On the Role of Multiply Sectioned Bayesian Networks 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, , 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 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!

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.

Introduction and Background 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.

Introduction and Background

a, b hypernode hyperlink

Introduction and Background 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.

Introduction and Background 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.

Introduction and Background Note: DAGs in MSDAG tree may be multiply connected.

Introduction and Background 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.

Introduction and Background

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

Introduction and Background Communication Graph Cluster Graph Junction Graph Junction Tree

Introduction and Background Cluster Separator

Introduction and Background 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

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 subdomains 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 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 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 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 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

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.