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On the Role of MSBN to Cooperative Multiagent Systems By Y. Xiang and V. Lesser

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

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On the Role of MSBN to Cooperative Multiagent Systems By Y. Xiang and V. Lesser

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  1. On the Role of MSBN to Cooperative Multiagent SystemsBy Y. Xiang and V. Lesser Presented by: Jingshan Huang and Sharon Xi

  2. 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 ?

  3. Motivation --- cont. • MSBNs (Multiply Sectioned Bayesian Networks) provide a solution • An effective and exact framework • But a set of constraints exist

  4. 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

  5. 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

  6. d a a a,b e c b b Hypertree Original Graph d a e c b Background knowledge --- cont. Hypernode • Definition of hypertree structure • 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) Hyperlink

  7. 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 if • Junction Graph A triplet (V, Ω, E) V is a non-empty set (the generating set) Ω is a subset of 2V s.t. , each element Q is called a cluster E is defined as

  8. Background knowledge --- cont. • Cluster Graph Let (V, Ω ,E) be a junction graph and , 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

  9. d,e d,e,i d,f d,f,h d d d d d d,h d d,i b,c,d b,c,d,i d,g d,g,h d d a,b a,e (a)Strong Degenerate Loop (b) Weak Degenerate Loop a e b b,c,d c,e c (c) Strong Nondegenerate Loop a,b,f a,e,f a,f e,f b,f b,c,d,f c,e,f c,f (d) Week Nondegenerate Loop Background knowledge --- cont.

  10. 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

  11. Seven Constraints • Each agent’s belief is represented by Bayesian probability • The domain is decomposed into subdomains with RIP • Subdomains are organized into a hyptertree structure • The dependency structure of each subdomain is represented by a DAG • The union of DAGs for all subdomains is a connected DAG • Each hyperlink is a d-sepset • The JPD can be expressed as in definition of MSBN

  12. 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

  13. 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’

  14. 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

  15. 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 • Each agent’s belief is represented by Bayesian probability • The domain is decomposed into subdomains with RIP • Subdomains are organized into a hyptertree structure • The dependency structure of each subdomain is represented by a DAG • The union of DAGs for all subdomains is a connected DAG • Each hyperlink is a d-sepset • The JPD can be expressed as in definition of MSBN

  16. 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 • Each agent’s belief is represented by Bayesian probability • The domain is decomposed into subdomains with RIP • Subdomains are organized into a hyptertree structure • The dependency structure of each subdomain is represented by a DAG • The union of DAGs for all subdomains is a connected DAG • Each hyperlink is a d-sepset • The JPD can be expressed as in definition of MSBN

  17. 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 • Each agent’s belief is represented by Bayesian probability • The domain is decomposed into subdomains with RIP • Subdomains are organized into a hyptertree structure • The dependency structure of each subdomain is represented by a DAG • The union of DAGs for all subdomains is a connected DAG • Each hyperlink is a d-sepset • The JPD can be expressed as in definition of MSBN

  18. A0 a,b a a,c A1 b c A2 b,c,d a b c d Figure 1 • 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) Mas3: a multiagent system of 3 agents.

  19. 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: For P2(b|d0)=P2’(b|d0), we have: Similarly, Because P2’(d|b,c) has4independent parameters but is constrained by only two equations, it has infinitely many solutions.

  20. 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’: 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???

  21. A0 a,b a a,c A1 b c A2 b,c,d Figure 1 Theorem 11: Message passing in Mas3 cannot be coherent in general, no matter how it is performed • Proof: • 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 • 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 • 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)

  22. 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

  23. Another conclusion without proof: A cluster graph with only degenerateloops can always be treated by first breaking the loops at appropriate separators. The resultant is a clustertree 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

  24. 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 • Each agent’s belief is represented by Bayesian probability • The domain is decomposed into subdomains with RIP • Subdomains are organized into a hyptertree structure • The dependency structure of each subdomain is represented by a DAG • The union of DAGs for all subdomains is a connected DAG • Each hyperlink is a d-sepset • The JPD can be expressed as in definition of MSBN

  25. 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 • Each agent’s belief is represented by Bayesian probability • The domain is decomposed into subdomains with RIP • Subdomains are organized into a hyptertree structure • The dependency structure of each subdomain is represented by a DAG • The union of DAGs for all subdomains is a connected DAG • Each hyperlink is a d-sepset • The JPD can be expressed as in definition of MSBN

  26. 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

  27. 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 • Each agent’s belief is represented by Bayesian probability • The domain is decomposed into subdomains with RIP • Subdomains are organized into a hyptertree structure • The dependency structure of each subdomain is represented by a DAG • The union of DAGs for all subdomains is a connected DAG • Each hyperlink is a d-sepset • The JPD can be expressed as in definition of MSBN

  28. 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

  29. Proofof 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)

  30. 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 • Each agent’s belief is represented by Bayesian probability • The domain is decomposed into subdomains with RIP • Subdomains are organized into a hyptertree structure • The dependency structure of each subdomain is represented by a DAG • The union of DAGs for all subdomains is a connected DAG • Each hyperlink is a d-sepset • The JPD can be expressed as in definition of MSBN

  31. 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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