content: """Private utilities."""
  
  import warnings
  
  import numpy as np
  
  
  def logdet(a):
      sign, logdet = np.linalg.slogdet(a)
      if (sign < 0).any():
          warnings.warn("invalid value encountered in log", RuntimeWarning)
          return np.where(sign < 0, np.nan, logdet)
      else:
          return logdet
  
  
  def split_X_lengths(X, lengths):
      if lengths is None:
          return [X]
      else:
          cs = np.cumsum(lengths)
          n_samples = len(X)
          if cs[-1] != n_samples:
              raise ValueError(
                  f"lengths array {lengths} doesn't sum to {n_samples} samples")
          return np.split(X, cs)[:-1]
  
  
  # Copied from scikit-learn 0.19.
  def _validate_covars(covars, covariance_type, n_components):
      """Do basic checks on matrix covariance sizes and values."""
      from scipy import linalg
      if covariance_type == 'spherical':
          if len(covars) != n_components:
              raise ValueError("'spherical' covars have length n_components")
          elif np.any(covars <= 0):
              raise ValueError("'spherical' covars must be positive")
      elif covariance_type == 'tied':
          if covars.shape[0] != covars.shape[1]:
              raise ValueError("'tied' covars must have shape (n_dim, n_dim)")
          elif (not np.allclose(covars, covars.T)
                or np.any(linalg.eigvalsh(covars) <= 0)):
              raise ValueError("'tied' covars must be symmetric, "
                               "positive-definite")
      elif covariance_type == 'diag':
          if len(covars.shape) != 2:
              raise ValueError("'diag' covars must have shape "
                               "(n_components, n_dim)")
          elif np.any(covars <= 0):
              raise ValueError("'diag' covars must be positive")
      elif covariance_type == 'full':
          if len(covars.shape) != 3:
              raise ValueError("'full' covars must have shape "
                               "(n_components, n_dim, n_dim)")
          elif covars.shape[1] != covars.shape[2]:
              raise ValueError("'full' covars must have shape "
                               "(n_components, n_dim, n_dim)")
          for n, cv in enumerate(covars):
              if (not np.allclose(cv, cv.T)
                      or np.any(linalg.eigvalsh(cv) <= 0)):
                  raise ValueError("component %d of 'full' covars must be "
                                   "symmetric, positive-definite" % n)
      else:
          raise ValueError("covariance_type must be one of " +
                           "'spherical', 'tied', 'diag', 'full'")
  
  
  # Copied from scikit-learn 0.19.
  def distribute_covar_matrix_to_match_covariance_type(
          tied_cv, covariance_type, n_components):
      """Create all the covariance matrices from a given template."""
      if covariance_type == 'spherical':
          cv = np.tile(tied_cv.mean() * np.ones(tied_cv.shape[1]),
                       (n_components, 1))
      elif covariance_type == 'tied':
          cv = tied_cv
      elif covariance_type == 'diag':
          cv = np.tile(np.diag(tied_cv), (n_components, 1))
      elif covariance_type == 'full':
          cv = np.tile(tied_cv, (n_components, 1, 1))
      else:
          raise ValueError("covariance_type must be one of " +
                           "'spherical', 'tied', 'diag', 'full'")
      return cv
  