text: """
  Various bayesian regression
  """
  
  # Authors: V. Michel, F. Pedregosa, A. Gramfort
  # License: BSD 3 clause
  
  from math import log
  import numpy as np
  from scipy import linalg
  
  from ._base import LinearModel, _rescale_data
  from ..base import RegressorMixin
  from ._base import _deprecate_normalize
  from ..utils.extmath import fast_logdet
  from scipy.linalg import pinvh
  from ..utils.validation import _check_sample_weight
  from ..utils.validation import _deprecate_positional_args
  
  
  ###############################################################################
  # BayesianRidge regression
  
  class BayesianRidge(RegressorMixin, LinearModel):
      """Bayesian ridge regression.
  
      Fit a Bayesian ridge model. See the Notes section for details on this
      implementation and the optimization of the regularization parameters
      lambda (precision of the weights) and alpha (precision of the noise).
  
      Read more in the :ref:`User Guide <bayesian_regression>`.
  
      Parameters
      ----------
      n_iter : int, default=300
          Maximum number of iterations. Should be greater than or equal to 1.
  
      tol : float, default=1e-3
          Stop the algorithm if w has converged.
  
      alpha_1 : float, default=1e-6
          Hyper-parameter : shape parameter for the Gamma distribution prior
          over the alpha parameter.
  
      alpha_2 : float, default=1e-6
          Hyper-parameter : inverse scale parameter (rate parameter) for the
          Gamma distribution prior over the alpha parameter.
  
      lambda_1 : float, default=1e-6
          Hyper-parameter : shape parameter for the Gamma distribution prior
          over the lambda parameter.
  
      lambda_2 : float, default=1e-6
          Hyper-parameter : inverse scale parameter (rate parameter) for the
          Gamma distribution prior over the lambda parameter.
  
      alpha_init : float, default=None
          Initial value for alpha (precision of the noise).
          If not set, alpha_init is 1/Var(y).
  
              .. versionadded:: 0.22
  
      lambda_init : float, default=None
          Initial value for lambda (precision of the weights).
          If not set, lambda_init is 1.
  
              .. versionadded:: 0.22
  
      compute_score : bool, default=False
          If True, compute the log marginal likelihood at each iteration of the
          optimization.
  
      fit_intercept : bool, default=True
          Whether to calculate the intercept for this model.
          The intercept is not treated as a probabilistic parameter
          and thus has no associated variance. If set
          to False, no intercept will be used in calculations
          (i.e. data is expected to be centered).
  
      normalize : bool, default=False
          This parameter is ignored when ``fit_intercept`` is set to False.
          If True, the regressors X will be normalized before regression by
          subtracting the mean and dividing by the l2-norm.
          If you wish to standardize, please use
          :class:`~sklearn.preprocessing.StandardScaler` before calling ``fit``
          on an estimator with ``normalize=False``.
  
          .. deprecated:: 1.0
              ``normalize`` was deprecated in version 1.0 and will be removed in
              1.2.
  
      copy_X : bool, default=True
          If True, X will be copied; else, it may be overwritten.
  
      verbose : bool, default=False
          Verbose mode when fitting the model.
  
  
      Attributes
      ----------
      coef_ : array-like of shape (n_features,)
          Coefficients of the regression model (mean of distribution)
  
      intercept_ : float
          Independent term in decision function. Set to 0.0 if
          ``fit_intercept = False``.
  
      alpha_ : float
         Estimated precision of the noise.
  
      lambda_ : float
         Estimated precision of the weights.
  
      sigma_ : array-like of shape (n_features, n_features)
          Estimated variance-covariance matrix of the weights
  
      scores_ : array-like of shape (n_iter_+1,)
          If computed_score is True, value of the log marginal likelihood (to be
          maximized) at each iteration of the optimization. The array starts
          with the value of the log marginal likelihood obtained for the initial
          values of alpha and lambda and ends with the value obtained for the
          estimated alpha and lambda.
  
      n_iter_ : int
          The actual number of iterations to reach the stopping criterion.
  
      X_offset_ : float
          If `normalize=True`, offset subtracted for centering data to a
          zero mean.
  
      X_scale_ : float
          If `normalize=True`, parameter used to scale data to a unit
          standard deviation.
  
      Examples
      --------
      >>> from sklearn import linear_model
      >>> clf = linear_model.BayesianRidge()
      >>> clf.fit([[0,0], [1, 1], [2, 2]], [0, 1, 2])
      BayesianRidge()
      >>> clf.predict([[1, 1]])
      array([1.])
  
      Notes
      -----
      There exist several strategies to perform Bayesian ridge regression. This
      implementation is based on the algorithm described in Appendix A of
      (Tipping, 2001) where updates of the regularization parameters are done as
      suggested in (MacKay, 1992). Note that according to A New
      View of Automatic Relevance Determination (Wipf and Nagarajan, 2008) these
      update rules do not guarantee that the marginal likelihood is increasing
      between two consecutive iterations of the optimization.
  
      References
      ----------
      D. J. C. MacKay, Bayesian Interpolation, Computation and Neural Systems,
      Vol. 4, No. 3, 1992.
  
      M. E. Tipping, Sparse Bayesian Learning and the Relevance Vector Machine,
      Journal of Machine Learning Research, Vol. 1, 2001.
      """
      @_deprecate_positional_args
      def __init__(self, *, n_iter=300, tol=1.e-3, alpha_1=1.e-6, alpha_2=1.e-6,
                   lambda_1=1.e-6, lambda_2=1.e-6, alpha_init=None,
                   lambda_init=None, compute_score=False, fit_intercept=True,
                   normalize='deprecated', copy_X=True, verbose=False):
          self.n_iter = n_iter
          self.tol = tol
          self.alpha_1 = alpha_1
          self.alpha_2 = alpha_2
          self.lambda_1 = lambda_1
          self.lambda_2 = lambda_2
          self.alpha_init = alpha_init
          self.lambda_init = lambda_init
          self.compute_score = compute_score
          self.fit_intercept = fit_intercept
          self.normalize = normalize
          self.copy_X = copy_X
          self.verbose = verbose
  
      def fit(self, X, y, sample_weight=None):
          """Fit the model
  
          Parameters
          ----------
          X : ndarray of shape (n_samples, n_features)
              Training data
          y : ndarray of shape (n_samples,)
              Target values. Will be cast to X's dtype if necessary
  
          sample_weight : ndarray of shape (n_samples,), default=None
              Individual weights for each sample
  
              .. versionadded:: 0.20
                 parameter *sample_weight* support to BayesianRidge.
  
          Returns
          -------
          self : returns an instance of self.
          """
          self._normalize = _deprecate_normalize(
              self.normalize, default=False,
              estimator_name=self.__class__.__name__
          )
  
          if self.n_iter < 1:
              raise ValueError('n_iter should be greater than or equal to 1.'
                               ' Got {!r}.'.format(self.n_iter))
  
          X, y = self._validate_data(X, y, dtype=np.float64, y_numeric=True)
  
          if sample_weight is not None:
              sample_weight = _check_sample_weight(sample_weight, X,
                                                   dtype=X.dtype)
  
          X, y, X_offset_, y_offset_, X_scale_ = self._preprocess_data(
              X, y, self.fit_intercept, self._normalize, self.copy_X,
              sample_weight=sample_weight)
  
          if sample_weight is not None:
              # Sample weight can be implemented via a simple rescaling.
              X, y = _rescale_data(X, y, sample_weight)
  
          self.X_offset_ = X_offset_
          self.X_scale_ = X_scale_
          n_samples, n_features = X.shape
  
          # Initialization of the values of the parameters
          eps = np.finfo(np.float64).eps
          # Add `eps` in the denominator to omit division by zero if `np.var(y)`
          # is zero
          alpha_ = self.alpha_init
          lambda_ = self.lambda_init
          if alpha_ is None:
              alpha_ = 1. / (np.var(y) + eps)
          if lambda_ is None:
              lambda_ = 1.
  
          verbose = self.verbose
          lambda_1 = self.lambda_1
          lambda_2 = self.lambda_2
          alpha_1 = self.alpha_1
          alpha_2 = self.alpha_2
  
          self.scores_ = list()
          coef_old_ = None
  
          XT_y = np.dot(X.T, y)
          U, S, Vh = linalg.svd(X, full_matrices=False)
          eigen_vals_ = S ** 2
  
          # Convergence loop of the bayesian ridge regression
          for iter_ in range(self.n_iter):
  
              # update posterior mean coef_ based on alpha_ and lambda_ and
              # compute corresponding rmse
              coef_, rmse_ = self._update_coef_(X, y, n_samples, n_features,
                                                XT_y, U, Vh, eigen_vals_,
                                                alpha_, lambda_)
              if self.compute_score:
                  # compute the log marginal likelihood
                  s = self._log_marginal_likelihood(n_samples, n_features,
                                                    eigen_vals_,
                                                    alpha_, lambda_,
                                                    coef_, rmse_)
                  self.scores_.append(s)
  
              # Update alpha and lambda according to (MacKay, 1992)
              gamma_ = np.sum((alpha_ * eigen_vals_) /
                              (lambda_ + alpha_ * eigen_vals_))
              lambda_ = ((gamma_ + 2 * lambda_1) /
                         (np.sum(coef_ ** 2) + 2 * lambda_2))
              alpha_ = ((n_samples - gamma_ + 2 * alpha_1) /
                        (rmse_ + 2 * alpha_2))
  
              # Check for convergence
              if iter_ != 0 and np.sum(np.abs(coef_old_ - coef_)) < self.tol:
                  if verbose:
                      print("Convergence after ", str(iter_), " iterations")
                  break
              coef_old_ = np.copy(coef_)
  
          self.n_iter_ = iter_ + 1
  
          # return regularization parameters and corresponding posterior mean,
          # log marginal likelihood and posterior covariance
          self.alpha_ = alpha_
          self.lambda_ = lambda_
          self.coef_, rmse_ = self._update_coef_(X, y, n_samples, n_features,
                                                 XT_y, U, Vh, eigen_vals_,
                                                 alpha_, lambda_)
          if self.compute_score:
              # compute the log marginal likelihood
              s = self._log_marginal_likelihood(n_samples, n_features,
                                                eigen_vals_,
                                                alpha_, lambda_,
                                                coef_, rmse_)
              self.scores_.append(s)
              self.scores_ = np.array(self.scores_)
  
          # posterior covariance is given by 1/alpha_ * scaled_sigma_
          scaled_sigma_ = np.dot(Vh.T,
                                 Vh / (eigen_vals_ +
                                       lambda_ / alpha_)[:, np.newaxis])
          self.sigma_ = (1. / alpha_) * scaled_sigma_
  
          self._set_intercept(X_offset_, y_offset_, X_scale_)
  
          return self
  
      def predict(self, X, return_std=False):
          """Predict using the linear model.
  
          In addition to the mean of the predictive distribution, also its
          standard deviation can be returned.
  
          Parameters
          ----------
          X : {array-like, sparse matrix} of shape (n_samples, n_features)
              Samples.
  
          return_std : bool, default=False
              Whether to return the standard deviation of posterior prediction.
  
          Returns
          -------
          y_mean : array-like of shape (n_samples,)
              Mean of predictive distribution of query points.
  
          y_std : array-like of shape (n_samples,)
              Standard deviation of predictive distribution of query points.
          """
          y_mean = self._decision_function(X)
          if return_std is False:
              return y_mean
          else:
              if self._normalize:
                  X = (X - self.X_offset_) / self.X_scale_
              sigmas_squared_data = (np.dot(X, self.sigma_) * X).sum(axis=1)
              y_std = np.sqrt(sigmas_squared_data + (1. / self.alpha_))
              return y_mean, y_std
  
      def _update_coef_(self, X, y, n_samples, n_features, XT_y, U, Vh,
                        eigen_vals_, alpha_, lambda_):
          """Update posterior mean and compute corresponding rmse.
  
          Posterior mean is given by coef_ = scaled_sigma_ * X.T * y where
          scaled_sigma_ = (lambda_/alpha_ * np.eye(n_features)
                           + np.dot(X.T, X))^-1
          """
  
          if n_samples > n_features:
              coef_ = np.linalg.multi_dot([Vh.T,
                                           Vh / (eigen_vals_ + lambda_ /
                                                 alpha_)[:, np.newaxis],
                                           XT_y])
          else:
              coef_ = np.linalg.multi_dot([X.T,
                                           U / (eigen_vals_ + lambda_ /
                                                alpha_)[None, :],
                                           U.T, y])
  
          rmse_ = np.sum((y - np.dot(X, coef_)) ** 2)
  
          return coef_, rmse_
  
      def _log_marginal_likelihood(self, n_samples, n_features, eigen_vals,
                                   alpha_, lambda_, coef, rmse):
          """Log marginal likelihood."""
          alpha_1 = self.alpha_1
          alpha_2 = self.alpha_2
          lambda_1 = self.lambda_1
          lambda_2 = self.lambda_2
  
          # compute the log of the determinant of the posterior covariance.
          # posterior covariance is given by
          # sigma = (lambda_ * np.eye(n_features) + alpha_ * np.dot(X.T, X))^-1
          if n_samples > n_features:
              logdet_sigma = - np.sum(np.log(lambda_ + alpha_ * eigen_vals))
          else:
              logdet_sigma = np.full(n_features, lambda_,
                                     dtype=np.array(lambda_).dtype)
              logdet_sigma[:n_samples] += alpha_ * eigen_vals
              logdet_sigma = - np.sum(np.log(logdet_sigma))
  
          score = lambda_1 * log(lambda_) - lambda_2 * lambda_
          score += alpha_1 * log(alpha_) - alpha_2 * alpha_
          score += 0.5 * (n_features * log(lambda_) +
                          n_samples * log(alpha_) -
                          alpha_ * rmse -
                          lambda_ * np.sum(coef ** 2) +
                          logdet_sigma -
                          n_samples * log(2 * np.pi))
  
          return score
  
  
  ###############################################################################
  # ARD (Automatic Relevance Determination) regression
  
  
  class ARDRegression(RegressorMixin, LinearModel):
      """Bayesian ARD regression.
  
      Fit the weights of a regression model, using an ARD prior. The weights of
      the regression model are assumed to be in Gaussian distributions.
      Also estimate the parameters lambda (precisions of the distributions of the
      weights) and alpha (precision of the distribution of the noise).
      The estimation is done by an iterative procedures (Evidence Maximization)
  
      Read more in the :ref:`User Guide <bayesian_regression>`.
  
      Parameters
      ----------
      n_iter : int, default=300
          Maximum number of iterations.
  
      tol : float, default=1e-3
          Stop the algorithm if w has converged.
  
      alpha_1 : float, default=1e-6
          Hyper-parameter : shape parameter for the Gamma distribution prior
          over the alpha parameter.
  
      alpha_2 : float, default=1e-6
          Hyper-parameter : inverse scale parameter (rate parameter) for the
          Gamma distribution prior over the alpha parameter.
  
      lambda_1 : float, default=1e-6
          Hyper-parameter : shape parameter for the Gamma distribution prior
          over the lambda parameter.
  
      lambda_2 : float, default=1e-6
          Hyper-parameter : inverse scale parameter (rate parameter) for the
          Gamma distribution prior over the lambda parameter.
  
      compute_score : bool, default=False
          If True, compute the objective function at each step of the model.
  
      threshold_lambda : float, default=10 000
          threshold for removing (pruning) weights with high precision from
          the computation.
  
      fit_intercept : bool, default=True
          whether to calculate the intercept for this model. If set
          to false, no intercept will be used in calculations
          (i.e. data is expected to be centered).
  
      normalize : bool, default=False
          This parameter is ignored when ``fit_intercept`` is set to False.
          If True, the regressors X will be normalized before regression by
          subtracting the mean and dividing by the l2-norm.
          If you wish to standardize, please use
          :class:`~sklearn.preprocessing.StandardScaler` before calling ``fit``
          on an estimator with ``normalize=False``.
  
          .. deprecated:: 1.0
              ``normalize`` was deprecated in version 1.0 and will be removed in
              1.2.
  
      copy_X : bool, default=True
          If True, X will be copied; else, it may be overwritten.
  
      verbose : bool, default=False
          Verbose mode when fitting the model.
  
      Attributes
      ----------
      coef_ : array-like of shape (n_features,)
          Coefficients of the regression model (mean of distribution)
  
      alpha_ : float
         estimated precision of the noise.
  
      lambda_ : array-like of shape (n_features,)
         estimated precisions of the weights.
  
      sigma_ : array-like of shape (n_features, n_features)
          estimated variance-covariance matrix of the weights
  
      scores_ : float
          if computed, value of the objective function (to be maximized)
  
      intercept_ : float
          Independent term in decision function. Set to 0.0 if
          ``fit_intercept = False``.
  
      X_offset_ : float
          If `normalize=True`, offset subtracted for centering data to a
          zero mean.
  
      X_scale_ : float
          If `normalize=True`, parameter used to scale data to a unit
          standard deviation.
  
      Examples
      --------
      >>> from sklearn import linear_model
      >>> clf = linear_model.ARDRegression()
      >>> clf.fit([[0,0], [1, 1], [2, 2]], [0, 1, 2])
      ARDRegression()
      >>> clf.predict([[1, 1]])
      array([1.])
  
      Notes
      -----
      For an example, see :ref:`examples/linear_model/plot_ard.py
      <sphx_glr_auto_examples_linear_model_plot_ard.py>`.
  
      References
      ----------
      D. J. C. MacKay, Bayesian nonlinear modeling for the prediction
      competition, ASHRAE Transactions, 1994.
  
      R. Salakhutdinov, Lecture notes on Statistical Machine Learning,
      http://www.utstat.toronto.edu/~rsalakhu/sta4273/notes/Lecture2.pdf#page=15
      Their beta is our ``self.alpha_``
      Their alpha is our ``self.lambda_``
      ARD is a little different than the slide: only dimensions/features for
      which ``self.lambda_ < self.threshold_lambda`` are kept and the rest are
      discarded.
      """
      @_deprecate_positional_args
      def __init__(self, *, n_iter=300, tol=1.e-3, alpha_1=1.e-6, alpha_2=1.e-6,
                   lambda_1=1.e-6, lambda_2=1.e-6, compute_score=False,
                   threshold_lambda=1.e+4, fit_intercept=True,
                   normalize='deprecated', copy_X=True, verbose=False):
          self.n_iter = n_iter
          self.tol = tol
          self.fit_intercept = fit_intercept
          self.normalize = normalize
          self.alpha_1 = alpha_1
          self.alpha_2 = alpha_2
          self.lambda_1 = lambda_1
          self.lambda_2 = lambda_2
          self.compute_score = compute_score
          self.threshold_lambda = threshold_lambda
          self.copy_X = copy_X
          self.verbose = verbose
  
      def fit(self, X, y):
          """Fit the ARDRegression model according to the given training data
          and parameters.
  
          Iterative procedure to maximize the evidence
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
              Training vector, where n_samples in the number of samples and
              n_features is the number of features.
          y : array-like of shape (n_samples,)
              Target values (integers). Will be cast to X's dtype if necessary
  
          Returns
          -------
          self : returns an instance of self.
          """
          self._normalize = _deprecate_normalize(
              self.normalize, default=False,
              estimator_name=self.__class__.__name__
          )
  
          X, y = self._validate_data(X, y, dtype=np.float64, y_numeric=True,
                                     ensure_min_samples=2)
  
          n_samples, n_features = X.shape
          coef_ = np.zeros(n_features)
  
          X, y, X_offset_, y_offset_, X_scale_ = self._preprocess_data(
              X, y, self.fit_intercept, self._normalize, self.copy_X)
  
          self.X_offset_ = X_offset_
          self.X_scale_ = X_scale_
  
          # Launch the convergence loop
          keep_lambda = np.ones(n_features, dtype=bool)
  
          lambda_1 = self.lambda_1
          lambda_2 = self.lambda_2
          alpha_1 = self.alpha_1
          alpha_2 = self.alpha_2
          verbose = self.verbose
  
          # Initialization of the values of the parameters
          eps = np.finfo(np.float64).eps
          # Add `eps` in the denominator to omit division by zero if `np.var(y)`
          # is zero
          alpha_ = 1. / (np.var(y) + eps)
          lambda_ = np.ones(n_features)
  
          self.scores_ = list()
          coef_old_ = None
  
          def update_coeff(X, y, coef_, alpha_, keep_lambda, sigma_):
              coef_[keep_lambda] = alpha_ * np.linalg.multi_dot([
                  sigma_, X[:, keep_lambda].T, y])
              return coef_
  
          update_sigma = (self._update_sigma if n_samples >= n_features
                          else self._update_sigma_woodbury)
          # Iterative procedure of ARDRegression
          for iter_ in range(self.n_iter):
              sigma_ = update_sigma(X, alpha_, lambda_, keep_lambda)
              coef_ = update_coeff(X, y, coef_, alpha_, keep_lambda, sigma_)
  
              # Update alpha and lambda
              rmse_ = np.sum((y - np.dot(X, coef_)) ** 2)
              gamma_ = 1. - lambda_[keep_lambda] * np.diag(sigma_)
              lambda_[keep_lambda] = ((gamma_ + 2. * lambda_1) /
                                      ((coef_[keep_lambda]) ** 2 +
                                       2. * lambda_2))
              alpha_ = ((n_samples - gamma_.sum() + 2. * alpha_1) /
                        (rmse_ + 2. * alpha_2))
  
              # Prune the weights with a precision over a threshold
              keep_lambda = lambda_ < self.threshold_lambda
              coef_[~keep_lambda] = 0
  
              # Compute the objective function
              if self.compute_score:
                  s = (lambda_1 * np.log(lambda_) - lambda_2 * lambda_).sum()
                  s += alpha_1 * log(alpha_) - alpha_2 * alpha_
                  s += 0.5 * (fast_logdet(sigma_) + n_samples * log(alpha_) +
                              np.sum(np.log(lambda_)))
                  s -= 0.5 * (alpha_ * rmse_ + (lambda_ * coef_ ** 2).sum())
                  self.scores_.append(s)
  
              # Check for convergence
              if iter_ > 0 and np.sum(np.abs(coef_old_ - coef_)) < self.tol:
                  if verbose:
                      print("Converged after %s iterations" % iter_)
                  break
              coef_old_ = np.copy(coef_)
  
              if not keep_lambda.any():
                  break
  
          if keep_lambda.any():
              # update sigma and mu using updated params from the last iteration
              sigma_ = update_sigma(X, alpha_, lambda_, keep_lambda)
              coef_ = update_coeff(X, y, coef_, alpha_, keep_lambda, sigma_)
          else:
              sigma_ = np.array([]).reshape(0, 0)
  
          self.coef_ = coef_
          self.alpha_ = alpha_
          self.sigma_ = sigma_
          self.lambda_ = lambda_
          self._set_intercept(X_offset_, y_offset_, X_scale_)
          return self
  
      def _update_sigma_woodbury(self, X, alpha_, lambda_, keep_lambda):
          # See slides as referenced in the docstring note
          # this function is used when n_samples < n_features and will invert
          # a matrix of shape (n_samples, n_samples) making use of the
          # woodbury formula:
          # https://en.wikipedia.org/wiki/Woodbury_matrix_identity
          n_samples = X.shape[0]
          X_keep = X[:, keep_lambda]
          inv_lambda = 1 / lambda_[keep_lambda].reshape(1, -1)
          sigma_ = pinvh(
              np.eye(n_samples) / alpha_ + np.dot(X_keep * inv_lambda, X_keep.T)
          )
          sigma_ = np.dot(sigma_, X_keep * inv_lambda)
          sigma_ = - np.dot(inv_lambda.reshape(-1, 1) * X_keep.T, sigma_)
          sigma_[np.diag_indices(sigma_.shape[1])] += 1. / lambda_[keep_lambda]
          return sigma_
  
      def _update_sigma(self, X, alpha_, lambda_, keep_lambda):
          # See slides as referenced in the docstring note
          # this function is used when n_samples >= n_features and will
          # invert a matrix of shape (n_features, n_features)
          X_keep = X[:, keep_lambda]
          gram = np.dot(X_keep.T, X_keep)
          eye = np.eye(gram.shape[0])
          sigma_inv = lambda_[keep_lambda] * eye + alpha_ * gram
          sigma_ = pinvh(sigma_inv)
          return sigma_
  
      def predict(self, X, return_std=False):
          """Predict using the linear model.
  
          In addition to the mean of the predictive distribution, also its
          standard deviation can be returned.
  
          Parameters
          ----------
          X : {array-like, sparse matrix} of shape (n_samples, n_features)
              Samples.
  
          return_std : bool, default=False
              Whether to return the standard deviation of posterior prediction.
  
          Returns
          -------
          y_mean : array-like of shape (n_samples,)
              Mean of predictive distribution of query points.
  
          y_std : array-like of shape (n_samples,)
              Standard deviation of predictive distribution of query points.
          """
          y_mean = self._decision_function(X)
          if return_std is False:
              return y_mean
          else:
              if self._normalize:
                  X = (X - self.X_offset_) / self.X_scale_
              X = X[:, self.lambda_ < self.threshold_lambda]
              sigmas_squared_data = (np.dot(X, self.sigma_) * X).sum(axis=1)
              y_std = np.sqrt(sigmas_squared_data + (1. / self.alpha_))
              return y_mean, y_std
  