(b,c|d 0 ).

On the Role of MSBN to
Cooperative Multiagent
Systems
By Y. Xiang and V. Lesser
Presented by:
Jingshan Huang and Sharon Xi
Motivation

A common task in multiagent systems
Agents need to estimate the state of an uncertain domain so
that they can act accordingly

Constraints

Each agent only has partial knowledge about the domain
 Only local observations are available
 Limited amount of communication
Solution ?
Motivation --- cont.

MSBNs (Multiply Sectioned Bayesian
Networks) provide a solution
 An
effective and exact framework
 But a set of constraints exist
Structure of the presentation




Introduction of the background
knowledge
Detail information about the constraints
A small set of high level choices
How those choices logically imply all
the constraints
Background knowledge
 Definition
of MSBN
A MSBN M is a triplet (V, G, P):



V is the union domain from all agents
G is the structure, i.e., hypertree MSDAG
 Hypertree structure
 D-sepset concept
P is the JPD (Joint Probability Distribution) over G
P( x | π(x)) is assigned to exactly one occurrence of x,
and uniform potential to all other occurrences
Background knowledge --- cont.

Definition of hypertree structure
d
d
a
a
e
c
b
Original Graph
Hypernode
a
a,b
c
b
e
b
Hypertree
Hyperlink

Each node (hypernode) is a DAG
 Each link (hyperlink) between two nodes is an non-empty interface
 RIP (Running Intersection Property)

D-sepset concept

An interface I is a d-sepset if every x∈I is a d-sepnode

A node x contained in more than one subgraph with its parents π(x) is a
d-sepnode if there exists at least one subgraph that contains π(x)
Background knowledge --- cont.
Some useful definitions

Communication Graph
In a graph with n hypernodes, associate each node with an agent
Ai and label it by Vi
Connect each pair of nodes Vi and Vj by a link labeled by I  Vi  Vj
if I  

Junction Graph
A triplet (V, Ω, E)
V is a non-empty set (the generating set)
Ω is a subset of 2V s.t. Q  V, each element Q is called a cluster
E is defined as { Q1, Q2 | Q1, Q2  , Q1  Q2 , Q1  Q2  }
Background knowledge --- cont.

Cluster Graph
Let (V, Ω ,E) be a junction graph and E'  E , then (V,
Ω ,E’) is a cluster graph over V

Degenerate Loop and Nondegenerate Loop
Let ρ be a loop in a cluster graph H. If there exists a
separator S on ρ that is contained in every other
separator on ρ, then ρ is a degenerate loop. Otherwise,
ρ is a nondegenerate loop
Background knowledge --- cont.
d,e
d,f
d
d
d
b,c,d
d
d
d
d,g
(a) Strong Degenerate Loop
d,e,i
d
d,i
b,c,d,i
d
a,b
a,e
e
b
b,c,d
c
c,e
(c) Strong Nondegenerate Loop
d,f,h
a,b,f
d,h
b,f
d,g,h
b,c,d,f
(b) Weak Degenerate Loop
a
a,f
a,e,f
e,f
c,f
c,e,f
(d) Week Nondegenerate Loop
Structure of the presentation




Introduction of the background
knowledge
Detail information about the constraints
A small set of high level choices
How those choices logically imply all
the constraints
Seven Constraints
1.
Each agent’s belief is represented by Bayesian probability
2.
The domain is decomposed into subdomains with RIP
3.
Subdomains are organized into a hyptertree structure
4.
The dependency structure of each subdomain is represented by
a DAG
5.
The union of DAGs for all subdomains is a connected DAG
6.
Each hyperlink is a d-sepset
7.
The JPD can be expressed as in definition of MSBN
Structure of the presentation




Introduction of the background
knowledge
Detail information about the constraints
A small set of high level choices
How those choices logically imply all
the constraints
High Level Choices (Basic Commitments)

BC1: Each agent’s belief is represented by Bayesian probability

BC2: Ai and Aj can communicate directly only with their
intersecting variables

BC3: A simpler agent organization, i.e., tree, is preferred when
degenerate loops exist in the CG

BC4: A DAG is used to structure each individual agent’s
knowledge

BC5: Within each agent’s subdomain, the JPD is consistent with
the agent’s belief. For shared nodes, the JPD supplements each
agent’s knowledge with others’
Structure of the presentation




Introduction of the background
knowledge
Detail information about the constraints
A small set of high level choices
How those choices logically imply all
the constraints
Proof of the logical implication
Five Basic Commitments

BC1: Each agent’s belief is represented
by Bayesian probability

BC2: Ai and Aj can communicate
directly only with their intersecting
variables



BC3: A simpler agent organization, i.e.,
tree, is preferred when degenerate
loops exist in the CG
BC4: A DAG is used to structure each
individual agent’s knowledge
BC5: Within each agent’s subdomain,
the JPD is consistent with the agent’s
belief. For shared nodes, the JPD
supplements each agent’s knowledge
with others’
Seven Constraints
1.
Each agent’s belief is
represented by Bayesian
probability
2.
The domain is decomposed into
subdomains with RIP
3.
Subdomains are organized into a
hyptertree structure
4.
The dependency structure of
each subdomain is represented
by a DAG
5.
The union of DAGs for all
subdomains is a connected DAG
6.
Each hyperlink is a d-sepset
7.
The JPD can be expressed as in
definition of MSBN
Five Basic Commitments

BC1: Each agent’s belief is represented
by Bayesian probability

BC2: Ai and Aj can communicate
directly only with their intersecting
variables



BC3: A simpler agent organization, i.e.,
tree, is preferred when degenerate
loops exist in the CG
BC4: A DAG is used to structure each
individual agent’s knowledge
BC5: Within each agent’s subdomain,
the JPD is consistent with the agent’s
belief. For shared nodes, the JPD
supplements each agent’s knowledge
with others’
Seven Constraints
1.
Each agent’s belief is
represented by Bayesian
probability
2.
The domain is decomposed into
subdomains with RIP
3.
Subdomains are organized into a
hyptertree structure
4.
The dependency structure of
each subdomain is represented
by a DAG
5.
The union of DAGs for all
subdomains is a connected DAG
6.
Each hyperlink is a d-sepset
7.
The JPD can be expressed as in
definition of MSBN
Five Basic Commitments

BC1: Each agent’s belief is represented
by Bayesian probability

BC2: Ai and Aj can communicate
directly only with their intersecting
variables



BC3: A simpler agent organization, i.e.,
tree, is preferred when degenerate
loops exist in the CG
BC4: A DAG is used to structure each
individual agent’s knowledge
BC5: Within each agent’s subdomain,
the JPD is consistent with the agent’s
belief. For shared nodes, the JPD
supplements each agent’s knowledge
with others’
Seven Constraints
1.
Each agent’s belief is
represented by Bayesian
probability
2.
The domain is decomposed into
subdomains with RIP
3.
Subdomains are organized into a
hyptertree structure
4.
The dependency structure of
each subdomain is represented
by a DAG
5.
The union of DAGs for all
subdomains is a connected DAG
6.
Each hyperlink is a d-sepset
7.
The JPD can be expressed as in
definition of MSBN

Lemma 9: Let s be a strictly positive initial state of Mas3.
There exists an infinite set S. Each element s’∈S is an initial
state of Mas3 identical to s in P(a), P(b|a), P(c|a) but distinct
in P(d|b,c) such that the message P2(b|d=d0) produced from s’
is identical to that produced from s, and so is the message
P2(c|d=d0)
A0
a
a
a,b
b
b
a,c
c
c
A2
d
b,c,d
Figure 1
Mas3: a multiagent
system of 3 agents.
A1
Proof: Denote P2(b=b0|d=d0) from state s by P2(b0|d0), P2’(b=b0|d=d0) from state s’ by
P2’(b0|d0). P2(b0|d0) can be expanded as:
 P (b , d ) 
P2 (b 0 , d 0 )
P2 (b 0 | d 0 ) 
 1  2 1 0 
P2 (b 0 , d 0 )  P2 (b1 , d 0 )  P2 (b 0 , d 0 ) 
 P (b , c , d )  P2 (b1 , c1 , d 0 ) 
 1  2 1 0 0

 P2 (b 0 , c 0 , d 0 )  P2 (b 0 , c1 , d 0 ) 
1
1

P (d | b , c )P (b , c )  P2 (d 0 | b1 , c1 )P2 (b1 , c1 ) 
 1  2 0 1 0 2 1 0

 P2 (d 0 | b 0 , c 0 )P2 (b 0 , c 0 )  P2 (d 0 | b 0 , c1 )P2 (b 0 , c1 ) 
1

P2' (d 0 | b1 , c 0 )P2 (b1 , c 0 )  P2' (d 0 | b1 , c1 )P2 (b1 , c1 ) 
'
P2 (b 0 | d 0 )  1  '

'
P
(
d
|
b
,
c
)
P
(
b
,
c
)

P
(
d
|
b
,
c
)
P
(
b
,
c
)
2
0
0
0
2
0
0
2
0
0
1
2
0
1 

1
For P2(b|d0)=P2’(b|d0), we have:
P2' (d 0 | b1 , c 0 )P2 (b1 , c 0 )  P2' (d 0 | b1 , c1 )P2 (b1 , c1 )
P2 (b1 , d 0 )

P2' (d 0 | b 0 , c 0 )P2 (b 0 , c 0 )  P2' (d 0 | b 0 , c1 )P2 (b 0 , c1 )
P2 (b 0 , d 0 )
Similarly,
P2' (d 0 | b 0 , c1 )P2 (b 0 , c1 )  P2' (d 0 | b1 , c1 )P2 (b1 , c1 )
P2 (c1 , d 0 )

P2' (d 0 | b 0 , c 0 )P2 (b 0 , c 0 )  P2' (d 0 | b1 , c 0 )P2 (b1 , c 0 ) P2 (c 0 , d 0 )
Because P2’(d|b,c) has 4 independent parameters but is constrained by only two
equations, it has infinitely many solutions.
 Lemma 10: Let P and P’ be strictly positive probability distributions over the
DAG of Figure 1 such that they are identical in P(a), P(b|a) and P(c|a) but
distinct in P(d|b,c). Then P(a|d=d0) is distinct from P’(a|d=d0) in general
Proof: The following can be obtained from P and P’:
P(a | d 0 )   P(a | b, c) P(b, c | d 0 )
P (a | d ) 
 P(a, b, c | d)
b,c
b. c
P' (a | d 0 )   P' (a | b, c) P' (b, c | d 0 )
P(a , b, c, d) P(a , b, c, d )P(b, c, d)

b. c
P (d )
P(b, c, d)P(d)
If P(b,c|d0) ≠ P’(b,c|d0), then in general  P(a | b, c, d)P(b, c | d)
 P(a | b, c)P(b, c | d )
P(a|d0) ≠P’(a|d0)
P(b, c | d 0 ) 
P(a , b, c | d ) 
P ( d 0 | b, c ) P (b, c )
P( d 0 | b, c ) P(b, c )

P(d 0 )
 P(d 0 | b, c) P(b, c)
b ,c
P' (b, c | d 0 ) 
P' ( d 0 | b, c ) P (b, c )
P' ( d 0 | b, c ) P(b, c )

P' (d 0 )
 P' (d 0 | b, c) P(b, c)
b ,c
Because P(d|b,c) ≠P’(d|b,c), in general, it is the case that P(b,c|d0) ≠P’(b,c|d0).
Do you agree???
Theorem 11: Message passing in Mas3 cannot be coherent in general, no matter
how it is performed
Proof:
1.
By Lemma 9, P2(b|d=d0) and P2(c|d=d0) are insensitive to the initial states
and hence the posteriors P0(a|d=d0) computed from the messages can not
be sensitive to the initial states either
2. However, by Lemma 10, the posterior should be different in general given
different initial states
Hence, correct belief updating cannot be achieved in Mas3
Insight
A0
a
a,b
b
A2
a,c
c
b,c,d
Figure 1
A1


Correct inference requires P(b,c|d0)
However, nondegenerate loop results
in the passing of the marginals of
P(b,c|d0), i.e., P(b|d=d0) and P(c|d=d0)

We can generalize this analysis to an arbitrary, strong
nondegenerate loop of length 3

Further generalize this analysis to an arbitrary, strong
nondegenerate loop of length K ≥ 3
Conclusion
Corollary 12: Message passing in a cluster graph with
nondegenerate loops cannot be coherent in general, no
matter how it is performed

Another conclusion without proof:
A cluster graph with only degenerate loops can
always be treated by first breaking the loops at
appropriate separators. The resultant is a cluster
tree
Therefore, we have:
Proposition 13: Let a multiagent system be one
that observes BC 1 through BC 3. Then a tree
organization of agents should be used
Five Basic Commitments

BC1: Each agent’s belief is represented
by Bayesian probability

BC2: Ai and Aj can communicate
directly only with their intersecting
variables



BC3: A simpler agent organization, i.e.,
tree, is preferred when degenerate
loops exist in the CG
BC4: A DAG is used to structure each
individual agent’s knowledge
BC5: Within each agent’s subdomain,
the JPD is consistent with the agent’s
belief. For shared nodes, the JPD
supplements each agent’s knowledge
with others’
Seven Constraints
1.
Each agent’s belief is
represented by Bayesian
probability
2.
The domain is decomposed into
subdomains with RIP
3.
Subdomains are organized into a
hyptertree structure
4.
The dependency structure of
each subdomain is represented
by a DAG
5.
The union of DAGs for all
subdomains is a connected DAG
6.
Each hyperlink is a d-sepset
7.
The JPD can be expressed as in
definition of MSBN
Five Basic Commitments

BC1: Each agent’s belief is represented
by Bayesian probability

BC2: Ai and Aj can communicate
directly only with their intersecting
variables



BC3: A simpler agent organization, i.e.,
tree, is preferred when degenerate
loops exist in the CG
BC4: A DAG is used to structure each
individual agent’s knowledge
BC5: Within each agent’s subdomain,
the JPD is consistent with the agent’s
belief. For shared nodes, the JPD
supplements each agent’s knowledge
with others’
Seven Constraints
1.
Each agent’s belief is
represented by Bayesian
probability
2.
The domain is decomposed into
subdomains with RIP
3.
Subdomains are organized into a
hyptertree structure
4.
The dependency structure of
each subdomain is represented
by a DAG
5.
The union of DAGs for all
subdomains is a connected DAG
6.
Each hyperlink is a d-sepset
7.
The JPD can be expressed as in
definition of MSBN
Proposition 17: Let a multiagent system over V be
constructed following BC 1 through BC 4. Then
each subdomain Vi is structured as a DAG over
Vi and the union of these DAGs is a connected
DAG over V
Proof:
1.
The connectedness is implied by Proposition 6
2.
If the union of subdomain DAGs is not a DAG, then it
has a directed loop. This contradicts the acyclic
interpretation of dependence in individual DAG models
Five Basic Commitments

BC1: Each agent’s belief is represented
by Bayesian probability

BC2: Ai and Aj can communicate
directly only with their intersecting
variables



BC3: A simpler agent organization, i.e.,
tree, is preferred when degenerate
loops exist in the CG
BC4: A DAG is used to structure each
individual agent’s knowledge
BC5: Within each agent’s subdomain,
the JPD is consistent with the agent’s
belief. For shared nodes, the JPD
supplements each agent’s knowledge
with others’
Seven Constraints
1.
Each agent’s belief is
represented by Bayesian
probability
2.
The domain is decomposed into
subdomains with RIP
3.
Subdomains are organized into a
hyptertree structure
4.
The dependency structure of
each subdomain is represented
by a DAG
5.
The union of DAGs for all
subdomains is a connected DAG
6.
Each hyperlink is a d-sepset
7.
The JPD can be expressed as in
definition of MSBN
Theorem 18: Let Ψ be a hypertree over a directed graph
G=(V, E). For each hyperlink I which splits Ψ into 2
subtrees over U⊂V and W⊂V respectively, U \ I and W \
I are d-separated by I iff each hyperlink in Ψ is a dsepset
Proposition 14: Let a multiagent system be one that
observes BC 1 through BC 3. Then a junction tree
organization of agents must be used
Proposition 19: Let a multiagent system be constructed
following BC 1 through BC 4. Then it must be
structured as a hypertree MSDAG
Proof of Proposition 19:
From BC 1 through BC 4, it follows that each
subdomain should be structured as a DAG and
the entire domain should be structured as a
connected DAG (Proposition 17). The DAGs
should be organized into a hypertree
(Proposition 14). The interface between
adjacent DAGs on the hypertree should be a dsepset (Theorem 18). Hence, the multiagent
system should be structured as a hypertree
MSDAG (Definition 3)
Five Basic Commitments

BC1: Each agent’s belief is represented
by Bayesian probability

BC2: Ai and Aj can communicate
directly only with their intersecting
variables



BC3: A simpler agent organization, i.e.,
tree, is preferred when degenerate
loops exist in the CG
BC4: A DAG is used to structure each
individual agent’s knowledge
BC5: Within each agent’s subdomain,
the JPD is consistent with the agent’s
belief. For shared nodes, the JPD
supplements each agent’s knowledge
with others’
Seven Constraints
1.
Each agent’s belief is
represented by Bayesian
probability
2.
The domain is decomposed into
subdomains with RIP
3.
Subdomains are organized into a
hyptertree structure
4.
The dependency structure of
each subdomain is represented
by a DAG
5.
The union of DAGs for all
subdomains is a connected DAG
6.
Each hyperlink is a d-sepset
7.
The JPD can be expressed as in
definition of MSBN
Conclusion
Theorem 22: Let a multiagent system be
constructed following BC 1 through BC 5.
Then it must be represented as a MSBN
or some equivalent

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