code: def smoothed_edf(
      y_true,
      y_pred=None,
      *,
      prot_attr=None,
      pos_label=1,
      concentration=1.0,
      sample_weight=None,
  ):
      r"""Smoothed empirical differential fairness (EDF).
  
      .. math::
          e^{-\epsilon} \leq \frac{\sum_{A=s_i}{P(y|x)} + \alpha}{N_{s_i} + |R_Y|\alpha}
          \frac{N_{s_j} + |R_Y|\alpha}{\sum_{A=s_j}{P(y|x) + \alpha}} \leq e^\epsilon
  
      See [#foulds18]_ for more details.
  
      Note:
          If only y_true is provided, this will return the maximum epsilon for any
          two intersectional groups (smoothed EDF of the original dataset). If
          both y_true and y_pred are provided, only y_pred is used.
  
      Args:
          y_true (pandas.Series): Ground truth (correct) target values. If y_pred
              is provided, this is ignored.
          y_pred (array-like, optional): Estimated targets as returned by a
              classifier.
          prot_attr (array-like, keyword-only): Protected attribute(s). If
              ``None``, all protected attributes in y_true are used.
          pos_label (scalar, optional): The label of the positive class.
          concentration (scalar, optional): Dirichlet smoothing concentration
              parameter :math:`|R_Y|\alpha` (must be non-negative).
          sample_weight (array-like, optional): Sample weights.
  
      Returns:
          float: Smoothed EDF, :math:`\epsilon`. Lower is better.
  
      See also:
          :func:`intersection`, :func:`smoothed_base_rate`
  
      References:
          .. [#foulds18] J. R. Foulds, R. Islam, K. N. Keya, and S. Pan,
             "An Intersectional Definition of Fairness," arXiv preprint
             arXiv:1807.08362, 2018.
      """
      rate = (
          smoothed_base_rate
          if y_pred is None
          else smoothed_selection_rate
      )
      sbr = intersection(
          rate,
          y_true,
          y_pred,
          prot_attr=prot_attr,
          sample_weight=sample_weight,
          pos_label=pos_label,
          concentration=concentration,
      )
  
      logsbr = np.log(sbr)
      pos_ratio = max(abs(i - j) for i, j in permutations(logsbr, 2))
      lognegsbr = np.log(1 - np.array(sbr))
      neg_ratio = max(abs(i - j) for i, j in permutations(lognegsbr, 2))
      return max(pos_ratio, neg_ratio)
  
imports: from itertools import permutations
  from typing import Union
  import warnings
  import numpy
  import pandas
  from scipy.special import rel_entr
  from sklearn.metrics import make_scorer, recall_score, precision_score
  from sklearn.metrics import multilabel_confusion_matrix
  from sklearn.metrics._classification import _prf_divide, _check_zero_division
  from sklearn.neighbors import NearestNeighbors
  from sklearn.utils import check_X_y
  from aif360.metrics import ot_metric
  from aif360.sklearn.utils import check_inputs, check_groups
  from aif360.detectors.mdss.ScoringFunctions import BerkJones, Bernoulli
  from aif360.detectors.mdss.MDSS import MDSS
label_desc: Use image node_id 26 for calling a global function with example usage: smoothed_edf(y_true, y_pred) and returns: max