text: """Bayesian Gaussian Mixture Model."""
  # Author: Wei Xue <xuewei4d@gmail.com>
  #         Thierry Guillemot <thierry.guillemot.work@gmail.com>
  # License: BSD 3 clause
  
  import math
  import numpy as np
  from scipy.special import betaln, digamma, gammaln
  
  from ._base import BaseMixture, _check_shape
  from ._gaussian_mixture import _check_precision_matrix
  from ._gaussian_mixture import _check_precision_positivity
  from ._gaussian_mixture import _compute_log_det_cholesky
  from ._gaussian_mixture import _compute_precision_cholesky
  from ._gaussian_mixture import _estimate_gaussian_parameters
  from ._gaussian_mixture import _estimate_log_gaussian_prob
  from ..utils import check_array
  
  
  def _log_dirichlet_norm(dirichlet_concentration):
      """Compute the log of the Dirichlet distribution normalization term.
  
      Parameters
      ----------
      dirichlet_concentration : array-like of shape (n_samples,)
          The parameters values of the Dirichlet distribution.
  
      Returns
      -------
      log_dirichlet_norm : float
          The log normalization of the Dirichlet distribution.
      """
      return gammaln(np.sum(dirichlet_concentration)) - np.sum(
          gammaln(dirichlet_concentration)
      )
  
  
  def _log_wishart_norm(degrees_of_freedom, log_det_precisions_chol, n_features):
      """Compute the log of the Wishart distribution normalization term.
  
      Parameters
      ----------
      degrees_of_freedom : array-like of shape (n_components,)
          The number of degrees of freedom on the covariance Wishart
          distributions.
  
      log_det_precision_chol : array-like of shape (n_components,)
           The determinant of the precision matrix for each component.
  
      n_features : int
          The number of features.
  
      Return
      ------
      log_wishart_norm : array-like of shape (n_components,)
          The log normalization of the Wishart distribution.
      """
      # To simplify the computation we have removed the np.log(np.pi) term
      return -(
          degrees_of_freedom * log_det_precisions_chol
          + degrees_of_freedom * n_features * 0.5 * math.log(2.0)
          + np.sum(
              gammaln(0.5 * (degrees_of_freedom - np.arange(n_features)[:, np.newaxis])),
              0,
          )
      )
  
  
  class BayesianGaussianMixture(BaseMixture):
      """Variational Bayesian estimation of a Gaussian mixture.
  
      This class allows to infer an approximate posterior distribution over the
      parameters of a Gaussian mixture distribution. The effective number of
      components can be inferred from the data.
  
      This class implements two types of prior for the weights distribution: a
      finite mixture model with Dirichlet distribution and an infinite mixture
      model with the Dirichlet Process. In practice Dirichlet Process inference
      algorithm is approximated and uses a truncated distribution with a fixed
      maximum number of components (called the Stick-breaking representation).
      The number of components actually used almost always depends on the data.
  
      .. versionadded:: 0.18
  
      Read more in the :ref:`User Guide <bgmm>`.
  
      Parameters
      ----------
      n_components : int, default=1
          The number of mixture components. Depending on the data and the value
          of the `weight_concentration_prior` the model can decide to not use
          all the components by setting some component `weights_` to values very
          close to zero. The number of effective components is therefore smaller
          than n_components.
  
      covariance_type : {'full', 'tied', 'diag', 'spherical'}, default='full'
          String describing the type of covariance parameters to use.
          Must be one of::
  
              'full' (each component has its own general covariance matrix),
              'tied' (all components share the same general covariance matrix),
              'diag' (each component has its own diagonal covariance matrix),
              'spherical' (each component has its own single variance).
  
      tol : float, default=1e-3
          The convergence threshold. EM iterations will stop when the
          lower bound average gain on the likelihood (of the training data with
          respect to the model) is below this threshold.
  
      reg_covar : float, default=1e-6
          Non-negative regularization added to the diagonal of covariance.
          Allows to assure that the covariance matrices are all positive.
  
      max_iter : int, default=100
          The number of EM iterations to perform.
  
      n_init : int, default=1
          The number of initializations to perform. The result with the highest
          lower bound value on the likelihood is kept.
  
      init_params : {'kmeans', 'random'}, default='kmeans'
          The method used to initialize the weights, the means and the
          covariances.
          Must be one of::
  
              'kmeans' : responsibilities are initialized using kmeans.
              'random' : responsibilities are initialized randomly.
  
      weight_concentration_prior_type : str, default='dirichlet_process'
          String describing the type of the weight concentration prior.
          Must be one of::
  
              'dirichlet_process' (using the Stick-breaking representation),
              'dirichlet_distribution' (can favor more uniform weights).
  
      weight_concentration_prior : float or None, default=None
          The dirichlet concentration of each component on the weight
          distribution (Dirichlet). This is commonly called gamma in the
          literature. The higher concentration puts more mass in
          the center and will lead to more components being active, while a lower
          concentration parameter will lead to more mass at the edge of the
          mixture weights simplex. The value of the parameter must be greater
          than 0. If it is None, it's set to ``1. / n_components``.
  
      mean_precision_prior : float or None, default=None
          The precision prior on the mean distribution (Gaussian).
          Controls the extent of where means can be placed. Larger
          values concentrate the cluster means around `mean_prior`.
          The value of the parameter must be greater than 0.
          If it is None, it is set to 1.
  
      mean_prior : array-like, shape (n_features,), default=None
          The prior on the mean distribution (Gaussian).
          If it is None, it is set to the mean of X.
  
      degrees_of_freedom_prior : float or None, default=None
          The prior of the number of degrees of freedom on the covariance
          distributions (Wishart). If it is None, it's set to `n_features`.
  
      covariance_prior : float or array-like, default=None
          The prior on the covariance distribution (Wishart).
          If it is None, the emiprical covariance prior is initialized using the
          covariance of X. The shape depends on `covariance_type`::
  
                  (n_features, n_features) if 'full',
                  (n_features, n_features) if 'tied',
                  (n_features)             if 'diag',
                  float                    if 'spherical'
  
      random_state : int, RandomState instance or None, default=None
          Controls the random seed given to the method chosen to initialize the
          parameters (see `init_params`).
          In addition, it controls the generation of random samples from the
          fitted distribution (see the method `sample`).
          Pass an int for reproducible output across multiple function calls.
          See :term:`Glossary <random_state>`.
  
      warm_start : bool, default=False
          If 'warm_start' is True, the solution of the last fitting is used as
          initialization for the next call of fit(). This can speed up
          convergence when fit is called several times on similar problems.
          See :term:`the Glossary <warm_start>`.
  
      verbose : int, default=0
          Enable verbose output. If 1 then it prints the current
          initialization and each iteration step. If greater than 1 then
          it prints also the log probability and the time needed
          for each step.
  
      verbose_interval : int, default=10
          Number of iteration done before the next print.
  
      Attributes
      ----------
      weights_ : array-like of shape (n_components,)
          The weights of each mixture components.
  
      means_ : array-like of shape (n_components, n_features)
          The mean of each mixture component.
  
      covariances_ : array-like
          The covariance of each mixture component.
          The shape depends on `covariance_type`::
  
              (n_components,)                        if 'spherical',
              (n_features, n_features)               if 'tied',
              (n_components, n_features)             if 'diag',
              (n_components, n_features, n_features) if 'full'
  
      precisions_ : array-like
          The precision matrices for each component in the mixture. A precision
          matrix is the inverse of a covariance matrix. A covariance matrix is
          symmetric positive definite so the mixture of Gaussian can be
          equivalently parameterized by the precision matrices. Storing the
          precision matrices instead of the covariance matrices makes it more
          efficient to compute the log-likelihood of new samples at test time.
          The shape depends on ``covariance_type``::
  
              (n_components,)                        if 'spherical',
              (n_features, n_features)               if 'tied',
              (n_components, n_features)             if 'diag',
              (n_components, n_features, n_features) if 'full'
  
      precisions_cholesky_ : array-like
          The cholesky decomposition of the precision matrices of each mixture
          component. A precision matrix is the inverse of a covariance matrix.
          A covariance matrix is symmetric positive definite so the mixture of
          Gaussian can be equivalently parameterized by the precision matrices.
          Storing the precision matrices instead of the covariance matrices makes
          it more efficient to compute the log-likelihood of new samples at test
          time. The shape depends on ``covariance_type``::
  
              (n_components,)                        if 'spherical',
              (n_features, n_features)               if 'tied',
              (n_components, n_features)             if 'diag',
              (n_components, n_features, n_features) if 'full'
  
      converged_ : bool
          True when convergence was reached in fit(), False otherwise.
  
      n_iter_ : int
          Number of step used by the best fit of inference to reach the
          convergence.
  
      lower_bound_ : float
          Lower bound value on the likelihood (of the training data with
          respect to the model) of the best fit of inference.
  
      weight_concentration_prior_ : tuple or float
          The dirichlet concentration of each component on the weight
          distribution (Dirichlet). The type depends on
          ``weight_concentration_prior_type``::
  
              (float, float) if 'dirichlet_process' (Beta parameters),
              float          if 'dirichlet_distribution' (Dirichlet parameters).
  
          The higher concentration puts more mass in
          the center and will lead to more components being active, while a lower
          concentration parameter will lead to more mass at the edge of the
          simplex.
  
      weight_concentration_ : array-like of shape (n_components,)
          The dirichlet concentration of each component on the weight
          distribution (Dirichlet).
  
      mean_precision_prior_ : float
          The precision prior on the mean distribution (Gaussian).
          Controls the extent of where means can be placed.
          Larger values concentrate the cluster means around `mean_prior`.
          If mean_precision_prior is set to None, `mean_precision_prior_` is set
          to 1.
  
      mean_precision_ : array-like of shape (n_components,)
          The precision of each components on the mean distribution (Gaussian).
  
      mean_prior_ : array-like of shape (n_features,)
          The prior on the mean distribution (Gaussian).
  
      degrees_of_freedom_prior_ : float
          The prior of the number of degrees of freedom on the covariance
          distributions (Wishart).
  
      degrees_of_freedom_ : array-like of shape (n_components,)
          The number of degrees of freedom of each components in the model.
  
      covariance_prior_ : float or array-like
          The prior on the covariance distribution (Wishart).
          The shape depends on `covariance_type`::
  
              (n_features, n_features) if 'full',
              (n_features, n_features) if 'tied',
              (n_features)             if 'diag',
              float                    if 'spherical'
  
      n_features_in_ : int
          Number of features seen during :term:`fit`.
  
          .. versionadded:: 0.24
  
      See Also
      --------
      GaussianMixture : Finite Gaussian mixture fit with EM.
  
      References
      ----------
  
      .. [1] `Bishop, Christopher M. (2006). "Pattern recognition and machine
         learning". Vol. 4 No. 4. New York: Springer.
         <https://www.springer.com/kr/book/9780387310732>`_
  
      .. [2] `Hagai Attias. (2000). "A Variational Bayesian Framework for
         Graphical Models". In Advances in Neural Information Processing
         Systems 12.
         <http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.36.2841&rep=rep1&type=pdf>`_
  
      .. [3] `Blei, David M. and Michael I. Jordan. (2006). "Variational
         inference for Dirichlet process mixtures". Bayesian analysis 1.1
         <https://www.cs.princeton.edu/courses/archive/fall11/cos597C/reading/BleiJordan2005.pdf>`_
  
      Examples
      --------
      >>> import numpy as np
      >>> from sklearn.mixture import BayesianGaussianMixture
      >>> X = np.array([[1, 2], [1, 4], [1, 0], [4, 2], [12, 4], [10, 7]])
      >>> bgm = BayesianGaussianMixture(n_components=2, random_state=42).fit(X)
      >>> bgm.means_
      array([[2.49... , 2.29...],
             [8.45..., 4.52... ]])
      >>> bgm.predict([[0, 0], [9, 3]])
      array([0, 1])
      """
  
      def __init__(
          self,
          *,
          n_components=1,
          covariance_type="full",
          tol=1e-3,
          reg_covar=1e-6,
          max_iter=100,
          n_init=1,
          init_params="kmeans",
          weight_concentration_prior_type="dirichlet_process",
          weight_concentration_prior=None,
          mean_precision_prior=None,
          mean_prior=None,
          degrees_of_freedom_prior=None,
          covariance_prior=None,
          random_state=None,
          warm_start=False,
          verbose=0,
          verbose_interval=10,
      ):
          super().__init__(
              n_components=n_components,
              tol=tol,
              reg_covar=reg_covar,
              max_iter=max_iter,
              n_init=n_init,
              init_params=init_params,
              random_state=random_state,
              warm_start=warm_start,
              verbose=verbose,
              verbose_interval=verbose_interval,
          )
  
          self.covariance_type = covariance_type
          self.weight_concentration_prior_type = weight_concentration_prior_type
          self.weight_concentration_prior = weight_concentration_prior
          self.mean_precision_prior = mean_precision_prior
          self.mean_prior = mean_prior
          self.degrees_of_freedom_prior = degrees_of_freedom_prior
          self.covariance_prior = covariance_prior
  
      def _check_parameters(self, X):
          """Check that the parameters are well defined.
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
          """
          if self.covariance_type not in ["spherical", "tied", "diag", "full"]:
              raise ValueError(
                  "Invalid value for 'covariance_type': %s "
                  "'covariance_type' should be in "
                  "['spherical', 'tied', 'diag', 'full']"
                  % self.covariance_type
              )
  
          if self.weight_concentration_prior_type not in [
              "dirichlet_process",
              "dirichlet_distribution",
          ]:
              raise ValueError(
                  "Invalid value for 'weight_concentration_prior_type': %s "
                  "'weight_concentration_prior_type' should be in "
                  "['dirichlet_process', 'dirichlet_distribution']"
                  % self.weight_concentration_prior_type
              )
  
          self._check_weights_parameters()
          self._check_means_parameters(X)
          self._check_precision_parameters(X)
          self._checkcovariance_prior_parameter(X)
  
      def _check_weights_parameters(self):
          """Check the parameter of the Dirichlet distribution."""
          if self.weight_concentration_prior is None:
              self.weight_concentration_prior_ = 1.0 / self.n_components
          elif self.weight_concentration_prior > 0.0:
              self.weight_concentration_prior_ = self.weight_concentration_prior
          else:
              raise ValueError(
                  "The parameter 'weight_concentration_prior' "
                  "should be greater than 0., but got %.3f."
                  % self.weight_concentration_prior
              )
  
      def _check_means_parameters(self, X):
          """Check the parameters of the Gaussian distribution.
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
          """
          _, n_features = X.shape
  
          if self.mean_precision_prior is None:
              self.mean_precision_prior_ = 1.0
          elif self.mean_precision_prior > 0.0:
              self.mean_precision_prior_ = self.mean_precision_prior
          else:
              raise ValueError(
                  "The parameter 'mean_precision_prior' should be "
                  "greater than 0., but got %.3f."
                  % self.mean_precision_prior
              )
  
          if self.mean_prior is None:
              self.mean_prior_ = X.mean(axis=0)
          else:
              self.mean_prior_ = check_array(
                  self.mean_prior, dtype=[np.float64, np.float32], ensure_2d=False
              )
              _check_shape(self.mean_prior_, (n_features,), "means")
  
      def _check_precision_parameters(self, X):
          """Check the prior parameters of the precision distribution.
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
          """
          _, n_features = X.shape
  
          if self.degrees_of_freedom_prior is None:
              self.degrees_of_freedom_prior_ = n_features
          elif self.degrees_of_freedom_prior > n_features - 1.0:
              self.degrees_of_freedom_prior_ = self.degrees_of_freedom_prior
          else:
              raise ValueError(
                  "The parameter 'degrees_of_freedom_prior' "
                  "should be greater than %d, but got %.3f."
                  % (n_features - 1, self.degrees_of_freedom_prior)
              )
  
      def _checkcovariance_prior_parameter(self, X):
          """Check the `covariance_prior_`.
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
          """
          _, n_features = X.shape
  
          if self.covariance_prior is None:
              self.covariance_prior_ = {
                  "full": np.atleast_2d(np.cov(X.T)),
                  "tied": np.atleast_2d(np.cov(X.T)),
                  "diag": np.var(X, axis=0, ddof=1),
                  "spherical": np.var(X, axis=0, ddof=1).mean(),
              }[self.covariance_type]
  
          elif self.covariance_type in ["full", "tied"]:
              self.covariance_prior_ = check_array(
                  self.covariance_prior, dtype=[np.float64, np.float32], ensure_2d=False
              )
              _check_shape(
                  self.covariance_prior_,
                  (n_features, n_features),
                  "%s covariance_prior" % self.covariance_type,
              )
              _check_precision_matrix(self.covariance_prior_, self.covariance_type)
          elif self.covariance_type == "diag":
              self.covariance_prior_ = check_array(
                  self.covariance_prior, dtype=[np.float64, np.float32], ensure_2d=False
              )
              _check_shape(
                  self.covariance_prior_,
                  (n_features,),
                  "%s covariance_prior" % self.covariance_type,
              )
              _check_precision_positivity(self.covariance_prior_, self.covariance_type)
          # spherical case
          elif self.covariance_prior > 0.0:
              self.covariance_prior_ = self.covariance_prior
          else:
              raise ValueError(
                  "The parameter 'spherical covariance_prior' "
                  "should be greater than 0., but got %.3f."
                  % self.covariance_prior
              )
  
      def _initialize(self, X, resp):
          """Initialization of the mixture parameters.
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
  
          resp : array-like of shape (n_samples, n_components)
          """
          nk, xk, sk = _estimate_gaussian_parameters(
              X, resp, self.reg_covar, self.covariance_type
          )
  
          self._estimate_weights(nk)
          self._estimate_means(nk, xk)
          self._estimate_precisions(nk, xk, sk)
  
      def _estimate_weights(self, nk):
          """Estimate the parameters of the Dirichlet distribution.
  
          Parameters
          ----------
          nk : array-like of shape (n_components,)
          """
          if self.weight_concentration_prior_type == "dirichlet_process":
              # For dirichlet process weight_concentration will be a tuple
              # containing the two parameters of the beta distribution
              self.weight_concentration_ = (
                  1.0 + nk,
                  (
                      self.weight_concentration_prior_
                      + np.hstack((np.cumsum(nk[::-1])[-2::-1], 0))
                  ),
              )
          else:
              # case Variationnal Gaussian mixture with dirichlet distribution
              self.weight_concentration_ = self.weight_concentration_prior_ + nk
  
      def _estimate_means(self, nk, xk):
          """Estimate the parameters of the Gaussian distribution.
  
          Parameters
          ----------
          nk : array-like of shape (n_components,)
  
          xk : array-like of shape (n_components, n_features)
          """
          self.mean_precision_ = self.mean_precision_prior_ + nk
          self.means_ = (
              self.mean_precision_prior_ * self.mean_prior_ + nk[:, np.newaxis] * xk
          ) / self.mean_precision_[:, np.newaxis]
  
      def _estimate_precisions(self, nk, xk, sk):
          """Estimate the precisions parameters of the precision distribution.
  
          Parameters
          ----------
          nk : array-like of shape (n_components,)
  
          xk : array-like of shape (n_components, n_features)
  
          sk : array-like
              The shape depends of `covariance_type`:
              'full' : (n_components, n_features, n_features)
              'tied' : (n_features, n_features)
              'diag' : (n_components, n_features)
              'spherical' : (n_components,)
          """
          {
              "full": self._estimate_wishart_full,
              "tied": self._estimate_wishart_tied,
              "diag": self._estimate_wishart_diag,
              "spherical": self._estimate_wishart_spherical,
          }[self.covariance_type](nk, xk, sk)
  
          self.precisions_cholesky_ = _compute_precision_cholesky(
              self.covariances_, self.covariance_type
          )
  
      def _estimate_wishart_full(self, nk, xk, sk):
          """Estimate the full Wishart distribution parameters.
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
  
          nk : array-like of shape (n_components,)
  
          xk : array-like of shape (n_components, n_features)
  
          sk : array-like of shape (n_components, n_features, n_features)
          """
          _, n_features = xk.shape
  
          # Warning : in some Bishop book, there is a typo on the formula 10.63
          # `degrees_of_freedom_k = degrees_of_freedom_0 + Nk` is
          # the correct formula
          self.degrees_of_freedom_ = self.degrees_of_freedom_prior_ + nk
  
          self.covariances_ = np.empty((self.n_components, n_features, n_features))
  
          for k in range(self.n_components):
              diff = xk[k] - self.mean_prior_
              self.covariances_[k] = (
                  self.covariance_prior_
                  + nk[k] * sk[k]
                  + nk[k]
                  * self.mean_precision_prior_
                  / self.mean_precision_[k]
                  * np.outer(diff, diff)
              )
  
          # Contrary to the original bishop book, we normalize the covariances
          self.covariances_ /= self.degrees_of_freedom_[:, np.newaxis, np.newaxis]
  
      def _estimate_wishart_tied(self, nk, xk, sk):
          """Estimate the tied Wishart distribution parameters.
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
  
          nk : array-like of shape (n_components,)
  
          xk : array-like of shape (n_components, n_features)
  
          sk : array-like of shape (n_features, n_features)
          """
          _, n_features = xk.shape
  
          # Warning : in some Bishop book, there is a typo on the formula 10.63
          # `degrees_of_freedom_k = degrees_of_freedom_0 + Nk`
          # is the correct formula
          self.degrees_of_freedom_ = (
              self.degrees_of_freedom_prior_ + nk.sum() / self.n_components
          )
  
          diff = xk - self.mean_prior_
          self.covariances_ = (
              self.covariance_prior_
              + sk * nk.sum() / self.n_components
              + self.mean_precision_prior_
              / self.n_components
              * np.dot((nk / self.mean_precision_) * diff.T, diff)
          )
  
          # Contrary to the original bishop book, we normalize the covariances
          self.covariances_ /= self.degrees_of_freedom_
  
      def _estimate_wishart_diag(self, nk, xk, sk):
          """Estimate the diag Wishart distribution parameters.
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
  
          nk : array-like of shape (n_components,)
  
          xk : array-like of shape (n_components, n_features)
  
          sk : array-like of shape (n_components, n_features)
          """
          _, n_features = xk.shape
  
          # Warning : in some Bishop book, there is a typo on the formula 10.63
          # `degrees_of_freedom_k = degrees_of_freedom_0 + Nk`
          # is the correct formula
          self.degrees_of_freedom_ = self.degrees_of_freedom_prior_ + nk
  
          diff = xk - self.mean_prior_
          self.covariances_ = self.covariance_prior_ + nk[:, np.newaxis] * (
              sk
              + (self.mean_precision_prior_ / self.mean_precision_)[:, np.newaxis]
              * np.square(diff)
          )
  
          # Contrary to the original bishop book, we normalize the covariances
          self.covariances_ /= self.degrees_of_freedom_[:, np.newaxis]
  
      def _estimate_wishart_spherical(self, nk, xk, sk):
          """Estimate the spherical Wishart distribution parameters.
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
  
          nk : array-like of shape (n_components,)
  
          xk : array-like of shape (n_components, n_features)
  
          sk : array-like of shape (n_components,)
          """
          _, n_features = xk.shape
  
          # Warning : in some Bishop book, there is a typo on the formula 10.63
          # `degrees_of_freedom_k = degrees_of_freedom_0 + Nk`
          # is the correct formula
          self.degrees_of_freedom_ = self.degrees_of_freedom_prior_ + nk
  
          diff = xk - self.mean_prior_
          self.covariances_ = self.covariance_prior_ + nk * (
              sk
              + self.mean_precision_prior_
              / self.mean_precision_
              * np.mean(np.square(diff), 1)
          )
  
          # Contrary to the original bishop book, we normalize the covariances
          self.covariances_ /= self.degrees_of_freedom_
  
      def _m_step(self, X, log_resp):
          """M step.
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
  
          log_resp : array-like of shape (n_samples, n_components)
              Logarithm of the posterior probabilities (or responsibilities) of
              the point of each sample in X.
          """
          n_samples, _ = X.shape
  
          nk, xk, sk = _estimate_gaussian_parameters(
              X, np.exp(log_resp), self.reg_covar, self.covariance_type
          )
          self._estimate_weights(nk)
          self._estimate_means(nk, xk)
          self._estimate_precisions(nk, xk, sk)
  
      def _estimate_log_weights(self):
          if self.weight_concentration_prior_type == "dirichlet_process":
              digamma_sum = digamma(
                  self.weight_concentration_[0] + self.weight_concentration_[1]
              )
              digamma_a = digamma(self.weight_concentration_[0])
              digamma_b = digamma(self.weight_concentration_[1])
              return (
                  digamma_a
                  - digamma_sum
                  + np.hstack((0, np.cumsum(digamma_b - digamma_sum)[:-1]))
              )
          else:
              # case Variationnal Gaussian mixture with dirichlet distribution
              return digamma(self.weight_concentration_) - digamma(
                  np.sum(self.weight_concentration_)
              )
  
      def _estimate_log_prob(self, X):
          _, n_features = X.shape
          # We remove `n_features * np.log(self.degrees_of_freedom_)` because
          # the precision matrix is normalized
          log_gauss = _estimate_log_gaussian_prob(
              X, self.means_, self.precisions_cholesky_, self.covariance_type
          ) - 0.5 * n_features * np.log(self.degrees_of_freedom_)
  
          log_lambda = n_features * np.log(2.0) + np.sum(
              digamma(
                  0.5
                  * (self.degrees_of_freedom_ - np.arange(0, n_features)[:, np.newaxis])
              ),
              0,
          )
  
          return log_gauss + 0.5 * (log_lambda - n_features / self.mean_precision_)
  
      def _compute_lower_bound(self, log_resp, log_prob_norm):
          """Estimate the lower bound of the model.
  
          The lower bound on the likelihood (of the training data with respect to
          the model) is used to detect the convergence and has to increase at
          each iteration.
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
  
          log_resp : array, shape (n_samples, n_components)
              Logarithm of the posterior probabilities (or responsibilities) of
              the point of each sample in X.
  
          log_prob_norm : float
              Logarithm of the probability of each sample in X.
  
          Returns
          -------
          lower_bound : float
          """
          # Contrary to the original formula, we have done some simplification
          # and removed all the constant terms.
          (n_features,) = self.mean_prior_.shape
  
          # We removed `.5 * n_features * np.log(self.degrees_of_freedom_)`
          # because the precision matrix is normalized.
          log_det_precisions_chol = _compute_log_det_cholesky(
              self.precisions_cholesky_, self.covariance_type, n_features
          ) - 0.5 * n_features * np.log(self.degrees_of_freedom_)
  
          if self.covariance_type == "tied":
              log_wishart = self.n_components * np.float64(
                  _log_wishart_norm(
                      self.degrees_of_freedom_, log_det_precisions_chol, n_features
                  )
              )
          else:
              log_wishart = np.sum(
                  _log_wishart_norm(
                      self.degrees_of_freedom_, log_det_precisions_chol, n_features
                  )
              )
  
          if self.weight_concentration_prior_type == "dirichlet_process":
              log_norm_weight = -np.sum(
                  betaln(self.weight_concentration_[0], self.weight_concentration_[1])
              )
          else:
              log_norm_weight = _log_dirichlet_norm(self.weight_concentration_)
  
          return (
              -np.sum(np.exp(log_resp) * log_resp)
              - log_wishart
              - log_norm_weight
              - 0.5 * n_features * np.sum(np.log(self.mean_precision_))
          )
  
      def _get_parameters(self):
          return (
              self.weight_concentration_,
              self.mean_precision_,
              self.means_,
              self.degrees_of_freedom_,
              self.covariances_,
              self.precisions_cholesky_,
          )
  
      def _set_parameters(self, params):
          (
              self.weight_concentration_,
              self.mean_precision_,
              self.means_,
              self.degrees_of_freedom_,
              self.covariances_,
              self.precisions_cholesky_,
          ) = params
  
          # Weights computation
          if self.weight_concentration_prior_type == "dirichlet_process":
              weight_dirichlet_sum = (
                  self.weight_concentration_[0] + self.weight_concentration_[1]
              )
              tmp = self.weight_concentration_[1] / weight_dirichlet_sum
              self.weights_ = (
                  self.weight_concentration_[0]
                  / weight_dirichlet_sum
                  * np.hstack((1, np.cumprod(tmp[:-1])))
              )
              self.weights_ /= np.sum(self.weights_)
          else:
              self.weights_ = self.weight_concentration_ / np.sum(
                  self.weight_concentration_
              )
  
          # Precisions matrices computation
          if self.covariance_type == "full":
              self.precisions_ = np.array(
                  [
                      np.dot(prec_chol, prec_chol.T)
                      for prec_chol in self.precisions_cholesky_
                  ]
              )
  
          elif self.covariance_type == "tied":
              self.precisions_ = np.dot(
                  self.precisions_cholesky_, self.precisions_cholesky_.T
              )
          else:
              self.precisions_ = self.precisions_cholesky_ ** 2
  