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Sensitivity Analysis

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.

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Sensitivity Analysis

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  1. 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

  2. Sensitivity Analysis Given a Bayesian network and evidence e, and some hypotheses • Sensitivity to evidence • Sensitivity to parameter

  3. Sensitivity to evidence • Which evidence is in favor of /against/irrelevant for • Which evidence discriminate from ?

  4. Sensitivity to evidenceHow to measure the sensitivity • Normalized likelihoods • Bayes factors • Fraction of achieved probability

  5. 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.

  6. 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

  7. 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.

  8. 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.

  9. 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. • Proof: Corollary 1 and fundamental rule

  10. 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

  11. Sensitivity Analysis in Our Project • Project Introduction

  12. Events Messages Tasks Massive Data Documents <Date>2002-09-20</Date> <Person>John Doe</Person> <Place>London</Place> … <Date>2002-09-27</Date> <Person>John Doe</Person> … Bayesian Networks Modified Text Tagged messages Instantiated Fragments Bayesian Reasoning Service BN Fragments Situation Specific Scenarios Value of Information Sensitivity Analyzer Surprise Detector Project Overview Matcher Composer

  13. 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. <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> Partially- Instantiated Bayesian Network Fragment BN FragmentRepository

  14. Bayesian Network Fragment Composition Example . . . . . + Fragments Situation-Specific Scenario

  15. Protégé overview • What is Protégé ? A tool which allows the user to: construct a domain ontology customize data entry forms enter data

  16. OpenCyc overview What is OpenCyc ? • The open source version of the Cyc technology • World's largest and most complete general knowledge base and commonsense reasoning engine

  17. OpenCyc overview --- cont. Where can we use OpenCyc ? • speech understanding • database integration • rapid development of an ontology in a vertical area • email prioritizing, routing, summarization, and annotating

  18. OpenCyc overview --- cont. What does OpenCyc look like ?

  19. OpenCyc overview --- cont. More Detail Here

  20. 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

  21. 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

  22. 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>

  23. 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"/>

  24. Protégé GUI—Class Design

  25. Protégé GUI—Instance View

  26. 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

  27. Example: Case Study #4Computing Sensitivity 2

  28. 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.

  29. 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

  30. 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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