class_docstr: Class for performing approximate inference using Max-Product Linear Programming method.
  
  We derive message passing updates that result in monotone decrease of the dual of the
  MAP LP Relaxation.
  
  Parameters
  ----------
  model: MarkovNetwork for which inference is to be performed.
  
  Examples
  --------
  >>> import numpy as np
  >>> from pgmpy.models import MarkovNetwork
  >>> from pgmpy.inference import Mplp
  >>> from pgmpy.factors.discrete import DiscreteFactor
  >>> student = MarkovNetwork()
  >>> student.add_edges_from([('A', 'B'), ('B', 'C'), ('C', 'D'), ('E', 'F')])
  >>> factor_a = DiscreteFactor(['A'], cardinality=[2], values=np.array([0.54577, 1.8323]))
  >>> factor_b = DiscreteFactor(['B'], cardinality=[2], values=np.array([0.93894, 1.065]))
  >>> factor_c = DiscreteFactor(['C'], cardinality=[2], values=np.array([0.89205, 1.121]))
  >>> factor_d = DiscreteFactor(['D'], cardinality=[2], values=np.array([0.56292, 1.7765]))
  >>> factor_e = DiscreteFactor(['E'], cardinality=[2], values=np.array([0.47117, 2.1224]))
  >>> factor_f = DiscreteFactor(['F'], cardinality=[2], values=np.array([1.5093, 0.66257]))
  >>> factor_a_b = DiscreteFactor(['A', 'B'], cardinality=[2, 2],
  ...                             values=np.array([1.3207, 0.75717, 0.75717, 1.3207]))
  >>> factor_b_c = DiscreteFactor(['B', 'C'], cardinality=[2, 2],
  ...                             values=np.array([0.00024189, 4134.2, 4134.2, 0.00024189]))
  >>> factor_c_d = DiscreteFactor(['C', 'D'], cardinality=[2, 2],
  ...                             values=np.array([0.0043227, 231.34, 231.34, 0.0043227]))
  >>> factor_d_e = DiscreteFactor(['E', 'F'], cardinality=[2, 2],
  ...                             values=np.array([31.228, 0.032023, 0.032023, 31.228]))
  >>> student.add_factors(factor_a, factor_b, factor_c, factor_d, factor_e, factor_f, factor_a_b,
  ...                     factor_b_c, factor_c_d, factor_d_e)
  >>> mplp = Mplp(student)
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docstr: This is the message-update method.
  
  Parameters
  ----------
  sending_cluster: The resulting messages are lambda_{c-->s} from the given
      cluster 'c' to all of its intersection_sets 's'.
      Here 's' are the elements of intersection_sets_for_cluster_c.
  
  References
  ----------
  Fixing Max-Product: Convergent Message-Passing Algorithms for MAP LP Relaxations
  by Amir Globerson and Tommi Jaakkola.
  Section 6, Page: 5; Beyond pairwise potentials: Generalized MPLP
  Later Modified by Sontag in "Introduction to Dual decomposition for Inference" Pg: 7 & 17
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code: def _update_message(self, sending_cluster):
      """
      This is the message-update method.
  
      Parameters
      ----------
      sending_cluster: The resulting messages are lambda_{c-->s} from the given
          cluster 'c' to all of its intersection_sets 's'.
          Here 's' are the elements of intersection_sets_for_cluster_c.
  
      References
      ----------
      Fixing Max-Product: Convergent Message-Passing Algorithms for MAP LP Relaxations
      by Amir Globerson and Tommi Jaakkola.
      Section 6, Page: 5; Beyond pairwise potentials: Generalized MPLP
      Later Modified by Sontag in "Introduction to Dual decomposition for Inference" Pg: 7 & 17
      """
  
      # The new updates will take place for the intersection_sets of this cluster.
      # The new updates are:
      # \delta_{f \rightarrow i}(x_i) = - \delta_i^{-f} +
      # 1/{\| f \|} max_{x_{f-i}}\left[{\theta_f(x_f) + \sum_{i' in f}{\delta_{i'}^{-f}}(x_i')} \right ]
  
      # Step. 1) Calculate {\theta_f(x_f) + \sum_{i' in f}{\delta_{i'}^{-f}}(x_i')}
      objective_cluster = self.objective[
          sending_cluster.cluster_variables
      ]
      for (
          current_intersect
      ) in sending_cluster.intersection_sets_for_cluster_c:
          objective_cluster += self.objective[current_intersect]
  
      updated_results = []
      objective = []
      for (
          current_intersect
      ) in sending_cluster.intersection_sets_for_cluster_c:
          # Step. 2) Maximize step.1 result wrt variables present in the cluster but not in the current intersect.
          phi = objective_cluster.maximize(
              list(
                  sending_cluster.cluster_variables - current_intersect
              ),
              inplace=False,
          )
  
          # Step. 3) Multiply 1/{\| f \|}
          intersection_length = len(
              sending_cluster.intersection_sets_for_cluster_c
          )
          phi *= 1 / intersection_length
          objective.append(phi)
  
          # Step. 4) Subtract \delta_i^{-f}
          # These are the messages not emanating from the sending cluster but going into the current intersect.
          # which is = Objective[current_intersect_node] - messages from the cluster to the current intersect node.
          updated_results.append(
              phi
              + -1
              * (
                  self.objective[current_intersect]
                  + -1
                  * sending_cluster.message_from_cluster[
                      current_intersect
                  ]
              )
          )
  
      # This loop is primarily for simultaneous updating:
      # 1. This cluster's message to each of the intersects.
      # 2. The value of the Objective for intersection_nodes.
      index = -1
      cluster_potential = copy.deepcopy(
          sending_cluster.cluster_potential
      )
      for (
          current_intersect
      ) in sending_cluster.intersection_sets_for_cluster_c:
          index += 1
          sending_cluster.message_from_cluster[
              current_intersect
          ] = updated_results[index]
          self.objective[current_intersect] = objective[index]
          cluster_potential += (-1) * updated_results[index]
  
      # Here we update the Objective for the current factor.
      self.objective[
          sending_cluster.cluster_variables
      ] = cluster_potential
  
imports: import copy
  import itertools
  import networkx
  import numpy
  from pgmpy.factors.discrete import DiscreteFactor
  from pgmpy.inference import Inference
  from pgmpy.models import MarkovNetwork
label_desc: Use image node_id 2 for calling the Mplp obj's underlying member method code with example usage: obj._update_message(sending_cluster) without return types