Uncertainty about the true source. A note on the likelihood ratio at the activity level

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Abstract

This paper focuses on likelihood ratio based evaluations of fibre evidence in cases in which there is uncertainty about whether or not the reference item available for analysis – that is, an item typically taken from the suspect or seized at his home – is the item actually worn at the time of the offence. A likelihood ratio approach is proposed that, for situations in which certain categorical assumptions can be made about additionally introduced parameters, converges to formula described in existing literature. The properties of the proposed likelihood ratio approach are analysed through sensitivity analyses and discussed with respect to possible argumentative implications that arise in practice.

Introduction

A model for the evaluation of the evidence through a likelihood ratio when the material is transferred during the commission of a crime has been proposed by Evett [8], and has since then be re-used in other publications [3], [6], [9]. More generally, this has led to the consideration that transfer and presence by chance of trace material on a receptor are fundamental factors when assessing evidence under so-called activity level propositions [4]. In many such contexts, it is reasonable to assume that recovered material has an undoubtable link with a given individual (which may be the suspect). That is to say, if there has been a direct and primary transfer,1 then DNA found on a crime scene provides a direct connection between the source of that DNA and the crime scene. The reason for this is that biological material is intimately related to each individual.

This may not be the case, however, with certain categories of evidence other than DNA, such as shoemarks or fibres. In particular, shoes or garments may have been worn by different individuals. Thus, one also needs to establish an association between a given donor item (such as a garment or a pair of shoes) and a given suspect in order to link that person to the action under investigation. This paper investigates the effect that the latter issue may have on the development of a likelihood ratio under activity level propositions. The argument will be illustrated through the use of probabilistic graphical models (Bayesian networks).

The paper is organized as follows: Section 2 starts presenting the likelihood ratio as developed originally in [8]. This development along with its assumptions are taken here as part of accepted theory on the topic. Its derivation thus is not repeated in full detail. Instead, its main components will be stated in order to clarify from where the paper here proposes a further development. Section 3 introduces probabilistic graphical models, notably Bayesian networks, which will be used in Section 4 to deal with scenarios of interest that point out the additional source of uncertainty that will be included in the existing approach. Section 5 develops the likelihood ratio formula guided by the graphical model previously presented in Section 4. The proposed likelihood ratio model is analysed through sensitivity analysis and discussed in 6 Properties of the proposed likelihood ratio, 7 Discussion and conclusions.

Section snippets

Likelihood ratios in scenarios involving DNA evidence: evidence left by the offender

Imagine the following scenario. A crime has been committed during which a young woman was attacked by an assailant. Examination of the victim's fingernails revealed the presence of bloodstains. From the position where the stains were found, as well as their apparent freshness and the fact that the DNA profile is different from that of the victim, investigators believe that the stains are relevant to the case. The crime sample is analysed in a forensic laboratory. A common kind of DNA typing

Bayesian networks

A Bayesian network (BN, for short) is a compact model representation for reasoning under uncertainty that formally combines elements of graph and probability theory. BNs allowing their user to define a pictorial representation of assumed probabilistic relationships5 among a set of variables, deemed to be relevant for a particular inferential problem.

In a BN, random variables are represented by nodes6

Graphical probabilistic models to deal with scenarios involving evidence other than DNA

The general approach for developing a likelihood ratio at the activity level can be applied to various categories of evidence other than DNA. Fibres and shoemarks are typical examples for kinds. Such evidence categories differ from DNA in the sense that they are not ‘intrinsic’ to a given individual. That is to say, leaving biological anomalies and special cases aside, a given individual has, as far as the current level of analytical detail is concerned, one and only one DNA profile. It cannot

The numerator of the likelihood ratio: Pr(y|Hp)

Start by considering that the probability of the features of the recovered fibres (node Y) depends, given Hp, on whether a transfer occurred (T) as well as on the characteristics of the true source (SS), that is the actual donor item. One can thus write:Pr(Y|Hp)=Pr(Y|T,SS,Hp)×Pr(T,SS|Hp)For the likelihood ratio, the outcome Y = y is of interest. In order to keep a compact notation, let us write, hereafter, Pr(y) shorthand for Pr(Y = y). In addition, assume that Pr(SS) and Pr(SS¯) is written

Categorical assumptions about key parameters

Eq. (6) contains some additional variables compared to the initially considered Eq. (3). It is thus necessary to examine if the implications of Eq. (6) appear reasonable.

One way to do this consist of setting categoric assumptions for key parameters. An example for this was considered towards the end of Section 5.1 where the probability that the suspect wore the item referred to as the known source was set to 1. It was found that this reduced the numerator back to its initial form.

Another

Discussion and conclusions

In order to set the evidential value of forensic fibre evidence appropriately in context, scientists should, whenever possible, try to address at least activity level propositions. This allows additional information to be incorporated in an assessment that would otherwise, such as in source level evaluations, be disregarded. Examples include phenomena of transfer and background presence of fibres. The underlying reality of how fibre evidence is produced can thus be approached more closely.

A

Acknowledgements

This research has been supported by The Swiss National Science Foundation (grant no. IZK0Z1-133250/1). The authors wish to thank the reviewers for their valuable comments.

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