# Quasi-continuous BN¶

In [1]:

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.

In [2]:

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

In [3]:

bn=gum.BayesNet("Quasi-Continuous")
print(bn)
gnb.showBN(bn)

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

In [4]:

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

In [5]:

# 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¶

In [6]:

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¶

In [7]:

c=bn.add(gum.RangeVariable("C","Another quasi continuous variable",0,NB-1))
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 $$P(C|B)$$).

In [8]:

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

In [9]:

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¶

In [10]:

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))

In [11]:

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


## Changing prior¶

In [12]:

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

• $$P(A | C=\frac{2*NB}{3})$$

• $$P(B | C=\frac{2*NB}{3})$$

In [13]:

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

Out[13]:

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

In [14]:

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

In [15]:

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).

In [16]:

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 $$P(A)$$¶

In [17]:

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 $$A=0$$.