machine intelligence thomas d. nielsen september...
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Bayesian networks: basicsMachine Intelligence
Thomas D. Nielsen
September 2008
Bayesian networks: basics September 2008 1 / 25
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Exact Inference
Message Passing in Join Trees
More sophisticated inference technique; used in most implemented Bayesian Network systems(e.g. Hugin).
Overview:
Given: Bayesian network for P(V)
⇓ Preprocessing
Construct a new internal representation for P(V) called a junction tree
⇓ Inference/Updating
Retrieve P(A | E = e) for single A ∈ V
Bayesian networks: basics September 2008 2 / 25
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Calculate a marginal
P(A4)
=X
A1
X
A2
X
A3
X
A5
X
A6
φ1(A1)φ2(A2,A1)φ3(A3,A1)φ4(A4,A2)φ5(A5,A2,A3)φ6(A6,A3)
Bayesian networks: basics September 2008 3 / 25
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Calculate a marginal
P(A4)
=X
A1
X
A2
X
A3
X
A5
X
A6
φ1(A1)φ2(A2,A1)φ3(A3,A1)φ4(A4,A2)φ5(A5,A2,A3)φ6(A6,A3)
=X
A1
φ1(A1)X
A2
φ2(A2,A1)φ4(A4,A2)X
A3
φ3(A3,A1)X
A5
φ5(A5,A2,A3)X
A6
φ6(A6,A3)
Bayesian networks: basics September 2008 3 / 25
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Calculate a marginal
P(A4)=P
A1φ1(A1)
P
A2φ2(A2,A1)φ4(A4,A2)
P
A3φ3(A3,A1)
P
A5φ5(A5,A2,A3)
P
A6φ6(A6,A3)
Bayesian networks: basics September 2008 4 / 25
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Calculate a marginal
Some of the calculations canbe simplified. E.g.
X
A6
P(A6 |A3) = 1
P(A4)=P
A1φ1(A1)
P
A2φ2(A2,A1)φ4(A4,A2)
P
A3φ3(A3,A1)
P
A5φ5(A5,A2,A3)
P
A6φ6(A6,A3)
Bayesian networks: basics September 2008 4 / 25
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Calculate a marginal
Assume evidence e = (e5, e6); represented as 0/1 potentials. Then:
P(A4, e) =X
A1
φ1(A1)X
A2
φ2(A2,A1)φ4(A4,A2)
X
A3
φ3(A3,A1)X
A5
φ5(A5,A2,A3)e5
X
A6
φ6(A6,A3)e6
and
P(A4 | e) =P(A4, e)
P
A4P(A4, e)
Bayesian networks: basics September 2008 5 / 25
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Calculate a marginal
Assume evidence e = (e5, e6); represented as 0/1 potentials. Then:
P(A4, e) =X
A1
φ1(A1)X
A2
φ2(A2,A1)φ4(A4,A2)
X
A3
φ3(A3,A1)X
A5
φ5(A5,A2,A3)e5
X
A6
φ6(A6,A3)e6
and
P(A4 | e) =P(A4, e)
P
A4P(A4, e)
The process is sufficiently general to handle all evidence scenarios!
⇒ We look for a general structure in which these calculations can be performed for all variables.
Bayesian networks: basics September 2008 5 / 25
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Domain graphs
The Bayesian network The domain graph
The link (A2,A3) is called a moral link.
Bayesian networks: basics September 2008 6 / 25
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Domain graphs
Moralization in general
For all nodes X : Connect pairwise all parents of X with undirected links.
Replace all original directed links by undirected ones.
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The Bayesian network
Bayesian networks: basics September 2008 7 / 25
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Domain graphs
Moralization in general
For all nodes X : Connect pairwise all parents of X with undirected links.
Replace all original directed links by undirected ones.
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The Bayesian network
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Step 1
Bayesian networks: basics September 2008 7 / 25
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Domain graphs
Moralization in general
For all nodes X : Connect pairwise all parents of X with undirected links.
Replace all original directed links by undirected ones.
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765
8
The Bayesian network
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Step 1
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Step 2
Bayesian networks: basics September 2008 7 / 25
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Eliminating a variable
Suppose that when calculating P(A4) we start off by eliminating A3:
φ′(A1,A2,A5,A6) =X
A3
φ3(A3,A1)φ6(A6,A3)φ5(A5,A2,A3).
Graphically:
The Bayesian network The domain graph The new domain graph
Bayesian networks: basics September 2008 8 / 25
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Eliminating a variable
Suppose that when calculating P(A4) we start off by eliminating A3:
φ′(A1,A2,A5,A6) =X
A3
φ3(A3,A1)φ6(A6,A3)φ5(A5,A2,A3).
Graphically:
The Bayesian network The domain graph The new domain graph
Perfect elimination sequences
If all variables can be eliminated without introducing fill-in edges, then the elimination sequence iscalled perfect. An example could be A5,A6,A3,A1,A2,A4.
Bayesian networks: basics September 2008 8 / 25
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Perfect elimination sequences
Perfect elimination sequences
All perfect elimination sequences produce the same domain set, namely the set of cliques of thedomain graph.
Clique
A set of nodes is complete if all nodes are pairwise linked. A complete set is a clique if it is not asubset of another complete set.
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Property
Any perfect elimination sequence ending with A is optimal w.r.t. calculating P(A).
Bayesian networks: basics September 2008 9 / 25
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Triangulated graphs
Definition
An undirected graph with a perfect elimination sequence is called a triangulated graph.
Example
not triangulated
triangulated
Bayesian networks: basics September 2008 10 / 25
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Triangulated graphs
Definition
An undirected graph with a perfect elimination sequence is called a triangulated graph.
Example
not triangulated
triangulated
Corollary
A graph is triangulated iff all nodes can successively be eliminated without introducing fill-in edges.
Bayesian networks: basics September 2008 10 / 25
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Triangulated graphs
Definition
An undirected graph with a perfect elimination sequence is called a triangulated graph.
Example
not triangulated
triangulated
Corollary
A graph is triangulated iff there does not exist a cycle of length ≥ 4 that is not cut by a cord.
Bayesian networks: basics September 2008 10 / 25
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Triangulated graphs
Definition
An undirected graph with a perfect elimination sequence is called a triangulated graph.
Example
not triangulated
triangulated
Corollary
In a triangulated graph, each variable A has a perfect elimination sequence ending with A.
Bayesian networks: basics September 2008 10 / 25
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Triangulated graphs
Definition
An undirected graph with a perfect elimination sequence is called a triangulated graph.
Example
not triangulated
triangulated
Assumption
For now we shall assume that the domain graph is triangulated!
Bayesian networks: basics September 2008 10 / 25
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Exact Inference
Join Tree (Junction Tree)
Let G be the set of cliques from an undirected graph, and let the cliques of G be organized in a treeT . Then T is a join tree if
a variable V that is contained in two nodes C,C′ also is contained in every node on the(unique) path connecting C and C′.
Example:
Bayesian networks: basics September 2008 11 / 25
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Exact Inference
Join Tree (Junction Tree)
Let G be the set of cliques from an undirected graph, and let the cliques of G be organized in a treeT . Then T is a join tree if
a variable V that is contained in two nodes C,C′ also is contained in every node on the(unique) path connecting C and C′.
Example:
(a) A join tree
(b) Not a join tree
Bayesian networks: basics September 2008 11 / 25
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Exact Inference
Join Tree (Junction Tree)
Let G be the set of cliques from an undirected graph, and let the cliques of G be organized in a treeT . Then T is a join tree if
a variable V that is contained in two nodes C,C′ also is contained in every node on the(unique) path connecting C and C′.
Example:
(a) A join tree
(b) Not a join tree
Theorem: An undirected graph G is triangulated if and only if the cliques of G can be organized ina join tree.
Bayesian networks: basics September 2008 11 / 25
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Join tree construction
From undirected graph to join graph
Define undirected graph (C, E∗):
C : Set of cliques in triangulated moral graphE∗ : {(C1,C2) | C1 ∩ C2 6= ∅}
Bayesian networks: basics September 2008 12 / 25
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Join tree construction
From undirected graph to join graph
Define undirected graph (C, E∗):
C : Set of cliques in triangulated moral graphE∗ : {(C1,C2) | C1 ∩ C2 6= ∅}
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Bayesian networks: basics September 2008 12 / 25
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Join tree construction
From undirected graph to join graph
Define undirected graph (C, E∗):
C : Set of cliques in triangulated moral graphE∗ : {(C1,C2) | C1 ∩ C2 6= ∅}
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3 4
765
8
Bayesian networks: basics September 2008 12 / 25
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Join tree construction
From undirected graph to join graph
Define undirected graph (C, E∗):
C : Set of cliques in triangulated moral graphE∗ : {(C1,C2) | C1 ∩ C2 6= ∅}
1 2
3 4
765
8
Bayesian networks: basics September 2008 12 / 25
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Join tree construction
From undirected graph to join graph
Define undirected graph (C, E∗):
C : Set of cliques in triangulated moral graphE∗ : {(C1,C2) | C1 ∩ C2 6= ∅}
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765
8 5,6,7,8
1,2,3 2,4,7
3,6,72,3,7
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Join tree construction
From join graph to join tree
Find maximal spanning tree: for edge e = (C1,C2) define weight w(e) := |C1 ∩ C2|. Let (C, E) bea maximal spanning tree of (C, E∗) with respect to edge weights.
Bayesian networks: basics September 2008 13 / 25
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Join tree construction
From join graph to join tree
Find maximal spanning tree: for edge e = (C1,C2) define weight w(e) := |C1 ∩ C2|. Let (C, E) bea maximal spanning tree of (C, E∗) with respect to edge weights.
5,6,7,8
1,2,3 2,4,7
3,6,72,3,7
Bayesian networks: basics September 2008 13 / 25
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Join tree construction
From join graph to join tree
Find maximal spanning tree: for edge e = (C1,C2) define weight w(e) := |C1 ∩ C2|. Let (C, E) bea maximal spanning tree of (C, E∗) with respect to edge weights.
1,2,3 2,4,7
3,6,72,3,7
5,6,7,8
1
1
1
2 2
2
2
1
Bayesian networks: basics September 2008 13 / 25
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Join tree construction
From join graph to join tree
Find maximal spanning tree: for edge e = (C1,C2) define weight w(e) := |C1 ∩ C2|. Let (C, E) bea maximal spanning tree of (C, E∗) with respect to edge weights.
1
1,2,3 2,4,7
3,6,72,3,7
5,6,7,8
1
1
1
2 2
2
2
Bayesian networks: basics September 2008 13 / 25
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Join tree construction
From join graph to join tree
Find maximal spanning tree: for edge e = (C1,C2) define weight w(e) := |C1 ∩ C2|. Let (C, E) bea maximal spanning tree of (C, E∗) with respect to edge weights.
1,2,3 2,4,7
3,6,72,3,7
5,6,7,8
Bayesian networks: basics September 2008 13 / 25
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Join tree construction
Another example
Bayesian networks: basics September 2008 14 / 25
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Join tree construction
Another example
A1
A2 A3
A4 A5 A6
Bayesian networks: basics September 2008 14 / 25
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Join tree construction
Another example
A1
A2 A3
A4 A5 A6
11
1 1
2A2,A4
A1,A2,A3
A2,A3,A5
A2,A3,A5
Bayesian networks: basics September 2008 14 / 25
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Join tree construction
Another example
A1
A2 A3
A4 A5 A6
11
1 1
2A2,A4
A1,A2,A3
A2,A3,A5
A2,A3,A5
A2,A4
A1,A2,A3
A2,A3,A5
A2,A3,A5
Bayesian networks: basics September 2008 14 / 25
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Junction tree construction
Correctness
Let (V,→) be a triangulated graph, (C, E∗). Every maximal spanning tree (C, E) of (C, E∗) is a jointree for V (Jensen,Jensen 1994).
If (V,→) is the triangulated moral graph of a Bayesian network, then there exists for every V ∈ V aclique C ∈ C with {V} ∪ pa(V ) ⊆ C.
Bayesian networks: basics September 2008 15 / 25
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Propagation in junction trees
Initialization
Let T be a join tree for the domain graph G with potentials Φ. A junction tree for G consists of Twith the additions:
each potential φ from Φ is assigned to a clique containing dom (φ).
each link has a separator attached.
each separator contains two mailboxes, one for each direction.
Example
Bayesian networks: basics September 2008 16 / 25
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Propagation in junction trees
Propagation
To calculate P(A4) we find a clique (V6) containing A4 and send messages to that clique.
Example
V4 : A1,A2,A3
↓ ↓
S4 : A2 S2 : A2,A3 S1 : A3
V6 : A2,A4 V2 : A2,A3,A5 V1 : A3,A6
φ1, φ2, φ3
↓
φ4 φ5 φ6
↑ ↑ ↑ψ4
ψ2 = φ↓S25 ψ1 = φ
↓S16
Bayesian networks: basics September 2008 17 / 25
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Propagation in junction trees
Propagation
To calculate P(A4) we find a clique (V6) containing A4 and send messages to that clique.
Example
1 : ψ1 = φ↓S16
↑1 : ψ2 = φ
↓S25
V4 : A1,A2,A3
↓ ↓
S4 : A2 S2 : A2,A3 S1 : A3
V6 : A2,A4 V2 : A2,A3,A5 V1 : A3,A6
φ1, φ2, φ3
↓
φ4 φ5 φ6
↑ ↑
Bayesian networks: basics September 2008 17 / 25
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Propagation in junction trees
Propagation
To calculate P(A4) we find a clique (V6) containing A4 and send messages to that clique.
Example
↓
↑ ↑ ↑1 : ψ1 = φ
↓S161 : ψ2 = φ
↓S25
V4 : A1,A2,A3
↓ ↓
S4 : A2 S2 : A2,A3 S1 : A3
V6 : A2,A4 V2 : A2,A3,A5 V1 : A3,A6
φ1, φ2, φ3
φ4 φ5 φ6
2 : ψ4 = . . .
ψ4 =P
A1φ1φ2
P
A3φ3ψ1ψ2
Now evidence has been collected to V4 and we get P(A4) =P
A2φ4ψ4.
Bayesian networks: basics September 2008 17 / 25
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Propagation in junction trees
Propagation
To calculate P(Ai ) for any Ai we send messages away from V6.
Example continued
↓
↑ ↑ ↑1 : ψ1 = φ
↓S161 : ψ2 = φ
↓S25
V4 : A1,A2,A3
↓ ↓
S4 : A2 S2 : A2,A3 S1 : A3
V6 : A2,A4 V2 : A2,A3,A5 V1 : A3,A6
φ1, φ2, φ3
φ4 φ5 φ6
2 : ψ4 = . . .
ψ4 =P
A1φ1φ2
P
A3φ3ψ1ψ2
Bayesian networks: basics September 2008 18 / 25
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Propagation in junction trees
Propagation
To calculate P(Ai ) for any Ai we send messages away from V6.
Example continued
↓
φ6
↑ ↑ ↑1 : ψ1 = φ
↓S161 : ψ2 = φ
↓S25
V4 : A1,A2,A3
↓ ↓
S4 : A2 S2 : A2,A3 S1 : A3
V6 : A2,A4 V2 : A2,A3,A5 V1 : A3,A6
φ1, φ2, φ3
φ4 φ5
2 : ψ4 = . . .
3 : ψ4 = φ↓A24
Bayesian networks: basics September 2008 18 / 25
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Propagation in junction trees
Propagation
To calculate P(Ai ) for any Ai we send messages away from V6.
Example continued
φ5V6 : A2,A4 V2 : A2,A3,A5 V1 : A3,A6
φ1, φ2, φ3
↓
φ4 φ6
↑ ↑ ↑1 : ψ1 = φ
↓S161 : ψ2 = φ
↓S25
V4 : A1,A2,A3
↓ ↓
S4 : A2 S2 : A2,A3 S1 : A3
4 : ψ2 = . . . 4 : ψ1 = . . .
ψ2 =P
A1ψ1φ1φ2φ3ψ
4 ∼ {ψ4,P
A1φ1φ2φ3, ψ1}
ψ1 =P
A2ψ2ψ4
P
A1φ1φ2φ3
2 : ψ4 = . . .
3 : ψ4 = φ↓A24
Bayesian networks: basics September 2008 18 / 25
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Propagation in junction trees
In general: Sending a message from C i to C j
CiCj
Cl
Cm
mij
mli
mmi
mij :=X
Cj\Ci
Y
φ∈ΦCi
φY
Ck∈nd(Ci )
mki
Bayesian networks: basics September 2008 19 / 25
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Exact Inference
Messages passing in general
R
Collect and distribute messages
i) Collect messages to a preselected root R
Bayesian networks: basics September 2008 20 / 25
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Exact Inference
Messages passing in general
R
1
11
Collect and distribute messages
i) Collect messages to a preselected root R
Bayesian networks: basics September 2008 20 / 25
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Exact Inference
Messages passing in general
R2
1
1
2
1
Collect and distribute messages
i) Collect messages to a preselected root R
Bayesian networks: basics September 2008 20 / 25
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Exact Inference
Messages passing in general
R2
1
1
2
1
3
3
Collect and distribute messages
i) Collect messages to a preselected root R
ii) Distribute messages away from the root R
Bayesian networks: basics September 2008 20 / 25
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Exact Inference
Messages passing in general
42
1
1
2
1
R
3
4
3
4
Collect and distribute messages
i) Collect messages to a preselected root R
ii) Distribute messages away from the root R
Bayesian networks: basics September 2008 20 / 25
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Propagation in junction trees
Theorem
Let the junction tree T represent the Bayesian network BN over the universe U and with evidencee. Assume that T is full.
1 Let V be a clique with set of potentials ΦV , and let S1, . . . ,Sk be V ’s neighboring separatorsand with V -directed messages Ψ1, . . . ,Ψk . Then
P(V , e) =Y
ΦV
Y
Ψ1 · . . . ·Y
Ψk .
2 Let S be a separator with the sets ΨS and ΨS in the mailboxes. Then
P(S, e) =Y
ΨS
Y
ΨS.
Bayesian networks: basics September 2008 21 / 25
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Propagation in junction trees
Theorem
Let the junction tree T represent the Bayesian network BN over the universe U and with evidencee. Assume that T is full.
1 Let V be a clique with set of potentials ΦV , and let S1, . . . ,Sk be V ’s neighboring separatorsand with V -directed messages Ψ1, . . . ,Ψk . Then
P(V , e) =Y
ΦV
Y
Ψ1 · . . . ·Y
Ψk .
2 Let S be a separator with the sets ΨS and ΨS in the mailboxes. Then
P(S, e) =Y
ΨS
Y
ΨS.
Evidence
Evidence is inserted by adding corresponding 0/1-potentials to the appropriate cliques.
Bayesian networks: basics September 2008 21 / 25
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Triangulation of graphs
So far we have assumed that the domain graph is triangulated. If this is not the case, then weembed it in a triangulated graph.
Example
Bayesian networks: basics September 2008 22 / 25
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Triangulation of graphs
So far we have assumed that the domain graph is triangulated. If this is not the case, then weembed it in a triangulated graph.
Example
A triangulated graph extending (b)
Bayesian networks: basics September 2008 22 / 25
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Triangulation of graphs
1 2
3 4
765
8
Usually there are many possible triangulations and finding the best one is NP-hard!
Bayesian networks: basics September 2008 23 / 25
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Triangulation of graphs
1 2
3 4
765
8
A heuristic
Repeatedly eliminate a simplicial node, and if this is not possible eliminate a node minimizing(fa(X) are noneliminated neighbors of X ):
|fa(X)| (fill-in-size ).
Eliminating a node
Make all its noneliminated neighbors complete and remove the node.Bayesian networks: basics September 2008 23 / 25
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Triangulation of graphs
1 2
3 4
765
8 8
1 2
3 4
765
A heuristic
Repeatedly eliminate a simplicial node, and if this is not possible eliminate a node minimizing(fa(X) are noneliminated neighbors of X ):
|fa(X)| (fill-in-size ).
Eliminating a node
Make all its noneliminated neighbors complete and remove the node.Bayesian networks: basics September 2008 23 / 25
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Triangulation of graphs
1 2
3 4
765
8 8
1 2
3 4
765
A heuristic
Repeatedly eliminate a simplicial node, and if this is not possible eliminate a node minimizing(fa(X) are noneliminated neighbors of X ):
|fa(X)| (fill-in-size ).
Eliminating a node
Make all its noneliminated neighbors complete and remove the node.Bayesian networks: basics September 2008 23 / 25
![Page 60: Machine Intelligence Thomas D. Nielsen September 2008people.cs.aau.dk/.../junction-tree-propagation.pdf · Construct a new internal representation for P(V) called a junction tree](https://reader035.vdocuments.net/reader035/viewer/2022081411/60a7f81f9c48c71c17161773/html5/thumbnails/60.jpg)
Triangulation of graphs
1 2
3 4
765
8
1 2
3 4
765
8
A heuristic
Repeatedly eliminate a simplicial node, and if this is not possible eliminate a node minimizing(fa(X) are noneliminated neighbors of X ):
|fa(X)| (fill-in-size ).
Eliminating a node
Make all its noneliminated neighbors complete and remove the node.Bayesian networks: basics September 2008 23 / 25
![Page 61: Machine Intelligence Thomas D. Nielsen September 2008people.cs.aau.dk/.../junction-tree-propagation.pdf · Construct a new internal representation for P(V) called a junction tree](https://reader035.vdocuments.net/reader035/viewer/2022081411/60a7f81f9c48c71c17161773/html5/thumbnails/61.jpg)
Triangulation of graphs
1 2
3 4
765
8
1 2
3 4
765
8
A heuristic
Repeatedly eliminate a simplicial node, and if this is not possible eliminate a node minimizing(fa(X) are noneliminated neighbors of X ):
|fa(X)| (fill-in-size ).
Eliminating a node
Make all its noneliminated neighbors complete and remove the node.Bayesian networks: basics September 2008 23 / 25
![Page 62: Machine Intelligence Thomas D. Nielsen September 2008people.cs.aau.dk/.../junction-tree-propagation.pdf · Construct a new internal representation for P(V) called a junction tree](https://reader035.vdocuments.net/reader035/viewer/2022081411/60a7f81f9c48c71c17161773/html5/thumbnails/62.jpg)
Triangulation of graphs
1 2
3 4
765
8
1 2
3 4
765
8
A heuristic
Repeatedly eliminate a simplicial node, and if this is not possible eliminate a node minimizing(fa(X) are noneliminated neighbors of X ):
|fa(X)| (fill-in-size ).
Eliminating a node
Make all its noneliminated neighbors complete and remove the node.Bayesian networks: basics September 2008 23 / 25
![Page 63: Machine Intelligence Thomas D. Nielsen September 2008people.cs.aau.dk/.../junction-tree-propagation.pdf · Construct a new internal representation for P(V) called a junction tree](https://reader035.vdocuments.net/reader035/viewer/2022081411/60a7f81f9c48c71c17161773/html5/thumbnails/63.jpg)
Triangulation of graphs
1 2
3 4
765
8
1 2
3 4
765
8
A heuristic
Repeatedly eliminate a simplicial node, and if this is not possible eliminate a node minimizing(fa(X) are noneliminated neighbors of X ):
|fa(X)| (fill-in-size ).
Eliminating a node
Make all its noneliminated neighbors complete and remove the node.Bayesian networks: basics September 2008 23 / 25
![Page 64: Machine Intelligence Thomas D. Nielsen September 2008people.cs.aau.dk/.../junction-tree-propagation.pdf · Construct a new internal representation for P(V) called a junction tree](https://reader035.vdocuments.net/reader035/viewer/2022081411/60a7f81f9c48c71c17161773/html5/thumbnails/64.jpg)
Triangulation of graphs
1 2
3 4
765
8
1 2
3 4
765
8
A heuristic
Repeatedly eliminate a simplicial node, and if this is not possible eliminate a node minimizing(fa(X) are noneliminated neighbors of X ):
|fa(X)| (fill-in-size ).
Eliminating a node
Make all its noneliminated neighbors complete and remove the node.Bayesian networks: basics September 2008 23 / 25
![Page 65: Machine Intelligence Thomas D. Nielsen September 2008people.cs.aau.dk/.../junction-tree-propagation.pdf · Construct a new internal representation for P(V) called a junction tree](https://reader035.vdocuments.net/reader035/viewer/2022081411/60a7f81f9c48c71c17161773/html5/thumbnails/65.jpg)
Triangulation of graphs
1 2
3 4
765
8
1 2
3 4
765
8
A heuristic
Repeatedly eliminate a simplicial node, and if this is not possible eliminate a node minimizing(fa(X) are noneliminated neighbors of X ):
|fa(X)| (fill-in-size ).
Eliminating a node
Make all its noneliminated neighbors complete and remove the node.Bayesian networks: basics September 2008 23 / 25
![Page 66: Machine Intelligence Thomas D. Nielsen September 2008people.cs.aau.dk/.../junction-tree-propagation.pdf · Construct a new internal representation for P(V) called a junction tree](https://reader035.vdocuments.net/reader035/viewer/2022081411/60a7f81f9c48c71c17161773/html5/thumbnails/66.jpg)
Triangulation of graphs
1 2
3 4
765
8
1 2
3 4
765
8
A heuristic
Repeatedly eliminate a simplicial node, and if this is not possible eliminate a node minimizing(fa(X) are noneliminated neighbors of X ):
|fa(X)| (fill-in-size ).| sp(fa(X)) | (clique-size).P
{Y ,Z}:{Y ,Z}∈nb(X)∧Z 6∈nb(Y ) | sp({Y , Z}) | (fill-in-weight)
Eliminating a node
Make all its noneliminated neighbors complete and remove the node.Bayesian networks: basics September 2008 23 / 25
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Summary
1 2
3 4
765
8
Bayesian network
Bayesian networks: basics September 2008 24 / 25
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Summary
1 2
3 4
765
8
1 2
3 4
765
8
Bayesian network Moral graph
Bayesian networks: basics September 2008 24 / 25
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Summary
1 2
3 4
765
8
1 2
3 4
765
8
1 2
3 4
765
8
Bayesian network Moral graph Triangulated graph
Bayesian networks: basics September 2008 24 / 25
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Summary
1 2
3 4
765
8
1 2
3 4
765
8
1 2
3 4
765
8
Bayesian network Moral graph Triangulated graph
5,6,7,8
1,2,3 2,4,7
3,6,72,3,7
Join graph
Bayesian networks: basics September 2008 24 / 25
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Summary
1 2
3 4
765
8
1 2
3 4
765
8
1 2
3 4
765
8
Bayesian network Moral graph Triangulated graph
5,6,7,8
1,2,3 2,4,7
3,6,72,3,7
1,2,3 2,4,7
3,6,72,3,7
5,6,7,8
Join graph Join tree
Bayesian networks: basics September 2008 24 / 25
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Summary
1 2
3 4
765
8
1 2
3 4
765
8
1 2
3 4
765
8
Bayesian network Moral graph Triangulated graph
5,6,7,8
1,2,3 2,4,7
3,6,72,3,7
1,2,3 2,4,7
3,6,72,3,7
5,6,7,8
6,7
1,2,3 2,4,7
3,6,72,3,7
5,6,7,8
2,3 2,7
3,7
Join graph Join tree Junction tree
Bayesian networks: basics September 2008 24 / 25
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Exact Inference
Summary
Given: Bayesian network BN for P(V)
Bayesian networks: basics September 2008 25 / 25
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Exact Inference
Summary
Given: Bayesian network BN for P(V)
⇓ Moralize
Domain graph G for BN (also called the moral graph)
Bayesian networks: basics September 2008 25 / 25
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Exact Inference
Summary
Given: Bayesian network BN for P(V)
⇓ Moralize
Domain graph G for BN (also called the moral graph)
⇓ Triangulate
Triangulated graph for G
Bayesian networks: basics September 2008 25 / 25
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Exact Inference
Summary
Given: Bayesian network BN for P(V)
⇓ Moralize
Domain graph G for BN (also called the moral graph)
⇓ Triangulate
Triangulated graph for G
⇓ Find cliques
Join graph
Bayesian networks: basics September 2008 25 / 25
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Exact Inference
Summary
Given: Bayesian network BN for P(V)
⇓ Moralize
Domain graph G for BN (also called the moral graph)
⇓ Triangulate
Triangulated graph for G
⇓ Find cliques
Join graph
⇓ Find maximal spanning tree
Join tree
Bayesian networks: basics September 2008 25 / 25
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Exact Inference
Summary
Given: Bayesian network BN for P(V)
⇓ Moralize
Domain graph G for BN (also called the moral graph)
⇓ Triangulate
Triangulated graph for G
⇓ Find cliques
Join graph
⇓ Find maximal spanning tree
Join tree
⇓
Perform belief updating by message passing.
Bayesian networks: basics September 2008 25 / 25