code: def forward(self, X: Tensor) -> MultivariateNormal:
      self._check_if_fitted()
      X = X.unsqueeze(MCMC_DIM)
  
      x_basic, task_idcs = self._split_inputs(X)
  
      mean_x = self.mean_module(x_basic)
      covar_x = self.covar_module(x_basic)
  
      tsub_idcs = task_idcs.squeeze(-3).squeeze(-1)
      latent_features = self.latent_features[:, tsub_idcs, :]
  
      if X.ndim > 3:
          # batch eval mode
          # for X (batch_shape x num_samples x q x d), task_idcs[:,i,:,] are the same
          # reshape X to (batch_shape x num_samples x q x d)
          latent_features = latent_features.permute(
              [-i for i in range(X.ndim - 1, 2, -1)]
              + [0]
              + [-i for i in range(2, 0, -1)]
          )
  
      # Combine the two in an ICM fashion
      covar_i = self.task_covar_module(latent_features)
      covar = covar_x.mul(covar_i)
      return MultivariateNormal(mean_x, covar)
  
imports: from typing import Any, Dict, List, Mapping, NoReturn, Optional, Tuple, Union
  import pyro
  import torch
  from botorch.acquisition.objective import PosteriorTransform
  from botorch.models.fully_bayesian import matern52_kernel, MIN_INFERRED_NOISE_LEVEL, PyroModel, reshape_and_detach, SaasPyroModel
  from botorch.models.multitask import MultiTaskGP
  from botorch.models.transforms.input import InputTransform
  from botorch.models.transforms.outcome import OutcomeTransform
  from botorch.posteriors.fully_bayesian import GaussianMixturePosterior, MCMC_DIM
  from botorch.utils.datasets import MultiTaskDataset, SupervisedDataset
  from gpytorch.distributions.multivariate_normal import MultivariateNormal
  from gpytorch.kernels import MaternKernel
  from gpytorch.kernels.kernel import Kernel
  from gpytorch.likelihoods.likelihood import Likelihood
  from gpytorch.means.mean import Mean
  from torch import Tensor
  from torch.nn.parameter import Parameter
label_desc: Use image node_id 10 for calling the SaasFullyBayesianMultiTaskGP obj's underlying member method code with example usage: obj.forward(X) and returns: MultivariateNormal