code: def test_batch_q_analytic_probability_of_improvement(self):
      for dtype in (torch.float, torch.double):
          # test batch mode
          mean = torch.tensor(
              [[0.0], [1.0]], device=self.device, dtype=dtype
          )
          cov = (
              torch.eye(n=1, device=self.device, dtype=dtype)
              .unsqueeze(0)
              .repeat(2, 1, 1)
          )
          mvn = MultivariateNormal(mean=mean, covariance_matrix=cov)
          posterior = GPyTorchPosterior(mvn)
          mm = MockModel(posterior)
          module = qAnalyticProbabilityOfImprovement(model=mm, best_f=0)
          X = torch.rand(2, 1, 1, device=self.device, dtype=dtype)
          pi = module(X)
          pi_expected = torch.tensor(
              [0.5, 0.8413], device=self.device, dtype=dtype
          )
          self.assertTrue(torch.allclose(pi, pi_expected, atol=1e-4))
  
          # test batched model and best_f values
          mean = torch.zeros(2, 1, device=self.device, dtype=dtype)
          cov = (
              torch.eye(n=1, device=self.device, dtype=dtype)
              .unsqueeze(0)
              .repeat(2, 1, 1)
          )
          mvn = MultivariateNormal(mean=mean, covariance_matrix=cov)
          posterior = GPyTorchPosterior(mvn)
          mm = MockModel(posterior)
          best_f = torch.tensor(
              [0.0, -1.0], device=self.device, dtype=dtype
          )
          module = qAnalyticProbabilityOfImprovement(
              model=mm, best_f=best_f
          )
          X = torch.rand(2, 1, 1, device=self.device, dtype=dtype)
          pi = module(X)
          pi_expected = torch.tensor(
              [[0.5, 0.8413]], device=self.device, dtype=dtype
          )
          self.assertTrue(torch.allclose(pi, pi_expected, atol=1e-4))
  
          # test batched model, output transform (single output)
          mean = torch.tensor(
              [[0.0], [1.0]], device=self.device, dtype=dtype
          )
          cov = (
              torch.eye(n=1, device=self.device, dtype=dtype)
              .unsqueeze(0)
              .repeat(2, 1, 1)
          )
          mvn = MultivariateNormal(mean=mean, covariance_matrix=cov)
          posterior = GPyTorchPosterior(mvn)
          mm = MockModel(posterior)
          weights = torch.tensor([0.5], device=self.device, dtype=dtype)
          transform = ScalarizedPosteriorTransform(weights)
          module = qAnalyticProbabilityOfImprovement(
              model=mm, best_f=0.0, posterior_transform=transform
          )
          X = torch.rand(2, 1, 2, device=self.device, dtype=dtype)
          pi = module(X)
          pi_expected = torch.tensor(
              [0.5, 0.8413], device=self.device, dtype=dtype
          )
          self.assertTrue(torch.allclose(pi, pi_expected, atol=1e-4))
  
          # test batched model, output transform (multiple output)
          mean = torch.tensor(
              [[[1.0, 1.0]], [[0.0, 1.0]]],
              device=self.device,
              dtype=dtype,
          )
          cov = (
              torch.eye(n=2, device=self.device, dtype=dtype)
              .unsqueeze(0)
              .repeat(2, 1, 1)
          )
          mvn = MultitaskMultivariateNormal(
              mean=mean, covariance_matrix=cov
          )
          posterior = GPyTorchPosterior(mvn)
          mm = MockModel(posterior)
          weights = torch.ones(2, device=self.device, dtype=dtype)
          transform = ScalarizedPosteriorTransform(weights)
          module = qAnalyticProbabilityOfImprovement(
              model=mm, best_f=0.0, posterior_transform=transform
          )
          X = torch.rand(2, 1, 2, device=self.device, dtype=dtype)
          pi = module(X)
          pi_expected = torch.tensor(
              [0.9214, 0.7602], device=self.device, dtype=dtype
          )
          self.assertTrue(torch.allclose(pi, pi_expected, atol=1e-4))
  
          # test bad posterior transform class
          with self.assertRaises(UnsupportedError):
              qAnalyticProbabilityOfImprovement(
                  model=mm,
                  best_f=0.0,
                  posterior_transform=IdentityMCObjective(),
              )
  
imports: import math
  import torch
  from botorch.acquisition import qAnalyticProbabilityOfImprovement
  from botorch.acquisition.analytic import _compute_log_prob_feas, _ei_helper, _log_ei_helper, AnalyticAcquisitionFunction, ConstrainedExpectedImprovement, ExpectedImprovement, LogConstrainedExpectedImprovement, LogExpectedImprovement, LogNoisyExpectedImprovement, LogProbabilityOfImprovement, NoisyExpectedImprovement, PosteriorMean, PosteriorStandardDeviation, ProbabilityOfImprovement, ScalarizedPosteriorMean, UpperConfidenceBound
  from botorch.acquisition.objective import IdentityMCObjective, ScalarizedPosteriorTransform
  from botorch.exceptions import UnsupportedError
  from botorch.models import SingleTaskGP
  from botorch.posteriors import GPyTorchPosterior
  from botorch.utils.testing import BotorchTestCase, MockModel, MockPosterior
  from gpytorch.distributions import MultitaskMultivariateNormal, MultivariateNormal
  from gpytorch.likelihoods.gaussian_likelihood import FixedNoiseGaussianLikelihood
label_desc: Use image node_id 2 for calling the TestqAnalyticProbabilityOfImprovement obj's underlying member method code with example usage: obj.test_batch_q_analytic_probability_of_improvement() without return types