class_docstr: Implementation of a Gaussian process model which out-integrates its hyperparameters by
  Markow-Chain-Monte-Carlo (MCMC). If you use this class make sure that you also use an integrated acquisition
  function to integrate over the GP's hyperparameter as proposed by Snoek et al.
  
  This code is based on the implementation of RoBO:
  
  Klein, A. and Falkner, S. and Mansur, N. and Hutter, F.
  RoBO: A Flexible and Robust Bayesian Optimization Framework in Python
  In: NIPS 2017 Bayesian Optimization Workshop
  
  Parameters
  ----------
  configspace : ConfigurationSpace
  kernel : Kernel
      Kernel which is used for the Gaussian process.
  n_mcmc_walkers : int, defaults to 20
      The number of hyperparameter samples. This also determines the number of walker for MCMC sampling as each
      walker will return one hyperparameter sample.
  chain_length : int, defaults to 50
      The length of the MCMC chain. We start `n_mcmc_walkers` walker for `chain_length` steps, and we use the last
      sample in the chain as a hyperparameter sample.
  burning_steps : int, defaults to 50
      The number of burning steps before the actual MCMC sampling starts.
  mcmc_sampler : str, defaults to "emcee"
      Choose a self-tuning MCMC sampler. Can be either ``emcee`` or ``nuts``.
  normalize_y : bool, defaults to True
      Zero mean unit variance normalization of the output values.
  instance_features : dict[str, list[int | float]] | None, defaults to None
      Features (list of int or floats) of the instances (str). The features are incorporated into the X data,
      on which the model is trained on.
  pca_components : float, defaults to 7
      Number of components to keep when using PCA to reduce dimensionality of instance features.
  seed : int
docstr: Returns the marginal log likelihood (+ the prior) for a hyperparameter configuration
  theta.
  
  Parameters
  ----------
  theta : np.ndarray
      Hyperparameter vector. Note that all hyperparameter are on a log scale.
docstr_tok: ["Returns","the","marginal","log","likelihood","(","+","the","prior",")","for","a","hyperparameter","configuration","theta",".","Parameters","--","--","--","--","--","theta",":","np.ndarray","Hyperparameter","vector",".","Note","that","all","hyperparameter","are","on","a","log","scale","."]
code: def _ll_w_grad(self, theta: np.ndarray) -> tuple[float, np.ndarray]:
      """Returns the marginal log likelihood (+ the prior) for a hyperparameter configuration
      theta.
  
      Parameters
      ----------
      theta : np.ndarray
          Hyperparameter vector. Note that all hyperparameter are on a log scale.
      """
      self._n_ll_evals += 1
  
      # Bound the hyperparameter space to keep things sane. Note that all hyperparameters live on a log scale.
      if (theta < -50).any():
          theta[theta < -50] = -50
      if (theta > 50).any():
          theta[theta > 50] = 50
  
      lml = 0.0
      grad = np.zeros(theta.shape)
  
      # Add prior
      for dim, priors in enumerate(self._all_priors):
          for prior in priors:
              lml += prior.get_log_probability(theta[dim])
              grad[dim] += prior.get_gradient(theta[dim])
  
      # Check if one of the priors is invalid, if so, no need to compute the log marginal likelihood
      if lml < -1e24:
          return -1e25, np.zeros(theta.shape)
  
      try:
          lml_, grad_ = self._gp.log_marginal_likelihood(
              theta, eval_gradient=True
          )
          lml += lml_
          grad += grad_
      except ValueError:
          return -1e25, np.zeros(theta.shape)
  
      # We add a minus here because scipy is minimizing
      if not np.isfinite(lml) or (~np.isfinite(grad)).any():
          return -1e25, np.zeros(theta.shape)
      else:
          return lml, grad
  
imports: from __future__ import annotations
  from typing import Any, Optional, TypeVar, cast
  import logging
  import warnings
  from copy import deepcopy
  import emcee
  import numpy
  from ConfigSpace import ConfigurationSpace
  from sklearn.gaussian_process import GaussianProcessRegressor
  from sklearn.gaussian_process.kernels import Kernel
  from smac.model.gaussian_process.abstract_gaussian_process import AbstractGaussianProcess
  from smac.model.gaussian_process.gaussian_process import GaussianProcess
  from smac.model.gaussian_process.priors.abstract_prior import AbstractPrior
label_desc: Use image node_id 7 for calling the MCMCGaussianProcess obj's underlying member method code with example usage: obj._ll_w_grad(theta) and returns: unknown,  np, unknown,  np, lml,  grad, unknown,  np