Bayesian Networks
Bayesian Networks
6. Exact Inference / Examples
Lars Schmidt-Thieme
Information Systems and Machine Learning Lab (ISMLL)Institute for Business Economics and Information Systems
& Institute for Computer ScienceUniversity of Hildesheim
http://www.ismll.uni-hildesheim.de
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 1/33
Bayesian Networks
1. Studfarm
2. Hailfinder
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 1/33
Bayesian Networks / 1. Studfarm
Studfarm / problem
K A B C L
F D E G
H I
J
Figure 1: Studfarm bayesian network.potentials:
(i) p(X) for X = A, B, C, K, L:p(X = aa) = 0.99, p(X = aA) = 0.01
(ii) p(X|Y, Z) for (X|Y, Z) = (D|A, B),(E|B, C), (F |A, K), (G|A, L),(H|F, D), (I|E,G):
father Y aa aAmother Z aa aA aa aA
aa 1 .5 .5 13
aA 0 .5 .5 23
(iii) and p(J |H, I):
father H aa aAmother I aa aA aa aA
aa 1 .5 .5 .25aA 0 .5 .5 .5AA 0 0 0 .25
Evidence is given, that John is sick, rep-resented by p(J):
p(J = AA) = 1
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 1/33
Bayesian Networks / 1. Studfarm
Studfarm / markov network
K A B C L
F D E G
H I
J
Figure 2: Studfarm bayesian network.
.
K A B C L
F D E G
H I
J
Figure 3: Studfarm markov network (moralgraph).
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 2/33
Bayesian Networks / 1. Studfarm
Studfarm / triangulation & cluster tree (MCS)
.
K A B C L
F D E G
H I
J
Figure 4: Triangulation of Studfarm markov net-work by MCS (fill-in 7).
A, G, L
A, E, G, IA, B, E, I
B, C, E
A, B, H, I, J
A, B, D, H
A, D, F, H
A, F, K
Figure 5: Cluster tree for the triangulation at theleft (total state space size 136).
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 3/33
Bayesian Networks / 1. Studfarm
Studfarm / triangulation & cluster tree (Minimal degree)
.
K A B C L
F D E G
H I
J
Figure 6: Triangulation of Studfarm markov net-work by Minimal Degree Heuristics (fill-in 5).
B, C, E
A, B, D, E
A, D, E, I
A, G, L
A, E, G, I
A, F, K
A, D, F, I
H, I, JD, F, H, I
Figure 7: Cluster tree for the triangulation at theleft (total state space size 116).
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 4/33
Bayesian Networks / 1. Studfarm
Studfarm / attach potentials
(i) p(A), p(B), p(C), p(K), p(L),
(ii) p(D|A, B), p(E|B, C), p(F |A, K),p(G|A, L), p(H|F, D), p(I|E,G),
(iii) and p(J |H, I),
(iv) and evidence p(J).
K A B C L
F D E G
H I
J
Figure 8: Studfarm bayesian network.AFK
ADFIDFHIHIJ
ADEI
AEGI
AGL
ABDE
BCE
p(A), p(K), p(F |A, K)
p(H|D, F )p(J |H, I), p(J)
p(I|E, G)
p(L), p(G|A, L)
p(D|A, B)
p(B), p(C), p(E|B, C)
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 5/33
Bayesian Networks / 1. Studfarm
Studfarm / collect evidence (1/8)
collect evidence:
qBCE→ABDE := (p(B) · p(C) · p(E|B, C))↓BE =E pure carrier
B = pure 0.985 0.005carrier 0.005 0.005
AFK
ADFIDFHIHIJ
ADEI
AEGI
AGL
ABDE
BCE
p(A), p(K), p(F |A, K)
p(H|D, F )p(J |H, I), p(J)
p(I|E, G)
p(L), p(G|A, L)
p(D|A, B)
p(B), p(C), p(E|B, C)
1
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 6/33
Bayesian Networks / 1. Studfarm
Studfarm / collect evidence (2/8)
qABDE→ADEI := (qBCE→ABDE · p(D|A, B))↓ADE
=
D pure carrierE pure carrier pure carrier
A = pure 0.9875 0.0075 0.0025 0.0025carrier 0.4942 0.0041 0.4958 0.0058
AFK
ADFIDFHIHIJ
ADEI
AEGI
AGL
ABDE
BCE
p(A), p(K), p(F |A, K)
p(H|D, F )p(J |H, I), p(J)
p(I|E, G)
p(L), p(G|A, L)
p(D|A, B)
p(B), p(C), p(E|B, C)
1
2
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 7/33
Bayesian Networks / 1. Studfarm
Studfarm / collect evidence (3/8)
qAGL→AEGI := (p(C) · p(G|A, L))↓AG =G pure carrier
A = pure 0.995 0.005carrier 0.4983 0.5017
AFK
ADFIDFHIHIJ
ADEI
AEGI
AGL
ABDE
BCE
p(A), p(K), p(F |A, K)
p(H|D, F )p(J |H, I), p(J)
p(I|E, G)
p(L), p(G|A, L)
p(D|A, B)
p(B), p(C), p(E|B, C)
1
2
3
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 8/33
Bayesian Networks / 1. Studfarm
Studfarm / collect evidence (4/8)
qAEGI→ADEI :=(qAGL→AEGI · p(I|E,G))↓AEI
=
E pure carrierI pure carrier
A = pure 0.9975 0.0025 0.4992 0.5008carrier 0.7492 0.2508 0.4164 0.5836
AFK
ADFIDFHIHIJ
ADEI
AEGI
AGL
ABDE
BCE
p(A), p(K), p(F |A, K)
p(H|D, F )p(J |H, I), p(J)
p(I|E, G)
p(L), p(G|A, L)
p(D|A, B)
p(B), p(C), p(E|B, C)
1
2
3
4
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 9/33
Bayesian Networks / 1. Studfarm
Studfarm / collect evidence (5/8)
qAFK→ADFI := (p(A) · p(K) · p(F |A, K))↓AF =F pure carrier
A = pure 0.985 0.005carrier 0.005 0.005
AFK
ADFIDFHIHIJ
ADEI
AEGI
AGL
ABDE
BCE
p(A), p(K), p(F |A, K)
p(H|D, F )p(J |H, I), p(J)
p(I|E, G)
p(L), p(G|A, L)
p(D|A, B)
p(B), p(C), p(E|B, C)
1
2
3
4
5
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 10/33
Bayesian Networks / 1. Studfarm
Studfarm / collect evidence (6/8)
qHIJ→DFHI := (p(J |H, I) · p(J))↓HI =I pure carrier
H = pure 0 0carrier 0 0.25
AFK
ADFIDFHIHIJ
ADEI
AEGI
AGL
ABDE
BCE
p(A), p(K), p(F |A, K)
p(H|D, F )p(J |H, I), p(J)
p(I|E, G)
p(L), p(G|A, L)
p(D|A, B)
p(B), p(C), p(E|B, C)
1
2
3
4
5
6
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 11/33
Bayesian Networks / 1. Studfarm
Studfarm / collect evidence (7/8)
qDFHI→ADFI := (qHIJ→DFHI · p(H|D, F ))↓DFI =
F pure carrierI pure carrier
D = pure 0 0 0 0.125carrier 0 0.125 0 0.1667
AFK
ADFIDFHIHIJ
ADEI
AEGI
AGL
ABDE
BCE
p(A), p(K), p(F |A, K)
p(H|D, F )p(J |H, I), p(J)
p(I|E, G)
p(L), p(G|A, L)
p(D|A, B)
p(B), p(C), p(E|B, C)
1
2
3
4
5
6 7
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 12/33
Bayesian Networks / 1. Studfarm
Studfarm / collect evidence (8/8)
qADFI→ADEI := (qDFHI→ADFI · qAFK→ADFI)↓ADI =
D pure carrierI pure carrier
A = pure 0 0.0006 0 0.124carrier 0 0.0006 0 0.0015
AFK
ADFIDFHIHIJ
ADEI
AEGI
AGL
ABDE
BCE
p(A), p(K), p(F |A, K)
p(H|D, F )p(J |H, I), p(J)
p(I|E, G)
p(L), p(G|A, L)
p(D|A, B)
p(B), p(C), p(E|B, C)
1
2
3
4
5
6 7
8
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 13/33
Bayesian Networks / 1. Studfarm
Studfarm / distribute evidence (1/8)distribute evidence:
qADEI→ADFI :=(qABDE→ADEI · qAEGI→ADEI)↓ADI
=
D pure carrierI pure carrier
A = pure 0.9888 0.0062 0.0037 0.0013carrier 0.372 0.1264 0.3739 0.1278
AFK
ADFIDFHIHIJ
ADEI
AEGI
AGL
ABDE
BCE
p(A), p(K), p(F |A, K)
p(H|D, F )p(J |H, I), p(J)
p(I|E, G)
p(L), p(G|A, L)
p(D|A, B)
p(B), p(C), p(E|B, C)
1
2
3
4
5
6 7
8
1
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 14/33
Bayesian Networks / 1. Studfarm
Studfarm / distribute evidence (2/8)
qADEI→AEGI :=(qABDE→ADEI · qADFI→ADEI)↓AEI
=
E pure carrierI pure carrier
A = pure 0 0.0009 0 0.0003carrier 0 0.001 0 0
AFK
ADFIDFHIHIJ
ADEI
AEGI
AGL
ABDE
BCE
p(A), p(K), p(F |A, K)
p(H|D, F )p(J |H, I), p(J)
p(I|E, G)
p(L), p(G|A, L)
p(D|A, B)
p(B), p(C), p(E|B, C)
1
2
3
4
5
6 7
8
1
2
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 15/33
Bayesian Networks / 1. Studfarm
Studfarm / distribute evidence (3/8)
qADEI→ABDE :=(qAEGI→ADEI · qADFI→ADEI)↓ADE
=
D pure carrierE pure carrier
A = pure 0 0.0003 0.0003 0.0621carrier 0.0002 0.0004 0.0004 0.0009
AFK
ADFIDFHIHIJ
ADEI
AEGI
AGL
ABDE
BCE
p(A), p(K), p(F |A, K)
p(H|D, F )p(J |H, I), p(J)
p(I|E, G)
p(L), p(G|A, L)
p(D|A, B)
p(B), p(C), p(E|B, C)
1
2
3
4
5
6 7
8
1
2
3
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 16/33
Bayesian Networks / 1. Studfarm
Studfarm / distribute evidence (4/8)
qADFI→DFHI :=(qADEI→ADFI · qAFK→ADFI)↓DFI
=
F pure carrierI pure carrier
D = pure 0.9759 0.0067 0.0068 0.0007carrier 0.0055 0.0019 0.0019 0.0006
AFK
ADFIDFHIHIJ
ADEI
AEGI
AGL
ABDE
BCE
p(A), p(K), p(F |A, K)
p(H|D, F )p(J |H, I), p(J)
p(I|E, G)
p(L), p(G|A, L)
p(D|A, B)
p(B), p(C), p(E|B, C)
1
2
3
4
5
6 7
8
1
2
3
4
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 17/33
Bayesian Networks / 1. Studfarm
Studfarm / distribute evidence (5/8)
qADFI→AFK := (qADEI→ADFI · qDFHI→ADFI)↓AF =
F pure carrierA = pure 0.0002 0.001
carrier 0.016 0.0371
AFK
ADFIDFHIHIJ
ADEI
AEGI
AGL
ABDE
BCE
p(A), p(K), p(F |A, K)
p(H|D, F )p(J |H, I), p(J)
p(I|E, G)
p(L), p(G|A, L)
p(D|A, B)
p(B), p(C), p(E|B, C)
1
2
3
4
5
6 7
8
1
2
3
4
5
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 18/33
Bayesian Networks / 1. Studfarm
Studfarm / distribute evidence (6/8)
qDFHI→HIJ := (qADFI→DFHI · p(H|D, F ))↓HI =I pure carrier
H = pure 0.9827 0.0082carrier 0.0074 0.0017
AFK
ADFIDFHIHIJ
ADEI
AEGI
AGL
ABDE
BCE
p(A), p(K), p(F |A, K)
p(H|D, F )p(J |H, I), p(J)
p(I|E, G)
p(L), p(G|A, L)
p(D|A, B)
p(B), p(C), p(E|B, C)
1
2
3
4
5
6 7
8
1
2
3
4
5
6
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 19/33
Bayesian Networks / 1. Studfarm
Studfarm / distribute evidence (7/8)
qAEGI→AGL := (qADEI→AEGI · p(I|E,G))↓AG =G pure carrier
A = pure 0.0002 0.0007carrier 0 0.0005
AFK
ADFIDFHIHIJ
ADEI
AEGI
AGL
ABDE
BCE
p(A), p(K), p(F |A, K)
p(H|D, F )p(J |H, I), p(J)
p(I|E, G)
p(L), p(G|A, L)
p(D|A, B)
p(B), p(C), p(E|B, C)
1
2
3
4
5
6 7
8
1
2
3
4
5
6
7
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 20/33
Bayesian Networks / 1. Studfarm
Studfarm / distribute evidence (8/8)
qABDE→BCE := (qADEI→ABDE · p(D|A, B))↓BE =E pure carrier
B = pure 0.0003 0.0009carrier 0.0005 0.0319
AFK
ADFIDFHIHIJ
ADEI
AEGI
AGL
ABDE
BCE
p(A), p(K), p(F |A, K)
p(H|D, F )p(J |H, I), p(J)
p(I|E, G)
p(L), p(G|A, L)
p(D|A, B)
p(B), p(C), p(E|B, C)
1
2
3
4
5
6 7
8
1
2
3
4
5
6
7
8
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 21/33
Bayesian Networks / 1. Studfarm
Studfarm / marginalize to target domains (1/5)marginalize to target domains:
pe(AFK) := qADFI→AFK · p(A) · p(K) · p(F |A, K)
pe(A) := pe(AFK)↓A =A = pure 0.3764
carrier 0.6236
pe(F ) := AFK↓F =F = pure 0.5517
carrier 0.4483
pe(K) := AFK↓K =K = pure 0.9797
carrier 0.0203
AFK
ADFIDFHIHIJ
ADEI
AEGI
AGL
ABDE
BCE
p(A), p(K), p(F |A, K)
p(H|D, F )p(J |H, I), p(J)
p(I|E, G)
p(L), p(G|A, L)
p(D|A, B)
p(B), p(C), p(E|B, C)
1
2
3
4
5
6 7
8
1
2
3
4
5
6
7
8
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 22/33
Bayesian Networks / 1. Studfarm
Studfarm / marginalize to target domains (2/5)
pe(AGL) := qAEGI→AGL · p(L) · p(G|A, L)
pe(G) := pe(AGL)↓G =G = pure 0.375
carrier 0.625
pe(L) := pe(AGL)↓L =L = pure 0.982
carrier 0.018
AFK
ADFIDFHIHIJ
ADEI
AEGI
AGL
ABDE
BCE
p(A), p(K), p(F |A, K)
p(H|D, F )p(J |H, I), p(J)
p(I|E, G)
p(L), p(G|A, L)
p(D|A, B)
p(B), p(C), p(E|B, C)
1
2
3
4
5
6 7
8
1
2
3
4
5
6
7
8
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 23/33
Bayesian Networks / 1. Studfarm
Studfarm / marginalize to target domains (3/5)
pe(BCE) := qABDE→BCE · p(B) · p(C) · p(E|B, C)
pe(B) := pe(BCE)↓B =B = pure 0.6192
carrier 0.3808
pe(C) := pe(BCE)↓C =C = pure 0.9812
carrier 0.0188
pe(E) := pe(BCE)↓E =E = pure 0.6138
carrier 0.3862
AFK
ADFIDFHIHIJ
ADEI
AEGI
AGL
ABDE
BCE
p(A), p(K), p(F |A, K)
p(H|D, F )p(J |H, I), p(J)
p(I|E, G)
p(L), p(G|A, L)
p(D|A, B)
p(B), p(C), p(E|B, C)
1
2
3
4
5
6 7
8
1
2
3
4
5
6
7
8
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 24/33
Bayesian Networks / 1. Studfarm
Studfarm / marginalize to target domains (4/5)
pe(HIJ) := qDFHI→HIJ · p(J |H, I) · p(J)
pe(H) := pe(HIJ)↓H =H = pure 0
carrier 1
pe(I) := pe(HIJ)↓I =I = pure 0
carrier 1
pe(J) := pe(HIJ)↓J =J = pure 0
carrier 0sick 1
AFK
ADFIDFHIHIJ
ADEI
AEGI
AGL
ABDE
BCE
p(A), p(K), p(F |A, K)
p(H|D, F )p(J |H, I), p(J)
p(I|E, G)
p(L), p(G|A, L)
p(D|A, B)
p(B), p(C), p(E|B, C)
1
2
3
4
5
6 7
8
1
2
3
4
5
6
7
8
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 25/33
Bayesian Networks / 1. Studfarm
Studfarm / marginalize to target domains (5/5)
pe(ADEI) := qABDE→ADEI · qAEGI→ADEI · qADFI→ADEI
pe(D) := pe(ADEI)↓D =D = pure 0.195
carrier 0.805
AFK
ADFIDFHIHIJ
ADEI
AEGI
AGL
ABDE
BCE
p(A), p(K), p(F |A, K)
p(H|D, F )p(J |H, I), p(J)
p(I|E, G)
p(L), p(G|A, L)
p(D|A, B)
p(B), p(C), p(E|B, C)
1
2
3
4
5
6 7
8
1
2
3
4
5
6
7
8
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 26/33
Bayesian Networks / 1. Studfarm
Studfarm / all single-variable marginals
2.2 Determining the Conditional Probabilities 49
K
JohnSie 0.04Cai 0.881Pur 99.081
FIGDRE 2.14. The stud farm model with initial probabilities.
K
JohnSie 100.00CaiPur
FIGDRE 2.15. Stud farm probabilities given that John is siek.Figure 9: Probabilities given evidence that John is sick (AA). [Jen01, p. 49]
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 27/33
Bayesian Networks / 1. Studfarm
Studfarm / higher marginals
Say, we are interested not only in single-variable marginals, but in joint marginalsof several variables, e.g., of A and B.
We proceed as follows:
(i) find a vertex potential that containsA and B,
(ii) compute that vertex potential fromthe link potentials,
(iii) marginalize that vertex potentialdown to the target domain {A, B}.
pe(A, B) =B pure carrier
A = pure 0.00718 0.36919carrier 0.61205 0.01159
If A and B are not conditionally indepen-dent given the evidence (here: J), thentheir joint potential is not the same asthe product of pe(A) and pe(B), e.g.,
pe(A) · pe(B) =A = pure 0.37636
carrier 0.62364
· B = pure 0.61923carrier 0.38077
=B pure carrier
A = pure 0.23305 0.14331carrier 0.38617 0.23746
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 28/33
Bayesian Networks
1. Studfarm
2. Hailfinder
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 29/33
Bayesian Networks / 2. Hailfinder
Hailfinder
(0,0)
Scenario
ScnRelPlFcst
SfcWndShfDis RHRatioScenRelAMIns WindFieldPln TempDis SynForcng MeanRH LowLLapse
ScenRel3_4
WindFieldMt WindAloft
AMInsWliScen
InsSclInScen
PlainsFcstMountainFcst
CurPropConv
N34StarFcst
LatestCINLLIW
Date
R5Fcst
AMDewptCalPl
(1,0)
MvmtFeatures MidLLapse ScenRelAMCIN DewpointsWindAloft
InsChange AMCINInScen
CapInScen
CapChange
CompPlFcst
AreaMoDryAir
CldShadeOth
InsInMt
AreaMeso_ALS
CombClouds
MorningCIN
CldShadeConv OutflowFrMt
WndHodograph
Boundaries
CombMoisture
LoLevMoistAd
MorningBound
AMInstabMt
CombVerMo SatContMoistRaoContMoist
LIfr12ZDENSd
VISCloudCov IRCloudCover
N0_7muVerMo SubjVertMoQGVertMotion
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 29/33
Bayesian Networks / 2. Hailfinder
Hailfinder
Hailfinder has
(i) 56 variables with
(ii) 2 – 11 states: 2 × 2, 25 × 3, 20 × 4,2× 5, 3× 6, 2× 7 and 2× 11
(iii) i.e., the hailfinder JPD has a totalstate space of size
119.307.129.289.316.404.700.753.638.195.200
(iv) the clique tree found by MCS has atotal state space size of 23050, thatfound by the minimal degree heuris-tics a total state space size of 9976.
(0,0)
Scenario
ScnRelPlFcst
SfcWndShfDis RHRatioScenRelAMIns WindFieldPln TempDis SynForcng MeanRH LowLLapse
ScenRel3_4
WindFieldMt WindAloft
AMInsWliScen
InsSclInScen
PlainsFcstMountainFcst
CurPropConv
N34StarFcst
LatestCINLLIW
Date
R5Fcst
AMDewptCalPl
(1,0)
MvmtFeatures MidLLapse ScenRelAMCIN DewpointsWindAloft
InsChange AMCINInScen
CapInScen
CapChange
CompPlFcst
AreaMoDryAir
CldShadeOth
InsInMt
AreaMeso_ALS
CombClouds
MorningCIN
CldShadeConv OutflowFrMt
WndHodograph
Boundaries
CombMoisture
LoLevMoistAd
MorningBound
AMInstabMt
CombVerMo SatContMoistRaoContMoist
LIfr12ZDENSd
VISCloudCov IRCloudCover
N0_7muVerMo SubjVertMoQGVertMotion
Figure 10: Hailfinder bayesian network (detailview).
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 30/33
Bayesian Networks / 2. Hailfinder
Hailfinder / clique tree (minimal degree)
[16, 45]
[14, 20, 36, 45]
[8, 14, 21, 36, 45]
[17, 45]
[45, 54]
[45, 53]
[45, 52]
[26, 45]
[45, 50]
[45, 49]
[27, 45]
[45, 47]
[28, 45]
[32, 45]
[39, 45]
[3, 10, 20]
[6, 9, 10, 14, 20]
[11, 18, 51]
[4, 5, 10, 11]
[4, 6, 9, 10, 14]
[12, 40, 41]
[4, 5, 12]
[31, 34, 38][20, 31, 34]
[20, 34, 36, 45]
[15, 23, 24]
[8, 15, 21, 36, 46][8, 21, 36, 45, 46]
[14, 19, 25]
[2, 14, 19, 21]
[2, 14, 21, 45]
[0, 30, 43][0, 43, 45]
[0, 8, 14, 45]
[1, 2, 22, 44]
[2, 44, 45]
[34, 36, 42, 45][13, 33, 37, 48]
[4, 13]
[6, 29, 35, 55] [6, 20, 35, 55]
[6, 9, 20, 55]
[0, 7, 8, 14]
Figure 11: Clique tree of Hailfinder bayesian network.Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 31/33
Bayesian Networks / 2. Hailfinder
Hailfinder / runtime
A full propagation of the Hailfinder net-work (including building the clique tree)takes
(i) 4s for BNJ (Lauritzen-Spiegelhalter),
(ii) 1.2s for cgnm-bn (Shafer-Shenoy;still not optimized)
on a "standard notebook".
Hailfinder is small compared to manyother published examples from the liter-ature as
(i) the Diabetes network with 413 vari-ables,
(ii) the Link network with 724 variables,or
(iii) the Munin4 network with 1041 vari-ables.
(0,0)
Scenario
ScnRelPlFcst
SfcWndShfDis RHRatioScenRelAMIns WindFieldPln TempDis SynForcng MeanRH LowLLapse
ScenRel3_4
WindFieldMt WindAloft
AMInsWliScen
InsSclInScen
PlainsFcstMountainFcst
CurPropConv
N34StarFcst
LatestCINLLIW
Date
R5Fcst
AMDewptCalPl
(1,0)
MvmtFeatures MidLLapse ScenRelAMCIN DewpointsWindAloft
InsChange AMCINInScen
CapInScen
CapChange
CompPlFcst
AreaMoDryAir
CldShadeOth
InsInMt
AreaMeso_ALS
CombClouds
MorningCIN
CldShadeConv OutflowFrMt
WndHodograph
Boundaries
CombMoisture
LoLevMoistAd
MorningBound
AMInstabMt
CombVerMo SatContMoistRaoContMoist
LIfr12ZDENSd
VISCloudCov IRCloudCover
N0_7muVerMo SubjVertMoQGVertMotion
Figure 12: Hailfinder bayesian network (detailview).
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 32/33
Bayesian Networks / 2. Hailfinder
References
[Jen01] Finn V. Jensen. Bayesian networks and decision graphs. Springer, New York, 2001.
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of HildesheimCourse on Bayesian Networks, summer term 2010 33/33