Credal Networks

Creative Commons License

aGrUM

interactive online version

In [1]:
import os

%matplotlib inline
from pylab import *
import matplotlib.pyplot as plt

In [2]:
import pyAgrum as gum
import pyAgrum.lib.notebook as gnb
gnb.configuration()
LibraryVersion
OSposix [darwin]
Python3.11.1 (main, Dec 23 2022, 09:28:24) [Clang 14.0.0 (clang-1400.0.29.202)]
IPython8.6.0
Matplotlib3.6.1
Numpy1.23.4
pyDot1.4.2
pyAgrum1.5.2.9
Tue Feb 14 00:20:37 2023 CET

Credal Net from BN

In [3]:
bn=gum.fastBN("A->B[3]->C<-D<-A->E->F")
bn_min=gum.BayesNet(bn)
bn_max=gum.BayesNet(bn)
for n in bn.nodes():
  x=0.4*min(bn.cpt(n).min(),1-bn.cpt(n).max())
  bn_min.cpt(n).translate(-x)
  bn_max.cpt(n).translate(x)

cn=gum.CredalNet(bn_min,bn_max)
cn.intervalToCredal()

gnb.flow.row(bn,bn.cpt("B"),cn,bn_min.cpt("B"),bn_max.cpt("B"),captions=["Bayes Net","CPT","Credal Net","CPTmin","CPTmax"])

G F F C C B B B->C E E E->F D D D->C A A A->B A->E A->D
Bayes Net
B
A
0
1
2
0
0.64670.14750.2057
1
0.31660.10900.5744

CPT
G F F C C B B B->C E E E->F D D D->C A A A->B A->E A->D
Credal Net
B
A
0
1
2
0
0.60310.10400.1621
1
0.27300.06540.5308

CPTmin
B
A
0
1
2
0
0.69030.19110.2493
1
0.36020.15260.6180

CPTmax

We can use LBP on CN (L2U) only for binary credal networks (here B is not binary). We then propose the classical binarization (but warn the user that this leads to approximation in the inference)

In [4]:
cn2=gum.CredalNet(bn_min,bn_max)
cn2.intervalToCredal()
cn2.approximatedBinarization()
cn2.computeBinaryCPTMinMax()

gnb.flow.row(cn,cn2,captions=["Credal net","Binarized credal net"])
G F F C C B B B->C E E E->F D D D->C A A A->B A->E A->D
Credal net
G B-b0 B-b0 B-v1 B-v1 B-b0->B-v1 C C B-b0->C B-v2 B-v2 B-b0->B-v2 B-v0 B-v0 B-b0->B-v0 B-b1 B-b1 B-b0->B-b1 F F E E E->F D D D->C B-b1->B-v1 B-b1->C B-b1->B-v2 B-b1->B-v0 A A A->B-b0 A->E A->D A->B-b1
Binarized credal net

Here, \(B\) becomes - \(B\)-b:math:i : the \(i\)-th bit of B - instrumental \(B\)-v:math:k : the indicator variable for each modality \(k\) of \(B\)

In [5]:
ie_mc=gum.CNMonteCarloSampling(cn)
ie2_lbp=gum.CNLoopyPropagation(cn2)
ie2_mc=gum.CNMonteCarloSampling(cn2)
In [6]:
gnb.sideBySide(gnb.getInference(cn,engine=ie_mc),
               gnb.getInference(cn2,engine=ie2_mc),
               gnb.getInference(cn2,engine=ie2_lbp))
structs Inference in 889.82ms A 2023-02-14T00:20:44.404746 image/svg+xml Matplotlib v3.6.1, https://matplotlib.org/ B 2023-02-14T00:20:44.468789 image/svg+xml Matplotlib v3.6.1, https://matplotlib.org/ A->B D 2023-02-14T00:20:44.532885 image/svg+xml Matplotlib v3.6.1, https://matplotlib.org/ A->D E 2023-02-14T00:20:44.557680 image/svg+xml Matplotlib v3.6.1, https://matplotlib.org/ A->E C 2023-02-14T00:20:44.491716 image/svg+xml Matplotlib v3.6.1, https://matplotlib.org/ B->C D->C F 2023-02-14T00:20:44.597978 image/svg+xml Matplotlib v3.6.1, https://matplotlib.org/ E->F
structs Inference in 819.43ms A 2023-02-14T00:20:45.620678 image/svg+xml Matplotlib v3.6.1, https://matplotlib.org/ B-b0 2023-02-14T00:20:45.652653 image/svg+xml Matplotlib v3.6.1, https://matplotlib.org/ A->B-b0 B-b1 2023-02-14T00:20:45.671230 image/svg+xml Matplotlib v3.6.1, https://matplotlib.org/ A->B-b1 D 2023-02-14T00:20:45.712955 image/svg+xml Matplotlib v3.6.1, https://matplotlib.org/ A->D E 2023-02-14T00:20:45.742624 image/svg+xml Matplotlib v3.6.1, https://matplotlib.org/ A->E B-b0->B-b1 C 2023-02-14T00:20:45.689109 image/svg+xml Matplotlib v3.6.1, https://matplotlib.org/ B-b0->C B-v0 2023-02-14T00:20:45.796443 image/svg+xml Matplotlib v3.6.1, https://matplotlib.org/ B-b0->B-v0 B-v1 2023-02-14T00:20:45.827230 image/svg+xml Matplotlib v3.6.1, https://matplotlib.org/ B-b0->B-v1 B-v2 2023-02-14T00:20:45.890167 image/svg+xml Matplotlib v3.6.1, https://matplotlib.org/ B-b0->B-v2 B-b1->C B-b1->B-v0 B-b1->B-v1 B-b1->B-v2 D->C F 2023-02-14T00:20:45.772907 image/svg+xml Matplotlib v3.6.1, https://matplotlib.org/ E->F
structs Inference in   0.31ms A 2023-02-14T00:20:46.108327 image/svg+xml Matplotlib v3.6.1, https://matplotlib.org/ B-b0 2023-02-14T00:20:46.135966 image/svg+xml Matplotlib v3.6.1, https://matplotlib.org/ A->B-b0 B-b1 2023-02-14T00:20:46.166280 image/svg+xml Matplotlib v3.6.1, https://matplotlib.org/ A->B-b1 D 2023-02-14T00:20:46.211148 image/svg+xml Matplotlib v3.6.1, https://matplotlib.org/ A->D E 2023-02-14T00:20:46.231379 image/svg+xml Matplotlib v3.6.1, https://matplotlib.org/ A->E B-b0->B-b1 C 2023-02-14T00:20:46.189865 image/svg+xml Matplotlib v3.6.1, https://matplotlib.org/ B-b0->C B-v0 2023-02-14T00:20:46.284732 image/svg+xml Matplotlib v3.6.1, https://matplotlib.org/ B-b0->B-v0 B-v1 2023-02-14T00:20:46.361167 image/svg+xml Matplotlib v3.6.1, https://matplotlib.org/ B-b0->B-v1 B-v2 2023-02-14T00:20:46.380563 image/svg+xml Matplotlib v3.6.1, https://matplotlib.org/ B-b0->B-v2 B-b1->C B-b1->B-v0 B-b1->B-v1 B-b1->B-v2 D->C F 2023-02-14T00:20:46.260751 image/svg+xml Matplotlib v3.6.1, https://matplotlib.org/ E->F
In [7]:
gnb.sideBySide(ie_mc.CN(),ie_mc.marginalMin("F"),ie_mc.marginalMax("F"),
               ie_mc.CN(),ie2_lbp.marginalMin("F"),ie2_lbp.marginalMax("F"),
              ncols=3)
print(cn)
G F F C C B B B->C E E E->F D D D->C A A A->B A->E A->D
F
0
1
0.37290.2947
F
0
1
0.70530.6271
G F F C C B B B->C E E E->F D D D->C A A A->B A->E A->D
F
0
1
0.37290.2947
F
0
1
0.70530.6271

A:Range([0,1])
<> : [[0.0837589 , 0.916241] , [0.195435 , 0.804565]]

B:Range([0,2])
<A:0> : [[0.60315 , 0.147547 , 0.249304] , [0.60315 , 0.19114 , 0.20571] , [0.646745 , 0.19114 , 0.162115] , [0.690339 , 0.147547 , 0.162115] , [0.646744 , 0.103952 , 0.249304] , [0.690339 , 0.103952 , 0.20571]]
<A:1> : [[0.272995 , 0.108987 , 0.618018] , [0.272995 , 0.152582 , 0.574423] , [0.316589 , 0.152582 , 0.530829] , [0.360184 , 0.108987 , 0.530829] , [0.31659 , 0.0653924 , 0.618018] , [0.360184 , 0.0653924 , 0.574424]]

C:Range([0,1])
<B:0|D:0> : [[0.500982 , 0.499018] , [0.513663 , 0.486337]]
<B:1|D:0> : [[0.418886 , 0.581114] , [0.431566 , 0.568434]]
<B:2|D:0> : [[0.114657 , 0.885343] , [0.127336 , 0.872664]]
<B:0|D:1> : [[0.977811 , 0.0221893] , [0.990491 , 0.00950872]]
<B:1|D:1> : [[0.143169 , 0.856831] , [0.155849 , 0.844151]]
<B:2|D:1> : [[0.62924 , 0.37076] , [0.641919 , 0.358081]]

D:Range([0,1])
<A:0> : [[0.593434 , 0.406566] , [0.783228 , 0.216772]]
<A:1> : [[0.142344 , 0.857656] , [0.332139 , 0.667861]]

E:Range([0,1])
<A:0> : [[0.209976 , 0.790024] , [0.489943 , 0.510057]]
<A:1> : [[0.389381 , 0.610619] , [0.669348 , 0.330652]]

F:Range([0,1])
<E:0> : [[0.251754 , 0.748246] , [0.479112 , 0.520888]]
<E:1> : [[0.602119 , 0.397881] , [0.82948 , 0.17052]]


Credal Net from bif files

In [8]:
cn=gum.CredalNet("res/cn/2Umin.bif","res/cn/2Umax.bif")
cn.intervalToCredal()
In [9]:
gnb.showCN(cn,"2")
../_images/notebooks_24-Models_credalNetworks_14_0.svg
In [10]:
ie=gum.CNMonteCarloSampling(cn)
ie.insertEvidenceFile("res/cn/L2U.evi")
In [11]:
ie.setRepetitiveInd(False)
ie.setMaxTime(1)
ie.setMaxIter(1000)

ie.makeInference()
In [12]:
cn
Out[12]:
G H H L L H->L F F F->H G G C C C->F B B E E B->E E->H D D D->F D->G A A A->E
In [13]:
gnb.showInference(cn,targets={"A","H","L","D"},engine=ie,evs={"L":[0,1],"G":[1,0]})
../_images/notebooks_24-Models_credalNetworks_18_0.svg

Comparing inference in credal networks

In [14]:
import pyAgrum as gum

def showDiffInference(model,mc,lbp):
    for i in model.current_bn().nodes():
        a,b=mc.marginalMin(i)[:]
        c,d=mc.marginalMax(i)[:]

        e,f=lbp.marginalMin(i)[:]
        g,h=lbp.marginalMax(i)[:]

        plt.scatter([a,b,c,d],[e,f,g,h])


cn=gum.CredalNet("res/cn/2Umin.bif","res/cn/2Umax.bif")
cn.intervalToCredal()

The two inference give quite the same result

In [ ]:
ie_mc=gum.CNMonteCarloSampling(cn)
ie_mc.makeInference()

cn.computeBinaryCPTMinMax()
ie_lbp=gum.CNLoopyPropagation(cn)
ie_lbp.makeInference()

showDiffInference(cn,ie_mc,ie_lbp)
../_images/notebooks_24-Models_credalNetworks_22_0.svg

but not when evidence are inserted

In [ ]:
ie_mc=gum.CNMonteCarloSampling(cn)
ie_mc.insertEvidenceFile("res/cn/L2U.evi")
ie_mc.makeInference()

ie_lbp=gum.CNLoopyPropagation(cn)
ie_lbp.insertEvidenceFile("res/cn/L2U.evi")
ie_lbp.makeInference()

showDiffInference(cn,ie_mc,ie_lbp)

Dynamical Credal Net

In [ ]:
cn=gum.CredalNet("res/cn/bn_c_8.bif","res/cn/den_c_8.bif")
cn.bnToCredal(0.8,False)
In [ ]:
ie=gum.CNMonteCarloSampling(cn)
ie.insertModalsFile("res/cn/modalities.modal")

ie.setRepetitiveInd(True)
ie.setMaxTime(5)
ie.setMaxIter(1000)

ie.makeInference()
In [ ]:
print(ie.dynamicExpMax("temp"))
In [ ]:
fig=figure()
ax=fig.add_subplot(111)
ax.fill_between(range(9),ie.dynamicExpMax("temp"),ie.dynamicExpMin("temp"))
In [ ]:
ie=gum.CNMonteCarloSampling(cn)
ie.insertModalsFile("res/cn/modalities.modal")

ie.setRepetitiveInd(False)
ie.setMaxTime(5)
ie.setMaxIter(1000)

ie.makeInference()
print(ie.messageApproximationScheme())
In [ ]:
fig=figure()
ax=fig.add_subplot(111)
ax.fill_between(range(9),ie.dynamicExpMax("temp"),ie.dynamicExpMin("temp"))
In [ ]:
ie=gum.CNMonteCarloSampling(cn)
ie.insertModalsFile("res/cn/modalities.modal")

ie.setRepetitiveInd(False)
ie.setMaxTime(5)
ie.setMaxIter(5000)

gnb.animApproximationScheme(ie)
ie.makeInference()
In [ ]:
fig=figure()
ax=fig.add_subplot(111)
ax.fill_between(range(9),ie.dynamicExpMax("temp"),ie.dynamicExpMin("temp"));
In [ ]: