Kullback-Leibler for Bayesian networks

import os

%matplotlib inline

from pylab import *
import matplotlib.pyplot as plt

import pyAgrum and pyAgrum.lib.notebook (for … notebooks :-) )

import pyAgrum as gum
import pyAgrum.lib.notebook as gnb

Create a first BN : bn

bn=gum.loadBN("res/asia.bif")
# randomly re-generate parameters for every Conditional Probability Table
bn.generateCPTs()
bn

Create a second BN : bn2

bn2=gum.loadBN("res/asia.bif")
bn2.generateCPTs()
bn2

bn vs bn2 : different parameters

gnb.flow.row(bn.cpt(3),bn2.cpt(3),
              captions=["a CPT in bn","same CPT in bn2 (with different parameters)"])

	positive_XraY
tuberculos_or_cancer	0	1
0	0.8859	0.1141
1	0.6475	0.3525

a CPT in bn

	positive_XraY
tuberculos_or_cancer	0	1
0	0.5543	0.4457
1	0.6152	0.3848

same CPT in bn2 (with different parameters)

Exact and (Gibbs) approximated KL-divergence

In order to compute KL-divergence, we just need to be sure that the 2 distributions are defined on the same domain (same variables, etc.)

Exact KL

g1=gum.ExactBNdistance(bn,bn2)
print(g1.compute())

{'klPQ': 7.443291682334482, 'errorPQ': 0, 'klQP': 4.9202095271714175, 'errorQP': 0, 'hellinger': 1.1555943226920256, 'bhattacharya': 1.1017144533663605, 'jensen-shannon': 0.7594743897090905}

If the models are not on the same domain :

bn_different_domain=gum.loadBN("res/alarm.dsl")

# g=gum.BruteForceKL(bn,bn_different_domain) # a KL-divergence between asia and alarm ... :(
#
# would cause
#---------------------------------------------------------------------------
#OperationNotAllowed                       Traceback (most recent call last)
#
#OperationNotAllowed: this operation is not allowed : KL : the 2 BNs are not compatible (not the same vars : visit_to_Asia?)

Gibbs-approximated KL

g=gum.GibbsBNdistance(bn,bn2)
g.setVerbosity(True)
g.setMaxTime(120)
g.setBurnIn(5000)
g.setEpsilon(1e-7)
g.setPeriodSize(500)

print(g.compute())
print("Computed in {0} s".format(g.currentTime()))

{'klPQ': 7.419965195684456, 'errorPQ': 0, 'klQP': 4.735746557327108, 'errorQP': 0, 'hellinger': 1.1462640194030809, 'bhattacharya': 1.0995583915974099, 'jensen-shannon': 0.747969168578439}
Computed in 1.2953540000000001 s

print("--")

print(g.messageApproximationScheme())
print("--")

print("Temps de calcul : {0}".format(g.currentTime()))
print("Nombre d'itérations : {0}".format(g.nbrIterations()))

--
stopped with epsilon=1e-07
--
Temps de calcul : 1.2953540000000001
Nombre d'itérations : 384000

p=plot(g.history(), 'g')

Animation of Gibbs KL

Since it may be difficult to know what happens during approximation algorithm, pyAgrum allows to follow the iteration using animated matplotlib figure

g=gum.GibbsBNdistance(bn,bn2)
g.setMaxTime(60)
g.setBurnIn(500)
g.setEpsilon(1e-7)
g.setPeriodSize(5000)

gnb.animApproximationScheme(g) # logarithmique scale for Y
g.compute()

{'klPQ': 7.438386018548863,
 'errorPQ': 0,
 'klQP': 4.93817439050127,
 'errorQP': 0,
 'hellinger': 1.1570436427400401,
 'bhattacharya': 1.095336026622168,
 'jensen-shannon': 0.7618873346309015}