Inference Algorithm for Similarity Networks

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Inference Algorithm for Similarity Networks

• Dan Geiger David Heckerman
• Presentation by Jingsong Wang
• USC CSE BN Reading Club
• 2008-03-17
• Contact wang82,mgv_at_cse.sc.edu

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The secured building story

• A guard of a secured building expects four types
  of persons to approach the building's entrance
  executives, regular workers, approved visitors,
  and spies. As a person approaches the building,
  the guard can note its gender, whether or not the
  person wears a badge, and whether or not the
  person arrives in a limousine. We assume that
  only executives arrive in limousines and that
  male and female executives wear badges just as do
  regular workers (to serve as role models).
  Furthermore, we assume that spies are mostly men.
  Spies always wear badges in an attempt to fool
  the guard. Visitors don't wear badges because
  they don't have one. Female-workers tend to wear
  badges more often than do male-workers.
• The task of the guard is to identify the type of
  person approaching the building.

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Definition of Similarity Network

• Distinguished Variable
• Hypothesis
• Cover
• A cover of a set of hypotheses H is a collection
  A1, . . . , Ak of nonempty subsets of H whose
  union is H.
• Each cover is a hypergraph, called a similarity
  hypergraph, where the Ai are hyperedges and the
  hypotheses are nodes.
• A cover is connected if the similarity hypergraph
  is connected.

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Definition of Similarity Network

• Similarity Network
• Let P(h, u1,. . . , un) be a probability
  distribution and A1,. . . , Ak be a connected
  cover of the values of h. A directed acyclic
  graph Di is called a local network of P
  associated with Ai if Di is a Bayesian network of
  P(h, v1,. . . , vm Ai) where v1,. . . ,
  vm is the set of all variables in u1,. . . ,
  un that help to discriminate the hypotheses in
  Ai. The set of k local networks is called a
  similarity network of P.

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A similarity network representation
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Definition of Similarity Network

• Subset Independence
• Hypothesis-specific Independence

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Definition of Similarity Network

• The practical solution for constructing the
  similarity hypergraph is to choose a connected
  cover by grouping together hypotheses that are
  similar'' to each other by some criteria under
  our control (e.g., spies and visitors).
• This choice tends to maximize the number of
  subset independence assertions encoded in a
  similarity network. Hence the name for this
  representation.

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Two Types of Similarity Networks

• helps to discriminate
• Related
• Relevant
• Define event e to be Ai
• A disjunction over a subset of the values of h

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Two Types of Similarity Networks

• Type 1
• A similarity network constructed by including in
  each local network Di only those variables u that
  satisfy related(u, h Ai) is said to be of
  type 1.
• Type 2
• relevant(u, h Ai)

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Two Types of Similarity Networks

• Theorem 1
• Let P(u1, un e ) be a probability
  distribution where U u1, un and e be a fixed
  event. Then, ui and uj are unrelated given e iff
  there exist a partition U1, U2 of U such that ui
  ? U1, uj ? U2, and P(U1, U2 e) P(U1 e)
  P(U2 e)

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Two Types of Similarity Networks

• Theorem 2
• Let P(u1,, un e) be a probability distribution
  where e is a fixed event. Then, for every ui and
  uj, relevant(ui, uj e) implies related(ui, uj
  e)

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Inference Using Similarity Networks

• The main task similarity networks are designed
  for is to compute the posterior probability of
  each hypothesis given a set of observations, as
  is the case in diagnosis.
• Under reasonable assumptions, the computation of
  the posterior probability of each hypothesis can
  be done in each local network and then be
  combined coherently according to the axioms of
  probability theory.

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Inference Using Similarity Networks

• Strictly Positive
• We will remove this assumption later at the cost
  of obtaining an inference algorithm that operates
  only on type 1 similarity networks and whose
  complexity is higher.

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Inference Using Similarity Networks

• The inference problem
• Compute P(hj v1,,vm)
• INFER procedure
• Two parameters a query, a BN

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Inference Using Similarity Networks
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Inference Using Similarity Networks

• Theorem 3
• Let P(h,u1,,un) be a probability distribution
  and A A1,, Ak be a partition of the values
  of h. Let S be a similarity network based on A.
  Let v1,,vm be a subset of variables whose value
  is given. There exists a single solution for the
  set of equations defined by Line 7 and 8 of the
  above algorithm and this solution determines
  uniquely the conditional probability
• P(h v1, , vm).
• Complexity

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Inferential And Diagnostic Completeness

• Inferential Complete
• Diagnostically Complete

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Inferential And Diagnostic Completeness

• Theorem 4
• (restricted inferential completeness)
• Theorem 5
• (Diagnostic completeness)

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Inferential And Diagnostic Completeness

• Hypothesis-specific Bayesian multinet of P
• Similarity network to Bayesian Multinet
  conversion

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Inferential And Diagnostic Completeness

• Hypothesis-specific Bayesian-Multinet Inference
  Algorithm
• For each hypothesis hi
• Bi INFER(P(v1,,vl hi), Mi)
• For each hypothesis hi
• Compute P(hi v1,,vl)