Sensitivity Analysis Reference n Bayesian Networks and Decision Graphs Finn V. Jensen n Expert Systems and Probabilistic Network Models Enrique Castillo, Jose Manuel Gutierrez, and Ali D. Hadi n Omniseer Project

Sensitivity Analysis Given a Bayesian network and evidence e, and some hypotheses n Sensitivity to evidence n Sensitivity to parameter

Sensitivity to evidence n Which evidence is in favor of /against/irrelevant for n Which evidence discriminate from ?

Sensitivity to evidence How to measure the sensitivity n Normalized likelihoods n Bayes factors n Fraction of achieved probability

Sensitivity to evidence Definition 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.

Sensitivity to parameters n how much the posterior probability of some event of interest changes with respect to the value of some parameter in the Bayesian network n 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 parameters Theorem and Corollaries n 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 n Proof: The probability of an instantiation (x1,…,xn) is n 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.

Sensitivity to parameters Theorem and Corollaries n 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 n Proof: let t=(x,y). From the previous theorem, we have n If we have more than two parameters, we let t=(x,y), where y is a set of parameters. And repeat the arguments above.

Sensitivity to parameters Theorem and Corollaries n 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. n Proof: Corollary 1 and fundamental rule

Sensitivity to parameters One-way sensitivity analysis n 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. n Let t0 be the initial value of t. n Let t1 be the second value of t n Combing Corollary 2, we have

Sensitivity Analysis in Our Project n Project Introduction

Value of Information Sensitivity Analyzer Surprise Detector Bayesian Reasoning Service Project Overview BN Fragments Matcher Composer Instantiated Fragments Situation Specific Scenarios Tagged messages Modified Text John Doe London … John Doe … Bayesian Networks Documents Messages Events Tasks Massive Data

Bayesian Network Fragment Matching Example 1) Report Date: 1 April, FBI: Abdul Ramazi is the owner of the Select Gourmet Foods shop in Springfield Mall. Springfield, VA. (Phone number ). First Union National Bank lists Select Gourmet Foods as holding account number 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 ….. …. BN Fragment Repository

Bayesian Network Fragment Composition Example Fragments Situation-Specific Scenario

Protégé overview n 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 technologyCyc 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 prioritizing, routing, summarization, and annotating

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

OpenCyc overview --- cont. More Detail Here

RDF overview What is RDF? n Stands for Resource Description Framework n Recommended by the World Wide Web Consortium (W3C) n 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 n RDFS file --- The schema file n RDF file --- The file containing all instances

RDF overview --- Cont. RDFS file <!DOCTYPE rdf:RDF [ ]>

RDF overview --- Cont. RDF file

Protégé GUI—Class Design

Protégé GUI—Instance View

Sensitivity Analysis in Our Project n 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 n 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 n 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 n 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 #4 Computing Sensitivity 2

Example: Case Study #4 Computing 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 n Let the evidence be a set of findings: n Let t be a parameter in the situation-specific scenario n Then, [Castillo et al., 1997; Jensen, 2000] n α and β can be determined by computing P(e) for two values of t n 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 n Recall that n Then, n 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

Algorithm and Implementation n Bucket Elimination n Goal-oriented Symbolic propagation n Differential Approach to Inference in BN A Differential Approach to Inference in Bayesian Networks Adnan Darwiche n 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