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Bayesian framework for the evaluation of fibre transfer evidence
Bayesian framework for the evailuation offibre transfer evidence
Introduction The interpretation of transfer evidence in forensic science is a fundamental problem to be solved in most scientific investigations. Bayesian theory has been put forward as a coherent model for interpreting forensic evidence [I]. In the field of fibre transfer, literature suggests using a Bayesian approach [2,3] but the Bayesian calculations published are case specific [4-61; they do not supply straightforward guidelines which could be applied in other fibre cases. Convinced that the evaluation process does not differ for both fibre and bloodstain evidence, the authors have attempted, in this paper, to extract a formal Bayesian framework from the current literature (generally focused on bloodstain evidence), that ought to be useful for the evaluation of fibre transfer cases. Some definitions will be proposed, and then the Bayesian framework will be developed and applied to various scenarios involving fibre evidence taken from car seats.
Definitions Contact actions may lead to multiple transfers. Even if a scientist faces multiple transfer cases it is easier for him to consider them individually, before evaluating the individual results in one final decision process. This paper will therefore provide tools to assess only single transfers, following previous work by Evett [7]. In 1984, Stoney [8] pointed out the need to distinguish the various objects or persons involved in forensic transfer cases, defining their relationships and asking the right questions. This idea will be developed here by using transfer traces as the focal point of the interpretation process. This point of view will produce some variations from the proposals of Stoney's paper, but take a standard approach to the relevant forensic characteristics and the relevant population. The following terms must first be defined: a contact trace (T) is a trace, or any material of forensic interest, recovered from an object or person, the receptor (R). Traces can have one or more control sources which produce material defined as control material (CS). Two examples illustrate these definitions and show that traces are not always related to the crime scene, but can be found in association with a suspect. The argument is also valid for the control source which is not always associated with the suspect; control material can come from the victim or be an object from the scene. In the first example, an offender, wearing a red pullover, entered the rear of a luxurious house through a hole which he had cut in a metal grille. He attempted to force an entry but failed; the security alarm went off and he left the scene. About ten minutes after the offence, a suspect was apprehended in the vicinity of the house after an eyewitness had
testified that he saw a man wearing a red pullover running away from the scene. At the scene, a tuft of red fibres was found on the jagged end of one of the cut edges of the grille. The trace is the tuft of fibres from the grille which is the receptor, whereas the control source is the suspect's pullover. Fibres taken from this pullover will constitute the control material. In the second, a victim came to police to report that she had been raped by one of her old friends. The suspect denied any recent contact with the victim. His T-shirt (the receptor) was taken for examination. Foreign fibres were collected on this garment, which constituted the traces. In this case, the victim's garments were the control source, which would produce the control material. After technical examination, the traces and control material can be described by their respective sets of attributes y and x. y: the sum of extrinsic features (physical attributes such as quantity, number, positions on the receptor, etc.) and intrinsic features (chemical or biochemical descriptors such as serological analysis results) [9]. x: as a general rule, intrinsic features contribute principally to this set of data, but, following control experiments of contact cases, extrinsic characteristics may be added. The relevant population, which includes all potential sources of the traces, is always defined by the hypothesis proposed by the defence [lo] and by background information relating to the case. In the first example, if the defence strategy is to argue that the suspect had never been present at the scene, then the potential perpetrators are defined as wearing a red upper garment (here we are admitting that the eyewitness has seen the perpetrators). If the defence proposes the hypothesis that the suspect has been correctly identified by the witness, but that he had never been in contact with the grille, then the relevant population is defined by the potential perpetrators without any distinction in respect to the colour of the garment (here we are admitting that the eyewitness has not necessarily seen the perpetrators). Hence, depending on the strategy of the defence, the relevant population can be modified. The influence of the background information relating to the case upon the relevant population can be illustrated by supposing the absence of any eyewitness. If the suspect has been apprehended following the observation by the forensic scientist that the tuft of fibres is red in colour, then the relevant population is defined by the potential perpetrators wearing a red garment. If the suspect has been apprehended independently of the forensic attributes of the tuft, then the relevant population is defined by the potential perpetrators without any distinction in respect to the colour of the garment. Science & Justice 1997; 37(2): 75-83
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'Grouping' approach for traces and control material In most cases of transfer, for example of fibre or glass, more than one single trace element is recovered. Hence, it is convenient to establish some groupings for the recovered traces. In the same way, groupings are possible for the control material originating in the control source. A group is defined as a set of materials (from the traces and the control material) which share the same forensic attributes. Moreover, for traces, a group is declared only if there is sufficient specificity in the shared features to link these traces reasonably with a unique source. Most of the time, these grouping decisions can be made only with difficulty through complete numerical demonstration, but there can, and should, be logically qualified opinions. For example, when a certain number of extraneous fibres are recovered on the seat of a stolen car, and the forensic analysis indicates an agreement between all fibres recovered, then the scientist can reasonably state that all the fibres came from the same source. Thus, a group can be declared and this entire group will be described by the set y. Adopting this grouping approach will facilitate the interpretation process in the sense that the comparison is made between groups. At each stage of his technical examination, the scientist compares groups from traces and groups from control material, considering the sets of forensic attributes x and y. Thus, in the investigation of a potential individual transfer, after technical examination of the items, the forensic scientist always faces evidence consisting of x and y. The forensic scientist must now assess the evidential value of the combined set x and y by first identifying, and then weighing, the different explanations of the evidence. Explanations of the evidence The court is interested in the explanations given for the evidence. Usually the prosecution will present the evidence as a result of a criminal contact between control source and receptor, hypothesis HI. Nevertheless, this event is rarely the only possible explanation of the evidence and the forensic scientist must also consider other explanations that could be provided, for example by the defence: hypotheses H2, H3,..., H, [9]. At a particular moment in a trial, the context is restricted to two hypotheses, the one proposed by the prosecution, and the one chosen by the defence. So the context of the interpretation of the evidence is not defined by the forensic scientist but by the prosecution and defence in relation to the specific circumstances of the case. For example, a set of explanations to the evidence suggested to a court could be presented as follows in reference to the previous rape case:
HI: The suspect is the aggressor.
HZ:The suspect is not the aggressor and has not seen the victim during the last three weeks. Science & Justice 1997; 37(2): 75-83
H3: The suspect is not the aggressor, but he went out dancing that night with the victim. The evidential value of the forensic examination lies in the assessment of the probabilities of the observations (x, y) under two competing hypotheses. This could be H, against H2, H1 against H3 or whatever other hypotheses have been expressed by the defence. This means that the interpretation of the evidence will change depending on the scenarios proposed by the opposing parties. Bayesian theory: a tool for assessing the evidence against competing hypotheses Bayesian theory supplies a probabilistic model allowing one to adapt a given prior probability ratio between two competing hypotheses (e.g., H, and H2) in the light of new information (here, forensic observations x, y), and to obtain a posterior probability ratio through simple multiplication, by a ratio called the likelihood ratio (LR). The prior ratio is evaluated with reference to some background information (I), such as police inquiry and witness testimony, other than the information related to the forensic evidence. Baves' theorem defines the respective roles of scientists and jurists; forensic scientists deal with the LR, while jurists are concerned with the prior and posterior probability ratios. The likelihood ratio represents the strength of the evidence to be considered under the two competing hypotheses, HI and H2, but the question of the relevance of the recovered material must also be incorporated. In some cases, it is clear that the recovered material is associated with the offence. In others, the material might either be associated with the offence or have been present for reasons unconnected with the offence. Consequently, two mutually exclusive events B and B can be defined, comparable to the intermediate association hypotheses suggested by Buckleton [quoted in 111:
B: in which recovered traces are associated with the offence have been transferred, have persisted, and have been successfullv recovered on the receDtor (R). L
x
,
B:
in which recovered traces are not associated with the offence. The LR, which is formally developed in the Appendix, is expressed by:
As will be shown below, however, the LR can be simplified under certain circumstances. Application to various scenarios An attempt will be made to apply this interpretation framework to different scenarios involving fibres in car seat cases. These scenarios have been chosen in order to illustrate the evolution of the parameters in different situations.
Bayesian framework for the evaluation offibre transfer evidence
Direct applicability of the model is becoming more practical with results from research that cover or measure the main variables (frequencies, transfer, persistence, relevance) [12-151. The use of Roux's results is beyond the scope of this presentation, whose aim is only to emphasize the general applicability of the reasoning, whatever transfer trace is considered. Scenario 1 A stolen car is used to perpetrate a robbery on the day of its theft. One hour later, the car is abandoned, and during the night is found by the police. On the polyester seats, which have recently been cleaned with a specific car vacuum cleaner, extraneous textile fibres are collected. The car owner lives alone and has never lent his vehicle to anyone. The owner wears nothing but cotton. The day following the robbery a suspect is apprehended, his red woollen pullover and his denim pants are confiscated and sent to the laboratory.
B : The fibres group G3 is not associated with the offence. In this type of transfer, the probability of the event B depends on the hypotheses considered (HI or H2), hence:
The probability of the relationship between the traces and the offence depends on the source of the traces. Here, the probabilities of B and B depend more on the sheddability of the garment than on variables in persistence or recovering methods. The sheddability is not only a factor of the composition of the garment (intrinsic characteristics analysed), but also depends on variables such as its construction [13]. Both these variables are assumed to be controlled in the hypothesis HI (indeed the police have the pullover of the suspect), but they are unknown if this pullover has never been in contact with the seat (H2). The LR equation then becomes:
On the driver's seat (receptor) three groups of foreign fibres have been collected. These consist of large numbers of: white cotton fibres (GI); blue cotton fibres (G2); and red wool fibres (G3). Following laboratory examination, the fibres from G1 and G2 are found to correspond to clothing of the owner; the presence of these fibres can be explained and therefore they will be ignored. Group G3 only will be considered for the interpretation. The forensic evidence can be formulated as follows:
where:
y: the group of traces is described by a set y of extrinsic (number, position on the seat) and intrinsic (colour, microscopic features, dye, etc.) characteristics.
tH1:the probability that these fibres G3 have been transferred from the offender's upper garment to the car seat, have persisted and have been recovered. This probability is estimated according to hypothesis HI.
x: the red woollen pullover of the suspect (control source) generates control fibres described by a set x of characteristics. Note that there is no bias (double counting error [lo], also called selection effect [16]) in considering the specificity of the colour in the assessment of the evidence, because it has not already been taken into account for the selection of the suspect's garment; the suspect has been apprehended independently of the evidence recovered. Considering the reported association between the recovered fibres and the suspect's pullover, the competing hypotheses could be as follows: HI: the suspect sat on this seat of the stolen car. H2: the suspect has never sat on this seat of the stolen car. The intermediate association hypotheses are simply:
B: the fibres group G3 is associated with the offence, it has been transferred from the offender's upper garment to the car seat, has persisted and has been recovered.
tH2:the probability that these fibres G3 have been transferred from the offender's upper garment (and not from the suspect's garments) to the car seat, have persisted and have been recovered. This probability is estimated according to hypothesis H2. f: estimated frequency of the compared characteristics from
y in similar sized extraneous groups of fibres found on car seats. bo: probability of no chance occurrence of group of foreign fibres (large number) on the driver's car seat. These fibres can be distinguished from the garments of the habitual user(s) of the car. bl: probability of having one chance occurrence of a group of foreign fibres (large number) on the driver's car seat. These fibres can be distinguished from the garments of the habitual user(s) of the car (bl I 1-bo). It is important to emphasize that bo and bl are mutually exclusive parts of a total event which could be called, 'having on the driver's seat, zero, one or more groups of Science & Justice 1997; 37(2): 75-83
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0025
0.125
0.225
0.325
0.425
0.525
0.625
0.725
0.825
0.925
Probability bo FIGURE 1 LR values calculated in function of bo, b, = 1-bo, t and f(O.O1). Probability t:- - - - - 0.9; -0.5 and; ........0.1.
extraneous fibres (large number) which can be distinguished from the garments of the habitual user(s) of the car'. Considering that the number of groups range from 0 to infinity:
particular circumstances of the case. The most favourable estimation (to the suspect) should be retained. Therefore tH2= 1. Thus, the simplified LR corresponds to the LR obtained in a classic 'one bloodstain on the scene' case [17]: 1 LR = f
Then:
In order to estimate bOand bl in this particular scenario, the vacuum cleaner bag used recently by the owner could be searched for extraneous groups made of a large number of fibres. But, as the stolen vehicle is regularly cleaned and the car is solely used by the owner wearing cotton-made garments, this probability can reasonably be assumed to be close to 1. Therefore bl is close to 0. Controlled experiments involving the suspect's pullover and the driver's seat would yield a valid estimation. Here, it will be assumed that on average a large number of fibres are transferred and recovered in the particular circumstances of the case. Therefore tH, = 1. Controlled experiments involving woollen garments and the driver's seat of the stolen car would also yield a valid estimation. Here, it will be assumed that on average a large number of fibres are transferred and recovered in the Science & Justice 1997; 37(2): 75-83
Scenario 2 The second scenario is very similar to the first, except that the number of recovered fibres is low compared to the number expected in this kind of contact. The parameters must therefore be reassessed.
As in scenario 1: LR = b o t ~+, b,f?l - t ~ , ) fIbOtH, + bldl - tHJl Figure 1 shows the evolution of the LR according to the variation of bo and t (estimated as equivalent for H1 or H2). The value f has been put at 0.01, the value of bl is equal to 1-bo. The LR will vary between 1 and 100 (110 depending on the values attributed to t and bO. This figure helps to emphasize the importance of factors other than the frequency f i n the evaluation of the LR. For the above simulation, bl was the strict complement of bO and the values b2 to bw were supposed to be equal to zero. This assumption might be acceptable for high values of bO,but as soon as bOdecreases, the probability bl would seem to be overestimated. Briggs' survey [18] for the 79
Bayesian framework for the el~aluationoffbre transfer evidence
TABLE 1 Observed and expected frequencies for respectively LSH (from Figure 6, surface), ME1 (from Figure 6,191 jackets plus 25 pullovers) and ME2 (from Figure 6,216 trousers). Probabilities of bi values Surveys
bo
bl
b2
b3
b4
Lambert et al. (LSH) [19] 0.41 0.27 0.12 0.09 0.05 LSH under Poisson distribution 0.41 0.37 0.16 0.05 0.01 McQui11an & Edgar (ME1) [20] 0.82 0.12 0.03 0.01 0.00 ME1 under Poisson distribution 0.82 0.16 0.02 0.00 0.00 McQuillan & Edgar (ME2) [20] 0.75 0.19 0.04 0.01 0.01 ME2 under Poisson distribution 0.75 0.22 0.03 0.00 0.00
presence of extraneous bloodstains by chance occurrence on clothing will illustrate this point. In this context, Evett [7] estimated bo > 0.95; hence b, < 0.05 and it would be reasonable to state that the values b2 to b, denote very rare events which can be neglected. Suppose now, that the survey had considered medical staff as the relevant population and that the only value at hand was bo which was equal to 0.05. In that case, taking bl = 1-bo would appear to be unrealistic because the possibility that the average number of bloodstains is greater than one cannot be excluded. Hence, it is proposed to modify the previous assumption (bl = 1-bo), by calculating bl in terms of bo following the hypothesis that the distribution of bi values follows a Poisson probability law of parameter h.
bO
b1
From each value of bo, the corresponding h is calculated: h = -ln(bo) From the parameter (h) the probability bl,b2,...,bi can be calculated using the equation. Some of the calculated distributions are presented in Figure 2. To support the hypothesis of a Poisson distribution of bi, calculated data under the hypothesis can be compared with observed data coming from previous surveys (available only for glass fragments) [19,20]. The results shown in Table 1 indicate that the Poisson distribution hypothesis is plausible and the model tends to overestimate bl. This deviation can be accepted because it will tend to a conservative LR. If the proposed model is assumed to be valid in the fibre scenario, the effect on the likelihood ratio can be seen in Figure 3 and compared with the evaluation of the LR under the previous assumption (bl = 1-bo). Logically, the first approach for calculating bl gives the most conservative estimate of the LR. For high values of transfer probabilities (i.e., t = 0.9), the difference between the two simulations is low if bo 2 0.425 (difference < 5%), but the difference rapidly increases when bo <0.425. Hence, the previous model (when bl = 1-bo) gives a large underestimation of the evidential value for small values of bo. For low values of transfer probabilities (i.e., t = 0.1), the difference between the conservative estimate of the LR and the
b5 b6 b7 b8 b9 b10 bi values FIGURE 2 Probability distributions under Poisson assumption for some distinct values of bo. -0bn=0.950,-Abn=0.500,--0-b0=0.250, bn=0.050. b2
b3
b4
+-
80
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0 0.050
0.450 Probability bo
0.950
0.050
0.450 Probability bo
0.950
FIGURE 3 LR values calculated in function of bo,bl, t(0.9), and f(O.O1) respectively, under two different models for calculating bl.
FIGURE 4 LR values calculated in function of b,, b,, t(0.1), and f(O.O1) respectively, under two different models for calculating.- bl. --
corrected one is even more significant. As soon as bo < 0.85, the difference exceeds 5%, as illustrated in Figure 4.
groups (in this case this value is estimated equal to 1). fl: estimated frequency of the compared characteristics of the set yl in the extraneous similar-sized groups of fibres found on car seats.
Scenario 3 This scenario is the same as the first, except that there are N number of foreign groups of fibres with a large number of fibres in each group, with data sets yl to yN. Nevertheless, only one group is compatible with the suspect's pullover (red wool).
f2,...,N:estimated frequency of the groups 2 to N among the foreign groups recovered on car seats. This simple result demonstrates that, in evaluating a match, the scientist must consider concording and non-concording elements as already suggested for bloodstain evidence in the classic 'two traces problem'[22]. Therefore, it is important not only to focus on the fibres that match the suspect's garments, but also to consider other groups of fibres compatible with the facts.
Here, for sake of simplicity, it will be assumed that the offender's upper garment is made of one type of fibre. In this case the LR is developed considering that the evidence is the set {xl, y I , y2...yN}and that the event B and B admit the following definitions: B: one of these recovered groups is associated with the offence and has been transferred from the offender's upper garment to the car seat, has persisted and has been recovered.
The introduction stressed the importance of the defence strategy which can modify the evaluation of the evidence, notably by modifying the relevant population. Here, further development of this scenario can demonstrate that the B: no recovered group is associated with the offence. defence strategy will dictate not only the relevant population but also the whole evaluation of the LR. Previously, H2 The LR equation is developed in the same way as proposed was implicitly defined as 'The suspect has never sat on this by Buckleton and Evett [21]: seat, but another unknown person has stolen the car'. I f2,...,~ 1 t~,=l negligible in this case Suppose now that the defence hypothesis changes as a result of a reliable eyewitness testimony. LR=P(YI,Y2.....YN~~l.~,~l). P(BJx~,H~) + P ( ~ ~ ~ ~ , . . . , ~P(BIx~,H,) ~~X~,B,H~). ~ )+ ~ B H 2 ) P ( Y I . Y ~ . . Y N IP~( BI/ .~ ~I , H. ~ ~p. ( ~ I ~ ~ l ~ ~ ~ N I x I p(Elxl,H2) H3: the suspect has never sat on this seat, but another man negligible in this case wearing a red pullover (as stated by the witness) has stolen b ~ - l. f~ . f 2 , . . . , ~. N! tHI=l the car.
---
-
-
-
4
The LR would remain formally the same (l/flN), but the specification off, and N would be modified as follows: where: bN-l: is as in scenario 1 but for a chance occurrence of N-1 group of foreign fibres (more than a hundred) on the driver's car seat. tH1and tH2:are as in scenario 1 but for one of these recovered Science & Justice 1997; 37(2): 75-83
f,: estimated frequency of the compared characteristics of the set yl in the extraneous similar-sized groups of red fibres found on car seats (modification of the relevant population). N: number of foreign groups, each containing a large number of red fibres. 81
Bayesian framework for the eljaluation offibre transfer evidence
Hence, if the group of red fibres is the only recovered group of red fibres on the seat, the LR is simply expressed by l/fl. Then, defence strategy can modify the scientist's approach not only during the evaluation stage of the evidence, but also at the laboratory stage. For example, under Hg the scientist does not need to count all the compatible extraneous groups of fibres (which can be time-consuming), but needs to consider only the groups of red fibres. All the other groups of extraneous fibres are not relevant for the evaluation of the evidence (their presence is explained as for the previous groups G1 and G2). Scenario 4
Another scenario can be considered where the car belongs to a man who is suspected of abducting a woman and attempting rape. The victim was wearing a red woollen pullover and denim pants. According to the suspect, nobody but his wife ever sits in the passenger's seat; moreover, the car seats have been recently vacuumed. On the passenger's seat three groups of foreign fibres have been collected. These groups consist of a large number of white cotton fibres (GI); blue cotton fibres ((32); and red wool fibres ((33). Following laboratory examinations, the fibres from G1 and G2 are in agreement with the wife's clothing. As the presence of these fibres is explained, they will be ignored. Then the forensic evidence can be formulated as follows: y: the recovered group G3 is described by a set y of characteristics (number, position on the seat, colour, microscopic features, dye). x: the red woollen pullover of the victim (control source) generates control fibres described by a set x of characteristics. Considering that the suspect denies that the victim has ever been in contact with this car, the competing hypotheses are: HI: the victim has sat on the passenger's seat of the suspect's car. H2: the victim has never sat on the passenger's seat of the suspect's car.
B
The associative events B and are defined as previously in Scenario 1. The LR is expressed by the general equation with the following parameters:
Logically, under the hypothesis H2 (meaning no offence), the probability that the traces are related to the offence is zero. Considering the circumstances of the offence, the number of recovered fibres and the sheddability of the pullover, it will be assumed that tHIis equal to 1: the LR becomes:
It is important to note that the denominator of the LR now includes bl: indeed, if H2 is accepted, then this foreign group is present by pure chance, bl being equal to, or smaller than, 1-bo; if bo is assessed as close to 1 (considering the use and the cleaning procedures of the car) then bl becomes smaller than 1 and the LR is strongly increased. If the number of fibres constituting the sample of extraneous fibres decreases, then tHI,b0 and bl must be further assessed. If the number of foreign groups recovered on this seat is equal to N (>2), the LR becomes (with tH1still estimated as equal to 1):
b ~f 1 .. f2,...,N . N!
bN
N.fl
N.fl
As quoted by Buckleton and Evett [21], according to Fong and Inami [23], when N is increased, bN is equal bN-l.Thus the LR becomes identical to that of scenario 3. Conclusion Following an explicit definition of objects and attributes involved in trace transfer, a general expression of the likelihood ratio has been derived in a Bayesian framework. This unique expression, which could be applied to the evaluation of numerous trace evidence, has permitted an approach to different case scenarios involving fibre evidence where the direction of the transfer and the amount of the recovered material were varied. The variations around the scenarios allows the identification and evaluation of the dominant parameters and their respective effect on the likelihood ratio. Moreover, it has been emphasized that these parameters are not only defined by the conditions of the contact, but also by the strategy chosen by the defence. Acknowledgments The authors are indebted to Dr Ian Evett and Dr John Buckleton for their fruitful comments, their patience and their encouragement. They wish also to thank Dr Colin Aitken for commenting on the manuscript and Dr Claude Roux for suggesting some scenarios.
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C CHAMPOD and F TARONI
Appendix - Development of the likelihood ratio The formal development of the LR is largely covered in recent literature [I]. Considering the definitions proposed in this paper, the LR becomes:
For sake of brevity, our equations will not take into account the background information I, thus yielding P ( X , ~ ~and H~) P ( ~ ,1Y~ 2 ) . The likelihood ratio then becomes:
P ( X ~ Hand ~ ) p(xlH2),being equivalent, cancel each other out. The likelihood ratio is thus reduced to:
The likelihood ratio is then developed by an 'extension of conversation' [24] taking into consideration the two events, that recovered traces are associated with, and not associated with, the offence, B and B respectively:
References 1 . Aitken CGG. Statistics and the Evaluation of Evidence for Forensic Scientists. Chichester: John Wiley & Sons, 1995. 2 . Grieve MC. Information Content: The Interpretation of Fibres Evidence. In: Robertson J, ed. Forensic Examination of Fibres. New York: Ellis Horwood, 1992: 239-262. 3 . Roux C and Margot P. L'estimation de la valeur indiciale des fibres textiles dCcouvertes en relation avec une affaire criminelle utopie ou rialitk? Revue internationale de criminologie et de police technique 1994; 67: 229-241. 4 . Evett IW, Cage PE and Aitken CGG. Evaluation of the likelihood ratio for fibre transfer evidence in criminal cases. Applied Statistics 1987; 36: 174-180. 5 . Grieve MC and Dunlop J. A practical aspect of the Bayesian interpretation of fibre evidence. Journal of the Forensic Science Society 1992: 32: 169-175.
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6 . Cook R, Evett IW, Jackson G. and Rogers M. A workshop approach to improving the understanding of the significance of fibres evidence. Journal of the Forensic Science Society 1993; 33: 149-152. 7 . Evett IW. A quantitative theory for interpreting transfer evidence in criminal cases. Applied Statistics 1984; 33: 25-32. 8 . Stoney DA. Evaluation of associative evidence: choosing the relevant question. Journal of the Forensic Science Society 1984; 24: 473482. 9 . Kind SS. Crime investigation and the criminal trial: A three chapter paradigm of evidence. Journal of the Forensic Science Society 1994; 34: 155-164. 10 . Robertson B and Vignaux GA. Interpreting Evidence - Evaluating Forensic Science in the Courtroom. Chichester: John Wiley & Sons, 1995. 1I Evett IW. Establishing the evidential value of small quantity of material found at a crime scene. Journal of the Forensic Science Society 1993; 33: 83-86. 12 Roux C. La valeur indiciale des fibres textiles dCcouvertes sur un sikge de voiture: Problkmes et solutions. Thkse de doctorat. Institut de Police Scientifique et de Criminologie - UniversitC de Lausanne, Switzerland, 1996. 13 . Roux C, Chable J and Margot P. Fibre transfer experiments onto car seats. Science & Justice 1996; 36: 143-151. 14 Roux C and Margot P. The population of textile fibres on car seats. Science & Justice 1997; 37: 25-30. 15 Roux C and Margot P. An attempt to assess the relevance of textile fibres recovered from car seats. Science & Justice. Submitted for publication. 16 . Thompson WC. Are juries competent to evaluate statistical evidence? Law and Contemporary Problems 1989; 52: 9 4 0 . 17 . Evett IW. What is the probability that this blood came from that person? A meaningful question? Journal of the Forensic Science Society 1983; 23: 1-23. 18 . Briggs TJ. The probative value of bloodstains on clothing. Medicine, Science and the Law 1978; 16: 79-83. 19 . Lambert JA, Satterthwaite MJ and Hanison PH. A survey of glass fragments recovered from clothing of persons suspected of involvement in crime. Science & Justice 1995; 35: 273-281. 2 0 . McQuillan J and Edgar K. A survey of the distribution of glass on clothing. Journal of the Forensic Science Society 1992; 32: 333-348. 21 . Buckleton J and Evett IW. Aspects of the Bayesian Interpretation of Fibre Evidence. CRSE Report 1989, No. 684. 22 . Evett IW. On meaningful questions: A two-trace problem. Journal of the Forensic Science Society 1987; 27: 375-381. 23 . Fong W and Inami H. Results of a study to determine the probability of chance match occurrences between fibers known to be from different sources. Journal of Forensic Sciences 1986; 31: 65-72. 24 . Lindley DV. Probability. In: Aitken CGG and Stoney DA, ed. The Use of Statistics in Forensic Science. New York: Ellis Horwood Ltd., 1991: 27-50.
Introduction The interpretation of transfer evidence in forensic science is a fundamental problem to be solved in most scientific investigations. Bayesian theory has been put forward as a coherent model for interpreting forensic evidence [I]. In the field of fibre transfer, literature suggests using a Bayesian approach [2,3] but the Bayesian calculations published are case specific [4-61; they do not supply straightforward guidelines which could be applied in other fibre cases. Convinced that the evaluation process does not differ for both fibre and bloodstain evidence, the authors have attempted, in this paper, to extract a formal Bayesian framework from the current literature (generally focused on bloodstain evidence), that ought to be useful for the evaluation of fibre transfer cases. Some definitions will be proposed, and then the Bayesian framework will be developed and applied to various scenarios involving fibre evidence taken from car seats.
Definitions Contact actions may lead to multiple transfers. Even if a scientist faces multiple transfer cases it is easier for him to consider them individually, before evaluating the individual results in one final decision process. This paper will therefore provide tools to assess only single transfers, following previous work by Evett [7]. In 1984, Stoney [8] pointed out the need to distinguish the various objects or persons involved in forensic transfer cases, defining their relationships and asking the right questions. This idea will be developed here by using transfer traces as the focal point of the interpretation process. This point of view will produce some variations from the proposals of Stoney's paper, but take a standard approach to the relevant forensic characteristics and the relevant population. The following terms must first be defined: a contact trace (T) is a trace, or any material of forensic interest, recovered from an object or person, the receptor (R). Traces can have one or more control sources which produce material defined as control material (CS). Two examples illustrate these definitions and show that traces are not always related to the crime scene, but can be found in association with a suspect. The argument is also valid for the control source which is not always associated with the suspect; control material can come from the victim or be an object from the scene. In the first example, an offender, wearing a red pullover, entered the rear of a luxurious house through a hole which he had cut in a metal grille. He attempted to force an entry but failed; the security alarm went off and he left the scene. About ten minutes after the offence, a suspect was apprehended in the vicinity of the house after an eyewitness had
testified that he saw a man wearing a red pullover running away from the scene. At the scene, a tuft of red fibres was found on the jagged end of one of the cut edges of the grille. The trace is the tuft of fibres from the grille which is the receptor, whereas the control source is the suspect's pullover. Fibres taken from this pullover will constitute the control material. In the second, a victim came to police to report that she had been raped by one of her old friends. The suspect denied any recent contact with the victim. His T-shirt (the receptor) was taken for examination. Foreign fibres were collected on this garment, which constituted the traces. In this case, the victim's garments were the control source, which would produce the control material. After technical examination, the traces and control material can be described by their respective sets of attributes y and x. y: the sum of extrinsic features (physical attributes such as quantity, number, positions on the receptor, etc.) and intrinsic features (chemical or biochemical descriptors such as serological analysis results) [9]. x: as a general rule, intrinsic features contribute principally to this set of data, but, following control experiments of contact cases, extrinsic characteristics may be added. The relevant population, which includes all potential sources of the traces, is always defined by the hypothesis proposed by the defence [lo] and by background information relating to the case. In the first example, if the defence strategy is to argue that the suspect had never been present at the scene, then the potential perpetrators are defined as wearing a red upper garment (here we are admitting that the eyewitness has seen the perpetrators). If the defence proposes the hypothesis that the suspect has been correctly identified by the witness, but that he had never been in contact with the grille, then the relevant population is defined by the potential perpetrators without any distinction in respect to the colour of the garment (here we are admitting that the eyewitness has not necessarily seen the perpetrators). Hence, depending on the strategy of the defence, the relevant population can be modified. The influence of the background information relating to the case upon the relevant population can be illustrated by supposing the absence of any eyewitness. If the suspect has been apprehended following the observation by the forensic scientist that the tuft of fibres is red in colour, then the relevant population is defined by the potential perpetrators wearing a red garment. If the suspect has been apprehended independently of the forensic attributes of the tuft, then the relevant population is defined by the potential perpetrators without any distinction in respect to the colour of the garment. Science & Justice 1997; 37(2): 75-83
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'Grouping' approach for traces and control material In most cases of transfer, for example of fibre or glass, more than one single trace element is recovered. Hence, it is convenient to establish some groupings for the recovered traces. In the same way, groupings are possible for the control material originating in the control source. A group is defined as a set of materials (from the traces and the control material) which share the same forensic attributes. Moreover, for traces, a group is declared only if there is sufficient specificity in the shared features to link these traces reasonably with a unique source. Most of the time, these grouping decisions can be made only with difficulty through complete numerical demonstration, but there can, and should, be logically qualified opinions. For example, when a certain number of extraneous fibres are recovered on the seat of a stolen car, and the forensic analysis indicates an agreement between all fibres recovered, then the scientist can reasonably state that all the fibres came from the same source. Thus, a group can be declared and this entire group will be described by the set y. Adopting this grouping approach will facilitate the interpretation process in the sense that the comparison is made between groups. At each stage of his technical examination, the scientist compares groups from traces and groups from control material, considering the sets of forensic attributes x and y. Thus, in the investigation of a potential individual transfer, after technical examination of the items, the forensic scientist always faces evidence consisting of x and y. The forensic scientist must now assess the evidential value of the combined set x and y by first identifying, and then weighing, the different explanations of the evidence. Explanations of the evidence The court is interested in the explanations given for the evidence. Usually the prosecution will present the evidence as a result of a criminal contact between control source and receptor, hypothesis HI. Nevertheless, this event is rarely the only possible explanation of the evidence and the forensic scientist must also consider other explanations that could be provided, for example by the defence: hypotheses H2, H3,..., H, [9]. At a particular moment in a trial, the context is restricted to two hypotheses, the one proposed by the prosecution, and the one chosen by the defence. So the context of the interpretation of the evidence is not defined by the forensic scientist but by the prosecution and defence in relation to the specific circumstances of the case. For example, a set of explanations to the evidence suggested to a court could be presented as follows in reference to the previous rape case:
HI: The suspect is the aggressor.
HZ:The suspect is not the aggressor and has not seen the victim during the last three weeks. Science & Justice 1997; 37(2): 75-83
H3: The suspect is not the aggressor, but he went out dancing that night with the victim. The evidential value of the forensic examination lies in the assessment of the probabilities of the observations (x, y) under two competing hypotheses. This could be H, against H2, H1 against H3 or whatever other hypotheses have been expressed by the defence. This means that the interpretation of the evidence will change depending on the scenarios proposed by the opposing parties. Bayesian theory: a tool for assessing the evidence against competing hypotheses Bayesian theory supplies a probabilistic model allowing one to adapt a given prior probability ratio between two competing hypotheses (e.g., H, and H2) in the light of new information (here, forensic observations x, y), and to obtain a posterior probability ratio through simple multiplication, by a ratio called the likelihood ratio (LR). The prior ratio is evaluated with reference to some background information (I), such as police inquiry and witness testimony, other than the information related to the forensic evidence. Baves' theorem defines the respective roles of scientists and jurists; forensic scientists deal with the LR, while jurists are concerned with the prior and posterior probability ratios. The likelihood ratio represents the strength of the evidence to be considered under the two competing hypotheses, HI and H2, but the question of the relevance of the recovered material must also be incorporated. In some cases, it is clear that the recovered material is associated with the offence. In others, the material might either be associated with the offence or have been present for reasons unconnected with the offence. Consequently, two mutually exclusive events B and B can be defined, comparable to the intermediate association hypotheses suggested by Buckleton [quoted in 111:
B: in which recovered traces are associated with the offence have been transferred, have persisted, and have been successfullv recovered on the receDtor (R). L
x
,
B:
in which recovered traces are not associated with the offence. The LR, which is formally developed in the Appendix, is expressed by:
As will be shown below, however, the LR can be simplified under certain circumstances. Application to various scenarios An attempt will be made to apply this interpretation framework to different scenarios involving fibres in car seat cases. These scenarios have been chosen in order to illustrate the evolution of the parameters in different situations.
Bayesian framework for the evaluation offibre transfer evidence
Direct applicability of the model is becoming more practical with results from research that cover or measure the main variables (frequencies, transfer, persistence, relevance) [12-151. The use of Roux's results is beyond the scope of this presentation, whose aim is only to emphasize the general applicability of the reasoning, whatever transfer trace is considered. Scenario 1 A stolen car is used to perpetrate a robbery on the day of its theft. One hour later, the car is abandoned, and during the night is found by the police. On the polyester seats, which have recently been cleaned with a specific car vacuum cleaner, extraneous textile fibres are collected. The car owner lives alone and has never lent his vehicle to anyone. The owner wears nothing but cotton. The day following the robbery a suspect is apprehended, his red woollen pullover and his denim pants are confiscated and sent to the laboratory.
B : The fibres group G3 is not associated with the offence. In this type of transfer, the probability of the event B depends on the hypotheses considered (HI or H2), hence:
The probability of the relationship between the traces and the offence depends on the source of the traces. Here, the probabilities of B and B depend more on the sheddability of the garment than on variables in persistence or recovering methods. The sheddability is not only a factor of the composition of the garment (intrinsic characteristics analysed), but also depends on variables such as its construction [13]. Both these variables are assumed to be controlled in the hypothesis HI (indeed the police have the pullover of the suspect), but they are unknown if this pullover has never been in contact with the seat (H2). The LR equation then becomes:
On the driver's seat (receptor) three groups of foreign fibres have been collected. These consist of large numbers of: white cotton fibres (GI); blue cotton fibres (G2); and red wool fibres (G3). Following laboratory examination, the fibres from G1 and G2 are found to correspond to clothing of the owner; the presence of these fibres can be explained and therefore they will be ignored. Group G3 only will be considered for the interpretation. The forensic evidence can be formulated as follows:
where:
y: the group of traces is described by a set y of extrinsic (number, position on the seat) and intrinsic (colour, microscopic features, dye, etc.) characteristics.
tH1:the probability that these fibres G3 have been transferred from the offender's upper garment to the car seat, have persisted and have been recovered. This probability is estimated according to hypothesis HI.
x: the red woollen pullover of the suspect (control source) generates control fibres described by a set x of characteristics. Note that there is no bias (double counting error [lo], also called selection effect [16]) in considering the specificity of the colour in the assessment of the evidence, because it has not already been taken into account for the selection of the suspect's garment; the suspect has been apprehended independently of the evidence recovered. Considering the reported association between the recovered fibres and the suspect's pullover, the competing hypotheses could be as follows: HI: the suspect sat on this seat of the stolen car. H2: the suspect has never sat on this seat of the stolen car. The intermediate association hypotheses are simply:
B: the fibres group G3 is associated with the offence, it has been transferred from the offender's upper garment to the car seat, has persisted and has been recovered.
tH2:the probability that these fibres G3 have been transferred from the offender's upper garment (and not from the suspect's garments) to the car seat, have persisted and have been recovered. This probability is estimated according to hypothesis H2. f: estimated frequency of the compared characteristics from
y in similar sized extraneous groups of fibres found on car seats. bo: probability of no chance occurrence of group of foreign fibres (large number) on the driver's car seat. These fibres can be distinguished from the garments of the habitual user(s) of the car. bl: probability of having one chance occurrence of a group of foreign fibres (large number) on the driver's car seat. These fibres can be distinguished from the garments of the habitual user(s) of the car (bl I 1-bo). It is important to emphasize that bo and bl are mutually exclusive parts of a total event which could be called, 'having on the driver's seat, zero, one or more groups of Science & Justice 1997; 37(2): 75-83
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0025
0.125
0.225
0.325
0.425
0.525
0.625
0.725
0.825
0.925
Probability bo FIGURE 1 LR values calculated in function of bo, b, = 1-bo, t and f(O.O1). Probability t:- - - - - 0.9; -0.5 and; ........0.1.
extraneous fibres (large number) which can be distinguished from the garments of the habitual user(s) of the car'. Considering that the number of groups range from 0 to infinity:
particular circumstances of the case. The most favourable estimation (to the suspect) should be retained. Therefore tH2= 1. Thus, the simplified LR corresponds to the LR obtained in a classic 'one bloodstain on the scene' case [17]: 1 LR = f
Then:
In order to estimate bOand bl in this particular scenario, the vacuum cleaner bag used recently by the owner could be searched for extraneous groups made of a large number of fibres. But, as the stolen vehicle is regularly cleaned and the car is solely used by the owner wearing cotton-made garments, this probability can reasonably be assumed to be close to 1. Therefore bl is close to 0. Controlled experiments involving the suspect's pullover and the driver's seat would yield a valid estimation. Here, it will be assumed that on average a large number of fibres are transferred and recovered in the particular circumstances of the case. Therefore tH, = 1. Controlled experiments involving woollen garments and the driver's seat of the stolen car would also yield a valid estimation. Here, it will be assumed that on average a large number of fibres are transferred and recovered in the Science & Justice 1997; 37(2): 75-83
Scenario 2 The second scenario is very similar to the first, except that the number of recovered fibres is low compared to the number expected in this kind of contact. The parameters must therefore be reassessed.
As in scenario 1: LR = b o t ~+, b,f?l - t ~ , ) fIbOtH, + bldl - tHJl Figure 1 shows the evolution of the LR according to the variation of bo and t (estimated as equivalent for H1 or H2). The value f has been put at 0.01, the value of bl is equal to 1-bo. The LR will vary between 1 and 100 (110 depending on the values attributed to t and bO. This figure helps to emphasize the importance of factors other than the frequency f i n the evaluation of the LR. For the above simulation, bl was the strict complement of bO and the values b2 to bw were supposed to be equal to zero. This assumption might be acceptable for high values of bO,but as soon as bOdecreases, the probability bl would seem to be overestimated. Briggs' survey [18] for the 79
Bayesian framework for the el~aluationoffbre transfer evidence
TABLE 1 Observed and expected frequencies for respectively LSH (from Figure 6, surface), ME1 (from Figure 6,191 jackets plus 25 pullovers) and ME2 (from Figure 6,216 trousers). Probabilities of bi values Surveys
bo
bl
b2
b3
b4
Lambert et al. (LSH) [19] 0.41 0.27 0.12 0.09 0.05 LSH under Poisson distribution 0.41 0.37 0.16 0.05 0.01 McQui11an & Edgar (ME1) [20] 0.82 0.12 0.03 0.01 0.00 ME1 under Poisson distribution 0.82 0.16 0.02 0.00 0.00 McQuillan & Edgar (ME2) [20] 0.75 0.19 0.04 0.01 0.01 ME2 under Poisson distribution 0.75 0.22 0.03 0.00 0.00
presence of extraneous bloodstains by chance occurrence on clothing will illustrate this point. In this context, Evett [7] estimated bo > 0.95; hence b, < 0.05 and it would be reasonable to state that the values b2 to b, denote very rare events which can be neglected. Suppose now, that the survey had considered medical staff as the relevant population and that the only value at hand was bo which was equal to 0.05. In that case, taking bl = 1-bo would appear to be unrealistic because the possibility that the average number of bloodstains is greater than one cannot be excluded. Hence, it is proposed to modify the previous assumption (bl = 1-bo), by calculating bl in terms of bo following the hypothesis that the distribution of bi values follows a Poisson probability law of parameter h.
bO
b1
From each value of bo, the corresponding h is calculated: h = -ln(bo) From the parameter (h) the probability bl,b2,...,bi can be calculated using the equation. Some of the calculated distributions are presented in Figure 2. To support the hypothesis of a Poisson distribution of bi, calculated data under the hypothesis can be compared with observed data coming from previous surveys (available only for glass fragments) [19,20]. The results shown in Table 1 indicate that the Poisson distribution hypothesis is plausible and the model tends to overestimate bl. This deviation can be accepted because it will tend to a conservative LR. If the proposed model is assumed to be valid in the fibre scenario, the effect on the likelihood ratio can be seen in Figure 3 and compared with the evaluation of the LR under the previous assumption (bl = 1-bo). Logically, the first approach for calculating bl gives the most conservative estimate of the LR. For high values of transfer probabilities (i.e., t = 0.9), the difference between the two simulations is low if bo 2 0.425 (difference < 5%), but the difference rapidly increases when bo <0.425. Hence, the previous model (when bl = 1-bo) gives a large underestimation of the evidential value for small values of bo. For low values of transfer probabilities (i.e., t = 0.1), the difference between the conservative estimate of the LR and the
b5 b6 b7 b8 b9 b10 bi values FIGURE 2 Probability distributions under Poisson assumption for some distinct values of bo. -0bn=0.950,-Abn=0.500,--0-b0=0.250, bn=0.050. b2
b3
b4
+-
80
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0 0.050
0.450 Probability bo
0.950
0.050
0.450 Probability bo
0.950
FIGURE 3 LR values calculated in function of bo,bl, t(0.9), and f(O.O1) respectively, under two different models for calculating bl.
FIGURE 4 LR values calculated in function of b,, b,, t(0.1), and f(O.O1) respectively, under two different models for calculating.- bl. --
corrected one is even more significant. As soon as bo < 0.85, the difference exceeds 5%, as illustrated in Figure 4.
groups (in this case this value is estimated equal to 1). fl: estimated frequency of the compared characteristics of the set yl in the extraneous similar-sized groups of fibres found on car seats.
Scenario 3 This scenario is the same as the first, except that there are N number of foreign groups of fibres with a large number of fibres in each group, with data sets yl to yN. Nevertheless, only one group is compatible with the suspect's pullover (red wool).
f2,...,N:estimated frequency of the groups 2 to N among the foreign groups recovered on car seats. This simple result demonstrates that, in evaluating a match, the scientist must consider concording and non-concording elements as already suggested for bloodstain evidence in the classic 'two traces problem'[22]. Therefore, it is important not only to focus on the fibres that match the suspect's garments, but also to consider other groups of fibres compatible with the facts.
Here, for sake of simplicity, it will be assumed that the offender's upper garment is made of one type of fibre. In this case the LR is developed considering that the evidence is the set {xl, y I , y2...yN}and that the event B and B admit the following definitions: B: one of these recovered groups is associated with the offence and has been transferred from the offender's upper garment to the car seat, has persisted and has been recovered.
The introduction stressed the importance of the defence strategy which can modify the evaluation of the evidence, notably by modifying the relevant population. Here, further development of this scenario can demonstrate that the B: no recovered group is associated with the offence. defence strategy will dictate not only the relevant population but also the whole evaluation of the LR. Previously, H2 The LR equation is developed in the same way as proposed was implicitly defined as 'The suspect has never sat on this by Buckleton and Evett [21]: seat, but another unknown person has stolen the car'. I f2,...,~ 1 t~,=l negligible in this case Suppose now that the defence hypothesis changes as a result of a reliable eyewitness testimony. LR=P(YI,Y2.....YN~~l.~,~l). P(BJx~,H~) + P ( ~ ~ ~ ~ , . . . , ~P(BIx~,H,) ~~X~,B,H~). ~ )+ ~ B H 2 ) P ( Y I . Y ~ . . Y N IP~( BI/ .~ ~I , H. ~ ~p. ( ~ I ~ ~ l ~ ~ ~ N I x I p(Elxl,H2) H3: the suspect has never sat on this seat, but another man negligible in this case wearing a red pullover (as stated by the witness) has stolen b ~ - l. f~ . f 2 , . . . , ~. N! tHI=l the car.
---
-
-
-
4
The LR would remain formally the same (l/flN), but the specification off, and N would be modified as follows: where: bN-l: is as in scenario 1 but for a chance occurrence of N-1 group of foreign fibres (more than a hundred) on the driver's car seat. tH1and tH2:are as in scenario 1 but for one of these recovered Science & Justice 1997; 37(2): 75-83
f,: estimated frequency of the compared characteristics of the set yl in the extraneous similar-sized groups of red fibres found on car seats (modification of the relevant population). N: number of foreign groups, each containing a large number of red fibres. 81
Bayesian framework for the eljaluation offibre transfer evidence
Hence, if the group of red fibres is the only recovered group of red fibres on the seat, the LR is simply expressed by l/fl. Then, defence strategy can modify the scientist's approach not only during the evaluation stage of the evidence, but also at the laboratory stage. For example, under Hg the scientist does not need to count all the compatible extraneous groups of fibres (which can be time-consuming), but needs to consider only the groups of red fibres. All the other groups of extraneous fibres are not relevant for the evaluation of the evidence (their presence is explained as for the previous groups G1 and G2). Scenario 4
Another scenario can be considered where the car belongs to a man who is suspected of abducting a woman and attempting rape. The victim was wearing a red woollen pullover and denim pants. According to the suspect, nobody but his wife ever sits in the passenger's seat; moreover, the car seats have been recently vacuumed. On the passenger's seat three groups of foreign fibres have been collected. These groups consist of a large number of white cotton fibres (GI); blue cotton fibres ((32); and red wool fibres ((33). Following laboratory examinations, the fibres from G1 and G2 are in agreement with the wife's clothing. As the presence of these fibres is explained, they will be ignored. Then the forensic evidence can be formulated as follows: y: the recovered group G3 is described by a set y of characteristics (number, position on the seat, colour, microscopic features, dye). x: the red woollen pullover of the victim (control source) generates control fibres described by a set x of characteristics. Considering that the suspect denies that the victim has ever been in contact with this car, the competing hypotheses are: HI: the victim has sat on the passenger's seat of the suspect's car. H2: the victim has never sat on the passenger's seat of the suspect's car.
B
The associative events B and are defined as previously in Scenario 1. The LR is expressed by the general equation with the following parameters:
Logically, under the hypothesis H2 (meaning no offence), the probability that the traces are related to the offence is zero. Considering the circumstances of the offence, the number of recovered fibres and the sheddability of the pullover, it will be assumed that tHIis equal to 1: the LR becomes:
It is important to note that the denominator of the LR now includes bl: indeed, if H2 is accepted, then this foreign group is present by pure chance, bl being equal to, or smaller than, 1-bo; if bo is assessed as close to 1 (considering the use and the cleaning procedures of the car) then bl becomes smaller than 1 and the LR is strongly increased. If the number of fibres constituting the sample of extraneous fibres decreases, then tHI,b0 and bl must be further assessed. If the number of foreign groups recovered on this seat is equal to N (>2), the LR becomes (with tH1still estimated as equal to 1):
b ~f 1 .. f2,...,N . N!
bN
N.fl
N.fl
As quoted by Buckleton and Evett [21], according to Fong and Inami [23], when N is increased, bN is equal bN-l.Thus the LR becomes identical to that of scenario 3. Conclusion Following an explicit definition of objects and attributes involved in trace transfer, a general expression of the likelihood ratio has been derived in a Bayesian framework. This unique expression, which could be applied to the evaluation of numerous trace evidence, has permitted an approach to different case scenarios involving fibre evidence where the direction of the transfer and the amount of the recovered material were varied. The variations around the scenarios allows the identification and evaluation of the dominant parameters and their respective effect on the likelihood ratio. Moreover, it has been emphasized that these parameters are not only defined by the conditions of the contact, but also by the strategy chosen by the defence. Acknowledgments The authors are indebted to Dr Ian Evett and Dr John Buckleton for their fruitful comments, their patience and their encouragement. They wish also to thank Dr Colin Aitken for commenting on the manuscript and Dr Claude Roux for suggesting some scenarios.
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C CHAMPOD and F TARONI
Appendix - Development of the likelihood ratio The formal development of the LR is largely covered in recent literature [I]. Considering the definitions proposed in this paper, the LR becomes:
For sake of brevity, our equations will not take into account the background information I, thus yielding P ( X , ~ ~and H~) P ( ~ ,1Y~ 2 ) . The likelihood ratio then becomes:
P ( X ~ Hand ~ ) p(xlH2),being equivalent, cancel each other out. The likelihood ratio is thus reduced to:
The likelihood ratio is then developed by an 'extension of conversation' [24] taking into consideration the two events, that recovered traces are associated with, and not associated with, the offence, B and B respectively:
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