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Quantitative Provenance. Using Bayesian Networks to Help Quantify the Weight of Evidence In Fine Arts Investigations A Case Study: Red Black and Silver. Outline. Probability Theory and Bayes’ Theorem Likelihood Ratios and the Weight of Evidence - PowerPoint PPT Presentation

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Quantitative Provenance

Using Bayesian Networks to Help Quantify the Weight of Evidence In Fine Arts Investigations

A Case Study: Red Black and Silver

OutlineProbability Theory and Bayes TheoremLikelihood Ratios and the Weight of EvidenceDecision Theory and its implementation: Bayesian NetworksSimple example of a BN: Why is the grass wet?Taroni Bayesian Network for trace evidenceThe Bayesian Network for Red, Black and SilverStress testing: Sensitivity analysisRecommendation for RBSProbability Theory

The actual science of logic is conversant at present only with things either certain [or] impossible. Therefore the true logic for this world is the calculus of Probabilities, which takes account of the magnitude of the probability which is in a reasonable mans mind. James Clerk Maxwell, 1850CProbability theory is nothing but common sense reduced to calculation. Laplace, 1819LProbability Theory

Probability: A particular scale on which degrees of plausibility can be measured.They are a means of describing the information given in the statement of a problem E.T. Jaynes, 1996JProbability theory forms the rules of reasoningUsing probability theory we can explore the logical consequences of our propositions

Probabilities can be updated in light of new evidence via Bayes theorem.

Probability Theory

Bayesian StatisticsThe basic Bayesian philosophy:Prior Knowledge Data =Updated KnowledgeA better understanding of the worldPrior Data = Posterior

The Bayesian Framework

Bayes Theorem to Compare Theories:Ha = Theory A (the prosecutions hypothesisAT)Hb = Theory B (the defences hypothesisAT) E = any evidenceI = any background information

Odds form of Bayes Rule:Posterior Odds = Likelihood Ratio Prior Odds{{{Posterior odds in favour of Theory ALikelihood RatioPrior odds in favour of Theory A

The Bayesian FrameworkThe likelihood ratio has largely come to be the main quantity of interest in the forensic statistics literature:

The Bayesian FrameworkA measure of how much weight or support the evidence gives to Theory A relative to Theory BATLikelihood ratio ranges from 0 to infinity

The Bayesian FrameworkPoints of interest on the LR scale:LRJeffreys ScaleJ< 1Evidence supports for Theory B1 to 3Evidence barely supports Theory A3 to 10Evidence substantially supports Theory A10 to 30Evidence strongly supports Theory A30 to 100Evidence very strongly supports Theory A> 100Evidence decisively supports Theory ALRKass-Raftery ScaleKR< 1Evidence supports for Theory B1 to 3Evidence barely supports Theory A3 to 20Evidence positively supports Theory A20 to 150Evidence strongly supports Theory A> 150Evidence very strongly supports Theory A

Decision TheoryFrame decision problem (scenario)List possibilities and optionsQuantify the uncertainty with available informationDomain specific expertiseHistorical data if availableCombine information respecting the laws of probability to arrive at a decision/recommendation

Bayesian NetworksA scenario is represented by a joint probability functionContains variables relevant to a situation which represent uncertain informationContain dependencies between variables that describe how they influence each other.

A graphical way to represent the joint probability function is with nodes and directed linesCalled a Bayesian NetworkPearl

Bayesian Networks(A Very!!) Simple exampleWiki:What is the probability the Grass is Wet?Influenced by the possibility of RainInfluenced by the possibility of Sprinkler actionSprinkler action influenced by possibility of RainConstruct joint probability function to answer questions about this scenario:Pr(Grass Wet, Rain, Sprinkler)

Bayesian NetworksSprinkler:was onwas onwas offwas offRain:yesnoyesnoGrass Wet:yes99%90%80%0%no1%10%80%100%Rain:yesnoSprinkler:was on40%1%was off60%99%Rain:yes20%no80%Pr(Sprinkler | Rain)Pr(Rain)Pr(Grass Wet | Rain, Sprinkler)

Pr(Sprinkler)Pr(Rain)Pr(Grass Wet)

Bayesian NetworksPr(Sprinkler)Pr(Rain)Pr(Grass Wet)

You observegrass is wet.Other probabilitiesare adjusted given the observation

Bayesian NetworksLikelihood Ratio can be obtained from the BN once evidence is enteredUse the odds form of Bayes Theorem:

Probabilities of the theories before we entered the evidenceProbabilities of the theories after we entered the evidence

Bayesian NetworksAreas where Bayesian Networks are usedMedical recommendation/diagnosisIBM/Watson, Massachusetts General Hospital/DXplainImage processingBusiness decision supportBoeing, Intel, United Technologies, Oracle, PhilipsInformation search algorithms and on-line recommendation enginesSpace vehicle diagnosticsNASASearch and rescue planningUS MilitaryRequires software. Some free stuff:GeNIe (University of Pittsburgh)G, SamIam (UCLA)SHugin (Free only for a few nodes)HgR R-packagesgR

Taroni Model for Trace EvidenceTaroni et al. have prescribed a general BN fragment that can model trace evidence transfer scenariosT: H: Theory (Hypothesis) nodeX: Trace associated with (a) suspect nodeTS: Mediating node to allow for chance match between suspects trace and trace from an alternative sourceT: Trace transfer nodeY: Trace associated with the crime scene node

Trace Evidence BN for RBS caseUse a Taroni fragment for each of:Group of wool carpet fibersHuman hairPolar bear hairTheories are that Pollock or someone else associated with him in summer 1956 made the paintingThe are two suspectsUse a modified Taroni fragment (no suspect node) for each of:Beach grass seedsGarnet

Trace Evidence BN for RBS caseLink the garnet and seeds fragment together directlyThey a very likely to co-occurLink all the fragments together with the Theory (Painter) node and a Location node

Trace Evidence BN for RBS caseEnter the evidence:

Local sensitivityC Posteriors sensitivity to small changes in the models parameters. Sensitivity Analysis

Threshold > 1 Global sensitivityC Posteriors sensitivity to large changes in the models parameters. Sensitivity Analysis

Parameter 24 is: the probability of a transfer of polar bear hair, given the painting was made outside of Springs by Pollock and he had little potential of shedding the hair.

Threshold < 0.1 Considering the Likelihood ratio calculated with the Red, Black and Silver trace evidence network coupled with the sensitivity analysis results:Conservative Recommendation

The physical evidence is more in support of the theory that Pollock made RBS vs. someone else made RBS: Strongly Very Strongly (Kass-Raftery Scale) Very Strongly Decisively (Jeffreys Scale)

ReferencesC Lewis Campbell. The Life of James Clerk Maxwell: With Selections from His Correspondence and Occasional Writings, Nabu Press, 2012.L Pierre Simon Laplace. Thorie Analytique des Probabilits. Nabu Press, 2010.J E. T. Jaynes. Probability Theory: The Logic of Science. Cambridge University Press, 2003.AT C. G. G. Aitken, F. Taroni. Statistics and the Evaluation of Evidence for Forensic Scientists. 2nd ed. Wiley, 2004.J Harold Jeffreys. Theory of Probability. 3rd ed. Oxford University Press, 1998.KR R. Kass, A. Raftery. Bayes Factors. J Amer Stat Assoc 90(430) 773-795, 1995.P Judea Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann Publishers, San Mateo, California, 1988.Wiki http://en.wikipedia.org/wiki/Bayesian_network T F. Taroni, A. Biedermann, S. Bozza, P. Garbolino, C. G. G. Aitken. Bayesian Networks for Probabilistic Inference and Decision Analysis in Forensic Science. 2nd ed. Wiley, 2014.C Veerle M. H. Coupe, Finn V. Jensen, Uffe Kjaerulff, and Linda C. van der Gaag. A computational architecture for n-way sensitivity analysis of Bayesian networks. Technical report, people.cs.aau.dk/~uk/papers/coupe-etal-00.ps.gz, 2000.G http://genie.sis.pitt.edu/ S http://reasoning.cs.ucla.edu/samiam/ H http://www.hugin.com/ gR Claus Dethlefsen, Sren Hjsgaard. A Common Platform for Graphical Models in R: The gRbase Package. J Stat Soft http://www.jstatsoft.org/v14/i17/, 2005.Fin

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