# MPLP¶

class pgmpy.inference.mplp.Mplp(model)[source]

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=, values=np.array([0.54577, 1.8323]))
>>> factor_b = DiscreteFactor(['B'], cardinality=, values=np.array([0.93894, 1.065]))
>>> factor_c = DiscreteFactor(['C'], cardinality=, values=np.array([0.89205, 1.121]))
>>> factor_d = DiscreteFactor(['D'], cardinality=, values=np.array([0.56292, 1.7765]))
>>> factor_e = DiscreteFactor(['E'], cardinality=, values=np.array([0.47117, 2.1224]))
>>> factor_f = DiscreteFactor(['F'], cardinality=, 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)
```
class Cluster(intersection_set_variables, cluster_potential)[source]

Inner class for representing a cluster. A cluster is a subset of variables.

Parameters
• set_of_variables (tuple) – This is the set of variables that form the cluster.

• intersection_set_variables (set containing frozensets.) – collection of intersection of all pairs of cluster variables. For eg: {{C_1 cap C_2}, {C_2 cap C_3}, {C_3 cap C_1} } for clusters C_1, C_2 & C_3.

• cluster_potential (DiscreteFactor) – Each cluster has a initial probability distribution provided beforehand.

find_triangles()[source]

Finds all the triangles present in the given model

Examples

```>>> from pgmpy.models import MarkovNetwork
>>> from pgmpy.factors.discrete import DiscreteFactor
>>> from pgmpy.inference import Mplp
>>> mm = MarkovNetwork()
>>> mm.add_nodes_from(['x1', 'x2', 'x3', 'x4', 'x5', 'x6', 'x7'])
>>> mm.add_edges_from([('x1', 'x3'), ('x1', 'x4'), ('x2', 'x4'),
...                    ('x2', 'x5'), ('x3', 'x6'), ('x4', 'x6'),
...                    ('x4', 'x7'), ('x5', 'x7')])
>>> phi = [DiscreteFactor(edge, [2, 2], np.random.rand(4)) for edge in mm.edges()]
>>> mplp = Mplp(mm)
>>> mplp.find_triangles()
```
get_integrality_gap()[source]
Returns the integrality gap of the current state of the Mplp algorithm. The lesser it is, the closer we are

towards the exact solution.

Examples

```>>> from pgmpy.models import MarkovNetwork
>>> from pgmpy.factors.discrete import DiscreteFactor
>>> from pgmpy.inference import Mplp
>>> mm = MarkovNetwork()
>>> mm.add_nodes_from(['x1', 'x2', 'x3', 'x4', 'x5', 'x6', 'x7'])
>>> mm.add_edges_from([('x1', 'x3'), ('x1', 'x4'), ('x2', 'x4'),
...                    ('x2', 'x5'), ('x3', 'x6'), ('x4', 'x6'),
...                    ('x4', 'x7'), ('x5', 'x7')])
>>> phi = [DiscreteFactor(edge, [2, 2], np.random.rand(4)) for edge in mm.edges()]
>>> mplp = Mplp(mm)
>>> mplp.map_query()
>>> int_gap = mplp.get_integrality_gap()
```
map_query(init_iter=1000, later_iter=20, dual_threshold=0.0002, integrality_gap_threshold=0.0002, tighten_triplet=True, max_triplets=5, max_iterations=100, prolong=False)[source]

MAP query method using Max Product LP method. This returns the best assignment of the nodes in the form of a dictionary.

Parameters
• init_iter (integer) – Number of maximum iterations that we want MPLP to run for the first time.

• later_iter (integer) – Number of maximum iterations that we want MPLP to run for later iterations

• dual_threshold (double) – This sets the minimum width between the dual objective decrements. If the decrement is lesser than the threshold, then that means we have stuck on a local minima.

• integrality_gap_threshold (double) – This sets the threshold for the integrality gap below which we say that the solution is satisfactory.

• tighten_triplet (bool) – set whether to use triplets as clusters or not.

• max_triplets (integer) – Set the maximum number of triplets that can be added at once.

• max_iterations (integer) – Maximum number of times we tighten the relaxation. Used only when tighten_triplet is set True.

• prolong (bool) – If set False: The moment we exhaust of all the triplets the tightening stops. If set True: The tightening will be performed max_iterations number of times irrespective of the triplets.

References

Section 3.3: The Dual Algorithm; Tightening LP Relaxation for MAP using Message Passing (2008) By Sontag Et al.

Examples

```>>> from pgmpy.models import MarkovNetwork
>>> from pgmpy.factors.discrete import DiscreteFactor
>>> from pgmpy.inference import Mplp
>>> import numpy as np
>>> student = MarkovNetwork()
>>> student.add_edges_from([('A', 'B'), ('B', 'C'), ('C', 'D'), ('E', 'F')])
>>> factor_a = DiscreteFactor(['A'], cardinality=, values=np.array([0.54577, 1.8323]))
>>> factor_b = DiscreteFactor(['B'], cardinality=, values=np.array([0.93894, 1.065]))
>>> factor_c = DiscreteFactor(['C'], cardinality=, values=np.array([0.89205, 1.121]))
>>> factor_d = DiscreteFactor(['D'], cardinality=, values=np.array([0.56292, 1.7765]))
>>> factor_e = DiscreteFactor(['E'], cardinality=, values=np.array([0.47117, 2.1224]))
>>> factor_f = DiscreteFactor(['F'], cardinality=, 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, 0.0002418]))
>>> 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)
>>> result = mplp.map_query()
>>> result
{'B': 0.93894, 'C': 1.121, 'A': 1.8323, 'F': 1.5093, 'D': 1.7765, 'E': 2.12239}
```