Post on 06-Jan-2016
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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.
Fin