CSCE 330 Programming Language Structures

UNIVERSITY OF SOUTH CAROLINA

Department of Computer Science and Engineering

Conflicts in Bayesian Networks

January 23, 2007

Marco Valtorta

mgv@cse.sc.edu

UNIVERSITY OF SOUTH CAROLINA

Department of Computer Science and Engineering

Example: Case Study #4Bayesian Network Fragment Matching

1) Report Date: 1April, 2003. FBI: AbdulRamazi is the owner ofthe Select GourmetFoods shop inSpringfield Mall.Springfield, VA. (Phonenumber 703-659.2317).First Union NationalBank lists SelectGourmet Foods asholding account number1070173749003. Sixchecks totaling $35,000have been deposited inthis account in the pastfour months and arerecorded as having beendrawn on accounts at thePyramid Bank of Cairo,Egypt and the CentralBank of Dubai, UnitedArab Emirates. Both ofthese banks have justbeen listed as possibleconduits in moneylaundering schemes.

Partially-InstantiatedBayesianNetworkFragment

<Protege:Personrdf:about="&Protege;Omniseer_00135"…..Protege:familyName="Ramazi"Protege:givenName="Abdulla“rrdfs:label="Abdulla Ramazi"/>

…..

<Protege:Bankrdf:about="&Protege;Omniseer_00614"Protege:alternateName="Pyramid Bank of Cairo" rdfs:label="Pyramid Bank of Cairo">

<Protege:addressrdf:resource="&Protege;Omniseer_00594"/>

<Protege:noterdf:resource="&Protege;Omniseer_00625"/>

</Protege:Bank>

….

<Protege:Reportrdf:about="&Protege;Omniseer_00626" Protege:abstract="Ramazi's deposit in the past 4months (1)" rdfs:label="Ramazi's deposit in the past 4 months(1)">

<Protege:reportedFromrdf:resource="&Protege;Omniseer_00501"/>

<Protege:detailrdf:resource="&Protege;Omniseer_00602"/>

<Protege:detailrdf:resource="&Protege;Omniseer_00612"/>

</Protege:Report>

</rdf:RDF>

BN FragmentRepository

UNIVERSITY OF SOUTH CAROLINA

Department of Computer Science and Engineering

Example: Case Study #4Bayesian Network Fragment Composition

. . . . .

Fragments

Situation-Specific Scenario

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Department of Computer Science and Engineering

Value of Information

•An item of information is useful if acquiring it leads to a betterdecision, that is, to a more useful action

•An item of information is useless if the actions that are takenafter acquiring it are no more useful than before acquiring it

•In particular, information is useless if the actions that are takenafter acquiring it are the same as before acquiring it

•In the absence of a detailed model of the utility of actions, thedecrease in uncertainty about a variable of interest is taken tobe a proxy for the increase in utility: the best item ofinformation to acquire is the one that reduces the most theuncertainty about a variable of interest

•Since the value of the new item of information is not known, weaverage over its possible values

•Uncertainty is measured by entropy. Reduction in uncertainty ismeasured by reduction in entropy

UNIVERSITY OF SOUTH CAROLINA

Department of Computer Science and Engineering

Example: Case Study #4Computing Value of Information and Surprise

This is the output

of the VOI program on a

situation-specific scenario

for Case Study #4 (Sign

of the Crescent).

Variable Travel (which

represents suspicious

travel) is significant

for determining the

state of variable Suspect

(whether Ramazi is

a terrorist), even

when it is already

known that Ramazi

has performed suspicious

banking transactions.

Ramazi performedillegal bankingtransactions

Is Ramazi a terrorist?

Would it help to know whether he traveled to sensitive locations?

Yes

UNIVERSITY OF SOUTH CAROLINA

Department of Computer Science and Engineering

Value of Information: Formal Definition

•Let V be a variable whose value affects the actions to be takenby an analyst. For example, V indicates whether a bomb isplaced on a particular airliner

•Let p(v) be the probability that variable V has value v.

•The entropy of V is:

•Let T be a variable whose value we may acquire (byexpending resources). For example, T indicates whether apassenger is a known terrorist.

•The entropy of V given that T has value t is:

•The expected entropy of V given T is:

•The value of information is:

UNIVERSITY OF SOUTH CAROLINA

Department of Computer Science and Engineering

Surprise Detection

•Surprise is the situation in which evidence (a set of findings)and a situation-specific scenario are incompatible

•Since situation-specific scenarios are Bayesian networks, itis very unusual for an outright inconsistency to occur

•In some cases, however, the evidence is very unlikely in agiven scenario; this may be because a rare case has beenfound, or because the scenario cannot explain the evidence

•To distinguish these two situations, we compare theprobability of the evidence in the situation-specific scenarioto the probability of the evidence in a scenario in which allevents are probabilistically independent and occur with thesame prior probability as in the situation-specific scenario

UNIVERSITY OF SOUTH CAROLINA

Department of Computer Science and Engineering

The VALUE OF INFORMATION of the test node C for the target node A is 0.0

Parsing the XMLBIF file 'ssn.xml' ...

done!

PROBABILITY FOR JOINT FINDINGS = 5.0E-4

Prior probability for NODE: Suspicious Person=yes is 0.01

Prior probability for NODE: Unusual Activities=yes is 0.0656

Prior probability for NODE: Stolen Weapons=yes is 0.05

PROBABILITY FOR INDIVIDUAL FINDINGS = 3.28E-5

No conflict was detected.

This shows the output of the surprise detection program. In this case,

the user is informed that no conflict is detected, i.e., the scenario is

likely to be a good interpretive model for the evidence received

Example: Case Study #4Computing Surprise

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Department of Computer Science and Engineering

Surprise Detection: Formal Definition

•Let the evidence be a set of findings:

•The probability of the evidence in the situation-specific scenariois where is the distribution represented in thesituation-specific scenario

•The probability of the evidence in the model in which allvariables are independent is

•The evidence is surprising if

•The conflict index is defined as

•The probability under that is greater than is

•Proof [Laskey, 1991]:

•If the conflict index is high, it is unlikely that the findings couldhave been generated by sampling the situation-specific scenario

•It is reasonable to inform the analyst that no good explanatorymodel of the findings exists, and we are in the presence of anovel or surprising situation

UNIVERSITY OF SOUTH CAROLINA

Department of Computer Science and Engineering

The Independent Straw Model

•In the absence of conflict, the joint probability of all evidence variables isgreater than the product of the probabilities of each evidence variable. This isnormally the case, because P(x|y) > P(x), and P(x,y) = P(x|y)P(y).

UNIVERSITY OF SOUTH CAROLINA

Department of Computer Science and Engineering

Straw Models in Diagnosis

A bipartite straw model is obtained by the elimination of some variablesfrom a given model. In diagnosis by heuristic classification, one candivide variables into three sets: Target, Evidence, and Other

UNIVERSITY OF SOUTH CAROLINA

Department of Computer Science and Engineering

How to Compute the Conflict Index (I)

•The marginalprobability ofeach finding isthe normalresult of anyprobabilitycomputationalgorithm

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Department of Computer Science and Engineering

How to Compute the Conflict Index (II)

•The probabilityof the evidenceis a bi-productof probabilityupdatecomputed usingthe variableelimination orjunction treealgorithms

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Department of Computer Science and Engineering

P(e) from the Variable Elimination AlgorithmP(e) from the Variable Elimination Algorithm

Bucket :

Bucket :

Bucket :

Bucket :

Bucket :

Bucket :

Bucket :

Bucket :

P(|)

P(|)*P(), =“yes”

P(|,)

P(|,), =“yes”

P(| =“yes”, =“yes”) = X\ {} (P(|)* P(|)* P(|,)* P(|,)* P()*P(|)*P(|)*P())

P(|)

P(|)*P()

H()

H()

H(,)

H(,,)

H(,,)

H()

H(,)

P(| =“yes”, =“yes”)

Hn(u)=xnПji=1Ci(xn,usi)

P (e) = 1- k, where k isa normalizing constant

















UNIVERSITY OF SOUTH CAROLINA

Department of Computer Science and Engineering

Sensitivity Analysis

•Sensitivity analysis assesses how much the posteriorprobability of some event of interest changes with respectto the value of some parameter in the model

•We assume that the event of interest is the value of atarget variable. The parameter is either a conditionalprobability or an unconditional prior probability

•If the sensitivity of the target variable having a particularvalue is low, then the analyst can be confident in the results,even if the analyst is not very confident in the precise valueof the parameter

•If the sensitivity of the target variable to a parameter isvery high, it is necessary to inform the analyst of the needto qualify the conclusion reached or to expend moreresources to become more confident in the exact value ofthe parameter

UNIVERSITY OF SOUTH CAROLINA

Department of Computer Science and Engineering

Example: Case Study #4Computing Sensitivity

This is the output of the Sensitivity Analysis program on a situation-specific scenario for Case Study#4.

In the context of the information already acquired, i.e., travel to dangerous places, large transfers ofmoney, etc., the parameter that links financial irregularities to being a suspect is much moreimportant for assessing the belief in Ramazi being a terrorist than the parameter that links dangeroustravel to being a suspect. The analyst may want to concentrate on assessing the first parameterprecisely.

UNIVERSITY OF SOUTH CAROLINA

Department of Computer Science and Engineering

Sensitivity Analysis: Formal Definition

•Let the evidence be a set of findings:

•Let t be a parameter in the situation-specific scenario

•Then, [Castillo et al., 1997; Jensen, 2000]

•α and β can be determined by computing P(e) for two values of t

•More generally, if t is a set of parameters, then P(e)(t) is a linearfunction in each parameter in t, i.e., it is a multi-linear function of t

•Recall that

•Then,

•We can therefore compute the sensitivity of a target variable V to aparameter t by repeating the same computation with two values for theevidence set, viz. e and