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Bayesian Methods What they are and how they fit into Forensic Science

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Outline Bayes’ Rule Bayesian Statistics (Briefly!) Conjugates General parametric using BUGS/MC software Bayesian Networks Some software: GeNIe, SamIam, Hugin, gR R-packages Bayesian Hypothesis Testing The “Bayesian Framework” in Forensic Science Likelihood Ratios with Bayesian Network software

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A little about conditional probability Bayes’ Rule:

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Frequency: ratio of the number of observations of interest (n i ) to the total number of observations (N) It is EMPIRICAL! Probability (frequentist): frequency of observation i in the limit of a very large number of observations Probability

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Belief: A “Bayesian’s” interpretation of probability. An observation (outcome, event) is a “measure of the state of knowlege” Jaynes. Bayesian-probabilities reflect degree of belief and can be assigned to any statement Beliefs (probabilities) can be updated in light of new evidence (data) via Bayes theorem. Probability

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

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Bayesian Statistics Bayesian-ism can be a lot like a religion Different “sects” of (dogmatic) Bayesians don’t believe other “sects” are “true Bayesians” The major Bayesian “churches” Parametric BUGS (Bayesian Using Gibbs Sampling) MCMC (Markov-Chain Monte Carlo) Andrew Gelman (Columbia) David Speigelhalter (Cambridge) Bayes Nets Graphical Models Steffan Lauritzen (Oxford) Judea Pearl (UCLA) Empirical Bayes Data-driven Brad Efron (Stanford)

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Bayesian Statistics What’s a Bayesian…?? Someone who adheres ONLY to belief interpretation of probability? Someone who uses Bayesian methods? Only uses Bayesian methods? Usually likes to beat-up on frequentist methodology…

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Bayesian Statistics Actually DOING Bayesian statistics is hard! Why? Parametric Bayesian Methods We will up-date this prior belief later All probability functions are “parameterized”

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Bayesian Statistics We have a prior “belief” for the value of the mean We observe some data What can we say about the mean now? We need Bayes’ rule: YUK! And this is for an “easy” problem

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Bayesian Statistics So what can we do???? Now there is software to evaluate the integrals. Some free stuff: MCMC: WinBUGS, OpenBUGS, JAGS HMC: Stan Until ~ 1990, get lucky….. Sometimes we can work out the integrals by hand Sometimes you get posteriors that are the same form as the priors (Conjugacy)

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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

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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)

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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)

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

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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

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Bayesian Statistics Bayesian network for the provenance of a painting given trace evidence found on that painting

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Bayesian Statistics Frequentist hypothesis testing: Assume/derive a “null” probability model for a statistic E.g.: Sample averages follow a Gaussian curve Say sample statistic falls here “Wow”! That’s an unlikely value under the null hypothesis (small p-value)

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Bayesian Statistics Bayesian hypothesis testing: Assume/derive a “null” probability model for a statistic Assume an “alternative” probability model p(x|null) p(x|alt) Say sample statistic falls here

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The “Bayesian Framework” Bayes’ Rule Aitken, Taroni : H p = the prosecution’s hypothesis H d = the defences’ hypothesis E = any evidence I = any background information

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

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The likelihood ratio has largely come to be the main quantity of interest in their literature: The “Bayesian Framework” A measure of how much “weight” or “support” the “evidence” gives to one hypothesis relative to the other Here, H p relative to H d Major Players: Evett, Aitken, Taroni, Champod Influenced by Dennis Lindley

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Likelihood ratio ranges from 0 to infinity The “Bayesian Framework” Points of interest on the LR scale: LR = 0 means evidence TOTALLY DOES NOT SUPPORT H p in favour of H d LR = 1 means evidence does not support either hypothesis more strongly LR = ∞ means evidence TOTALLY SUPPORTS H p in favour of H d

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The “Bayesian Framework” A standard verbal scale of LR “weight of evidence” IS IN NO WAY, SHAPE OR FORM, SETTLED IN THE STATISTICS LITERATURE! A popular verbal scale is due to Jefferys but there are others READ British R v. T footwear case!

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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

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The “Bayesian Framework” Computing the LR from our painting provenance example:

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An I.D. is output for each questioned toolmark This is a computer “match” What’s the probability the tool is truly the source of the toolmark? Similar problem in genomics for detecting disease from microarray data They use data and Bayes’ theorem to get an estimate No disease genomics = Not a true “match” toolmarks How good of a “match” is it? Efron Empirical Bayes’

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Empirical Bayes’ We use Efron’s machinery for “empirical Bayes’ two-groups model” Efron Surprisingly simple! Use binned data to do a Poisson regression Some notation: S -, truly no association, Null hypothesis S +, truly an association, Non-null hypothesis z, a score derived from a machine learning task to I.D. an unknown pattern with a group z is a Gaussian random variate for the Null

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Empirical Bayes’ From Bayes’ Theorem we can get Efron : Estimated probability of not a true “match” given the algorithms' output z-score associated with its “match” Names: Posterior error probability (PEP) Kall Local false discovery rate (lfdr) Efron Suggested interpretation for casework: We agree with Gelaman and Shalizi Gelman : = Estimated “believability” of machine made association “…posterior model probabilities …[are]… useful as tools for prediction and for understanding structure in data, as long as these probabilities are not taken too seriously.”

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Empirical Bayes’ Bootstrap procedure to get estimate of the KNM distribution of “Platt-scores” Platt,e1071 Use a “Training” set Use this to get p-values/z-values on a “Validation” set Inspired by Storey and Tibshirani’s Null estimation method Storey z-score From fit histogram by Efron’s method get: “mixture” density We can test the fits to and ! What’s the point?? z-density given KNM => Should be Gaussian Estimate of prior for KNM Use SVM to get KM and KNM “Platt-score” distributions Use a “Validation” set

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Posterior Association Probability: Believability Curve 12D PCA-SVM locfdr fit for Glock primer shear patterns +/- 2 standard errors

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Bayes Factors/Likelihood Ratios In the “Forensic Bayesian Framework”, the Likelihood Ratio is the measure of the weight of evidence. LRs are called Bayes Factors by most statistician LRs give the measure of support the “evidence” lends to the “prosecution hypothesis” vs. the “defense hypothesis” From Bayes Theorem:

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Bayes Factors/Likelihood Ratios Once the “fits” for the Empirical Bayes method are obtained, it is easy to compute the corresponding likelihood ratios. o Using the identity: the likelihood ratio can be computed as:

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Bayes Factors/Likelihood Ratios Using the fit posteriors and priors we can obtain the likelihood ratios Tippett, Ramos Known match LR values Known non-match LR values

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Acknowledgements Professor Chris Saunders (SDSU) Professor Christophe Champod (Lausanne) Alan Zheng (NIST) Research Team: Dr. Martin Baiker Ms. Helen Chan Ms. Julie Cohen Mr. Peter Diaczuk Dr. Peter De Forest Mr. Antonio Del Valle Ms. Carol Gambino Dr. James Hamby Ms. Alison Hartwell, Esq. Dr. Thomas Kubic, Esq. Ms. Loretta Kuo Ms. Frani Kammerman Dr. Brooke Kammrath Mr. Chris Luckie Off. Patrick McLaughlin Dr. Linton Mohammed Mr. Nicholas Petraco Dr. Dale Purcel Ms. Stephanie Pollut Dr. Peter Pizzola Dr. Graham Rankin Dr. Jacqueline Speir Dr. Peter Shenkin Ms. Rebecca Smith Mr. Chris Singh Mr. Peter Tytell Ms. Elizabeth Willie Ms. Melodie Yu Dr. Peter Zoon

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