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  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) d bc a e d bc a b a e a,b Original Graph Hypertree Hypernode Hyperlink

Background knowledge --- cont. Some useful definitions  Communication Graph In a graph with n hypernodes, associate each node with an agent A i and label it by V i Connect each pair of nodes V i and V j 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 2 V s.t., each element Q is called a cluster E is defined as

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

Background knowledge --- cont. d,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

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

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’ Proof of the logical implication

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’

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 P 2 (b|d=d 0 ) produced from s’ is identical to that produced from s, and so is the message P 2 (c|d=d 0 ) a,ba,c b,c,d a b c A0A0 A2A2 A1A1 Figure 1 Mas3: a multiagent system of 3 agents. a bc d

Proof: Denote P 2 (b=b 0 |d=d 0 ) from state s by P 2 (b 0 |d 0 ), P 2 ’(b=b 0 |d=d 0 ) from state s’ by P 2 ’(b 0 |d 0 ). P 2 (b 0 |d 0 ) can be expanded as: For P 2 (b|d 0 )=P 2 ’(b|d 0 ), we have: Similarly, Because P 2 ’(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=d 0 ) is distinct from P’(a|d=d 0 ) in general Proof: The following can be obtained from P and P’: If P(b,c|d 0 ) ≠ P’(b,c|d 0 ), then in general P(a|d 0 ) ≠P’(a|d 0 ) Because P(d|b,c) ≠P’(d|b,c), in general, it is the case that P(b,c|d 0 ) ≠P’(b,c|d 0 ). 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, P 2 (b|d=d 0 ) and P 2 (c|d=d 0 ) are insensitive to the initial states and hence the posteriors P 0 (a|d=d 0 ) 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,ba,c b,c,d a b c A0A0 A2A2 A1A1 Figure 1  Correct inference requires P(b,c|d 0 )  However, nondegenerate loop results in the passing of the marginals of P(b,c|d 0 ), i.e., P(b|d=d 0 ) and P(c|d=d 0 ) 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 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’

Proposition 17: Let a multiagent system over V be constructed following BC 1 through BC 4. Then each subdomain V i is structured as a DAG over V i 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

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’

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)

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’

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