A probabilistic approach to the joint evaluation of firearm evidence and gunshot residues

Forensic Science International 163 (2006) 18–33 www.elsevier.com/locate/forsciint

A probabilistic approach to the joint evaluation of firearm evidence and gunshot residues A. Biedermann a,b,*, F. Taroni a,c a

Ecole des Sciences Criminelles, Institut de Police Scientifique, The University of Lausanne, le Batochime, 1015 Lausanne-Dorigny, Switzerland b Federal Office of Police, 3003 Berne, Switzerland c Institut de Me´decine Le´gale, The University of Lausanne, 21 Rue de Bugnon, 1005 Lausanne, Switzerland Received 12 May 2005; received in revised form 1 November 2005; accepted 1 November 2005 Available online 5 December 2005

Abstract The present paper addresses issues that affect both the separate as well as the joint evaluation of firearm evidence (i.e., marks) and gunshot residues (GSR). Mark evidence will be used as a basis to discriminate among barrels through which a bullet in question might have been shot whereas GSR will be used to draw inferences about the distance of firing. Particular attention is drawn to the coherent handling of uncertainties associated with the various parameters considered within each item of evidence. The proposed analysis relies on a probabilistic viewpoint that uses graphical models (i.e., Bayesian networks) as an aid to cope with the complexity induced by the number of variables considered. The paper discusses how an approach based on a probabilistic network environment can be used for the formal analysis and construction of arguments. Emphasis is made on the gain of insight into structural dependencies that may be uncovered when the evaluative process is extended beyond single items of scientific evidence. # 2005 Elsevier Ireland Ltd. All rights reserved. Keywords: Evaluation and combination of evidence; Mark evidence; Gunshot residues (GSR); Bayes’ theorem; Bayesian networks

1. Introduction During investigative proceedings of incidents involving the use of firearms, as well as at trial, forensic expertise can provide key elements for decision makers to reach an opinion. In such contexts, particular attention is usually drawn to evaluative issues associated with firearms (i.e., marks) and related evidence, such as gunshot residues (GSR), two aspects on which the present paper will focus in further detail. * Corresponding author. Tel.: +41 31 324 76 29. E-mail addresses: [email protected], [email protected] (A. Biedermann), [email protected] (F. Taroni).

The former topic is typically concerned with situations where it is believed that a bullet found on the scene of a crime has been fired by a weapon found in possession of a suspect. A firearm examiner may be asked to compare the bullet from the scene with bullets test-fired by the suspect’s weapon. It is now widely held as acceptable that the results of such comparative examinations are amenable to support conclusions with respect to the propositions that the bullet in question was or was not fired from the barrel of the suspect’s weapon. Besides characterizing the link that may exist between a suspect’s gun and an incriminated bullet, firearm examiners may also be called on to conduct range of fire evaluations; a second aspect that the present paper will consider in further detail. Roughly speaking, in a range of fire evaluation the firearm specialist may examine the distribution pattern of

0379-0738/$ – see front matter # 2005 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.forsciint.2005.11.001

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certain gunshot residues in the area of the entrance hole. It is thought that for a given weapon and ammunition, there is a dependence between the distribution of GSR and the distance from which the firearm was discharged [1]. Based on test-firing experiments at known distances, scientists may gather information so as to provide estimates about the distance from which a pattern in question was shot [2]. However, before proceeding to such range of fire evaluations, scientists usually require the existence of a certain (subjective) degree of belief that some specified weapon – for instance that of a suspect – is in fact the one used for firing an incriminated pattern. It is with this (deterministic) requirement that there can appear potentially conflicting situations, notably when GSR and mark evidence need to be considered within the same case. The difficulty that may arise is that scientists may face the question of how they can, on the one hand, justify an assumption about the absence of uncertainty about the truth or otherwise of the proposition relating to the firing weapon, while on the other hand, consider the same proposition in a probabilistic way when evaluating mark evidence. Notice that we will not, for the time being, discuss whether or not scientists should directly express an opinion about a proposition stating that a bullet in question was fired from a particular weapon; it shall suffice as a reminder that thoughtful literature considers that a scientific approach should take the latter proposition to rely on probabilistic arguments [3]. It thus seems that mark and GSR evidence appear to be evaluated somewhat independently from one another. This gives rise to a series of interesting questions, such as: ‘‘What, if any, sources of uncertainty are associated with mark and GSR evidence?’’ ‘‘How should uncertainties relating to one of these types of evidence affect the assessment of the other, and vice versa?’’ Questions of this kind are closely linked to an issue currently occupying researchers in the field of evidence evaluation. That issue is concerned with the fact that many evaluative frameworks proposed in forensic literature tend to focus on single items of scientific evidence. As noted by Lindley ([4], p. xxiv), for example, a ‘problem that arises in a courtroom, affecting both lawyers, witnesses and jurors, is that several pieces of evidence have to be put together before a reasoned judgement can be reached’. Lindley has also pointed out that ‘probability is designed to effect such combinations but the accumulation of simple rules can produce complicated procedures. Methods of handling sets of evidence have been developed; for example, Bayes nets (. . .)’ ([4], p. xxiv). Following these ideas, the present paper proposes to study the combination of evidence with regards to mark and GSR evidence. Attention will notably be drawn to the potential (probabilistic) relations that may exist among different parameters that are deemed to be relevant when assessing these types of evidence. Issues affecting the coherent evaluation of mark and GSR evidence will first be studied separately. In a second step, ways will be sought

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in order to operate a logical joint evaluation of both evidence types. Throughout the paper, the construction of probabilistic arguments will be assisted through the use of graphical models (i.e., Bayesian networks). Pictorial representations yielded by these models greatly facilitate the formal discussion of the parameters considered as well as their assumed relationships. In addition, probabilistic calculus can be performed in a rigorous and efficient way. The next section provides details on the concept of Bayesian networks as well as on probabilistic evidence evaluation in general. Readers already acquainted with these topics may prefer to skip that part and continue directly with Section 3. Much of the formal derivation of the Bayesian network discussed throughout the text is confined to Appendix A.

2. Bayesian networks Bayesian networks (BNs) are graphical models with an underlying probabilistic framework. They play an increasingly important role in the evaluation of scientific evidence, notably where multiple and complicated interrelated issues need to be considered [5–9]. Let us imagine an item of evidence E used to infer something about H, some sort of proposition. Hp and Hd may refer to positions held by, respectively, the prosecution and the defense. In a Bayesian perspective to evidence evaluation, beliefs about H are revised through learning of E [10]. Bayes’ theorem formally describes how knowledge about E affects the respective probability of Hp compared Hd:

(1)

The strength of the evidence is measured by a likelihood ratio, denoted V here (shown in the center of Eq. (1)). For example, an item of evidence E would be said to support Hp over Hd when Pr(EjHp) > Pr(EjHd). When Pr(EjHp) < Pr(EjHd), the evidence supports Hd over Hp, and when Pr(EjHp) = Pr(EjHd), the evidence does not allow discrimination between Hp and Hd, i.e., it would be considered to be neutral. In a Bayesian network, nodes are used to represent variables, and directed edges express assumed dependencies among nodes. So, when it is thought that the outcomes of one variable influence the outcomes of another variable, then there would be a directed arc from the latter to the former. In forensic contexts, for example, scientists regularly consider the relative probabilities with which competing hypotheses H imply some evidence E. In terms of a Bayesian network, one would thus have two nodes H and E connected such that H ! E. Note that the variable E may be referred to as a ‘child variable’ as it has an entering arc

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originating from H. Conversely, H may be referred to as a ‘parent’ (of E). The nature and the strength of the relationship between nodes are modelled through probability tables associated to each node. For a node that does not depend on any other variables, e.g., H, there is a table containing unconditional probabilities Pr(H). For a node with entering arcs from other variables there is a table containing conditional probabilities for all possible parental configurations. For example, the node table of E would contain conditional ¯ PrðEjHÞ ¯ ¯ HÞ. ¯ probabilities Pr(EjH), PrðEjHÞ, and PrðEj Nodes and variables are connected such as to form a directed acyclic graph (DAG). In a Bayesian network, newly acquired information about one or more variables can be used to re-evaluate the truthstate of other variables. Forensic literature has demonstrated the possibility of constructing Bayesian networks that faithfully represent existing and accepted probabilistic solutions for the evaluation of certain kinds of forensic evidence (e.g., [7,11]). Further details on the construction of Bayesian networks in the context of forensic science can be found, for example, in [9] and references therein. In the forthcoming sections, Bayesian networks will be used to study parameters thought to be relevant to the assessment of firearm and GSR evidence. The discussion will mostly be based on likelihood ratios. More formal explanations of the relationship between the likelihood ratios and the proposed Bayesian networks are given in Appendix A.

3. Marks present on fired bullets Let us imagine a case in which an individual died due to a bullet fired in the center of the thorax. Neither cartridges nor a firearm were found on the scene. During subsequent investigations, a suspect was apprehended on the basis of information completely unrelated to the firearm evidence. A weapon found in possession of the suspect was seized and submitted to a forensic laboratory for comparative examinations. Generally speaking, a firearm examiner’s work will cover two main stages. Initially, marks left by a weapon’s manufacturing features will be examined. These may include parameters such as the number, width, angle and twist of grooves. When these parameters are compatible with the characteristics observable on bullets test-fired by the suspect’s weapon, further examinations may be undertaken. The examiner may then compare microscopic marks generated during the bullet’s passage through the barrel. Note that the inner surface of a barrel holds an unique set of characteristics originating from, for example, its manufacture, extent of use, cleaning habits, and storing conditions. A bullet fired through a barrel of a firearm thus acquires highly characteristic surface marks which are amenable for comparison purposes.

3.1. Parameters affecting the evaluation of mark evidence Consider first some issues that scientists need to consider when assessing mark evidence. When two bullets have been fired through the same barrel, scientists may expect to find a certain degree of agreement in microscopic detail. However, there are some factors that frequently tend to compromise the evaluation of such correspondences.
The condition of the crime bullet: As a result of striking surfaces of different kind, bullets recovered on crime scenes are likely to be deformed, or even incomplete.
The condition of the suspect’s weapon: Considerable time may have elapsed between the crime and the test-firing of a suspect’s firearm. Meanwhile, the suspect’s weapon may have been fired and cleaned a number of times. In addition, the weapon may have been stored in unfavorable conditions, e.g., humidity. Consequently, considerable differences may be observed between a bullet in question and test-fired bullets even though they were fired from the same barrel. This surely is an incomplete summary of all the subtleties that firearm examination may entail. However, it appears sufficient to illustrate that practicing forensic scientists face a difficult evaluative task: they are required to account for different sets of observations – features of manufacture and acquired characteristics – each of which being described by similarities and differences. Moreover, scientists need to consider their findings in the light of a unique set of circumstantial information, denoted I occasionally. 3.2. A Bayesian network for mark evidence Consider the Bayesian network shown in Fig. 1. This model allows one to assess the value of features of manufacture and acquired characteristics observed on an incriminated bullet and bullets test-fired by a suspect’s weapon. The probabilistic approach underlying this model is in agreement with a framework previously described by [12] in the context of footwear mark evidence. This is more formally stated in Appendix A. The definition of the network’s nodes together with their probabilistic dependencies is discussed below with reference to the firearm scenario introduced at the beginning of this section.
Node H: This variable is concerned with the truth or otherwise of the proposition according to which the suspect is the offender, i.e., the person who fired the incriminated bullet. Let Hp and Hd denote the positions held by the prosecution and the defense, respectively. Hp: the suspect is the offender, Hd: some unknown person is the offender.

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Fig. 1. A Bayesian network for evaluating mark evidence.

Generally, the assessment of prior probabilities for these propositions lies outside the scientist’s area of competence. An assessment of the likelihood ratio does not require such priors to be specified. The reason for specifying these probabilities here is purely of technical nature: they are needed to run the model. For the purpose of illustration, let Hp = Hd = 0.5.
Node F: The variable F stands for the proposition that the incriminated bullet was fired by the suspect’s weapon. Pr(FjHp), abbreviated by w, denotes the probability that the suspect’s weapon was used to fire the incriminated bullet, given the suspect is the offender. This probability depends, on the one hand, on the number of weapons in possession of the suspect, or the number of weapons to which the suspect had access at the time the crime happened, and on the other hand, his eventual preferences in using one or another of these weapons. Imagine that no other weapons were found in possession of the suspect and that there is information that the suspect did not have access to a weapon other than the one being seized. The parameter w could then be taken to be 0.99, for example. The probability that the suspect’s weapon was used for firing the incriminated bullet, given the suspect is not the offender, Pr(FjHd), is dependent in part on the number of persons that had access to the suspect’s weapon during the time the crime happened. For the scenario considered here, assume that it does not appear conceivable that someone other than the suspect could have used the suspect’s weapon: Pr(FjHd) = 0.
Node ym: This variable refers to the set of marks that originate from the features of manufacture of the weapon from which the incriminated bullet was fired. The outcomes of this variable depend on whether the suspect’s weapon fired the incriminated bullet (F), and if so, the features of manufacture of that weapon. The latter are assessed via marks present on test-fired bullets, summarised by the variable xm. So, given the characteristics of manufacture of the suspect’s weapon (xm) and given that this weapon fired the incriminated bullet (F), there is some chance of there being corresponding marks on this bullet (ym). Stability of features of manufacture can reasonably be assumed so that a value of 1 may be chosen for Pr(ymjxm, F). If the suspect’s weapon did not fire the incriminated bullet, then its characteristics ¯ are irrelevant for the assessment of ym: Prðym jxm ; FÞ

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¯ ¼ Prðym jFÞ. ¯ Given F, ¯ the incriminated ¼ Prðym j¯xm ; FÞ bullet was fired by a weapon different from that of the suspect. The probability of ym, thus, is dependent on the frequency of the configuration of manufacturing features among weapons of a relevant population. Assume ¯ that the scientist assigns a value of 0.05 to Pr(ymjF). Finally, if the incriminated bullet was fired by the suspect’s weapon, and the manufacturing characteristics of that weapon produce marks described as x¯ m , then it can be considered impossible to have marks on the incriminated bullet described as ym: arguably, Prðym j¯xm ; FÞ is set to 0.
Node ya: This variable is concerned with the marks that originate from acquired characteristics of the barrel through which the incriminated bullet was fired. Given that the suspect’s weapon was not immediately seized after the crime happened, but only a few months later, there is some chance that this weapon now leaves a pattern of marks that may differ from the kind of marks left a the time the crime happened. As noted in Section 3.1, this may be due to firing and cleaning of the weapon as well as storage conditions. So, when there are differences in observations relating to ya and xa, the probability of ya may be reduced. For the purpose of illustration, let Pr(yajxa,F) be 0.2. Consideration of xa can be omitted if F¯ is true, and the probability of observing marks described as ya given that another weapon fired the incriminated bullet can be assumed to be reasonably ¯ ¼ 0:001. The remaining paralow: thus, let Prðya jFÞ meter, Prðya j¯xa ; FÞ, can, as was done for ym, be considered impossible.
Nodes xm and xa: The prior probabilities required for these variables can be chosen on the basis of the estimated frequencies of the manufacturing and acquired features in a relevant population of weapons. Different sources of information may be used here, such as databases or expert knowledge. Let Pr(xm) and Pr(xa) be 0.01 and 0.001, respectively. Notice, however, that the evaluation of the likelihood ratio is not necessarily influenced by the values that have actually been chosen: when evaluating the numerator of the likelihood ratio, the nodes xm and xa are set to ‘known’, i.e., assumed to be certain, and when evaluating the denominator, knowledge about the truthstate of xm and xa may be irrelevant. The latter is notably the case when Hp implies F¯ with certainty and ¯ ¼ Prðyj¯x; FÞ ¯ holds. Prðyjx; FÞ The above mentioned parameters are an integral part of the likelihood ratio, which provides a measure for the value of observations made on the incriminated bullet (described by ym and ya), in the light of the propositions Hp and Hd and given knowledge about the characteristics of bullets fired by the suspect’s weapon (expressed by xm and xa). Formally, the likelihood ratio associated with the Bayesian network shown in Fig. 1 writes (see also Appendix A):

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(2)

Vm and Va represent what is sometimes termed ‘the value of the evidence’ [10] as contributed by marks left by the firing weapon’s manufacturing and acquired features. 3.3. Evaluation of parameter uncertainties When a Bayesian network is set up with the values specified in the previous section, then, the value of the likelihood ratio is given by: Prðym ; ya jxm ; xa ; Hp Þ Prðym ; ya jHd Þ 1 0:2  þ ð1  0:99Þ  4000: ¼ 0:99  0:05 0:001

V¼

One may have natural questions about this result. It is desirable that one be aware that the result is due to a series of assumptions that are implied, in part, by the network’s graphical structure,i.e., the number of variables togetherwiththeir definitions and structural dependencies. Quantitative assessments play an essential role as well, and it is about the latter that scientists most often disagree. Occasionally, it also happens that apparent inability of so-called ‘‘accurate’’ or ‘‘precise’’ quantiative assessments is forwarded as an argument against the use probabilistic approaches in forensic science. For the purpose of the current discussion, we insist that the values actually chosen are not of primary importance. A concise summary of such a viewpoint has been given, for instance, by Jackson ([13], p. 85): I am not implying that a scientist should provide a magic number for the value of the evidence—that would require a degree of precision to the evaluation of the LR [likelihood ratio] that is just not possible in many situations. Rather, the scientist should evaluate broadly the magnitude for the LR and translate that into a verbal equivalent. We have found that the most important part of adopting an LR approach to evaluation is that of improving the clarity of thinking—we should not, in the first instance, become too focused on the ‘numbers’. [emph. by the authors] Thus, the aim of the present discussion is to point out that models can be constructed so as to encode rigorously one’s beliefs about a specific problem of interest. The proposed models can handle parameters that are considered relevant for a problem under study, but the availability of hard numerical data is not a necessary requirement to perform probabilistic reasoning [14]. Forensic science is concerned with unique real-world events that happened in the past. As these may vary infinitely, scientists cannot be expected to have appropriate

estimates in all cases. Scientists may be asked, however, to inform about the degree by which their overall conclusions are affected by uncertainties in the value of certain parameters thought to be relevant for the evaluative process. Hereafter, this will be studied in further detail. Let us assume that acute uncertainties exist, for example, ¯ and Prðya jFÞ. ¯ Let us with respect to the parameters Prðym jFÞ ¯ recall that Prðym jFÞ is concerned with the probability of the incriminated bullet bearing marks described by ym if it was ¯ is fired by a weapon other than that of the suspect. Prðya jFÞ defined analogously but refers to acquired characteristics described by ya. Instead of providing exact point probabilities for these parameters, one may consider a range of values and evaluate their effect on the likelihood ratio. Such an operation can be considered a sensitivity analysis, a technique that allows one to investigate the properties of a mathematical model. Initially, it may often be useful to remain as general as possible. Thus, let us start by considering values for ¯ and Prðya jFÞ ¯ covering the whole range from zero Prðym jFÞ to one. This is a two-way sensitivity analysis that can be represented graphically in the form of contour lines (see Fig. 2). Contour lines connect combinations of values for ¯ and Prðya jFÞ ¯ that result in the same value for the Prðym jFÞ likelihood ratio V. Fig. 2 shows contour lines for likelihood ratios of 2, 5, 10, 20, 40, 100 and 1000. One can observe that the largest distances between the contour lines are located in the upper right part of the graph. This indicates that the likelihood ratio is relatively insensitive for high values of the target parameters. On the other hand, whenever the values of

Fig. 2. Two-way sensitivity analysis of the likelihood ratio applic¯ able for the evaluation of mark evidence. The parameters Prðym jFÞ ¯ are varied simultaneously. Each contour line represents and Prðya jFÞ a likelihood ratio of a specific value. The dashed parallel lines delimit hypothetical plausible intervals.

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the parameters decrease, contour lines become closer: low ¯ and Prðya jFÞ ¯ tend to lead to greater values of Prðym jFÞ changes in the value of V. The reader can thus realise that an evaluation of the likelihood ratio may be obtained even though no specific assessment is provided for certain parameters. Next, let us imagine that a scientist is willing to express his uncertainty about the actual value of a target parameter by means of a plausible interval. Such an interval defines a range of values within which the ‘true’ probability is thought to lie [15]. The limits of hypothetical plausible intervals are shown in Fig. 2 in the form of dashed, parallel lines. These provide an indication of the plausible effect that the scientist’s parameter uncertainties have on the magnitude of the likelihood ratio. This is highlighted by the grey shaded area: the number of contour lines that fall within that area together with their values allow for an appreciation of the magnitude of the plausible effect. Fig. 2 further illustrates that, depending on the location of the area in which an uncertain parameter is thought to lie, the likelihood ratio may vary considerably even though the plausible interval is narrow. On the other hand, there are areas where plausible intervals may be large and the likelihood ratio is not subjected to great variation.

4. Gunshot residues and range of firing Elicitation of cases involving the discharge of a firearm frequently requires information about the range of firing. Based on the observation that the way in which GSR are deposited around the entrance hole is varying in part with the distance of firing, scientists can conduct experiments under controlled conditions, i.e., firings at known distances, so as to assist in evaluating the distance from which a GSR pattern in question was shot [2]. Generally, firearms examiners would use some terminology to characterize the range of firing [16]. A distant shot, for example, refers to a distance such that no residues would reach the target. Close-range shots refer to distances close enough for GSR particles to reach the target. When a shot was fired at a range of 1 in. (2.5 cm) or less, it may be referred to as near-contact shot and when the shot was fired with the muzzle in contact with the target surface, it may be referred to as contact-shot [16]. Certain kinds of observations are typically encountered with near-contact and contact shots, such as the singeing of hairs or textile fibres. The examination of gunshot wounds during autopsy may also reveal distinctive defects, such as the splitting of tissue. Notice also that when GSR patterns are poorly visible, e.g., when present on a dark target medium, various techniques may be applied for visualizing residue patterns. From the wide range of observations that scientists may gather, particular aspects will be chosen hereafter in order to discuss ways in which such evidence may serve as a basis for

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constructing inductive arguments to propositions relating to distances of firing. The analysis will consider different levels of detail and focus on selected sources of uncertainty that may affect the evaluative process. 4.1. A basic approach Let us start by considering the process of evaluating gunshot residue patterns at a rather general level. One may imagine that an examiner test-fires a suspect’s weapon into appropriate targets arranged at different distances dx, where x denotes a particular distance expressed in cm. For each distance dx, the scientist would usually perform several firings in order to gain an approximate idea about the variability of the deposited GSR pattern. A widely used pragmatic approach to distance evaluation is a consideration of the group of test-firings that yielded GSR patterns resembling closest a pattern in question. Such examinations are essentially based on a visual inspection during which parameters such as the size and density of GSR patterns are compared. Such a procedure appears intuitively attractive. It provides a description as to how scientists should compare patterns in question and known patterns. However, the procedure tells little about how to draw an inference to the distance from which the pattern in question was shot. Imagine, for example, that a scientist found closest similarities between the pattern in question and patterns obtained from controlled firings at a certain distance dx. Should such observations be taken to suggest that the pattern in question was ‘probably’ shot from a distance similar to dx? It is tempting to accept this idea. But doing so would imply a direct passage from observations to conclusions, which obviously violates some inferential rules currently held acceptable for evidence evaluation in forensic science. In addition, it is rather well known that GSR particle deposition may, even under carefully controlled laboratory conditions, be subjected to considerable variation. It should also be remembered that it may be difficult to simulate in laboratory those conditions that are thought to have been present during the firing of the bullet in question. Things can become even more complicated if substantial doubts persist about the actual circumstances of the crime. It is thus argued here that the sole information given by observed similarities between characteristics of a pattern in question and those of firings from known distances is insufficient to infer something about the distance of firing. Notably, scientists should also consider the degree to which they would expect to observe a given similarity if the true distance of firing were different from dx. This amounts to considering an alternative proposition. For the purpose of illustration, let us consider again a hypothetical case, earlier described at the beginning of Section 3. Let D denote a proposition referring to the distance of firing, with dx being a particular outcome of D. Variable E relates to a proposition that provides for a

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description of the pattern in question in terms such as density and spread. In order to be general enough, E could also be taken to refer to the overall set of observations that have been made. Let us assume that the scientist is asked to evaluate the potential of the evidence to discriminate among the propositions according to which the distances of firing are, for example, 30 cm (d30) and 90 cm (d90), respectively. The latter may be scenarios forwarded by the prosecution and the defense, respectively. Following Eq. (1), the scientist could approach such a question by considering a likelihood ratio V of the following form: V¼

PrðEjD ¼ d30 Þ : PrðEjD ¼ d90 Þ

(3)

Note that the purpose of such a likelihood ratio, and the current discussion in general, is not to focus on any numerical value. What is important is that Eq. (3) provides assistance in asking the relevant questions when the aim is to evaluate which (if any) of the two propositions d30 and d90 is supported by the evidence. Now let us imagine that the scientist has performed testfirings at known distances and that he has gained a rather clear idea as to how the GSR pattern shot at distances of 30 and 90 cm look. He may then consider the incriminated pattern, described by E, and try to answer the following questions: (1) Do the characteristics of the pattern in question (E) occur as I would expect them to, if the distance of firing was 30 cm? (2) To what degree would I expect to see the pattern E if the distance of firing were 90 cm? Answers to these questions, expressed in terms of probabilities, represent values useable for, respectively, the numerator and denominator of Eq. (3). The reader can realise that if one’s answer to the first question is, for example, greater than that for the second, then the evidence supports d30 over d90. Notice also that the proposition d30 does not necessarily need to offer a ‘perfect’ explanation for E, i.e., Pr(EjD = d30) may assume values smaller than 1. It is solely of interest how the value of Pr(EjD = d30) compares with Pr(EjD = d90). Thus, for evaluating the direction of inference, a comparison of the relative magnitude of the two probability assessments, i.e., the numerator and the denominator, is sufficient. In legal literature, such qualitative forms of inference have been described, for example, by Garbolino [17]. Generally, scientists tend to be well acquainted with handling qualitative beliefs, and often they are even forced to do so, notably in situations where the overall available evidence may be difficult to capture and to describe entirely in numerical terms. GSR patterns are a good example of this.

4.2. An attempt to quantify GSR patterns In the previous section, a general framework was proposed to use observations made on GSR patterns as evidence to discriminate between propositions relating to different distances of firing. It was argued that the reasoning process does not necessarily require numerical assessments in order to comply with the rules of probability theory. However, it happens in practice that scientists may be asked to explain and justify the grounds of their beliefs. What is it that makes the scientist believe that Pr(EjD = d30) is, for example, greater that Pr(EjD = d90)? When a scientist looks at and compares patterns in question with known patterns, what are the observed characteristics to which attention is drawn, and in what does an assessment of similarity consist? Such questions, if taken seriously, may be hard to answer. Hereafter, an attempt will be made to describe GSR patterns in terms of the number of GSR particles detected on either a target in question or a known target. This is a measure that, on the one hand, can be represented in a simple numerical form, and on the other hand, can be regarded as sufficiently variable with the distance of firing. The authors are aware that this is a subjective choice. Other characteristics may be selected, such as density or spread, but modelling of these may rapidly become more complex. The development presented below aims to evaluate the degree to which an expression of the number of detected GSR particles is amenable for informing the numerator and the denominator of the likelihood ratio (Eq. (3)), for which purely qualitative expressions of subjective beliefs have been used so far (Section 4.1). The current discussion will focus on GSR patterns that have been treated with a Modified Griess Test (MGT), a technique useable for visualizing nitrite compounds. These are formed when smokeless powder burns or partially burns [18]. Nitrite residues will appear in orange colour subsequent to the application of the MGT. It is the number of visible orange particles that will be used to describe patterns in question and known patterns. Some may object that the counting of large numbers of residues is too tedious a task to be performed in practice. However, one could imagine the development of certain scanning tools and image treatment procedures that could be of assistance for scientists1 (tools of this kind already exist and are used, for example, by forensic fibre analysts). Let us now assume a scientist test-firing a suspect’s weapon at distances of, for example, 30, 50, 70 and 90 cm. Usually, several firings would be made for each distance in order to get an idea of the pattern’s variability. 1

A preliminary study conducted at the IPS (University of Lausanne) indicates that this is feasible (IPS firearm group, personal communication). The set of data used later in the text is inspired by these tests.

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Table 1 Mean and standard deviation of the number of visualised nitrite residues for firings at known distances Distance (cm)

Mean Standard deviation

30

50

70

90

500 80

350 60

120 35

55 20

Conditions for test-firing would be chosen so as to reproduce as closely as possible the circumstances at the moment the pattern in question was shot. A hypothetical set of data could be as shown in Table 1. For the sake of simplicity, the theoretical probability density functions for the number of particles are assumed to be Normal with parameters m and s2, denoted N(m,s2). The parameters are estimated by the sample mean and variance. For example, let D30 denote the number of particles found for firings made from a distance of 30 cm. The distribution may be written (D30j500,802)  N(500,802), or D30  N(500,802) for short. One may proceed analogously for the other distances. Y represents a pattern in question with y denoting a particular outcome, i.e., a certain number of visualised particles. For the purpose of the current example, let y be 400. It is accepted here that the approximation of the distribution of a discrete random variable, i.e., the number of particles by a continuous random variable, is reasonable because of the large number of particles. Next, let us imagine that a scientist seeks to evaluate the pattern in question given specific distributive assumptions. For example, the scientist may consider to what degree the finding of 400 GSR particles in a pattern in question is what may be expected if the distance of firing was 30 cm. The scientist would also have to consider the evidence given some alternative proposition, e.g., the distance of firing being 50 cm. Thus, the evidence is evaluated given the following pair of propositions:
Hp: the pattern in question was shot form a distance of 30 cm.
Hd: the pattern in question was shot from a distance of 50 cm. A likelihood ratio V can then be proposed in order to express the degree to which the evidence favours one proposition over another. For the scenario considered here, V is written as a ratio of two probability densities: V¼

f ðyj¯xD30 ; s 2D30 Þ f ð400j500; 802 Þ 0:0023 ¼ ¼ 2 f ð400j350; 602 Þ 0:0047 f ðyj¯xD50 ; s D50 Þ

 0:49:

Fig. 3. Graphical representation of the distributions D30  N(500,802) and D50  N(350,602). The dotted vertical line indicates the probability densities at the point 400 for each of the two distributive assumptions.

the point y when the distribution is D50  N(350,602). In the current example, the evidence supports, however weakly, Hd, the proposition according to which the pattern in question was shot from a distance of 50 cm. The likelihood ratio supporting Hd is V1, or approximately 2. Generally, the likelihood ratio for some evidence Y = y (assuming Y being Normally distributed), and hypotheses H1 : Nðm1 ; s 21 Þ and H2 : Nðm2 ; s 22 Þ can be written: "

 # s2 1 y  m2 2 y  m1 2 exp  : (5) 2 s1 s2 s1 The evaluation of an alternative scenario shows that the value of V depends crucially on the specified propositions, the parameters of the associated probability density functions, as well as on the characteristics of the evidence (i.e., the number of visualised GSR particles). One can then realise, for example, why the likelihood ratio found via Eq. (4) is relatively low. The distributions of D30 and D50 partially overlap, and it is in that region where the number of GSR particles of the pattern in question lies. Fig. 3 provides a graphical representation for this. For illustration, contrast this finding with the value of the likelihood ratio in the following settings:
The number of GSR particles found in the pattern in question is 250. The two propositions compared are as defined above. The likelihood ratio becomes:

V¼ (4)

The numerator represents the probability density at the point y when the distribution is D30  N(500,802). Analogously, the denominator is given by the probability density at

¼

f ð250j¯xD30 ; s 2D30 Þ f ð250j500; 802 Þ ¼ 2 f ð250j350; 602 Þ f ð250j¯xD50 ; s D50 Þ 3:78  105 ¼ 0:0228: 1:66  103

(6)

Here, the evidence still supports Hd, but the V has increased to approximately 44.

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Table 2 Means and variances of the probability density functions assigned to the variable Y D

Mean Variance

D30

D50

D70

D90

500 6400

350 3600

120 1225

55 400


When the propositions considered are D50 and D70, for example, V becomes:

V¼ ¼

f ð250j¯xD50 ; s D50 Þ f ð250j350; 602 Þ ¼ f ð250j¯xD70 ; s D70 Þ f ð250j120; 352 Þ 1:66  103 ¼ 144: 1:15  105

(7)

This result provides relatively strong support for the proposition D50. Notice that it is not always necessary to have two discrete propositions for comparison [19]. However, if the propositions do not refer to a specific value, the evaluation of the respective components of the likelihood ratio requires additional mathematical derivations that are beyond the scope of this paper; further details on such derivations can be found, for example, in [20]. 4.3. A Bayesian network for evaluating GSR evidence

proposed: D30, D50, D70 and D90, denoting distances of firing of, respectively, 30, 50, 70 and 90 cm.
Node Y: This variable is continuous and is used to represent the quantity of particles observed in a pattern in question. The probabilities assigned to the node D are unconditional and represent prior beliefs held about the distance from which the pattern in question may have been shot. Let us assume that, a priori, there is no preference for any of the four distances so that Pr(D30) = Pr(D50) = Pr(D70) = Pr(D90) = 0.25. Note that this is a discrete probability distribution. For the node Y, means and variances need to be specified given each of the possible states of the respective parent node D. These parameters can be adopted from Table 1. A summary is provided by Table 2. As the quantity of visualised GSR particles is thought to depend on the distance of firing, the node D is chosen as a parental variable for the node Y. A graphical representation is provided by Fig. 4(i). Fig. 4(ii) displays the Bayesian network in its initial state, i.e., when no evidence is entered. Each of the four possible states of the variable D are equally likely. The variable Y indicates the mean number of GSR particles that may be expected to be found, given the specified prior beliefs. This value is given by: EðYÞ ¼ m30 PrðD30 Þ þ m50 PrðD50 Þ þ m70 PrðD70 Þ þ m90 PrðD90 Þ:

The previously described procedure for evaluating GSR can be implemented within a Bayesian network. For this purpose, two variables are defined as follows:
Node D: This variable is discrete and represents the various propositions relating to the distance from which a pattern in question may have been shot. Four states are

(8)

This is a particular example of a general result that, given two dependent random variables Y and X, then E(Y) = E(E(YjX)). Consider some of the properties of the proposed Bayesian network when entering evidence. When the variable D is set to any of its four possible states, the node E would

Fig. 4. Bayesian network for evaluating GSR: (i) abstract representation of the graphical structure (a rectangle is used to represent a continuous variable); (ii) initial state of the numerically specified network; (iii and iv) states of the network after entering evidence at the node E, i.e., observation of, respectively, 400 and 250 GSR.

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display the parameters of the respective Normal distribution specified in Table 2. For example, if Pr(D = D90) = 1, then the node E indicates a Normal distribution with a mean of 55 and a variance of 400. As may be seen, the value 55 may also be obtained via Eq. (8), which reduces to E(Y) = m90 = 55 (due to Pr(D = D90) = 1). Next, consider an inference in the opposite direction. Fig. 4(iii) shows a situation where 400 GSR particles are observed. This evidence is entered at the node Y. The node D displays the posterior probability of each hypothesis given the evidence. For example, the probability associated with the state D30 is the posterior probability of the distance of firing being 30 cm, given that 400 particles have been observed, formally written Pr(D = D30jy = 400). The latter probability can be obtained as follows: f ðyjD30 ÞPrðD30 Þ ; PrðD ¼ D30 jy ¼ 400Þ ¼ P i f ðyjDi ÞPrðDi Þ with i ¼ ð30; 50; 70; 90Þ and f(yjDi) being the probability density function for y at distance Di. The value for Pr(D = D30jy = 400) thus obtained is 0.3270. Analogously, the posterior probabilities for D50, D70 and D90 may be obtained. Consider, for example, the ratio of the posterior probabilities of, respectively, D30 and D50, given y, which yields 0.327/0.673 = 0.49. This value corresponds to the likelihood ratio earlier obtained in Section 4.2. Notice, however, that this agreement requires equal prior prob- abilities for D30 and D50. In Section 4.2, likelihood ratios have also been proposed for situations in which the number of visualised GSR is 250. An evaluation of this scenario is represented by Fig. 4(iv).

5. The joint evaluation of GSR and marks evidence In the previous sections, a procedure was described for discriminating among a discrete set of hypotheses. Roughly speaking, the procedure combines observed data – the quantity of visualised GSR particles – with previous knowledge and can be seen as an aid to rank hypotheses according to their credibility. We will now continue to study the scenario earlier introduced in Section 3. A forensic firearm specialist is needed to conduct certain examinations. A bullet has been extracted from a dead body. It is assumed that only one bullet has been fired and that the GSR around the entrance hole on the victim’s clothing originates from that firing. A suspect’s gun is available and comparisons have been made between both bullets fired from that weapon and the incriminated bullet. For the time being, let us suppose that the scientist judges the quality of the observations to be sufficient for reaching the (subjective) opinion that the incriminated bullet has been fired from the suspect’s gun.

27

In a subsequent step, the scientist is asked to conduct an estimation of the range of fire. Ammunition is available that is thought to come from the same lot as that used to fire the incriminated pattern. The suspect’s weapon, reasonably believed to have fired the pattern in question, is used for firings at known distances and under controlled conditions. The GSR patterns thus obtained are compared with the pattern surrounding the entrance hole on the victim’s clothing. A procedure as described in Section 4 may be used to assist the inferential task associated with these experiments. So far, the evaluation of the mark evidence has been considered quite independently from the evaluation of the GSR evidence, and vice versa. Usually, the scientist would not proceed to a range of fire estimation until he has attained a certain subjective degree of belief that some specified weapon is the one used to fire an incriminated pattern. However, this opinion is probabilistic in nature, meaning that some uncertainty may usually exist about whether that weapon is in fact the one used during the shooting incident. Hence, it may be felt that this uncertainty should be accounted for in a range of fire estimation. This is an interesting aspect of the joint evaluation of mark and GSR evidence. Below, previously developed Bayesian networks will be extended and combined in order to cope with that issue. Let us start by considering the Bayesian network described in Section 4.3 (Fig. 4(i)). For each hypothesis relating to a specified distance of firing (node D), a probability distribution for modelling the number of GSR particles visualised on the target is defined (node Y). An implicit assumption of this approach is that the evidence is assessed only under varying propositions for distances of firing. For example, the evidence y is assessed under the proposition that the distance of firing was 30 cm (D = D30) and under the proposition that the distance of firing was 50 cm (D = D50). However, in both settings the suspect’s weapon is taken to be the one having fired the pattern in question, an assumption that may not necessarily reflect the defense’s position. This inconvenience may be overcome by adopting a further node, named F for instance. F is a binary node which takes two states F s and F¯ s , defined as ‘the bullet was or was not fired by the suspect’s weapon’. Notice that one assumes the GSR to be the result of firing the incriminated bullet. The node F is chosen as a parental variable for the node Y so that the Bayesian network shown in Fig. 4(i) now becomes D ! Y F (Fig. 5(i)). Consequently, the probability table of the node Y needs to be extended as shown in Table 3. When the first four columns of that table are specified in the same way as Table 2, then, with F set to F s, the Bayesian network D ! Y F yields strictly the same results as the network D ! Y (Fig. 4(i)). The parameters specified for Y, given F¯ s , characterise the expected Normal distributions for Y when a firearm other than that of the suspect was used to fire the incriminated bullet. Note that these parameters are labelled with a prime

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Table 3 Parameters of the Normal distributions associated with the node Y F¯ s

Fs D30 Mean Variance

m30 s 230

D50 m50 s 250

D70 m70 s 270

D90

D30

D50

D70

D90

m90 s 290

m030 s 0302

m050 s 0502

m070 s 0702

m090 s 0902

(0 ) as they may be different from the parameters assumed for Y given F s. The parameters for Y given F¯ s tend to affect the evaluation of some observation y if, for a given distance Di of firing, the parameters of the Normal distributions specified for Y given F s and F¯ s differ, and F s is not true with certainty. If F s were known to be true, then the evaluation is not affected by whatever value is specified for Y given F¯ s . Let the prior probabilities for F s and F¯ s be denoted f 0 and f 1, respectively. Then the mean number of GSR particles expected to be found, given beliefs specified for the variables D and Y, is given by: EðYÞ ¼ EðEðYjDÞÞ f0 þ EðEðYjDÞÞ f1 X X ¼ mi PrðDi Þ f0 þ m0i PrðDi Þ f1 i

¼

X

i

PrðDi Þðmi f0 þ m0i f1 Þ;

(9)

i

for i ¼ ð30; 50; 70; 90Þ: If it can be accepted that uncertainty may exist with respect to the variable F, which is the proposition that the suspect’s weapon has fired the incriminated bullet, then it may be desirable to consider one’s belief in the truth, or otherwise, of that variable, based on other evidence. The mark evidence appears to be suitable for this task. This then is the instance where the reader can realise that the evaluation of GSR and mark evidence are interrelated. Thus, imagine that comparative examinations have been performed between the incriminated bullet and bullets fired through the barrel of the suspect’s weapon, under controlled conditions. During the examination process, a specific set of similarities and differences is observed. Evaluation of such evidence at the source level can be considered with a Bayesian network constructed and discussed earlier in Section 3.2 (Fig. 5(ii)).

The variables Y and X in the model shown in Fig. 5(ii) refer to the observations made on, respectively, the incriminated bullet and the bullets from a known source (e.g., the suspect’s weapon). The subscripts m and a indicate that the observed marks originate from the firearm’s features of manufacture and acquired characteristics, respectively. Fig. 5(iii) depicts a logical combination of the networks for evaluating GSR and mark evidence. The two network fragments are linked via the node F, defined as ‘the incriminated bullet was fired by the suspect’s weapon’. As mentioned before, the importance of incorporating F when evaluating GSR evidence is that one may account for the uncertainty about whether the weapon used for the testfirings at known distances, i.e., the suspect’s weapon is in fact the one used to fire the incriminated pattern. An advantage of the Bayesian network shown in Fig. 5(iii) then is that one’s beliefs about the truthstate of F can be more than just a guess: they may coherently be informed by knowledge available from mark evidence. The Bayesian network shown in Fig. 5(iii) can approach quite different positions that may be held by the prosecution and the defense. For example, the prosecution’s case may be that the suspect’s weapon fired the incriminated bullet (F s) and that the distance of firing was about 30 cm. The defense may argue that a weapon other than that of the suspect was used (F¯ s ) and that the distance of firing was about 70 cm. In such a setting, the likelihood ratio for some observation Y = y is given by: "

 # s 070 1 y  m070 2 y  m30 2 : exp  s 30 s 070 s 30 2 The proposed Bayesian network clarifies the potential scenarios as to how the pattern in question may have been produced, and the assumptions made by the expert during the evaluative process. A difficulty, however, is that experts may not have knowledge of the nature of the outcomes in all scenarios. For example, parameters assigned to Y given F¯ s in Table 3, i.e., the kind of GSR patterns produced by firearms other than that of the suspect. This is not about putting scientists into uncomfortable situations. The aim is to illustrate that the evaluation of such evidence requires certain assumptions which can be made explicit in a rigorous and efficient way. So, either the scientist can make reasonable assumptions such as values for m0 and s0 , i.e., about the nature of the GSR deposition of an alternative weapon (e.g., because such a weapon is available for testfiring), or he may need to make the explicit assumption that it was the suspect’s weapon that was used to fire the incriminated pattern.

6. Discussion Fig. 5. Bayesian networks for (i) evaluating GSR evidence, (ii) evaluating marks evidence at the source level, and (iii) the joint evaluation of GSR and marks evidence.

Within the field of firearm and toolmark examination there is currently considerable debate about how evidence

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pertaining to this discipline is appropriately evaluated. Much of that debate concentrates on what should be considered as so-called ‘criteria of identification’ and whether or not these can enable scientists to express an opinion about a proposition according to which a specific firearm has shot a bullet in question. Counted consecutive matching striations (CMS) is a statistic frequently referred to in this context. CMS has been suggested by in [21,22]. For a review of this method, see [23]. General reviews of criteria for firearm and toolmark identification can be found in [24,25]. There are fundamental disagreements about the existence of a certain CMS threshold above which a barrel may be ‘identified’ as being the source of marks found on a bullet in question. A CMS threshold approach is notably supported by Moran [26–28] whereas Champod et al. [3] argue with the logic of inductive inference, as provided by an approach based on a likelihood ratio. Although the authors here favour the latter, the proposed probabilistic framework discussed throughout this paper can accommodate both positions equally well: in a Bayesian network, the user is free to consider, on the one hand, the probability of a proposition of interest, given specific evidence, or on the other hand, the probability of the evidence given a proposition of interest. Also, the scientist is authorised to specify, for example, any probability he or she personally considers appropriate for the evidence to occur given an alternative proposition. This is interesting insofar as the latter probabilities are recurrent instances of disagreement. Notice also that the proposed Bayesian network can readily be extended to handle CMS data in a way that agrees with [23] (see Appendix A for further details). The above mentioned debate in firearm and toolmark identification is important and one worthy of being continued. However, there are still a number of issues that have not yet been addressed in detail. The results of the analysis provided here are intended to shed light on some of these instances.

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a bullet in question is severely damaged and acquired features are altered and/or difficult to observe;
considerable time has passed between the firing of the incriminated bullet and the seizure of a suspect’s firearm; time during which the acquired characteristics of barrel may have changed due to, for example, repeated use, cleaning and storing conditions. But even if acquired features are available, marks left by a weapon’s manufacturing features should be assessed and incorporated in the scientist’s evaluation. The strength of the link between a suspect’s weapon and an incriminated bullet may be underestimated if focus is restricted to acquired characteristics. Fig. 2 proposes a graphical illustration of the interaction between different aspects of mark evidence and the effect on the magnitude of the likelihood ratio. Thus, there is a clear need for scientists to adopt a viewpoint sufficiently large to account for the different aspects and levels of detail associated with mark evidence. For this purpose, the present paper has considered a distinction between marks that originate from, respectively, a weapon’s features of manufacture and acquired characteristics. The probabilistic development is one that follows an approach earlier proposed by Evett et al. [12] in the context of shoemark evidence (see Appendix A). Bayesian networks have been used here with the aim of clarifying the assumptions underlying the formal development. These assumptions are primarily concerned with the believed relationships among the different aspects of the evidence and the way in which these are used to construct an argument to source level propositions, and by extension, to crime level propositions. The proposed framework expresses the value of the evidence in terms of a likelihood ratio. The respective evidential value of marks left by a weapon’s manufacturing features and acquired characteristics are explicitly incorporated in the overall evaluative process. This is illustrated by Eq. (A.11) where distinct factors Vm and Va are used to account for different aspects of the evidence.

6.1. A comprehensive description of the evidence? 6.2. Beyond evaluation of single items of evidence Recurrent discussions on evaluation of firearm and toolmark evidence gravitate around (a) concepts, such as CMS counts, intended as descriptors of acquired features, and (b) the evidential significance of such features. Often, however, correspondences in manufacturing features do not seem to be considered from the same point of view. They are taken into account at an early stage of examination where an agreement in manufacturing characteristics is required before proceeding to comparative analyses of acquired features. No explicit assessment is generally made of the potential of manufacturing characteristics, such as caliber, number of lands and grooves, their twist, etc., for reducing the population of suspected firearms. This is not satisfactory because there are many situations in which an expression of the value of such information may be of particular importance, for example when:

In real-world circumstances, evidence rarely stands alone. When considering one item of evidence, scientists should assure that their evaluative framework is amenable, on the one hand, for building upon existing knowledge, based on particular evidence, and on the other hand, for logical combination with forthcoming evidence, i.e., evidence that has not yet been considered. In addition, one must be aware that sources of uncertainty associated with the evaluation of one item of evidence may be relevant when considering another piece of evidence. In legal contexts, the joint evaluation of several items of evidence has attracted scholars for quite some time. For an early graphical approach, however, essentially non-probabilistic, see [29,30], or reworks of it as given in [31]. Graphical probability models for the combination of evidence have, in

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the general context of fact analysis, been studied in [32]. An application to a famous criminal case has been provided in [33]. With the exception of [6], studies on the combination of scientific evidence in forensic science are much rarer. In the present paper, issues in combining evidence have been studied and illustrated in the context of mark and GSR evidence. The joint evaluation of these kinds of evidence revealed a somewhat particular form of combining evidence. Usually, two or more items of evidence are used to infer something about the same proposition. For example, a fingermark and a bloodstain found on the scene of a crime may, under certain assumptions, serve as evidence for the proposition that the suspect was at the scene of the crime. The scenario involving mark and GSR evidence has another structure: the two items of evidence are used to draw inferences to different propositions, but the joint evaluation is nevertheless interrelated. The GSR evidence is used to reason about the distance of firing. The mark evidence is used to infer something about the proposition according to which the suspect’s weapon fired an incriminated bullet. Uncertainty about the latter proposition can be relevant when assessing GSR evidence. In an attempt to capture the relationship between mark and GSR evidence, a formal development using Bayesian networks can assist scientists in analyzing and reasoning probabilistically about the problem of interest. The development has, however, not gone as far as to suggest any precise numerical figures. As far as mark evidence is concerned, a CMS count is a particular concept that could potentially be used to inform one’s belief about a target parameter (see Appendix A) and it is clear from the proposed analyses that further considerations are necessary for logically evaluating such evidence in a wider context. Thus, the primary aim has rather been to clarify and to obtain a concise representation of the structural dependencies assumed to hold among different aspects of each item of evidence. The proposed framework facilitates the locating and formal articulating of relevant parameters which can aid scientists to bring in a more secure position whenever they are required, for example, to explain the foundations of their reasoning and to evaluate the effect of specific parameter uncertainties. Note also that it is not the intention of the authors to propose here, at the moment, such models for use in written reports or for presentation before trial. Working with Bayesian networks in forensic science is largely concerned with thinking about the way in which scientists assess evidence in the light of propositions relevant for a customer. Something worthwhile has been gained if Bayesian networks can increase the level of insight in the inductive nature of this thought process.

7. Conclusions Interpretation of evidence in forensic science is based on a logical approach, a core part of which is represented by

Bayes’ theorem. Considerable research has been devoted in order to explore the potential of this approach for assessing scientific evidence as well as sources of uncertainty associated thereto. The present paper has explored the potential of a Bayesian viewpoint for approaching evaluative issues associated with the separate as well as the joint evaluation of mark and GSR evidence. Bayesian networks have been used as an aid to cope with the complexity induced by the increased number of variables that have been incorporated in the analyses. Discourses on how to assess the evidential value of mark and GSR evidence should draw attention to Bayesian networks as a means to convey a clear idea of the sources of uncertainty considered relevant, the level at which they intervene and how they should affect one’s beliefs about propositional variables of interest. The availability of hard numerical data is, however, not a necessary requirement for the use of the proposed probabilistic framework. Scientists can, for instance, consider the network environment as an assistance in deriving and formally articulating parameters that should be considered when evaluating such evidence. In addition, sensitivity analyses may be performed in order to gain an idea as to the relative importance of various target parameters. If data, or at least, qualitative expressions of subjective beliefs, are available, then the approach can provide guidance in revising one’s beliefs in a way that agrees with the laws of probability theory, a recognized standard for handling uncertainty in knowledge.

Acknowledgements The authors are grateful to two anonymous referees for their fruitful comments as well as the firearm group of the Institut de Police Scientifique (University of Lauanne) for information on tests conducted on shooting range evaluation.

Appendix A A.1. Likelihood ratio with crime-level propositions Consider a variable H covering the following pair of crime-level propositions:
Hp: the suspect is the offender
Hd: some unknown person is the offender Here, the offender is taken to be the individual that fired the incriminated bullet. X is a weapon found in possession of the suspect and observations made on test-fired bullets from that weapon are denoted x. Likewise, y denotes observations made on an incriminated bullet Y. For the time being, attention is solely drawn to the sets of observations x and

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y without distinguishing between features of manufacture and acquired characteristics. The likelihood ratio (V) can then be written: V¼

Prðx; yjHp Þ : Prðx; yjHd Þ

(A.1)

Prðyjx; Hp Þ PrðxjHp Þ  : Prðyjx; Hd Þ PrðxjHd Þ

¯ p Þ:  PrðFjH (A.2)

It appears reasonable to assume that, in the absence of knowledge of y, uncertainty about x is not affected by the truth or otherwise of H: Pr(xjHp) = Pr(xjHd). In addition, if the marks on the incriminated bullet do not originate from the weapon X, then knowledge about x is irrelevant for assessing the probability of y: Pr(yjx,Hd) = Pr(yjHd). Eq. (A.2) then reduces to: V¼

Prðyjx; Hp Þ : PrðyjHd Þ

fired by the suspect’s weapon, is already known. So Pr(yjx,F,Hp) can be reduced to Pr(yjx,F). In the absence of knowledge about y, uncertainty about F is not affected by knowledge about x, so Pr(Fjx,Hp) can be written as Pr(FjHp). Eq. (A.4) now writes: ¯ Prðyjx; Hp Þ ¼ Prðyjx; FÞ  PrðFjHp Þ þ Prðyjx; FÞ

Applying the third law of probability, V becomes: V¼

31

(A.3)

A.2. Deriving the likelihood ratio from a Bayesian network Fig. 6(i) shows a Bayesian network useable for evaluating mark evidence. With respect to Eq. (A.3), an additional (binary) variable is considered:
F: the incriminated bullet was fired by the suspect’s weapon. F may be referred to as a source-level proposition. A derivation of the likelihood ratio (V) associated with the Bayesian network is given below. First, consider the numerator of V: Pr(yjx,Hd). This parameter is concerned with the probability of the observations made on the incriminated bullet, given the observations made on the test-fired bullets and given that the suspect is the offender. Considering uncertainty in relation to F, the numerator becomes: Prðyjx; Hp Þ ¼ Prðyjx; F; Hp Þ  PrðFjx; Hp Þþ ¯ Hp Þ  PrðFjx; ¯ Hp Þ Prðyjx; F; It can be assumed that knowledge about H should not change one’s belief in y if F, the incriminated bullet being

(A.5)

The denominator is concerned with the probability of y given Hd and x. Considering again that F ‘screens off’ y from H, the denominator can be written as: ¯ Prðyjx; Hd Þ ¼ Prðyjx; FÞ  PrðFjHd Þ þ Prðyjx; FÞ ¯ d Þ:  PrðFjH

(A.6)

The assumption is made here that if the suspect is not the shooter (Hd), then the suspect’s weapon cannot have fired the incriminated bullet: Pr(FjHd) = 0. This is equivalent to saying that no individual other than the suspect had access to the suspect’s weapon at the moment the crime happened. The denominator can thus be simplified considerably and rewritten as: ¯ Prðyjx; Hd Þ ¼ Prðyjx; FÞ:

(A.7)

Through combination of Eqs. (A.5) and (A.7), the likelihood ratio obtained is: V¼

¯  PrðFjH ¯ pÞ Prðyjx; FÞ  PrðFjHp Þ þ Prðyjx; FÞ : ¯ Prðyjx; FÞ

(A.8) This result can further be simplified by assuming the following: (1) Pr(FjHp) is the probability that the suspect – if he were in fact the offender – was using the weapon X. Let this parameter be abbreviated by w. Note that its comple¯ p Þ, is 1  w. ment, PrðFjH ¯ ¼ PrðyjFÞ ¯ ¼ pðyÞ: (2) Prðyjx; FÞ Applying these transformations to Eq. (A.8) yields: V¼

Prðyjx; FÞ  w þ pðyÞ  ð1  wÞ : pðyÞ

(A.9)

This can further be simplified to: Prðyjx; FÞ þ ð1  wÞ: (A.10) pðyÞ This result is a simplified version of a likelihood ratio earlier described by Evett et al. ([12], Eq. (5)) in the context of evaluating footwear mark evidence in forensic casework. The sole difference is that Evett’s formula considers a further variable that takes the role of addressing the relevance of the recovered footwear mark, allowing for the possibility that the mark under consideration was not left

V ¼w

Fig. 6. Bayesian networks for evaluating mark evidence.

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by the offender. In the context of the firearm scenarios considered here, the incriminated bullet was selected because it caused a specific damage, e.g., a gunshot wound. Since the incriminated bullet is necessarily relevant for the event that caused it being fired, no relevance is adopted here. In cases where this assumption should not hold, a relevance node could be added by specifying an additional parental node for F.

A.3. Consideration of distinct components So far, y and x have been considered as global assignments for observable characteristics on the incriminated and the test-fired bullets, respectively. As noted in Section 3, observations may be partitioned into: (a) a component relating to the traits that originate from the weapon’s features of manufacture and (b) a component relating to traits that originate from the weapon’s acquired features. Let us denote these components, respectively, ym and ya when referring to observations made on the incriminated bullet. Analogously, xm and xa denote the characteristics observed on the test-fired bullets. A logical extension from y and x to ym, ya and xm, xa can be obtained by ‘duplication’ of the nodes y and x. Such a network is shown in Fig. 6(ii). Notice that the network shown in Fig. 6(ii) is, to some degree, a minimal representation, providing room for possible extensions. For example, the following may be considered:
The nodes with subscripts m and a denote the true presence of traits originating from manufactured and acquired features, respectively. Thus no distinction is made between the observation of a characteristic and a characteristic itself. Distinct observational nodes may be adopted.
The absence of an arc between ym and ya assumes independence between a weapon’s features of manufacture and acquired characteristics. As noted by [12], the validity of such an assumption depends on the type of acquired characteristic and the way in which it has been described. The formula for V that corresponds to the Bayesian network shown in Fig. 6(ii) is an extension of Eq. (10): V ¼ w  Vm  Va þ ð1  wÞ:

(A.11)

The parameters Vm and Va represent the following ratios: Prðym jxm ; FÞ Vm ¼ ¯ ; Prðym jFÞ and Va ¼

Prðya jxa ; FÞ ¯ : Prðya jFÞ

A.4. A Bayesian network for analysis of CMS data Scientists may prefer to be more specific in describing acquired features and rely, for instance, on consecutive matching striations (CMS). The Bayesian network shown in Fig. 6(ii) can, through a slight modification, be used to analyse CMS data in a way described by [23]. Part of the network, xa ! ya, needs to be replaced by a single node C (see Fig. 6(iii)). C is a discrete chance node that has states numbered 0, 1, . . ., 10. A probability needs to be assigned to ¯ Sample each of these states given, respectively, F and F. values from a Poisson distribution can be found, for example, in [23] (also reproduced in [10]). Notice that certain Bayesian network software packages, e.g., HUGIN [34], support the automated specification of probabilities for node tables through special functionalities. In HUGIN language, for example, the conditional probabilities for C ¯ can be defined, using data from [23], given F (and F) through the expression Poisson (3.91) (and Poisson (1.325)).

References [1] W. Lichtenberg, Methods for the determination of shooting distance, For. Sci. Rev. 2 (1990) 37–62. [2] T.A. Warlow, Firearms, the Law and Forensic Ballistics, Taylor & Francis, 1996. [3] C. Champod, C. Baldwin, F. Taroni, J. Buckleton, Firearms and tool marks identification: the Bayesian approach, AFTE J. 35 (2003) 307–316. [4] D.V. Lindley, Foreword, in: C.G.G. Aitken, F. Taroni (Eds.), Statistics and the Evaluation of Evidence for Forensic Scientists, second ed., John Wiley & Sons, New York, 2004. [5] C.G.G. Aitken, A. Gammerman, Probabilistic reasoning in evidential assessment, J. For. Sci. Soc. 29 (1989) 303–316. [6] A.P. Dawid, I.W. Evett, Using a graphical method to assist the evaluation of complicated patterns of evidence, J. For. Sci. 42 (1997) 226–231. [7] P. Garbolino, F. Taroni, Evaluation of scientific evidence using Bayesian networks, For. Sci. Int. 125 (2002) 149–155. [8] I.W. Evett, P.D. Gill, G. Jackson, J. Whitaker, C. Champod, Interpreting small quantities of DNA: the hierarchy of propositions and the use of Bayesian networks, J. For. Sci. 47 (2002) 520–530. [9] F. Taroni, A. Biedermann, P. Garbolino, C.G.G. Aitken, A general approach to Bayesian networks for the interpretation of evidence, For. Sci. Int. 139 (2004) 5–16. [10] C.G.G. Aitken, F. Taroni, Statistics and the Evaluation of Evidence for Forensic Scientists, second ed., John Wiley & Sons, New York, 2004. [11] A.P. Dawid, J. Mortera, V.L. Pascali, D. van Boxel, Probabilistic expert systems for forensic inference from genetic markers, Scand. J. Stat. 29 (2002) 577–595. [12] I.W. Evett, J.A. Lambert, J.S. Buckleton, A Bayesian approach to interpreting footwear marks in forensic casework, Sci. Justice 38 (4) (1998) 241–247. [13] G. Jackson, The scientist and the scales of justice, Sci. Justice 40 (2000) 81–85.

A. Biedermann, F. Taroni / Forensic Science International 163 (2006) 18–33 [14] A. Biedermann, F. Taroni, Bayesian networks and probabilistic reasoning about scientific evidence when there is a lack of data, For. Sci. Int. 157 (2–3) (2006) 163–167. [15] V.M.H. Coupe´, L.C. Van der Gaag, J.D.F. Habbema, Sensitivity analysis: an aid for belief-network quantification, Knowl. Eng. Rev. 15 (3) (2000) 215–232. [16] W.F. Rowe, Range, in: J.H. Siegel, P.J. Saukko, G.C. Knupfer (Eds.), Encyclopedia of Forensic Science, vol. 2, Academic Press, San Diego, 2000, pp. 949–953. [17] P. Garbolino, Explaining relevance, Cardozo Law Rev. 22 (2001) 1503–1521. [18] J.H. Dillon, The Modified Griess test: a chemically specific chromophoric test for nitrite compounds in gunshot residues, AFTE J. 22 (3) (1990) 243–250. [19] F. Taroni, C.G.G. Aitken, G. Garbolino, A. Biedermann, Bayesian Networks and Probabilistic Inference in Forensic Science, John Wiley & Sons, Chichester, in press. [20] P.M. Lee, Bayesian Statistics, An Introduction, third ed., Arnold, London, 2004. [21] A.A. Biasotti, A statistical study of the individual characteristics of fired bullets, J. For. Sci. 4 (1959) 34–50. [22] A.A. Biasotti, J. Murdock, Firearms and toolmark identification: the scientific basis of firearms and toolmark identification., in: L. Faigman, D.H. Kaye, M.J. Saks, J. Sanders (Eds.), Modern Scientific Evidence: The Law and Science of Expert Testimony, West Publishing, St. Paul, MN, 1997, pp. 144–150. [23] S.G. Bunch, Consecutive matching striation criteria: a general critique, J. For. Sci. 45 (5) (2000) 955–962.

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[24] R.G. Nichols, Firearms and toolmark identification criteria: a review of the literature, J. For. Sci. 42 (3) (1997) 466–474. [25] R.G. Nichols, Firearms and toolmark identification criteria: a review of the literature, Part II, J. For. Sci. 48 (2) (2003) 318– 327. [26] B. Moran, Toolmark criteria for identification: pattern match, CMS, or Bayesian? INTERface 28 (2001) 9–10. [27] B. Moran, A report on the AFTE theory of identification and range of conclusions for tool mark identification and resulting approaches to casework, AFTE J. 34 (2) (2002) 227–235. [28] B. Moran, Comments and clarification of responses from a member of the AFTE 2001 criteria for identification of toolmarks discussion panel, AFTE J. 35 (2003) 55–65. [29] J.H. Wigmore, The problem of proof, Illinois Law Rev. 8 (1913) 77–103. [30] J.H. Wigmore, The Science of Judicial Proof: As Given by Logic, Psychology, and General Experience and Illustrated in Judicial Trials, third ed., Little, Brown, Boston, 1937. [31] T. Anderson, W. Twining, Analysis of Evidence: How to do Things with Facts Based on Wigmore’s Science of Judicial Proof, Northwestern University Press, Evanston, 1998. [32] D.A. Schum, Evidential Foundations of Probabilistic Reasoning, John Wiley & Sons, Inc., New York, 1994. [33] J.B. Kadane, D.A. Schum, A Probabilistic Analysis of the Sacco and Vanzetti Evidence, John Wiley & Sons, New York, 1996. [34] HUGIN, Free demonstration version available at http:// www.hugin.com.