Skip to main content
SpringerLink
Log in

Modeling the forensic two-trace problem with Bayesian networks

  • Published:
Artificial Intelligence and Law Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Notes

  1. Two propositions are mutually exclusive if they cannot both be true at the same time.

  2. A set of propositions is exhaustive if it covers all scenarios, so that at least one of its propositions is always true.

  3. Note that Eq. (7) gives the simplified form of the value of the evidence, so that the numerator and denominator of this ratio do not represent the probabilities forming the numerator and denominator in Eq. (1).

  4. Note that, by definition, pair \({H_1^{\prime}}\) consists of two nonexhaustive propositions. This is not problematic in this situation, because the evidence introduced later on will render the remaining proposition impossible.

  5. Note that the Bayesian network presented here models the probability of Y 1 and Y 2 as independent of the suspect’s sample given ‘proposition 2’. This makes the probability of Y 1 and Y 2 when ‘proposition 2’ and X are instantiated identical to the probability of Y 1 and Y 2 when only ‘proposition 2’ is instantiated, so that the instantiation of X is not absolutely necessary in this case.

References

  • Aitken CGG, Gammerman A (1989) Probabilistic reasoning in evidential assessment. J Forensic Sci Soc 29:303–316

    Article  Google Scholar 

  • Aitken CGG, Taroni F (2004) Statistics and the evaluation of evidence for forensic scientists, 2nd ed. Wiley, Chichester

    Book  MATH  Google 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 147:49–57

    Article  Google 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 147:59–69

    Article  Google Scholar 

  • Biedermann A, Taroni F (2006) A probabilistic approach to the joint evaluation of firearm evidence and gunshot residues. Forensic Sci Int 163:18–33

    Article  Google 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 191:24–35

    Article  Google 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 38:231–239

    Article  Google 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 3:65–71

  • Dawid AP, Mortera J, Pascali V, van Boxel D (2002) Probabilistic expert systems for forensic inference from genetic markers. Scand J Stat 29:577–595

    Article  MATH  Google Scholar 

  • Dawid AP, Mortera J, Vicard P (2007) Object-oriented Bayesian networks for complex forensic DNA profiling problems. Forensic Sci Int 169:195–205

    Article  Google Scholar 

  • Edwards W (1991) Influence diagrams, Bayesian imperialism, and the Collins case: an appeal to reason. Cardozo Law Rev 13:1025–1079

    Google Scholar 

  • Evett IW (1987) On meaningful questions: a two-trace transfer problem. J Forensic Sci Soc 27:375–381

    Article  Google Scholar 

  • Evett IW (1998) Towards a uniform framework for reporting opinions in forensic science casework. Sci Just 38:198–202

    Article  Google Scholar 

  • Fenton NE, Neil M (2011) Avoiding probabilistic reasoning fallacies in legal practice using Bayesian networks. Aust J Leg Philos 36:114–151

    Google 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 22:1503–1521

    Google Scholar 

  • Garbolino P, Taroni F (2002) Evaluation of scientific evidence using Bayesian networks. Forensic Sci Int 125:149–155

    Article  Google 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 57:1199–1216

    Article  Google 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 4:62–67

    Article  Google Scholar 

  • Jensen F (2001) Bayesian networks and decision graphs. Springer, New York

    MATH  Google Scholar 

  • Kadane JB, Schum DA (1996) A probabilistic analysis of the Sacco and Vanzetti evidence. Wiley, New York

    Google Scholar 

  • Kjaerulff UB, Madsen AL (2008) Bayesian networks and influence diagrams: a guide to construction and analysis. Springer, New York

    MATH  Google Scholar 

  • Lindley DV (1977) Probability and the law. Statistician 26:203–220

    Article  Google Scholar 

  • Lindley DV (2000) The philosophy of statistics. Statistician 49:293–337

    Google Scholar 

  • Meester R, Sjerps M (2003) The evidential value in the DNA database search controversy and the two-stain problem. Biometrics 59:727–732

    Article  MathSciNet  MATH  Google Scholar 

  • Meester R, Sjerps M (2004a) Why the effect of prior odds should accompany the likelihood ratio when reporting DNA evidence. Law Prob Risk 3:51–62

    Article  Google 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 3:83–86

  • Mortera J, Dawid AP, Lauritzen S (2003) Probabilistic expert systems for DNA mixture profiling. Theor Popul Biol 63:191–205

    Article  MATH  Google Scholar 

  • Robertson B, Vignaux GA (1995) Interpreting evidence. Wiley, Chichester

    Google Scholar 

  • Schum DA (1994) The evidential foundations of probabilistic reasoning. Wiley, New York

    Google Scholar 

  • Taroni F, Aitken CGG, Garbolino P, Biedermann A (2006) Bayesian networks and probabilistic inference in forensic science. Wiley, Chichester

    Book  MATH  Google Scholar 

  • Thompson WC, Taroni F, Aitken CGG (2003) How the probability of a false positive affects the value of DNA evidence. J Forensic Sci 48:47–54

    Google Scholar 

  • Triggs C, Buckleton JS (2003) The two trace transfer problem re-examined. Sci Just 43:127–134

    Article  Google Scholar 

  • Weir BS (2000) Statistical analysis. In: Siegel J, Saukko P, Knupfer G (eds) Encyclopedia of forensic science. Academic Press, San Diego, pp 545–550

    Chapter  Google Scholar 

Download references

Acknowledgments

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

Author information

Authors and Affiliations

  1. Institut de Police Scientifique, Ecole des Sciences Criminelles, Université de Lausanne, Quartier Sorge: le Batochime, 1015, Lausanne, Switzerland

    Simone Gittelson, Alex Biedermann & Franco Taroni

  2. Dipartimento di Economia, Università Ca’ Foscari di Venezia, San Giobbe, Cannaregio 873, 30121, Venezia, Italy

    Silvia Bozza

Authors
  1. Simone Gittelson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alex Biedermann
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Silvia Bozza
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Franco Taroni
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Simone Gittelson.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gittelson, S., Biedermann, A., Bozza, S. et al. Modeling the forensic two-trace problem with Bayesian networks. Artif Intell Law 21, 221–252 (2013). https://doi.org/10.1007/s10506-012-9136-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10506-012-9136-5

Keywords

Search

Navigation