Credal Networks

In [1]:

import matplotlib.pyplot as plt

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

gnb.configuration()

Library	Version
OS	posix [darwin]
Python	3.14.6 (main, Jun 10 2026, 10:03:53) [Clang 21.0.0 (clang-2100.0.123.102)]
IPython	9.15.0
Matplotlib	3.11.0
Numpy	2.5.1
pyDot	4.0.1
pyAgrum	3.0.0

Thu Jul 16 18:15:08 2026 CEST

Credal Net from BN

In [2]:

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

Out[2]:

inference on Credal Net

In [3]:

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

Out[3]:

Bayes Net

	B
A	0	1	2
0	0.4917	0.2835	0.2248
1	0.1314	0.4469	0.4217

CPT

Credal Net

	B
A	0	1	2
0	0.4391	0.2310	0.1722
1	0.0788	0.3944	0.3691

CPTmin

	B
A	0	1	2
0	0.5443	0.3361	0.2773
1	0.1840	0.4995	0.4742

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)

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"])

Out[4]:

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 \(B\)

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)
)

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)

F
0	1
0.2231	0.5097

F
0	1
0.4903	0.7769

F
0	1
0.2231	0.5097

F
0	1
0.4903	0.7769


A:Range([0,1])
<> : [[0.238616 , 0.761384] , [0.556772 , 0.443228]]

B:Range([0,2])
<A:0> : [[0.439142 , 0.283518 , 0.277339] , [0.439142 , 0.336079 , 0.224778] , [0.491702 , 0.336079 , 0.172219] , [0.544262 , 0.283519 , 0.172219] , [0.491702 , 0.230958 , 0.277339] , [0.544262 , 0.230958 , 0.224779]]
<A:1> : [[0.078841 , 0.446938 , 0.474221] , [0.078841 , 0.499497 , 0.421662] , [0.131401 , 0.499497 , 0.369102] , [0.18396 , 0.446938 , 0.369102] , [0.1314 , 0.394379 , 0.474221] , [0.18396 , 0.394379 , 0.421661]]

C:Range([0,1])
<B:0|D:0> : [[0.334328 , 0.665672] , [0.518771 , 0.481229]]
<B:1|D:0> : [[0.332627 , 0.667373] , [0.517068 , 0.482932]]
<B:2|D:0> : [[0.677225 , 0.322775] , [0.861668 , 0.138332]]
<B:0|D:1> : [[0.249843 , 0.750157] , [0.434288 , 0.565712]]
<B:1|D:1> : [[0.311744 , 0.688256] , [0.496187 , 0.503813]]
<B:2|D:1> : [[0.560863 , 0.439137] , [0.745305 , 0.254695]]

D:Range([0,1])
<A:0> : [[0.305218 , 0.694782] , [0.461071 , 0.538929]]
<A:1> : [[0.727259 , 0.272741] , [0.883111 , 0.116889]]

E:Range([0,1])
<A:0> : [[0.17662 , 0.82338] , [0.412114 , 0.587886]]
<A:1> : [[0.415819 , 0.584181] , [0.651313 , 0.348687]]

F:Range([0,1])
<E:0> : [[0.555642 , 0.444358] , [0.678365 , 0.321635]]
<E:1> : [[0.0920422 , 0.907958] , [0.214765 , 0.785235]]

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_15_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]:

In [13]:

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

../_images/notebooks_24-Models_credalNetworks_19_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()

Inference with no evidence

The two inference give quite the same result

In [15]:

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_23_0.svg

The problem of evidence

When evidence are inserted, there are some divergence.

In [16]:

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)

../_images/notebooks_24-Models_credalNetworks_25_0.svg

Dynamical Credal Net

In [17]:

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

In [18]:

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

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

ie.makeInference()

In [19]:

print(ie.dynamicExpMax("temp"))

(14.203404648293022, 11.864395265918773, 12.173214728164902, 12.017742181033096, 12.004198572496797, 12.008328195599265, 12.007694250617146, 12.007688078235908, 12.00772208700393)

In [20]:

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

../_images/notebooks_24-Models_credalNetworks_30_0.svg

In [21]:

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

In [22]:

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

../_images/notebooks_24-Models_credalNetworks_32_0.svg

In [23]:

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

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

gnb.animApproximationScheme(ie)
ie.makeInference()

../_images/notebooks_24-Models_credalNetworks_33_0.svg

In [24]:

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

../_images/notebooks_24-Models_credalNetworks_34_0.svg

In [ ]: