Evidence evaluation for discrete data Evidence evaluation for discrete data Evidence evaluation for discrete data
Evidence evaluation for discrete data Evidence evaluation for discrete data Colin Aitken School of Mathematics and Maxwell Institute, The University of Edinburgh Bayesian Biometrics for Forensics Network (BBFOR2) http://cls.ru.nl/projects/bbfor2 British Academy / Leverhulme Foundation (eGAP2): ‘Modelling features for forensic speaker comparison’. Erica Gold Language and Linguistic Science The University of York. c.g.g.aitken@ed.ac.uk Forensic Science International, 2013, 230, 147-155. Evidence evaluation for discrete data
Control and recovered evidence Two situations: control evidence is associated with the crime scene and recovered evidence is associated with a suspect and vice versa . Evidence evaluation for discrete data Evidence evaluation for discrete data
Control and recovered evidence Two situations: control evidence is associated with the crime scene and recovered evidence is associated with a suspect and vice versa . Evidence evaluation for Control evidence may be found at the scene of a crime and recovered discrete data evidence associated with a suspect. For example, in a burglary, glass fragments found below a broken window at a crime scene may be assumed to come from that window. Glass fragments found on a suspect’s clothing may or may not have come from the broken window at the crime scene. Evidence evaluation for discrete data
Control and recovered evidence Two situations: control evidence is associated with the crime scene and recovered evidence is associated with a suspect and vice versa . Evidence evaluation for Control evidence may be found at the scene of a crime and recovered discrete data evidence associated with a suspect. For example, in a burglary, glass fragments found below a broken window at a crime scene may be assumed to come from that window. Glass fragments found on a suspect’s clothing may or may not have come from the broken window at the crime scene. Control evidence may be found on a suspect and recovered evidence at a crime scene. For example, in an assault, blood which cannot be matched to the victim may be found at the crime scene and could be assumed to come from the criminal, whose identity is unknown. A suspect is identified and a DNA swab taken. The source of this DNA is known and is control evidence. Evidence evaluation for discrete data
Control and recovered evidence Two situations: control evidence is associated with the crime scene and recovered evidence is associated with a suspect and vice versa . Evidence evaluation for Control evidence may be found at the scene of a crime and recovered discrete data evidence associated with a suspect. For example, in a burglary, glass fragments found below a broken window at a crime scene may be assumed to come from that window. Glass fragments found on a suspect’s clothing may or may not have come from the broken window at the crime scene. Control evidence may be found on a suspect and recovered evidence at a crime scene. For example, in an assault, blood which cannot be matched to the victim may be found at the crime scene and could be assumed to come from the criminal, whose identity is unknown. A suspect is identified and a DNA swab taken. The source of this DNA is known and is control evidence. In forensic phonetics, one scenario would be an audio recording of a telephone message thought to be from the criminal, not identified, and an audio recording taken from a suspect, identified. The telephone message would be the recovered evidence, the audio recording from the suspect would be the control evidence. Evidence evaluation for discrete data
Evidence evaluation - two-stage approach Evidence evaluation for discrete data Evidence evaluation for discrete data
Evidence evaluation - two-stage approach Evidence Similarity: Assess similarity of control and recovered evidence by evaluation for some measure, such as a t -test for continuous measurements, or a discrete data chi-squared test for discrete measurements. Evidence evaluation for discrete data
Evidence evaluation - two-stage approach Evidence Similarity: Assess similarity of control and recovered evidence by evaluation for some measure, such as a t -test for continuous measurements, or a discrete data chi-squared test for discrete measurements. If the control and recovered evidence are not similar; the evidence is deemed to have different sources. By not ’similar’ is meant a significant result for a common mean in a t -test or for a common distribution for discrete measurements assessed by a chi-squared test. Evidence evaluation for discrete data
Evidence evaluation - two-stage approach Evidence Similarity: Assess similarity of control and recovered evidence by evaluation for some measure, such as a t -test for continuous measurements, or a discrete data chi-squared test for discrete measurements. If the control and recovered evidence are not similar; the evidence is deemed to have different sources. By not ’similar’ is meant a significant result for a common mean in a t -test or for a common distribution for discrete measurements assessed by a chi-squared test. If the control and recovered evidence are similar , the evidence is deemed to have a common source. By ’similar’ is meant a non-significant result for a common mean in a t -test or for a common distribution for discrete measurements assessed by a chi-squared test. The second stage is implemented. Evidence evaluation for discrete data
Evidence evaluation - two-stage approach Evidence Similarity: Assess similarity of control and recovered evidence by evaluation for some measure, such as a t -test for continuous measurements, or a discrete data chi-squared test for discrete measurements. If the control and recovered evidence are not similar; the evidence is deemed to have different sources. By not ’similar’ is meant a significant result for a common mean in a t -test or for a common distribution for discrete measurements assessed by a chi-squared test. If the control and recovered evidence are similar , the evidence is deemed to have a common source. By ’similar’ is meant a non-significant result for a common mean in a t -test or for a common distribution for discrete measurements assessed by a chi-squared test. The second stage is implemented. Rarity: Similarity in measurements which are rare in some sense is taken to be stronger evidence support of a common source then similarity in measurements which are common. Evidence evaluation for discrete data
Evidence evaluation Evidence In forensic statistics, evidence E is evaluated by its effect on the odds evaluation for in favour of a proposition put forward by the prosecution H p discrete data compared with a proposition put forward by the defence H d . Thus: Pr ( H d | E ) = Pr ( E | H p ) Pr ( H p | E ) Pr ( E | H d ) × Pr ( H p ) Pr ( H d ) . In general, consider E to have two components: one, X , is evidence whose source is known; this is control evidence, the other, Y , is evidence whose source is unknown: this is recovered evidence. The statistic used to evaluate the evidence is the likelihood ratio LR = Pr ( E | H p ) Pr ( E | H d ) = Pr ( X , Y | H p ) Pr ( X , Y | H d ) . Evidence evaluation for discrete data
Evidence evaluation - continued Evidence evaluation for discrete data Evidence evaluation for discrete data
Evidence evaluation - continued Likelihood ratios greater than one support the prosecution Evidence proposition. The evidence is more likely if the prosecution’s evaluation for discrete data proposition is true than if the defence proposition is true. Evidence evaluation for discrete data
Evidence evaluation - continued Likelihood ratios greater than one support the prosecution Evidence proposition. The evidence is more likely if the prosecution’s evaluation for discrete data proposition is true than if the defence proposition is true. The posterior odds for one piece of evidence E 1 are the prior odds for a second piece of evidence E 2 : Pr ( H p | E 1 , E 2 ) Pr ( H d | E 1 , E 2 ) = Pr ( E 2 | H p , E 1 ) Pr ( E 2 | H d , E 1 ) × Pr ( H p | E 1 ) Pr ( H d | E 1 ) . Evidence evaluation for discrete data
Evidence evaluation - continued Likelihood ratios greater than one support the prosecution Evidence proposition. The evidence is more likely if the prosecution’s evaluation for discrete data proposition is true than if the defence proposition is true. The posterior odds for one piece of evidence E 1 are the prior odds for a second piece of evidence E 2 : Pr ( H d | E 1 , E 2 ) = Pr ( E 2 | H p , E 1 ) Pr ( H p | E 1 , E 2 ) Pr ( E 2 | H d , E 1 ) × Pr ( H p | E 1 ) Pr ( H d | E 1 ) . With logarithms the updating process becomes additive: � Pr ( H p | E 1 , E 2 ) � Pr ( E 2 | H p , E 1 ) � Pr ( H p | E 1 ) � � � log = log +log . Pr ( H d | E 1 , E 2 ) Pr ( E 2 | H d , E 1 ) Pr ( H d | E 1 ) Evidence evaluation for discrete data
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