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