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Sensitivity Analysis
Reference Bayesian Networks and Decision Graphs
Finn V. Jensen Expert Systems and Probabilistic Network Models
Enrique Castillo, Jose Manuel Gutierrez, and Ali D. Hadi Omniseer Project
Sensitivity Analysis
Given a Bayesian network and evidence e, and some hypotheses
Sensitivity to evidence
Sensitivity to parameter
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Sensitivity to evidence
Which evidence is in favor of /against/irrelevant for
Which evidence discriminate from ?
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Sensitivity to evidenceHow to measure the sensitivity
Normalized likelihoods
Bayes factors
Fraction of achieved probability
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Sensitivity to evidenceDefinition
Let e be evidence and h a hypothesis. Suppose that we want to investigate how sensitive the result p(h|e) is to the particular set e.
We say that evidence is sufficient if p(h|e’) is almost equal to p(h|e’). We then also say that e\e’ is redundant. The term almost equal can be made precise by selecting a threshold and requiring that . Note that is the
fraction between the two likelihood ratios.
e’ is minimal sufficient if it is sufficient, but no proper subset of e’ is so.
e’ is crucial if it is a subset of any sufficient set.
e’ is important if the probability of h changes too much without it. To be more precise, if , where is some threshold.
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Sensitivity to parameters
how much the posterior probability of some event of interest changes with respect to the value of some parameter in the Bayesian network
We assume that the event of interest is the value of a target variable. The parameter is either a conditional probability or an unconditional prior probability
Sensitivity to parametersTheorem and Corollaries
Theorm 1: Let BN be a Bayesian network over the universe U. Let t be a parameter and let e be evidence entered in BN. Then, assuming proportional scaling, we have
Proof: The probability of an instantiation (x1,…,xn) is
Note that all the parameters appearing in the above product are associated with different variables, and some of them may be specified numerically. Thus p(x1,…,xn) is a monomial of degree less than or equal to the number of symbolic nodes.
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Sensitivity to parametersTheorem and Corollaries
Corollary 1: Let BN be a Bayesian network over the universe U. Let t be a set of parameter for different distributions, and let e be evidence entered into BN. Then, assuming proportional scaling, P(e)(t) is a multi-linear polynomial over t
Proof: let t=(x,y). From the previous theorem, we have
If we have more than two parameters, we let t=(x,y), where y is a set of parameters. And repeat the arguments above.
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Sensitivity to parametersTheorem and Corollaries
Corollary 2: Let BN be a Bayesian network over the universe U. Let t be a set of parameters for different distributions. Let a be a state of and let e be evidence. Then P(a|e)(t) is a fraction of two multi-linear polynomials over t.
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Sensitivity to parametersOne-way sensitivity analysis
Let t be a parameter for BN and let e be evidence. Let a be a state of the target node. In one-way sensitivity analysis, we wish to determine p(e) and p(a,e) as functions of t.
Let t0 be the initial value of t.
Let t1 be the second value of t
Combing Corollary 2, we have
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Sensitivity Analysis in Our Project
Project Introduction
Value of Information
Sensitivity Analyzer
Surprise Detector
Bayesian Reasoning Service
Project Overview
BN FragmentsMatcher
Composer
Instantiated Fragments
Situation SpecificScenarios
Tagged messages
Modified Text
<Date>2002-09-20</Date> <Person>John Doe</Person>
<Place>London</Place>
…<Date>2002-09-27</Date>
<Person>John Doe</Person>
…
Bayesian NetworksDocuments
Messages
Events
Tasks
MassiveData
Bayesian Network Fragment Matching Example
1) Report Date: 1 April, 2003. FBI: Abdul Ramazi is the owner of the Select Gourmet Foods shop in Springfield Mall. Springfield, VA. (Phone number 703-659.2317). First Union National Bank lists Select Gourmet Foods as holding account number 1070173749003. Six checks totaling $35,000 have been deposited in this account in the past four months and are recorded as having been drawn on accounts at the Pyramid Bank of Cairo, Egypt and the Central Bank of Dubai, United Arab Emirates. Both of these banks have just been listed as possible conduits in money laundering schemes.
Partially- Instantiated
Bayesian Network Fragment
<Protege:Person rdf:about="&Protege;Omniseer_00135"….. Protege:familyName="Ramazi"Protege:givenName="Abdulla“rrdfs:label="Abdulla Ramazi"/>
…..
<Protege:Bank rdf:about="&Protege;Omniseer_00614"Protege:alternateName="Pyramid Bank of Cairo" rdfs:label="Pyramid Bank of Cairo">
<Protege:address rdf:resource="&Protege;Omniseer_00594"/>
<Protege:note rdf:resource="&Protege;Omniseer_00625"/>
</Protege:Bank>
….
<Protege:Report rdf:about="&Protege;Omniseer_00626" Protege:abstract="Ramazi's deposit in the past 4 months (1)" rdfs:label="Ramazi's deposit in the past 4 months (1)">
<Protege:reportedFrom rdf:resource="&Protege;Omniseer_00501"/>
<Protege:detail rdf:resource="&Protege;Omniseer_00602"/>
<Protege:detail rdf:resource="&Protege;Omniseer_00612"/>
</Protege:Report>
</rdf:RDF>
BN FragmentRepository
Bayesian Network Fragment Composition Example
. . . . . +
FragmentsSituation-Specific Scenario
Protégé overview
What is Protégé ?
A tool which allows the user to:
construct a domain ontology
customize data entry forms
enter data
OpenCyc overview
What is OpenCyc ?
o The open source version of the Cyc technology
o World's largest and most complete general knowledge base and commonsense reasoning engine
OpenCyc overview --- cont.
Where can we use OpenCyc ?
o speech understanding o database integration o rapid development of an ontology in a vertical
area o email prioritizing, routing, summarization, and
annotating
OpenCyc overview --- cont.
What does OpenCyc look like ?
OpenCyc overview --- cont.
More Detail Here
RDF overview
What is RDF?
Stands for Resource Description Framework
Recommended by the World Wide Web Consortium (W3C)
Model meta-data about the resources of the web
RDF overview --- Cont.
What does RDF file look like?
Basically, there are two kinds of file in RDF system
RDFS file
--- The schema file
RDF file
--- The file containing all instances
RDF overview --- Cont.
RDFS file<?xml version='1.0' encoding='ISO-8859-1'?>
<!DOCTYPE rdf:RDF [
<!ENTITY rdf 'http://www.w3.org/1999/02/22-rdf-syntax-ns#'> <!ENTITY a 'http://protege.stanford.edu/system#'> <!ENTITY Protege 'http://protege.stanford.edu/Protege#'> <!ENTITY rdfs 'http://www.w3.org/TR/1999/PR-rdf-schema-19990303#'>]>
<rdf:RDF xmlns:rdf="&rdf;" xmlns:a="&a;" xmlns:Protege="&Protege;" xmlns:rdfs="&rdfs;">
<rdfs:Class rdf:about="&Protege;Action" rdfs:label="Action">
<rdfs:subClassOf rdf:resource="&rdfs;Resource"/>
</rdfs:Class>
<rdfs:Class rdf:about="&Protege;Address" rdfs:label="Address">
<rdfs:subClassOf rdf:resource="&Protege;Location"/>
</rdfs:Class>
<rdfs:Class rdf:about="&Protege;Agency" rdfs:label="Agency">
<rdfs:subClassOf rdf:resource="&Protege;Legal_Entity"/>
</rdfs:Class>
RDF overview --- Cont.
RDF file
<rdf:RDF xmlns:rdf="&rdf;" xmlns:Protege="&Protege;" xmlns:rdfs="&rdfs;">
<Protege:Landmines rdf:about="&Protege;Omniseer_00088" Protege:mines_type="arid mines" Protege:quantity="16" rdfs:label="arid mines"/>
<Protege:Military_Reservation rdf:about="&Protege;Omniseer_00114" Protege:name="Camp George West" rdfs:label="Camp George West"/>
<Protege:Date rdf:about="&Protege;Omniseer_00118" Protege:day="20" Protege:full_date="4/20/2003" Protege:month="4"
Protege:year="2003" rdfs:label="4/20/2003"/>
Protégé GUI—Class Design
Protégé GUI—Instance View
Sensitivity Analysis in Our Project
Sensitivity analysis assesses how much the posterior probability of some event of interest changes with respect to the value of some parameter in the model
We assume that the event of interest is the value of a target variable. The parameter is either a conditional probability or an unconditional prior probability
If the sensitivity of the target variable having a particular value is low, then the analyst can be confident in the results, even if the analyst is not very confident in the precise value of the parameter
If the sensitivity of the target variable to a parameter is very high, it is necessary to inform the analyst of the need to qualify the conclusion reached or to expend more resources to become more confident in the exact value of the parameter
Example: Case Study #4Computing Sensitivity 2
Example: Case Study #4Computing Sensitivity
In the context of the information already acquired, i.e., travel to dangerous places, large transfers of money, etc., the parameter that links financial irregularities to being a suspect is much more important for assessing the belief in Ramazi being a terrorist than the parameter that links dangerous travel to being a suspect. The analyst may want to concentrate on assessing the first parameter precisely.
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
linear function 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 a parameter t by repeating the same computation with two values for the evidence set, viz. e and
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Algorithm and Implementation
Bucket Elimination
Goal-oriented Symbolic propagation
Differential Approach to Inference in BN
A Differential Approach to Inference in Bayesian Networks
Adnan Darwiche A Computational Architecture for N-way Sensitivity
Analysis of Bayesian Networks
Veerle M. H. Coupé, Finn V. Jensen, Uffe Kjærulff & Linda C. van der Gaag
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