Y. Xiang

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Multiagent Distributed Interpretation With
Multiply Sectioned Bayesian Netowrks

• Y. Xiang
• Department of Computer Science
• University of Regina
• Regina, Saskatchewan, Canada

International Workshop on Multi-Agent Systems,
1997
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Recent Advances
Some slides containing formal results are labeled
with in the title fields. These slides may be
skipped if only an intuitive tutorial is desired.
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Common approaches in MAS

• Logic-based Wooldridge 94, Rao and
  Georgeff 95, Halpern et al. 96.
• Game-theoretic Rosenschein and Zlotkin 94.
• Economic Wellman 92.
• Decision-theoretic Gmytrasiewicz and
  Durfee 95.
• Truth maintenance (default-reasoning)-based
  Mason and Johnson 89, Huhns and Bridgeland 91.
• Distributed search-based Yokoo et al.
  92.

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MAS Area of Focus

• Task distributed interpretation
• Producing higher level descriptions of the
  environment from distributed sensing without
  centralizing the evidence.
• Examples of distributed interpretation systems
• Sensor networks.
• Medical diagnosis by multiple specialists.
• Trouble shooting complex artifacts.
• Distributed image interpretation.

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Recent Advance

• Foundationa probabilistic framework for
  multiagent distributed interpretation based on
  multiply sectioned Bayesian networks (MSBNs).
• Advance
• Distributed representation of uncertainty
  knowledge that is consistent with probability
  theory.
• Distributed inference that ensures global
  consistency.
• Agents can be developed by independent designers.
  
• Internal structure/knowledge of agents remains
  private.
• Controversial aspects
• Are cooperative MASs important?
• Is homogeneous knowledge representation necessary?

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Foundations
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Background

• Common approaches for distributed interpretation
  are based on logic (e.g., blackboard) or default
  reasoning (e.g., DATMS and DTMS).
• Logic-based approaches do not have coherent
  mechanism to deal with uncertain knowledge.
• Default reasoning treats uncertain knowledge as
  believed until there is reason to believe
  otherwise and not as believed to a certain
  degree.
• However, decisions often involve tradeoffs and
  comparison of strength of belief on states of
  world or outcomes of actions is thus necessary.

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Background

• Substantial progress has been made in uncertain
  inference using Bayesian networks (BNs) Pearl
  88.
• Dependencies of domain variables are represented
  by a DAG.
• Strength of dependencies is quantified by an
  associated jpd.
• The jpd is interpreted as the degree of belief of
  an agent.
• Many effective inference algorithms have been
  developed.
• A single-agent paradigm is commonly assumed
• A single processor accesses a single global BN,
  updates the jpd as evidence becomes available,
  and answers queries.
• This research advances the DTMS approach with a
  representation of agents degree of belief
  consistent with the probability theory, and
  advances the single-agent BN approach with a
  multiagent paradigm and distributed inference
  algorithm.

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Bayesian Networks (BN)

• Defi A BN is a triplet (N,D,P) where
• N is a set of random variables,
• D is a directed acyclic graph (DAG) whose nodes
  are labeled by elements of N,
• P is a jpd over N specified by probability
  distributions of each node x in D conditioned on
  its parents parn(x) in D.
• D expresses dependency relations among elements
  of N.
• A variable is independent of its non-descendents
  given its parents.
• Hence P can be expressed as
  P(N) ?
  x? N P(x parn(x)).

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A Trivial Example BN L S 88
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Multiply Sectioned Bayesian Networks (MSBNs)

• Single-agent oriented MSBN Xiang et al., CI93,
  AIM93
• A set of Bayesian subnets that collectively
  define a BN.
• Interface b/w subnets renders them conditionally
  independent.
• Top level structure is a hypertree.
• Compiled into a linked junction forest (LJF) for
  inference.
• Coherent inference operations are defined for a
  LJF.

A MSBN (left) and its LJF (right)
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The d-sepset Interface b/w Subnets in a MSBN

• Defi Let Di(Ni,Ei) (i1,2) be two DAGs such
  that their union D is a DAG. IN1?N2 is a
  d-sepset b/w D1 and D2 if for every x?I with its
  parents parn(x) in D, either parn(x)?N1 or
  parn(x) ?N2. D is said to be sectioned into
  D1,D2.
• Theorem Let a DAG D(N,E) be sectioned into
  D1,,Dk and IijNi ?Nj be the d-sepset b/w Di
  and Dj. Then for each i, ? j Iij d-separates
  Pearl88 Ni\ ?j Iij from N\Ni.
• Semantics If D represents the dependence
  relations among elements of N, then d-sepset
  ensures that variables in a subnet are
  independent of other variables given the
  d-sepsets of the subnet.

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Hypertree MSDAG Top Level Structure of a MSBN

• Defi Let D be the union of Di (i1,,n) where
  each Di is a connected DAG. D is a hypertree
  MSDAG if it is a DAG built by the following
  procedure
• Start with an empty graph. Recursively add a DAG
  Di called a hypernode, to the existing MSDAG
  subject to the constraints
• d-sepset For each Dj (jltk), Ijk Nj?Nk is a
  d-sepset when the two DAGs are isolated.
• Local covering There exists Di (Iltk) such that,
  for each Dj (jltkj?i), Ijk ? Ni . For such Di,
  Iik is called the hyperlink b/w hypernodes Di and
  Dk.
• Semantics Each hyperlink renders the two parts
  of the MSBN that it connects conditionally
  independent.

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Compilation of a MSBN into a Linked Junction
Forest

• Major steps in compiling a LJF
• Convert each DAG (hypernode) into a chordal graph
  such that all dependence relations are preserved.
• Express the chordal graph as a junction tree (JT)
  of cliques.
• Convert each hyperlink (d-sepset) into linkages,
  each of which is a subset of the d-sepest.
• Convert conditional probability distributions in
  each subnet to belief tables of the corresponding
  JT and d-sepset.
• Let B(Ni) be the belief table on a hypernode and
  B(Ij) be the belief table on a hyperlink. The
  joint system belief (JSB) of a LJF is ? i B(Ni) /
  ? j B(Ij).
• Theorem JSB of a LJF is equivalent to jpd of its
  MSBN.
• Since each subnet is organized as a tree, a LJF
  is an equivalent but more effective data
  structure for inference computation.

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Inference in a Single-agent Oriented MSBN

• Inference using LJF of a MSBN
• Queries of a single user are focused on a single
  JT at a time, where a query has the form what is
  the probability of event A given that B has
  occured?
• Evidence can be entered incrementally to the JT
  and queries are answered by local computation
  only .
• As the user shifts attention to another JT,
  belief propagation is performed only along the
  hyperpath to the target JT in the hypertree.
• Queries can then be entered at the target JT as
  above.
• Theorem After finite times of attention shifts,
  answers to queries computed locally are idential
  to what would be obtained from an equivalent
  homogeneous BN.

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What Can be Gained y Using a MSBN?

• If a domain consists of loosely coupled
  subdomains, then...
• Knowledge acquisition is natural and modular
  Subnet can be built one at a time.
• Inference requires only local computation.
  Attention shift uses only hyperpath. Hence
  computation is more efficient
• Answers to queries are the same as from a
  homogeneous BN.

Structure of a general MSBN (left) and the
corresponding hypertree
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Why Using MSBNs for Distributed Interpretation?

• Representation of MSBNs is modular.
• Inference in MSBNs is coherent.
• The framework of MSBNs is general.

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Major Issues in Extending MSBNs to MASs

• What is the semantics of a subnet?
• What is the semantics of the jpd of the MSBN?
• How do we build the system by multiple agent
  developers?
• How do we ensure a correct overall structure
  while protecting the know-how of each developer?
• How do we ensure a coherent inference?
• What difference does a multi-agent MSBN make
  relative to a MAS not organized into a MSBN?

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What is the semantics of a subnet?

• In a single-agent MSBN
• The MSBN represents multiple perspectives of a
  domain hold by a single agent.
• Each subnet represents one such perspective.
• In a multi-agent MSBN Xiang, AIJ96
• The MSBN represents multiple agents in a domain,
  each of which holds one perspective.
• Each subnet represents one agent's perspective of
  the domain.

An agent model
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What is the semantics of the jpd in a MSBN?

• If the distribution of each subnet represents one
  agent's belief, whose belief does the jpd of the
  MSBN represent?
• Example a computer system.
• It processes information coherently as a whole.
• Its components are supplied by different vendors.
• Observation
• As long as vendors follow a protocol in designing
  component interfaces, the system functions as if
  it follows a single will.

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What is the semantics of the jpd in a MSBN?

• Another example a patient sees a doctor.
• Patient tells what doctor needs to know for
  diagnosis.
• After doctor reaches a diagnosis, he prescribes a
  therapy which patient follows.
• Observation
• Doctor does not experience symptoms.
• Patient does not understand how diagnosis is
  reached.
• A coherent belief is demonstrated on symptoms
  (used by doctor to reach diagnosis) and the
  diagnosis (therapy is followed by patient).

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What is the semantics of the jpd in a MSBN?

• Due to the way the jpd of a MSBN is defined,
  there exists a unique jpd Xiang, AIJ96 such
  that
• its marginalization to each subnet is identical
  to the distribution of the subnet
• adjacent subnets are conditionally independent
  given their interface.
• Implication If agents are (1) cooperative, (2)
  independent conditioned on interface, and (3)
  initially consistent, then the jpd of the MSBN
  represents a unique collective belief
• identical to each agent's belief within its
  subdomain,
• and supplemental to its belief outside its
  subdomain.

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How do we ensure coherent inference?

• Issue arising
• In a single-agent MSBN, evidence is entered one
  subnet at a time.
• In a multi-agent MSBN, evidence are entered
  asynchronously at multiple subnets in parallel.
• Solution extended inference operations Xiang,
  AIJ96.

CommunicateBelief
CollectNewBelief
DistributeBelief
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How do we ensure coherent inference?

• CollectNewBelief Initiated at an agent to
  activate an inward propagation towards the agent.

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How do we ensure coherent inference?

• DistributeBelief Initiated at an agent to
  activate an outward propagation.

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How do we ensure coherent inference?

• Theorem After CommunicateBelief, answers to
  queries from any agent is identical to what is
  obtained from an equivalent homogeneous BN.
• Implication Distribution causes no loss of
  coherence.
• Complexity of inference computation
• Inference at one agent O(k 2m), where m is the
  maximal size of a clique and k is the number of
  cliques in the JT.
• CommunicateBelief O(t g k 2m), where t is the
  number of agents and g is the maximal number of
  linkages in a hyperlink.

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How do we ensure coherent inference?

• Theorem Between successive CommunicateBeliefs,
  answers to queries from any agent X is identical
  to what would be obtained in an equivalent
  homogeneous BN where only evidence in the bottom
  are entered.

Evidence to A
Evidence to A
...
...
Evidence to W
Evidence to W
Evidence to X
Evidence to X
Evidence to Y
Evidence to Y
Evidence to Z
Evidence to Z
t
CommunicateBelief
CommunicateBelief
t
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How to build a MSBN by multiple developers?

• How to ensure system coherence without disclosing
  structure and distribution of individual subnets?
• It is possible if
• the interface of each subnet renders it
  conditionally independent of others and
• adjacent agents agree on an initial belief of
  their interface.
• Solution Xiang, AI96
• A single integrater with the knowledge of agents
  interface puts agents into a hypertree.
• Agents negotiate to achieve initial belief on
  interface.

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Global structure vs each agent's know-how

• Structures of subnets in a MSBN collectively
  define a directed acyclic graph (DAG).
• Local acyclicity doesn't warrant global
  acyclicity.

• Algorithms to test acyclicity based on
  topological sorting are well known. However, a
  central representation of the graph is assumed.

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Global structure vs each agent's know-how

• If each subnets structure is unknown to others,
  how can we ensure acyclicity of the MSBN?
• A distributed algorithm has been developed that
  has the following features Xiang, FLAIRS96
• Each agent provides only info on whether a shared
  node has a parent or a child in its DAG, plus
  some flag info.
• The acyclicity of the MSBN can be correctly
  determined.

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What if agents are not organized into a MSBN?

• Belief propagation in a MSBN proceeds along
  hypertree in a regulated fashion.

• What happens otherwise?

• Circular evidence propagation causes no problem
  if agents are logical.
• But it causes false belief if agents knowledge
  is uncertain.

Agent X
Agent Y
Agent W
Agent Z

• Not knowing message from Y is based on evidence
  originated from itself, Z counts the same info
  twice.

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Prospects
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Prospects for distributed interpretation

• A framework is provided for tasks that rely on
  uncertain knowledge and distributed inference
  without sacrifice of coherence in the
  interpretation.
• The framework protects individual agent
  developers know-how and hence encourages
  cooperation of many agent developers in building
  MASs in large and complex domains.
• Ex. Systems for trouble-shooting complex
  artifacts.

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Prospects for distributed interpretation

• The framework suggests standardization of agent
  interfaces in large and complex domains where
  knowledge sources are naturally distributed and
  separately owned.
• The framework suggests future research
  directions
• Dynamic formulation of multiagent MSBNs.
• Incorporation of decision making.
• Incorporation of temporal inference.