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alt="images"/>, the recovered evidential material (i.e. evidential material whose source is unknown), and is the genotype of biological material (e.g. blood, saliva swab) taken from the suspect under controlled conditions (i.e. so‐called control material whose source is known). From Example 1.2, suppose glass is broken during the commission of a crime. would be the fragments of glass (the control material) found at the crime scene, would be fragments of glass (the recovered material) found on the clothing of a suspect, and would be the two sets of fragments.

      Qualities, such as genotypes, or measurements, such as the refractive indices of glass fragments, are taken from . Comparisons are made of the measurements made on recovered and control material. Denote these by and , respectively, and let denote the combined set. Comparison of and is to be made and the assessment of this comparison has to be quantified. The totality of the evidence is denoted and is such that .

      Statistics has developed as a subject in which one of its main concerns is the quantification of the assessments of comparisons. The performance of a new treatment, drug, or fertiliser has to be compared with that of an old treatment, drug, or fertiliser, for example. Two sets of materials, control and recovered, are to be compared. It seems natural that statistics and forensic science should come together, and this has been happening over the last 40 years after strong criticisms from some outstanding quarters. Recall Kirk and Kingston (1964). They remarked that

       When we claim that criminalistics is a science, we must be embarrassed, for no science is without some mathematical background, however meagre. This lack must be a matter of primary concern to the educator [ ]. Most, if not all, of the amateurish efforts of all of us to justify our own evidence interpretations have been deficient in mathematical exactness and philosophical understanding. (pp. 435–436)

      They concluded by affirming that

       It can be fairly stated that there is no form of evidence whose interpretation is so definite that statistical treatment is not needed or desirable. (p. 437)

      As discussed in Section 1.2, there have been several books describing the role of statistics in the law. Until the first edition of this book, there had been none concerned with statistics and the evaluation of scientific evidence. Two factors may have been responsible for this.

First, there was a lack of suitable data from relevant populations. There was a consequential lack of a baseline against which measures of typicality of any characteristics of interest may be determined. One exception are the reference data that have been available for many years on allele frequencies for DNA analysis amongst certain populations. Not only has it been possible to say that the DNA of a PoI corresponded⁴ to that of a stain found at the scene of a crime, but also that this profile is only present in, say, 0.01% of the population. Now these have been superceded by results of surveys of allele frequencies in various populations. Announcements of population data are published regularly in peer‐reviewed journals such as Forensic Science International: Genetics and the International Journal of Legal Medicine. Also, data collections exist for the refractive index of glass fragments found at random on clothing and for transfer and persistence parameters linked to glass evidence; see, for example, Curran et al. (2000), O'Sullivan et al. (2011), and Jackson et al. (2013). Contributions towards characterising the rarity of different fibre types have also been published since the late 1990s; for a review, see Palmer (2016).

      Secondly, the approach adopted by forensic scientists in the assessment of their evidence has been difficult to model. The approach has been one of comparison and significance. Characteristics of the control and recovered items are compared. If the examining scientists believe them to be similar, the typicality, and hence the significance of the similarity, of the characteristics is then assessed. This approach is what has been modelled by the two‐stage approach of Evett (1977), described briefly in Section 1.3.3 and in fuller detail in Chapter 3 . However, interpretation of the results provided by this approach is difficult.

Then, in a classic paper, Lindley (1977c) described an approach that was easy to justify, to implement, and to interpret. It combined the two parts of the two‐stage approach into one statistic and is discussed in detail in Section 7.4.3. The approach compares two probabilities, the probability of the evidence, assuming one proposition to be true (e.g. that a PoI is the source of the evidence), and the probability of the evidence, assuming another, mutually exclusive, proposition to be true (e.g. that the PoI is not the source of the evidence). Note that some people use the term hypothesis rather than proposition; the authors will endeavour to use the term proposition as they believe this reduces the risk of confusion of their ideas with the ideas of hypothesis testing associated with the alternative term. A proposition is interpreted here as an assertion or statement that, for example, a particular outcome has occurred or a particular state of nature occurs.

      This approach implies that it is not enough for a prosecutor to show that there is a low probability to observe the evidence if a PoI is innocent. It should also be more probable to observe the evidence if the PoI is truly guilty. Such an approach has a good historical pedigree (Good, 1950, and also Good, 1991, for a review) yet it had received very little attention in the forensic science literature, even though it was clearly proposed at the beginning of the twentieth century (Taroni et al. 1998; Champod et al. 1999), and earlier by Peirce (1878). It is also capable of extension beyond the particular type of example discussed by Lindley, as will be seen by the discussion throughout this book, for example, in Chapters 6 and 7 .

However, in order to proceed it is necessary to have some idea about how uncertainty can be measured. This is best done through probability (Lindley 1991, 1985, 2014). This central role for probability in evidence evaluation is supported by the ENFSI. In the ENFSI Guideline for evaluative reporting in forensic science,⁵ is reported, at page 6 (under point 2.3), that:

       Evaluation of forensic science findings in court uses probability as a measure of uncertainty. This is based upon the findings, associated data and expert knowledge, case specific propositions and conditioning information.

      where the term ‘findings’ denotes ‘evidence’ in our usage.

      1.7.2 A Standard for Uncertainty

      An excellent description of probability and its role in forensic science has been given by Lindley (1991). Lindley's

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