# Modeling the forensic two-trace problem with Bayesian networks

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## Abstract

The forensic two-trace problem is a perplexing inference problem introduced by Evett (J Forensic Sci Soc 27:375–381, 1987). Different possible ways of wording the competing pair of propositions (i.e., one proposition advanced by the prosecution and one proposition advanced by the defence) led to different quantifications of the value of the evidence (Meester and Sjerps in Biometrics 59:727–732, 2003). Here, we re-examine this scenario with the aim of clarifying the interrelationships that exist between the different solutions, and in this way, produce a global vision of the problem. We propose to investigate the different expressions for evaluating the value of the evidence by using a graphical approach, i.e. Bayesian networks, to model the rationale behind each of the proposed solutions and the assumptions made on the unknown parameters in this problem.

## Keywords

Evaluation of evidence Value of the evidence Graphical probability models Bayesian networks Two-trace problem## Notes

### Acknowledgments

This research was supported by the Swiss National Science Foundation grant n° 100014-122601/1.

## References

- Aitken CGG, Gammerman A (1989) Probabilistic reasoning in evidential assessment. J Forensic Sci Soc 29:303–316CrossRefGoogle Scholar
- Aitken CGG, Taroni F (2004) Statistics and the evaluation of evidence for forensic scientists, 2nd ed. Wiley, ChichesterzbMATHCrossRefGoogle Scholar
- Biedermann A, Taroni F, Delemont O, Semadeni C, Davison A (2005a) The evaluation of evidence in the forensic investigation of fire incidents (Part I): an approach using Bayesian networks. Forensic Sci Int 147:49–57CrossRefGoogle Scholar
- Biedermann A, Taroni F, Delemont O, Semadeni C, Davison A (2005b) The evaluation of evidence in the forensic investigation of fire incidents (Part II): practical examples of the use of Bayesian networks. Forensic Sci Int 147:59–69CrossRefGoogle Scholar
- Biedermann A, Taroni F (2006) A probabilistic approach to the joint evaluation of firearm evidence and gunshot residues. Forensic Sci Int 163:18–33CrossRefGoogle Scholar
- Biedermann A, Bozza S, Taroni F (2009) Probabilistic evidential assessment of gunshot residue particle evidence (Part I): likelihood ratio calculation and case pre-assessment using Bayesian networks. Forensic Sci Int 191:24–35CrossRefGoogle Scholar
- Cook R, Evett IW, Jackson G, Jones P, Lambert J (1998) A hierarchy of propositions: deciding which level to address in casework. Sci Just 38:231–239CrossRefGoogle Scholar
- Dawid AP (2004) Which likelihood ratio? Comment on ‘Why the effect of prior odds should accompany the likelihood ratio when reporting DNA evidence’ by R. Meester and M. Sjerps. Law Prob Risk 3:65–71Google Scholar
- Dawid AP, Mortera J, Pascali V, van Boxel D (2002) Probabilistic expert systems for forensic inference from genetic markers. Scand J Stat 29:577–595zbMATHCrossRefGoogle Scholar
- Dawid AP, Mortera J, Vicard P (2007) Object-oriented Bayesian networks for complex forensic DNA profiling problems. Forensic Sci Int 169:195–205CrossRefGoogle Scholar
- Edwards W (1991) Influence diagrams, Bayesian imperialism, and the
*Collins*case: an appeal to reason. Cardozo Law Rev 13:1025–1079Google Scholar - Evett IW (1987) On meaningful questions: a two-trace transfer problem. J Forensic Sci Soc 27:375–381CrossRefGoogle Scholar
- Evett IW (1998) Towards a uniform framework for reporting opinions in forensic science casework. Sci Just 38:198–202CrossRefGoogle Scholar
- Fenton NE, Neil M (2011) Avoiding probabilistic reasoning fallacies in legal practice using Bayesian networks. Aust J Leg Philos 36:114–151Google Scholar
- Fenton NE, Neil M, Lagnado D (2011) Modelling mutually exclusive causes in Bayesian networks. Working paper. http://www.eecs.qmul.ac.uk/~norman/papers/mutual_IEEE_format_version.pdf. Accessed 8 Oct 2012
- Garbolino P (2001) Explaining relevance. Cardozo Law Rev 22:1503–1521Google Scholar
- Garbolino P, Taroni F (2002) Evaluation of scientific evidence using Bayesian networks. Forensic Sci Int 125:149–155CrossRefGoogle Scholar
- Gittelson S, Biedermann A, Bozza S, Taroni F (2012) Bayesian networks and the value of the evidence for the forensic two-trace transfer problem. J Forensic Sci 57:1199–1216CrossRefGoogle Scholar
- Goray M, Eken E, Mitchell RJ, van Oorschot RAH (2010) Secondary DNA transfer of biological substances under varying test conditions. Forensic Sci Int Genet 4:62–67CrossRefGoogle Scholar
- Jensen F (2001) Bayesian networks and decision graphs. Springer, New YorkzbMATHGoogle Scholar
- Kadane JB, Schum DA (1996) A probabilistic analysis of the Sacco and Vanzetti evidence. Wiley, New YorkGoogle Scholar
- Kjaerulff UB, Madsen AL (2008) Bayesian networks and influence diagrams: a guide to construction and analysis. Springer, New YorkzbMATHGoogle Scholar
- Lindley DV (1977) Probability and the law. Statistician 26:203–220CrossRefGoogle Scholar
- Lindley DV (2000) The philosophy of statistics. Statistician 49:293–337Google Scholar
- Meester R, Sjerps M (2003) The evidential value in the DNA database search controversy and the two-stain problem. Biometrics 59:727–732MathSciNetzbMATHCrossRefGoogle Scholar
- Meester R, Sjerps M (2004a) Why the effect of prior odds should accompany the likelihood ratio when reporting DNA evidence. Law Prob Risk 3:51–62CrossRefGoogle Scholar
- Meester R, Sjerps M (2004b) Response to Dawid, Balding, Triggs and Buckleton (concerning: why the effect of prior odds should accompany the likelihood ratio when reporting DNA evidence. Law, Prob. and Risk 3:51–62). Law Prob Risk 3:83–86Google Scholar
- Mortera J, Dawid AP, Lauritzen S (2003) Probabilistic expert systems for DNA mixture profiling. Theor Popul Biol 63:191–205zbMATHCrossRefGoogle Scholar
- Robertson B, Vignaux GA (1995) Interpreting evidence. Wiley, ChichesterGoogle Scholar
- Schum DA (1994) The evidential foundations of probabilistic reasoning. Wiley, New YorkGoogle Scholar
- Taroni F, Aitken CGG, Garbolino P, Biedermann A (2006) Bayesian networks and probabilistic inference in forensic science. Wiley, ChichesterzbMATHCrossRefGoogle Scholar
- Thompson WC, Taroni F, Aitken CGG (2003) How the probability of a false positive affects the value of DNA evidence. J Forensic Sci 48:47–54Google Scholar
- Triggs C, Buckleton JS (2003) The two trace transfer problem re-examined. Sci Just 43:127–134CrossRefGoogle Scholar
- Weir BS (2000) Statistical analysis. In: Siegel J, Saukko P, Knupfer G (eds) Encyclopedia of forensic science. Academic Press, San Diego, pp 545–550CrossRefGoogle Scholar