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## Adversary/Defender Model of Risk of Terrorist Attacks using Belief

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**Adversary/Defender Modelof Risk of Terrorist Attacks using**Belief SAND 2006-1470C Risk Symposium 2006 Risk Analysis for Homeland Security and Defense: Theory and Application March 20, 2006 John Darby Sandia National Laboratories jldarby@sandia.gov 505-284-7862**Acknowledgments**• Linguistic Model for Adversary • Logic Evolved Decision (LED) Methodology developed at Los Alamos National Laboratory by Terry Bott and Steve Eisenhawer • Belief/Plausibility measure of uncertainty added • Numerical Model for Defender • Uses work and suggestions from Jon Helton, Arizona State University • Belief/Plausibility with Fuzzy Sets • 1986 paper by Ronald Yager jldarby@sandia.gov 505-284-7862**Goal**• Apply Belief/Plausibility Measure of Uncertainty to Evaluating Risk from Acts of Terrorism • Why Belief/Plausibility? • We have considerable epistemic (state of knowledge) uncertainty • Terrorists acts are not random events, but we have considerable epistemic uncertainty for evaluating them jldarby@sandia.gov 505-284-7862**Random vs. Intentional**• Random event: Earthquake • Magnitude of earthquake independent of structures exposed to earthquake • Event is a “dumb” failure • Intentional Event: Terrorist Blow Up Building • Are the Consequences worth the effort to gather the resources to be able to defeat any security systems in place and destroy the building • Event is a choice by a thinking, malevolent adversary with significant resources jldarby@sandia.gov 505-284-7862**Terrorist Risk (Intentional)**Safety Risk (Random) Likelihood Likelihood Maximum Risk Maximum Risk (Uncertainty) (Uncertainty) Consequence Consequence Safety Risk vs. Terrorist Risk jldarby@sandia.gov 505-284-7862**Belief/Plausibility for Epistemic Uncertainty**• Toss a fair coin • Uncertainty is aleatory (random) • Probability Heads is ½ • Probability Tails is ½ • But if we do not know coin is fair • May be two-headed or two-tailed • Epistemic (state of knowledge) uncertainty • Insufficient information to assign Probability to Heads and Tails • Belief/Plausibility for Heads is 0/1 • Belief/Plausibility for Tails is 0/1 • With more information (actually tossing the coin) we can reduce Epistemic Uncertainty • For Fair Coin we cannot reduce aleatory uncertainty jldarby@sandia.gov 505-284-7862**Belief and Plausibility**• Belief / Plausibility form a Lower / Upper Bound for Probability • Belief is what probability will be • Plausibility is what probability could be • Similar to a Confidence Interval for a Parameter of a probability distribution; a confidence measure that parameter is in interval, but exactly where in interval is not known • Belief/Plausibility both reduce to Probability if Evidence is Specific Plausibility Probability is somewhere in [Belief, Plausibility] Interval Belief jldarby@sandia.gov 505-284-7862**Fuzzy Sets for Vagueness**• Consequences (Deaths) are “Major” • “Major” is fuzzy: between about 500 and about 5000 deaths jldarby@sandia.gov 505-284-7862**Adversary/Defender**• Adversary (them) • Defender (us) • Adversary and Defender each have different goals and different states of knowledge • Risk = Threat x Vulnerability x Consequence • Defender goal: Minimize Risk with available resources • Adversary goal: Maximize Consequence with available resources (working assumption) • Adversary is the Threat • Epistemic Uncertainty for Vulnerability and Consequence • Defender knows Vulnerability and Consequence • Epistemic Uncertainty for Threat jldarby@sandia.gov 505-284-7862**Scenario and Dependence**• Scenario defined as: Specific Target, Adversary Resources and Attack Plan • Resources are: attributes (numbers weapons, etc.) and knowledge • Risk for a Scenario • Risk = f x P x C • f is frequency of scenario (number per year) • P is probability scenario is successful • C is consequence • P is conditional on scenario (adversary resources and security in place, both physical security and intelligence gathering) • C is conditional on scenario (target) • Risk is scenario dependent • Adversary has choice (not random event) • Risk must consider millions of scenarios jldarby@sandia.gov 505-284-7862**Defender Model for a Scenario**• Risk = f x P x C • f, P, and C are random variables with uncertainty • Degrees of Evidence to f, P, C based on state of knowledge • Convolution using Belief/Plausibility Measure of Uncertainty jldarby@sandia.gov 505-284-7862**Example Result from Defender Model**Probability assumes evidence for an interval uniformly distributed over interval jldarby@sandia.gov 505-284-7862**Worst**Scenarios: Ranked By Decreasing Expected Value Best 0 106 Expected Value of Deaths per Year: f*P*C Defender Ranking of Scenarios • For Belief/Plausibility Expected Value is an Interval [Elow, Ehigh]. Reduces to point (Mean) for Probability • Rank by Ehigh, Subrank by Elow Scenario jldarby@sandia.gov 505-284-7862**Expected Value of**Likelihood: f 0 Expected Value of Deaths: P*C 106 Next Level Of Detail for Defender Ranking jldarby@sandia.gov 505-284-7862**Adversary Model**• Use surrogate Adversary (Special Forces) • Adversary has Choice • All Variables of concern must be “OK” or we will pick another scenario • Recruit Insider? Not unless already placed • Large Team? Concern about being detected by Intelligence • Uncertainty? • Door was green yesterday, is red today…What else changed? • Variables for Adversary Decision are Not all Numeric • Consequence = Deaths x Economic Damage x Fear in Populace x Damage to National Security x Religious Significance x ….. • Deaths and Economic Damage are numeric • Fear in Populace, Damage to National Security, and Religious Significance are not numeric jldarby@sandia.gov 505-284-7862**Adversary Model**• Linguistic Model • Develop Fuzzy Sets for Each Variable • Develop Approximate Reasoning Rule Base for Linguistic Convolution of Variables to Reflect Scenario Selection Decision Process (LANL LED process) • We are not the Adversary, we try to think like the Adversary • Considerable Epistemic Uncertainty • Use Belief/Plausibility Measure of Uncertainty Propagated up the Rule Base jldarby@sandia.gov 505-284-7862**Adversary Model**• Assume Adversary Goal is Maximize Expected Consequence • Expected Consequence ≡ P x C • Expected Consequence is Adversary estimate of Consequence, C, weighted by Adversary estimate of Probability of Success, P jldarby@sandia.gov 505-284-7862**Example of Adversary Model**• Rule Base and Variables • Expected Consequence = Probability Of (Adversary) Success x Consequence • Probability Of Success = Probability Resources Required Gathered Without Detection x Probability Information Required Can Be Obtained x Probability Physical Security System can be Defeated • Consequence = Deaths x Damage To National Security • Fuzzy Sets • Expected Consequence = {No, Maybe, Yes} • Probability Of Success = {Low, Medium, High} • Consequence = {Small, Medium, Large} • Probability Resources Required Gathered Without Detection = • {Low, Medium, High} • Probability Information Required Can Be Obtained = {Low, Medium, High} • Probability Physical Security System can be Defeated = {Low, Medium, High} • Deaths = {Minor, Moderate, Major, Catastrophic} • Damage To National Security = {Insignificant, Significant, Very Significant} jldarby@sandia.gov 505-284-7862**Example of Adversary Model**• Part of Example Rule Base jldarby@sandia.gov 505-284-7862**Example of Adversary Model**• Focal Elements (Evidence) for Particular Scenario • Deaths: 0.8 for {Major, Catastrophic} • 0.2 for {Moderate, Major} • Damage To National Security: 0.1 to {Insignificant, Significant} 0.9 to {Significant, Very Significant} • Probability Resources Required Obtained Without Detection: 0.7 to {Medium} 0.3 to {Medium, High} • Probability Information Required can Be Obtained: 0.15 to {Medium} 0.85 to {Medium, High} • Probability Physical Security System can be Defeated: 1.0 to {Medium, High} jldarby@sandia.gov 505-284-7862**Example of Adversary Model**jldarby@sandia.gov 505-284-7862**Example of Adversary Model**jldarby@sandia.gov 505-284-7862**Example of Adversary Model**jldarby@sandia.gov 505-284-7862**Adversary Ranking of Scenarios**• Defender thinking like Adversary Ranks by Plausibility • Rank scenarios based on the plausibility for the worst fuzzy set for expected consequence, “Yes” in the prior example, sub-ranked by plausibility of the next-worst fuzzy sets, “Maybe” and “No” in the prior example • Note: Actual Adversary using the Model would Rank by Belief • “We will not attempt a scenario unless we believe it will succeed”… Osama jldarby@sandia.gov 505-284-7862**Software Tools**• Numerical Evaluation of Risk for Defender • BeliefConvolution Java code (written by author) • RAMAS RiskCalc • Linguistic Evaluation for Adversary • LinguisticBelief Java code (written by author) • LANL LEDTools jldarby@sandia.gov 505-284-7862**“Combinatorics” Issue for Convolution with**Belief/Plausibility • Convolution using Belief/Plausibility must be done at the focal element level • Convolution of 20 variables each with 3 focal elements results in a variable with 320 focal elements • Need to Condense or Aggregate Degrees of Evidence for Large Scale Problems • Must “condense” or “aggregate” focal elements with repeated convolution of variables to reduce number of focal elements to manageable size jldarby@sandia.gov 505-284-7862**Evidence**Aggregated Evidence * Bins (linear or log10) Aggregation in BeliefConvolution Code jldarby@sandia.gov 505-284-7862**Aggregation in LinguisticBelief Code**• Aggregation is “Automatic” per Rule Base • Focal Elements for Inputs are on Fuzzy Sets For Input variables • Focal Elements for Output are on Fuzzy Sets of Output Variable • Happiness = Health x Wealth x Outlook on Life • Assume Health, Wealth, and Outlook on Life each have 3 focal elements: implies Health x Wealth x Outlook on Life has 27 focal elements • Happiness = {Not So Good, OK, Good} • Evidence on Happiness is aggregated on subsets of {Not So Good, OK, Good} so Happiness has at most 23 = 8 Focal Elements jldarby@sandia.gov 505-284-7862**Assumptions**• Focal Elements for Different Variables are Noninteracting for Convolution with Belief/Plausibility (independent for Probability) • SAND2004-3072 “Dependence in Probabilistic Modeling, Dempster-Shafer Theory, and Probability Boxes”, Ferson et al, Oct. 2004 • Adversary Goal is to Maximize Expected Consequence • Don’t like this goal? Change the rule base! jldarby@sandia.gov 505-284-7862**Future Work**• How Consider Millions of Scenarios? • Binning and Screening Criteria • Can reason at any level if honest about uncertainty • What are # deaths from a terrorist attack? [0, 6x109] • Use binning/filtering process at successively finer levels of detail, but capture uncertainty at each step • Dependence? Risk will always be dependent on scenarios • Implement Monte-Carlo Sampling with Belief/Plausibility (work by Helton et al) • Eliminates problem of dependence from a repeated variable that cannot be factored • Automate evaluation of large scale problems jldarby@sandia.gov 505-284-7862**Summary**• Adversary and Defender have Different States of Knowledge • Defender wants to Minimize Risk • Adversary wants to Maximize Expected Consequence (working assumption) • Adversary selection of Scenarios(s) is not random • Large Epistemic Uncertainty for Defender • Use Belief/Plausibility Measure instead of Probabililty Measure • Adversary selects Scenario(s) with “high” ranking • Defender thinks like adversary using linguistics • Linguistic variables modeled with Fuzzy Sets • Adversary/Defender Models Developed • Implemented in Java software jldarby@sandia.gov 505-284-7862