Kullback-Leibler for Bayesian networks

%matplotlib inline

from pylab import *

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	0.6700	0.3300
1	0.4316	0.5684

a CPT in bn

	positive_XraY
tuberculos_or_cancer	0	1
0	0.2943	0.7057
1	0.4315	0.5685

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': 2.476584381649645, 'errorPQ': 0, 'klQP': 2.244520928404808, 'errorQP': 0, 'hellinger': 0.813592705605187, 'bhattacharya': 0.4019212130892461, 'jensen-shannon': 0.4136335100698562}

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': 2.475361213496724, 'errorPQ': 0, 'klQP': 2.1957241806099814, 'errorQP': 0, 'hellinger': 0.8105538873770256, 'bhattacharya': 0.3989042244816927, 'jensen-shannon': 0.411228001161747}
Computed in 1.338172 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 : 1.338172
Nombre d'itérations : 380500

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': 2.469542051480538,
 'errorPQ': 0,
 'klQP': 2.1146518638711105,
 'errorQP': 0,
 'hellinger': 0.8035301805274394,
 'bhattacharya': 0.404240337375901,
 'jensen-shannon': 0.40444581031110494}