Quantitative Provenance

1. Quantitative Provenance Using Bayesian Networks to Help Quantify the Weight of EvidenceIn Fine Arts Investigations A Case Study: Red Black and Silver

2. Outline • Probability Theory and Bayes’ Theorem • Likelihood Ratios and the Weight of Evidence • Decision Theory and its implementation: Bayesian Networks • Simple example of a BN: Why is the grass wet? • TaroniBayesian Network for trace evidence • The Bayesian Network for Red, Black and Silver • Stress testing: Sensitivity analysis • Recommendation for RBS

3. Probability 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 man’s mind.” — James Clerk Maxwell, 1850C Probability theory is nothing but common sense reduced to calculation.”— Laplace, 1819L

4. Probability 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, 1996J

5. Probability Theory • Probability theory forms the rules of reasoning • Using probability theory we can explore the logical consequences of our propositions • Probabilities can be updated in light of new evidence via Bayes theorem.

6. Bayesian Statistics • The basic Bayesian philosophy: Updated Knowledge Prior Knowledge × Data = A better understanding of the world Prior × Data = Posterior

7. The “Bayesian Framework” • Bayes’ Theorem to Compare Theories: • Ha = Theory A (the “prosecution’s” hypothesisAT) • Hb = Theory B (the “defence’s” hypothesisAT) • E = any evidence • I = any background information

8. The “Bayesian Framework” • Odd’s form of Bayes’ Rule: { { { Posterior odds in favour of Theory A Likelihood Ratio Prior odds in favour of Theory A Posterior Odds = Likelihood Ratio × Prior Odds

9. The “Bayesian Framework” • The likelihood ratio has largely come to be the main quantity of interest in the forensic statistics literature: • A measure of how much “weight” or “support” the “evidence” gives to Theory A relative to Theory BAT

10. The “Bayesian Framework” • Likelihood ratio ranges from 0 to infinity • Points of interest on the LR scale:

11. Decision Theory • Frame decision problem (scenario) • List possibilities and options • Quantify the uncertainty with available information • Domain specific expertise • Historical data if available • Combine information respecting the laws of probability to arrive at a decision/recommendation

12. Bayesian Networks • A “scenario” is represented by a joint probability function • Contains variables relevant to a situation which represent uncertain information • Contain “dependencies” between variables that describe how they influence each other. • A graphical way to represent the joint probability function is with nodes and directed lines • Called a Bayesian NetworkPearl

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

14. Bayesian Networks Pr(Sprinkler | Rain) Pr(Rain) Pr(Sprinkler) Pr(Rain) Pr(Grass Wet) Pr(Grass Wet | Rain, Sprinkler)

15. Bayesian Networks Pr(Rain) Pr(Sprinkler) Other probabilities are adjusted given the observation You observe grass is wet. Pr(Grass Wet)

16. Bayesian Networks • Likelihood Ratio can be obtained from the BN once evidence is entered • Use the odd’s form of Bayes’ Theorem: Probabilities of the theories after we entered the evidence Probabilities of the theories before we entered the evidence

17. Bayesian Networks • Areas where Bayesian Networks are used • Medical recommendation/diagnosis • IBM/Watson, Massachusetts General Hospital/DXplain • Image processing • Business decision support • Boeing, Intel, United Technologies, Oracle, Philips • Information search algorithms and on-line recommendation engines • Space vehicle diagnostics • NASA • Search and rescue planning • US Military • Requires software. Some free stuff: • GeNIe(University of Pittsburgh)G, • SamIam (UCLA)S • Hugin (Free only for a few nodes)H • gR R-packagesgR

18. Taroni Model for Trace Evidence • Taroni et al. have prescribed a general BN fragment that can model trace evidence transfer scenariosT: • H: Theory (Hypothesis) node • X: Trace associated with (a) “suspect” node • TS: Mediating node to allow for chance match between suspect’s trace and trace from an alternative source • T: Trace transfer node • Y: Trace associated with the “crime scene” node

19. Trace Evidence BN for RBS case • Theories are that Pollock or someone else associated with him in summer 1956 made the painting • The are two “suspects” • Use a Taroni fragment for each of: • Group of wool carpet fibers • Human hair • Polar bear hair • Use a modified Taroni fragment (no suspect node) for each of: • Beach grass seeds • Garnet

20. Trace Evidence BN for RBS case • Link the garnet and seeds fragment togetherdirectly • They a very likely to co-occur • Link all the fragments together with the Theory (Painter) node and a Location node

21. Trace Evidence BN for RBS case • Enter the evidence:

22. Sensitivity Analysis • Local sensitivityC • Posterior’s sensitivity to small changes in the • model’s parameters. Threshold > 1

23. Sensitivity Analysis • Global sensitivityC • Posterior’s sensitivity to large changes in the • model’s parameters. Threshold < 0.1 • 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”.

24. Conservative Recommendation • Considering the Likelihood ratio calculated with the “Red, Black and Silver” trace evidence network coupled with the sensitivity analysis results: • 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)

25. References • C Lewis Campbell. The Life of James Clerk Maxwell: With Selections from His Correspondence and Occasional Writings, Nabu Press, 2012. • L Pierre Simon Laplace. Théorie Analytique des Probabilités. 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 IntelligentSystems: 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, UffeKjaerulff, 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, Søren Højsgaard. A Common Platform for Graphical Models in R: The gRbase Package. J Stat Soft http://www.jstatsoft.org/v14/i17/, 2005.

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