Credal Networks

import matplotlib.pyplot as plt

import pyagrum as gum
import pyagrum.lib.notebook as gnb

gnb.configuration()

Library	Version
OS	posix [darwin]
Python	3.14.0 (main, Oct 7 2025, 09:34:52) [Clang 17.0.0 (clang-1700.3.19.1)]
IPython	9.6.0
Matplotlib	3.10.7
Numpy	2.3.4
pyDot	4.0.1
pyAgrum	2.3.0.9

Wed Oct 29 14:15:41 2025 CET

Credal Net from BN

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()
cn

inference on Credal Net

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

Bayes Net

	B
A	0	1	2
0	0.4546	0.3811	0.1643
1	0.1253	0.4896	0.3851

CPT

Credal Net

	B
A	0	1	2
0	0.4045	0.3309	0.1142
1	0.0752	0.4394	0.3350

CPTmin

	B
A	0	1	2
0	0.5048	0.4312	0.2144
1	0.1755	0.5397	0.4352

CPTmax

Binarization

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)

cn2 = gum.CredalNet(bn_min, bn_max)
cn2.intervalToCredal()
cn2.approximatedBinarization()
cn2.computeBinaryCPTMinMax()

gnb.flow.row(cn, cn2, captions=["Credal net", "Binarized credal net"])

Credal net

Binarized credal net

Here, $B$ becomes

$B$ -b $i$ : the $i$ -th bit of B
instrumental $B$ -v $k$ : the indicator variable for each modality $k$ of $B$

ie_mc = gum.CNMonteCarloSampling(cn)
ie2_lbp = gum.CNLoopyPropagation(cn2)
ie2_mc = gum.CNMonteCarloSampling(cn2)

gnb.sideBySide(
  gnb.getInference(cn, engine=ie_mc), gnb.getInference(cn2, engine=ie2_mc), gnb.getInference(cn2, engine=ie2_lbp)
)

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)

F
0	1
0.3642	0.3075

F
0	1
0.6925	0.6358

F
0	1
0.3642	0.3075

F
0	1
0.6925	0.6358

A:Range([0,1])
<> : [[0.564403 , 0.435597] , [0.813316 , 0.186684]]

B:Range([0,2])

<A:0> : [[0.404503 , 0.381073 , 0.214425] , [0.404503 , 0.431208 , 0.164289] , [0.454634 , 0.431208 , 0.114158] , [0.50477 , 0.381072 , 0.114158] , [0.454634 , 0.330941 , 0.214425] , [0.50477 , 0.330941 , 0.164289]] <A:1> : [[0.0752 , 0.489576 , 0.435224] , [0.0752 , 0.53971 , 0.38509] , [0.125334 , 0.53971 , 0.334956] , [0.175468 , 0.489575 , 0.334956] , [0.125335 , 0.439441 , 0.435224] , [0.175468 , 0.439441 , 0.385091]]

C:Range([0,1])

<B:0|D:0> : [[0.274878 , 0.725122] , [0.378155 , 0.621845]] <B:1|D:0> : [[0.394721 , 0.605279] , [0.497997 , 0.502003]] <B:2|D:0> : [[0.390519 , 0.609481] , [0.493797 , 0.506203]] <B:0|D:1> : [[0.373125 , 0.626875] , [0.476401 , 0.523599]] <B:1|D:1> : [[0.0774566 , 0.922543] , [0.180733 , 0.819267]] <B:2|D:1> : [[0.566948 , 0.433052] , [0.670225 , 0.329775]]

D:Range([0,1])

<A:0> : [[0.169429 , 0.830571] , [0.395333 , 0.604667]] <A:1> : [[0.192405 , 0.807595] , [0.41831 , 0.58169]]

E:Range([0,1])

<A:0> : [[0.487658 , 0.512342] , [0.594354 , 0.405646]] <A:1> : [[0.813281 , 0.186719] , [0.919977 , 0.0800234]]

F:Range([0,1])

<E:0> : [[0.494737 , 0.505263] , [0.768817 , 0.231183]] <E:1> : [[0.20556 , 0.79444] , [0.47964 , 0.52036]]

Credal Net from bif files

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

gnb.showCN(cn, "2")

svg

ie = gum.CNMonteCarloSampling(cn)
ie.insertEvidenceFile("res/cn/L2U.evi")

ie.setRepetitiveInd(False)
ie.setMaxTime(1)
ie.setMaxIter(1000)

ie.makeInference()

cn

gnb.showInference(cn, targets={"A", "H", "L", "D"}, engine=ie, evs={"L": [0, 1], "G": [1, 0]})

svg

Comparing inference in credal networks

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()

Inference with no evidence

The two inference give quite the same result

ie_mc = gum.CNMonteCarloSampling(cn)
ie_mc.makeInference()

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

showDiffInference(cn, ie_mc, ie_lbp)

svg

The problem of evidence

When evidence are inserted, there are some divergence.

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)

svg

Dynamical Credal Net

cn = gum.CredalNet("res/cn/bn_c_8.bif", "res/cn/den_c_8.bif")
cn.bnToCredal(0.8, False)

ie = gum.CNMonteCarloSampling(cn)
ie.insertModalsFile("res/cn/modalities.modal")

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

ie.makeInference()

print(ie.dynamicExpMax("temp"))

(14.20340464862347, 11.911090684366485, 12.0406461626149, 12.031555584857191, 12.003107180947513, 12.008870898650432, 12.007860641421736, 12.007682925808101, 12.007727248106775)

fig = plt.figure()
ax = fig.add_subplot(111)
ax.fill_between(range(9), ie.dynamicExpMax("temp"), ie.dynamicExpMin("temp"))
plt.show()

svg

ie = gum.CNMonteCarloSampling(cn)
ie.insertModalsFile("res/cn/modalities.modal")

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

ie.makeInference()
print(ie.messageApproximationScheme())

stopped with epsilon=0

fig = plt.figure()
ax = fig.add_subplot(111)
ax.fill_between(range(9), ie.dynamicExpMax("temp"), ie.dynamicExpMin("temp"))
plt.show()

svg

ie = gum.CNMonteCarloSampling(cn)
ie.insertModalsFile("res/cn/modalities.modal")

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

gnb.animApproximationScheme(ie)
ie.makeInference()

svg

fig = plt.figure()
ax = fig.add_subplot(111)
ax.fill_between(range(9), ie.dynamicExpMax("temp"), ie.dynamicExpMin("temp"))
plt.show()

svg