1. Quantitative Provenance Using Bayesian Networks to Help Quantify the Weight of Evidence In 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? Taroni Bayesian 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, 1850 C Probability theory is nothing but common sense reduced to calculation.”— Laplace, 1819 L

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, 1996 J

5. 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. Probability Theory

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

7. The “Bayesian Framework” Bayes’ Theorem to Compare Theories: H a = Theory A (the “prosecution’s” hypothesis AT ) H b = Theory B (the “defence’s” hypothesis AT ) E = any evidence I = any background information

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

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

10. Likelihood ratio ranges from 0 to infinity The “Bayesian Framework” Points of interest on the LR scale: LRJeffreys Scale J < 1Evidence supports for Theory B 1 to 3Evidence barely supports Theory A 3 to 10Evidence substantially supports Theory A 10 to 30Evidence strongly supports Theory A 30 to 100Evidence very strongly supports Theory A > 100Evidence decisively supports Theory A LRKass-Raftery Scale KR < 1Evidence supports for Theory B 1 to 3Evidence barely supports Theory A 3 to 20Evidence positively supports Theory A 20 to 150Evidence strongly supports Theory A > 150Evidence very strongly supports Theory A

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 Network Pearl

13. Bayesian Networks (A Very!!) Simple example Wiki : 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 Sprinkler:was on was off Rain:yesnoyesno Grass Wet:yes99%90%80%0% no1%10%80%100% Rain:yesno Sprinkler :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)

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

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 before we entered the evidence Probabilities of the theories after 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-packages gR

18. Taroni Model for Trace Evidence Taroni et al. have prescribed a general BN fragment that can model trace evidence transfer scenarios T : 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 Use a Taroni fragment for each of: Group of wool carpet fibers Human hair Polar bear hair Theories are that Pollock or someone else associated with him in summer 1956 made the painting The are two “suspects” 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 together directly 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. Local sensitivity C Posterior’s sensitivity to small changes in the model’s parameters. Sensitivity Analysis Threshold > 1

23. Global sensitivity C Posterior’s sensitivity to large changes in the model’s 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

24. 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)

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 Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann Publishers, San Mateo, California, 1988. Wiki http://en.wikipedia.org/wiki/Bayesian_networkhttp://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/http://genie.sis.pitt.edu/ S http://reasoning.cs.ucla.edu/samiam/http://reasoning.cs.ucla.edu/samiam/ H http://www.hugin.com/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.http://www.jstatsoft.org/v14/i17/

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