code: def test_posterior_latent_gp_and_likelihood_with_optimization(
      self, cuda=False, cg=False
  ):
      # This test throws a warning because the fixed noise likelihood gets the wrong input
      warnings.simplefilter("ignore", GPInputWarning)
  
      train_x, test_x, train_y, test_y = self._get_data(cuda=cuda)
      # We're manually going to set the hyperparameters to something they shouldn't be
      likelihood = FixedNoiseGaussianLikelihood(torch.ones(11) * 0.001)
      gp_model = ExactGPModel(train_x, train_y, likelihood)
      mll = gpytorch.ExactMarginalLogLikelihood(likelihood, gp_model)
      gp_model.rbf_covar_module.initialize(lengthscale=exp(1))
      gp_model.mean_module.initialize(constant=0)
  
      if cuda:
          gp_model.cuda()
          likelihood.cuda()
  
      # Find optimal model hyperparameters
      gp_model.train()
      likelihood.train()
  
      optimizer = optim.Adam(gp_model.parameters(), lr=0.1)
      optimizer.n_iter = 0
      with gpytorch.settings.debug(
          False
      ), gpytorch.settings.max_cholesky_size(0 if cg else 1e10):
          for _ in range(75):
              optimizer.zero_grad()
              output = gp_model(train_x)
              loss = -mll(output, train_y)
              loss.backward()
              optimizer.n_iter += 1
              optimizer.step()
  
          for param in gp_model.parameters():
              self.assertTrue(param.grad is not None)
              self.assertGreater(param.grad.norm().item(), 0)
          optimizer.step()
  
          # Test the model
          gp_model.eval()
          likelihood.eval()
          test_function_predictions = likelihood(gp_model(test_x))
          mean_abs_error = torch.mean(
              torch.abs(test_y - test_function_predictions.mean)
          )
  
      self.assertLess(mean_abs_error.squeeze().item(), 0.05)
  
imports: import os
  import random
  import unittest
  import warnings
  from math import exp, pi
  import torch
  from torch import optim
  import gpytorch
  from gpytorch.distributions import MultivariateNormal
  from gpytorch.kernels import RBFKernel, ScaleKernel
  from gpytorch.likelihoods import FixedNoiseGaussianLikelihood, GaussianLikelihood
  from gpytorch.means import ConstantMean
  from gpytorch.priors import SmoothedBoxPrior
  from gpytorch.test.utils import least_used_cuda_device
  from gpytorch.utils.warnings import GPInputWarning
label_desc: Use image node_id 4 for calling the TestWhiteNoiseGPRegression obj's underlying member method code with example usage: obj.test_posterior_latent_gp_and_likelihood_with_optimization(cuda, cg) without return types