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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

• 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

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

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

• 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

Bayesian Statistics

• The basic Bayesian philosophy:

Prior Knowledge × Data = Updated Knowledge

A better understanding of the world

Prior × Data = Posterior

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

• 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”

• 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 BAT

• Likelihood ratio ranges from 0 to infinity

The “Bayesian Framework”

• Points of interest on the LR scale:

LR Jeffreys ScaleJ

< 1 Evidence supports for Theory B

1 to 3 Evidence barely supports Theory A

3 to 10 Evidence substantially supports Theory A

10 to 30 Evidence strongly supports Theory A

30 to 100 Evidence very strongly supports Theory A

> 100 Evidence decisively supports Theory A

LR Kass-Raftery ScaleKR

< 1 Evidence supports for Theory B1 to 3 Evidence barely supports Theory A

3 to 20 Evidence positively supports Theory A20 to 150 Evidence strongly supports Theory A

> 150 Evidence very strongly supports Theory A

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

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

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)

Bayesian Networks

Sprinkler: was on was on was off was off  Rain: yes no yes no

Grass Wet: yes 99% 90% 80% 0%no 1% 10% 80% 100%

  Rain: yes noSprinkler: was on 40% 1%

was off 60% 99% Rain: yes 20%no 80%

Pr(Sprinkler | Rain)

Pr(Rain)

Pr(Grass Wet | Rain, Sprinkler)

Pr(Sprinkler) Pr(Rain)

Pr(Grass Wet)

Bayesian Networks

Pr(Sprinkler) Pr(Rain)

Pr(Grass Wet)

You observegrass is wet.

Other probabilitiesare adjusted given the observation

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

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

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

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

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

Trace Evidence BN for RBS case• Enter the evidence:

• Local sensitivityC

• Posterior’s sensitivity to small changes in the model’s parameters.

Sensitivity Analysis

Threshold > 1

• Global sensitivityC

• 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

• 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)

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_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, 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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