Quasi-continuous BN

from pylab import *
import matplotlib.pyplot as plt

aGrUM cannot (currently) deal with with continuous variables. However, a discrete variable with a large enough domain size is an approximation of such variables.

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

#nbr of states for quasi continuous variables. You can change the value
#but be careful of the quadratic behavior of both memory and time complexity
#in this example.
NB=300

bn=gum.BayesNet("Quasi-Continuous")
a=bn.add(gum.LabelizedVariable("A","A binary variable",2))
b=bn.add(gum.RangeVariable("B","A range variable",0,NB-1))
bn.addArc(a,b)
print(bn)
gnb.showBN(bn)

BN{nodes: 2, arcs: 1, domainSize: 600, dim: 602}

bn.cpt(a)[:]=[0.4, 0.6]
gnb.showProba(bn.cpt(a))

CPT for quasi-continuous variables (with parents)

Using python (and scipy), it is easy to find pdf for continuous variable

# we truncate a pdf, so we need to normalize
def normalize(rv,vmin,vmax,size):
    pdf=rv.pdf(linspace(vmin,vmax,size))
    return (pdf/sum(pdf))

from scipy.stats import norm,maxwell

plot(normalize(norm(),-4,1,NB)) # P(B|A=0) is a Gaussian distribution
plot(normalize(maxwell(),0.5,6,NB))  # P(B|A=1) is a Maxwell-Boltzmann distribution
title("P(B | A)")
legend(["P(B|A=0)","P(B|A=1)"],loc='best')

bn.cpt(b)[{'A':0}]=normalize(norm(),-4,1,NB)
bn.cpt(b)[{'A':1}]=normalize(maxwell(),0.5,6,NB)

Quasi =-continuous inference

ie=gum.LazyPropagation(bn)
ie.makeInference()
plot(ie.posterior(b)[:]) # the posterior for B
t=title("P(B)")

Quasi-continuous variable with quasi-continuous parent

c=bn.add(gum.RangeVariable("C","Another quasi continuous variable",0,NB-1))
bn.addArc(b,c)
gnb.showBN(bn) # B and C are quasi-continouous

Even if this BN is quite small (and linear), the size of nodes \(B\) et \(C\) are rather big and creates a complex model (NBxNB parameters in \(P(C|B)\)).

print("nombre de paramètres du bn : {0}".format(bn.dim()))
print("domaine du bn : 10^{0}".format(bn.log10DomainSize()))

nombre de paramètres du bn : 90299
domaine du bn : 10^5.2552725051033065

from scipy.stats import gamma
# cpt(c) is NB x NB matrix !
l=[]
for i in range(NB):
    # the size and the parameter of gamma depends on the parent value
    k=(i*30.0)/NB
    l.append(normalize(gamma(k+1),4,5+k,NB))

bn.cpt(c)[:]=l

#for instance
plot(bn.cpt(c)[{'B':0}])
plot(bn.cpt(c)[{'B':NB//4}])
plot(bn.cpt(c)[{'B':NB*2//3}])
plot(bn.cpt(c)[{'B':NB-1}])
title("gamma distributions for variable P(C | B=i)")
legend(["i=0","i={0}".format(NB/4),"i={0}".format(2*NB/3),"i={0}".format(NB-1)],
       loc='best');

Inference in quasi-continuous BN

import time

ts = time.time()
ie=gum.LazyPropagation(bn)
ie.makeInference()
te=time.time()

plot(ie.posterior(c)[:])
t=title("P(C) computed in {0:2.5f} sec".format(te-ts))

gnb.showPosterior(bn,target="C",evs={})

Changing prior

bn.cpt(a)[:]=[0.1,0.9]
ie=gum.LazyPropagation(bn)
ie.makeInference()
plot(ie.posterior(c)[:])
t=title("P(C) with prior p(A)=[0.1,0.9]")

gnb.showPosterior(bn,target="C",evs={})

inference with evidence in quasi-continuous BN

We want to compute

• \(P(A | C=\frac{2*NB}{3})\)

• \(P(B | C=\frac{2*NB}{3})\)

ie=gum.LazyPropagation(bn)
ie.setEvidence({'C':NB*2//3})
ie.makeInference()
plot(ie.posterior(b)[:])
title("P( B | C={0})".format(NB//3))

Text(0.5, 1.0, 'P( B | C=100)')

gnb.showPosterior(bn,target="B",evs={"C":NB*2//3})

gnb.showProba(ie.posterior(a))

Multiple inference : MAP DECISION between Gaussian and Maxwell-Boltzman distributions

What is the behaviour of \(P(A | B=i)\)   when \(i\) varies ? I.e. we perform a MAP decision between the two models (\(A=0\)  for the Gaussian distribution and \(A=1\)  for the Maxwell-Boltzman distribution).

bn.cpt(a)[:]=[0.1, 0.9]
ie=gum.LazyPropagation(bn)
p0=[]
p1=[]
for i in range(300):
    ie.setEvidence({'B':i})
    ie.makeInference()
    p0.append(ie.posterior(a)[0])
    p1.append(ie.posterior(a)[1])
plot(p0)
plot(p1)
title("P( A | B=i) with prior p(A)=[0.1,0.9]")
legend(["A=0","A=1"],loc='best')
inters=(transpose(p0)<transpose(p1)).argmin()
text(inters-2,p0[inters],
     "{0},{1:5.4f}  ".format(inters,p0[inters]),
     bbox=dict(facecolor='red', alpha=0.1),ha='right');

i.e. if \(i<164\) then \(A=1\) else \(A=0\)

Changing the prior \(P(A)\)

bn.cpt(a)[:]=[0.4, 0.6]
ie=gum.LazyPropagation(bn)
p0=[]
p1=[]
for i in range(300):
    ie.setEvidence({'B':i})
    ie.makeInference()
    p0.append(ie.posterior(a)[0])
    p1.append(ie.posterior(a)[1])
plot(p0)
plot(p1)
title("P( A | B=i) with prior p(A)=[0.4,0.6]")
legend(["A=0","A=1"],loc='best')
inters=(transpose(p0)<transpose(p1)).argmin()
text(inters+1.5,p0[inters],
     "{0},{1:5.4f}  ".format(inters,p0[inters]),
     bbox=dict(facecolor='red', alpha=0.1));

ie. with \(p(A)=[0.4,0.6]\), if \(i<140\) then \(A=1\) else \(A=0\).