code: def log_multivariate_normal_density(
      X, means, covars, covariance_type="diag"
  ):
      """Compute the log probability under a multivariate Gaussian distribution.
  
      Parameters
      ----------
      X : array_like, shape (n_samples, n_features)
          List of n_features-dimensional data points. Each row corresponds to a
          single data point.
  
      means : array_like, shape (n_components, n_features)
          List of n_features-dimensional mean vectors for n_components Gaussians.
          Each row corresponds to a single mean vector.
  
      covars : array_like
          List of n_components covariance parameters for each Gaussian. The shape
          depends on `covariance_type`:
              (n_components, n_features)      if 'spherical',
              (n_features, n_features)    if 'tied',
              (n_components, n_features)    if 'diag',
              (n_components, n_features, n_features) if 'full'
  
      covariance_type : string
          Type of the covariance parameters.  Must be one of
          'spherical', 'tied', 'diag', 'full'.  Defaults to 'diag'.
  
      Returns
      -------
      lpr : array_like, shape (n_samples, n_components)
          Array containing the log probabilities of each data point in
          X under each of the n_components multivariate Gaussian distributions.
      """
      log_multivariate_normal_density_dict = {
          "spherical": _log_multivariate_normal_density_spherical,
          "tied": _log_multivariate_normal_density_tied,
          "diag": _log_multivariate_normal_density_diag,
          "full": _log_multivariate_normal_density_full,
      }
      return log_multivariate_normal_density_dict[covariance_type](
          X, means, covars
      )
  
imports: import numpy
  from scipy import linalg
  from time import time
  from ..base import BaseEstimator
  from ..utils import check_random_state, check_array, deprecated
  from ..utils.extmath import logsumexp
  from ..utils.validation import check_is_fitted
  from ..None import cluster
  from sklearn.externals.six.moves import zip
label_desc: Use image node_id 1 for calling a global function with example usage: log_multivariate_normal_density(X, means, covars, covariance_type) and returns: log_multivariate_normal_density_dict