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	3.14.7 (main, Aug 5 2026, 10:29:49) [Clang 21.0.0 (clang-2100.1.1.101)]
IPython	9.16.1
Matplotlib	3.11.1
Numpy	2.5.2
pyDot	4.0.1
pyAgrum	3.1.0

Tue Aug 18 12:00:01 2026 CEST

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	0.4995	0.0521	0.4483
1	0.4235	0.5590	0.0175

CPT

Credal Net

	B
A	0	1	2
0	0.4925	0.0451	0.4414
1	0.4165	0.5520	0.0105

CPTmin

	B
A	0	1	2
0	0.5065	0.0591	0.4553
1	0.4305	0.5660	0.0245

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 0.3661 0.3428	F 0 1 0.6572 0.6339
	F 0 1 0.3359 0.3414	F 0 1 0.6586 0.6641

A:Range([0,1])
<> : [[0.467684 , 0.532316] , [0.771865 , 0.228135]]

B:Range([0,2])

<A:0> : [[0.492528 , 0.052124 , 0.455348] , [0.492528 , 0.0591216 , 0.448351] , [0.499528 , 0.0591216 , 0.44135] , [0.506527 , 0.0521224 , 0.44135] , [0.499528 , 0.0451237 , 0.455348] , [0.506527 , 0.0451237 , 0.448349]] <A:1> : [[0.416541 , 0.558962 , 0.0244969] , [0.416541 , 0.56596 , 0.0174983] , [0.423541 , 0.56596 , 0.0104987] , [0.430541 , 0.55896 , 0.0104987] , [0.423542 , 0.551961 , 0.0244969] , [0.430541 , 0.551961 , 0.0174984]]

C:Range([0,1])

<B:0|D:0> : [[0.405181 , 0.594819] , [0.41299 , 0.58701]] <B:1|D:0> : [[0.798842 , 0.201158] , [0.806649 , 0.193351]] <B:2|D:0> : [[0.504783 , 0.495217] , [0.512591 , 0.487409]] <B:0|D:1> : [[0.568387 , 0.431613] , [0.576196 , 0.423804]] <B:1|D:1> : [[0.469866 , 0.530134] , [0.477674 , 0.522326]] <B:2|D:1> : [[0.00585652 , 0.994143] , [0.0136646 , 0.986335]]

D:Range([0,1])

<A:0> : [[0.291036 , 0.708964] , [0.379995 , 0.620005]] <A:1> : [[0.0667203 , 0.93328] , [0.15568 , 0.84432]]

E:Range([0,1])

<A:0> : [[0.94357 , 0.0564304] , [0.975816 , 0.0241838]] <A:1> : [[0.417605 , 0.582395] , [0.449851 , 0.550149]]

F:Range([0,1])

<E:0> : [[0.396617 , 0.603383] , [0.684601 , 0.315399]] <E:1> : [[0.215986 , 0.784014] , [0.503971 , 0.496029]]

Credal Net from bif files

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

gnb.showCN(cn, "2")

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]})

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)

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)

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"))

(14.203404648293022, 11.817699847864338, 12.10019505553209, 11.99476087981647, 11.966313382958862, 11.964867852223103, 11.965031829300205, 11.965013837826506, 11.965015808981818)

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

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()

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

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

gnb.animApproximationScheme(ie)
ie.makeInference()

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