code: def predict_mean_std(
      self,
      trials: list[FrozenTrial],
  ) -> tuple[np.ndarray, np.ndarray]:
      assert self._gp is not None
  
      x, _ = _convert_trials_to_tensors(trials)
  
      with torch.no_grad(), gpytorch.settings.fast_pred_var():
          posterior = self._gp.posterior(x)
          mean = posterior.mean
          variance = posterior.variance
          std = variance.sqrt()
  
      return mean.detach().numpy().squeeze(
          -1
      ), std.detach().numpy().squeeze(-1)
  
file_path: optuna/optuna/terminator/improvement/gp/botorch.py
imports: from __future__ import annotations
  from typing import Optional
  import numpy
  from packaging import version
  from optuna._imports import try_import
  from optuna.distributions import _is_distribution_log
  from optuna.distributions import CategoricalDistribution
  from optuna.distributions import FloatDistribution
  from optuna.distributions import IntDistribution
  from optuna.search_space import intersection_search_space
  from optuna.terminator.improvement.gp.base import BaseGaussianProcess
  from optuna.trial._frozen import FrozenTrial
  from optuna.trial._state import TrialState
label_desc: Use image node_id 3 for calling the _BoTorchGaussianProcess obj's underlying member method code with example usage: obj.predict_mean_std(trials) and returns: mean,  std