class_docstr: Probabilistic Classifier Chains.
  
  The Probabilistic Classifier Chains (PCC) [^1] is a Bayes-optimal method
  based on the Classifier Chains (CC).
  
  Consider the concept of chaining classifiers as searching a path in a
  binary tree whose leaf nodes are associated with a label $y \in Y$. While
  CC searches only a single path in the aforementioned binary tree, PCC looks
  at each of the $2^l$ paths, where $l$ is the number of labels. This limits
  the applicability of the method to data sets with a small to moderate
  number of labels. The authors recommend no more than about 15 labels for
  real-world applications.
  
  Parameters
  ----------
  model
  
  Examples
  --------
  
  >>> from river import linear_model
  >>> from river import metrics
  >>> from river import multioutput
  >>> from river.datasets import synth
  
  >>> dataset = synth.Logical(seed=42, n_tiles=100)
  
  >>> model = multioutput.ProbabilisticClassifierChain(
  ...     model=linear_model.LogisticRegression()
  ... )
  
  >>> metric = metrics.multioutput.MicroAverage(metrics.Jaccard())
  
  >>> for x, y in dataset:
  ...    y_pred = model.predict_one(x)
  ...    y_pred = {k: y_pred.get(k, 0) for k in y}
  ...    metric.update(y, y_pred)
  ...    model.learn_one(x, y)
  
  >>> metric
  MicroAverage(Jaccard): 51.84%
  
  References
  ----------
  [^1]: Cheng, W., Hüllermeier, E., & Dembczynski, K. J. (2010).
        Bayes optimal multilabel classification via probabilistic classifier
        chains. In Proceedings of the 27th international conference on
        machine learning (ICML-10) (pp. 279-286).
code: def _payoff(self, x, y, **kwargs):
      # Calculate payoff for predicting y | x, under the chains model.
      p = {}
  
      x = copy.copy(x)
  
      for label in self.order:
          clf = self[label]
  
          y_pred = clf.predict_proba_one(x, **kwargs)
          # Extend features
          x[label] = y.get(label, 0)
          p[label] = y_pred.get(y[label], 0) if label in y else 0
  
      return prod(p.values()) if p else 0
  
imports: from __future__ import annotations
  import collections
  import copy
  import functools
  import random
  from river import base, linear_model
  from river.utils.math import minkowski_distance, prod
label_desc: Use image node_id 3 for calling the ProbabilisticClassifierChain obj's underlying member method code with example usage: obj._payoff(x, y) and returns: unknown