On the Role of Multiply Sectioned Bayesian Networks to Cooperative Multiagent Systems

On the Role of Multiply Sectioned BayesianNetworks to Cooperative Multiagent Systems

Presented By: Yasser EL-Manzalawy

Reference

• Y. Xiang and V. Lesser, On the Role of Multiply Sectioned Bayesian Networks to Cooperative Multiagent Systems. IEEE Trans. Systems, Man, and Cybernetics-Part A, Vol.33, No.4, 489-501, 2003

Structure of the presentation

• Motivation

• 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

Motivation

• What’s an agent?– Program that takes sensory input from the

environment, and produces output actions that affect it.

– If the agent works in uncertain environment, then the agent can represent its believes about the environment as a Bayesian Network.

Motivation

• What’s a Multi-Agent System (MAS)?– Multi-Agent System is a set of agents and the

environment they interact.

Agent

Agent

Agent

Agent

AgentAgent

environment

Motivation

• In MAS, each agent can only observe and reason about a subdomain.

• Agents are assumed to cooperate in order to achieve a common global goal.

• For uncertain domains, agent believes can be represented as a BN (subnet).

Several Issues Arise!

Motivation

• How should the domain be partitioned into subdomains? • How should each agent represent its knowledge about a

subdomain?• How should the knowledge of each agent relate to that of

others? • How should the agents be organized in their activities?• What information should they exchange and how, in order

to accomplish their task with a limited amount of communication?

• Can they achieve the same level of accuracy in estimating the state of the domain as that of a single centralized agent?

Motivation

• MSBN provides a solution to these issues.

• Applying MSBN implies some technical constraints.

Are these constraints necessary?

Example

Introduction and Background

• Definition: A Bayesian Network is a triplet (V,G,P) where V is a set of domain variables, G is a DAG whose nodes are labeled by elements of V , and P is a joint probability distribution (jpd) over V, specified in terms of a distribution for each variable conditioned on the parents

of in G.

Vx

)(x


• Definition: Let G = (V,E) be a connected graph sectioned into subgraphs . Let the subgraphs be organized into an undirected tree

where each node is uniquely labeled by a and each link between and is labeled by the non-empty interface such that for each

and , is contained in each subgraph on the path between and in . Then is a hypertree over G. Each is a hypernode and each interface is a hyperlink.

)},({ iii EVG

iG

kG mG

mk VV

ji VV i j

iGjG iG



a, b

hypernode

hyperlink


• Definition: Let G be a directed graph such that a hypertree over G exists. A node contained in more than one subgraph with its parents in G is a d-sepnode if there exists at least one subgraph that contains . An interface is a d-sepset if every is a d-sepnode.

x

I)(x

Ix

)(x


• Definition: A hypertree MSDAG , where each is a DAG, is a connected DAG such that (1) there exists a hypertree over , and (2) each hyperlink in is a d-sepset.

ii GG

iG


• Note: DAGs in MSDAG tree may be multiply connected.


• A potential over a set of variables is an non-negative distribution of at least one positive parameter.

• One can always convert a potential into a conditional probability by dividing each potential value with a proper sum: an operation termed normalization.

• A uniform potential is one with all its potential values being 1.



• Definition: An MSBN is a triplet (V,G,P). is the domain where each is a set of variables. (a hypertree MSDAG) is the structure where nodes of each DAG are labeled by elements of . Let be a variable and be all the parents of in G. For each , exactly one of its occurrences (in a containing ) is assigned , and each occurrence in other DAGs is assigned a uniform potential. is the jpd, where each is the product of the potentials associated with nodes in . A triplet is called a subnet of M. Two subnets and are said to be adjacent if and are adjacent on the hypertree MSDAG

iiVV

ii GG

iG

iG

iG

iG

jG

iV

iV

iP

iS jS

x

)(x

))(/( xxP

)(}{ xx

ii PP

x x

),,( iii PGV


• Communication Graph

• Cluster Graph

• Junction Graph

• Junction Tree


Cluster

Separator

Introduction and Backgroundd,e

b,c,d

d,f

d,g

d

d

dd

d

d

(a) Strong Degenerate Loop

d,e,i

b,c,d,i

d,f,h

d,g,h

d,i

d

d,h

d

(b) Weak Degenerate Loop

a,b

b,c,d

a,e

c,e

b

a

e

c

(c) Strong Nondegenerate Loop

a,b,f

b,c,d,f

a,e,f

c,e,f

b,f

a,f

e,f

c,f

(d) Week Nondegenerate Loop

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’

Seven Constraints

1. Each agent’s belief is represented by Bayesian probability

2. The domain is decomposed into subdomains3. Subdomains are organized into a hyptertree

structure4. The dependency structure of each subdomain is

represented by a DAG5. The union of DAGs for all subdomains is a

connected DAG6. Each hyperlink is a d-sepset7. 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)

a,b a,c

b,c,d

a

b c

A0

A2

A1

Figure 1

Mas3: a multiagent system of 3 agents.

a

b c

d

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:

1

102100'2002000

'2

112110'2012010

'2

00'2

1

10210020020002

11211020120102

1

01020002

01120012

1

002

012

012002

002002

)c,b(P)c,b|d(P)c,b(P)c,b|d(P

)c,b(P)c,b|d(P)c,b(P)c,b|d(P1)d|b(P


)c,b(P)c,b|d(P)c,b(P)c,b|d(P1

)d,c,b(P)d,c,b(P

)d,c,b(P)d,c,b(P1

)d,b(P

)d,b(P1

)d,b(P)d,b(P

)d,b(P)d|b(P

)d,b(P

)d,b(P



002

012

102100'2002000

'2

112110'2012010

'2

For P2(b|d0)=P2’(b|d0), we have:

Similarly, )d,c(P

)d,c(P



002

012

012010'2002000

'2

112110'2102100

'2

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’:

cb

cb

dcbPcbaPdaP

dcbPcbaPdaP

.00

.00

)|,('),|(')|('

)|,(),|()|(

cb

cb

cbPcbdP

cbPcbdP

dP

cbPcbdPdcbP

cbPcbdP

cbPcbdP

dP

cbPcbdPdcbP

,0

0

0

00

,0

0

0

00

),(),|('

),(),|('

)('

),(),|(')|,('

),(),|(

),(),|(

)(

),(),|()|,(

If P(b,c|d0) ≠ P’(b,c|d0), then in general P(a|d0) ≠P’(a|d0)

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???

)d|c,b(P)c,b|a(P

)d|c,b(P)d,c,b|a(P

)d(P)d,c,b(P

)d,c,b(P)d,c,b,a(P

)d(P

)d,c,b,a(P)d|c,b,a(P

)d|c,b,a(P)d|a(Pc,b

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

a,b a,c

b,c,d

a

b c

A0

A2

A1

Figure 1

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)

Insight

• 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

Seven Constraints1. 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 Constraints1. Each agent’s belief is represented by

Bayesian probability













• 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:

• The connectedness is implied by Proposition 6

• 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















• 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 d-sepset

• 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 d-sepset (Theorem 18). Hence, the multiagent system should be structured as a hypertree MSDAG (Definition 3)















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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On the Role of Multiply Sectioned Bayesian Networks to Cooperative Multiagent Systems