A new uniform framework of source attribution in forensic science

Scientific principles of forensic source identification have attracted widespread interest in recent years. Among those presented principles and theorems, the Bayes inference was regarded as one of the most scientific principles. In this paper, we argue that the Bayes theorem is in challenge when used as principal basis for forensic source identification. Furthermore, two novel concepts: feature-matching value and feature-matching identification value are proposed inspired by the basic ideas of information theory. Based on these two concepts, a new framework is established to describe the source identification principles of forensic science. The proposed source identification principle uses deduction logic structure, and unifies the three existing source identification paradigms. The newly proposed framework is expected to provide a solid scientific basis for the source attribution methods in forensic science.


"E
nsuring Scientific Validity of Feature-Comparison Methods" is the theme presented in the PCAST's 2016 Forensic Report (PCAST, 2016).In forensic science, the feature-comparison methods used for evidence source identification aim to determine whether the evidence (from the crime scene) and the suspect's sample share a common origin.The feature-comparison methods, also known as the source attribution (source identification) methods, have achieved a dominant position in forensic science.On the one hand, the source identification methods are widely used in evidences extraction such as DNA, fingerprints, footprints, bullets, and voiceprints.On the other hand, the source identification results obtained through these methods are the base of different-levels evidential reasoning in court trials (Cook et al., 1998a).Thus, the scientific principle of the evidence source identification methods is vital for its practical application in forensic science.
The assumption about uniqueness, proposed by Kirk in the paper "The ontogeny of criminalistics" (Kirk, 1963), is usually considered to be the pioneering work on the core principles of forensic science.This assumption believes that "A thing can be identical only with itself, never with any other object, since all objects in the universe are unique.If this were not true, there could be no identification in the sense used by the criminalist.",which was considered to be the principle assumption of traditional source identification (Inman and Rudin, 2001).
In recent years, traditional source identification has been criticized a lot.Saks presented the view: "Although obstacles exist both inside and out side forensic science, the time is ripe for the traditional forensic sciences to replace antiquated assumptions of uniqueness and perfection with a more defensible empirical and probabilistic foundation."(Saks and Koehler, 2005).In addition, Saks specially conducted a detailed analysis of the source identification (Saks and Koehler, 2008).He claimed that there is an insurmountable fallacy in the uniqueness assumption of the method.In this regard, there are different opinions.For example, Kaye points out that if the court trial needs to be effectively supported, uniqueness assumption is reasonable for the source identification process (Kaye, 2010).In 2009, the National Academy of Science (NAS) report even declared that "much of forensic science is not rooted in solid science" (U.S. National Research Council, 2009).Growns also thinks that forensic science is "at a crossroads" (Growns and Martire, 2020).These debates show that the current principle of source identification is not perfect.
Although there are many different source identification methods, we believe that they can be summarized into three typical types, including the traditional paradigm, probability paradigm, and extended probability paradigm.
Traditional paradigm.The traditional paradigm is based on Kirk's uniqueness assumption (Kirk, 1963).However, some scholars including Champod, Stoney, and Cole, pointed out that the direct use of this assumption have some defects in source identification (Champod and Evett, 2001;Stoney, 2001;Cole, 2009).Saks provided a uniqueness assumption description closer to the source identification: The traditional forensic individualization sciences rest on a central assumption that two indistinguishable marks must have been produced by a single object (Saks and Koehler, 2005).The following paradigm in Box 1 can be used to introduce fingerprint source identification: This paradigm plays the most directly supporting role for court evidence, and its conclusion is clear and unambiguous.The logical reasoning structure is a classic deductive reasoning syllogism (Meuwly, 2006).
Although there are many controversies (Saks and Koehler, 2008;Koehler and Saks, 2010), this paradigm has been widely adopted in court trials for decades, and it has also been recognized by some scholars to some extent (Inman and Rudin, 2001;Kaye, 2010).The argument "uniqueness is unproven" is quoted by many scholars as an important reason to negating the uniqueness (Cole, 2009;Meuwly, 2006).However, as described in the paper of Kaye, there is no reliable demonstrations to support the absolute exclusion of the uniqueness assumption (Kaye, 2010).
Probability paradigm.The probability paradigm presented clear opposition against the uniqueness assumption, and hold out the probabilistic assumption.This paradigm still works through feature matching, but the result is a probabilistic conclusion.One of the simple DNA identification example is in Box 2. For the sake of principle discussion, only simple DNA situations are involved in this discussion, and complex situations such as mixed DNA are not considered.

Box 2 | Probability paradigm example
The probability of matching between different individual DNA profiles in a population is the random matching probability RMP (probability assumption); A matching between the scene DNA pattern and the suspect's DNA pattern was observed; The likelihood ratio is 1/RMP, supporting that the DNA at the scene originated from the same individual as the suspect's DNA.The meaning of the likelihood ratio is the odds of following two probabilities: (1) The probability of the DNA profile matching if the prosecution's proposition is true.
(2) The probability of the DNA profile matching if the defense's proposition is true.Where: The prosecution's proposition is that the scene DNA and suspect's DNA come from common source; The defense's proposition is that the scene DNA and suspect's DNA come from different sources.
Sake declared that this is the new paradigm for forensic science (Saks and Koehler, 2005).It is also a paradigm accepted by many scholars such as Fenton, Nordgaard, and Aitken (Fenton et al., 2013;Nordgaard and Rasmusson, 2012;Aitken et al., 2010).The source identification of DNA evidence is the most typical and representative paradigm, and it has also been used in other evidences, such as fingerprints (Neumann et al., 2012).
Extended probability paradigm.Owing to the successful application of DNA identification, the probability assumptions and conclusion statements in the above probability paradigm have been used for reference by more evidences such as face, voice, handwriting and other comparative identification (Jacquet and Champod, 2019;Gonzalez-Rodriguez et al., 2005).However, the features of these evidences are not as good as those of DNA, so the probability paradigm of DNA cannot be used directly.Instead, a slightly modification of this paradigm is adopted, which is called the extended probability paradigm in this paper.One example is as follows (Box 3): In this paradigm, matching is replaced by matching degree, which may lead to greater controversy, but the concept of matching degree has its own value.In addition, the concept "feature similarity score" used in some papers is actually the feature-matching degree here.Rodriguez showed that using similarity scores in the comparison of frontal face images has better results (Rodriguez et al., 2020).
The challenges in feature-comparison methods.Among three paradigms, the probability paradigm is considered as the scientific validity method for the DNA comparison in particular (Evett, 2015).Some scholars claimed that the probability paradigm belongs to inductive reasoning.Evett declared that the core activity of science is inductive reasoning (Evett, 1996).Bayesian reasoning is an important and widely used form of inductive reasoning.In source identification, the central plank relies on Bayes theorem, which is accepted by many scholars and forensic institutions such as Taroni (Taroni et al., 1998), Fenton (Fenton et al., 2016), Aitken (Aitken et al., 2010), Evett (Evett, 2015), and ENFSI (ENFSI, 2015).And it is also considered to be a process of updating beliefs about hypothesis based on evidence.Since Bayes theorem is a theory that has been tested and proved for a long time, this reasoning seems very stable.Evett believes that "The nature of forensic science is now firmly founded in the Bayesian paradigm" (Evett, 2015).However, if we observe the Bayesian inference process more closely, there may be cracks in its seemingly solid foundation for evidence identification.We put a brief analysis of this point as follows.
Firstly, the natural resistance within the legal profession to the Bayes method is wildly existing.Tribe's highly influential paper skeptical of the potential use of Bayes theorem (Tribe, 1971).Allen systematically critics the probability account (include Bayes method) (Allen and Pardo, 2019).In court cases, the challenges came from the misleading presentation of evidence and complex presentation of Bayes (Adams, 1996(Adams, , 1998)), and other factors.As Fenton pointed out (Fenton et al., 2016), the role of probabilityand Bayes in particular-was dealt devastating and surprising blow in a 2013 UK appeal court case ruling (Nulty & Ors v. Milton Keynes Borough Council, 2013).Some debates in probability account remain unresolved.For example, the conjunction problem is questioned by Allen and Pardo (Allen and Pardo, 2007;Pardo and Allen, 2008), however, Dawid (Dawid, 1987), Schwartz (Schwartz and Sober, 2017), Clermont (Clermont, 2018) and Fenton (Fenton et al., 2016) believe there is no problem.
Secondly, the disputes over the likelihood ratio are increasing recently.Steven and Hari believe that a likelihood ratio value offered by the expert may differ from that of the decision makers (Steven and Hari, 2017).In a recently paper, Allen claim that "Both prior probabilities and likelihood ratios are literally just made up by the decision maker."(Allen and Pardo, 2019).
Thirdly, the logical process of likelihood ratio is indirectly.As claimed in the guideline of ENFSI (ENFSI, 2015): "The likelihood ratio measures the strength of support the findings provide to discriminate between propositions of interest."The question is: how the observed match supports the hypothesis?One way, according to Roberts (Roberts and Aitken, 2015), maybe the factfinder assess the probability of the hypothesis through Bayesian updating (by likelihood ratio) of the prior probability.It belongs to inductive reasoning, and rely on the fact-finder's agreement to the Bayes reasoning.However, the debating about the Bayes approach is strong (Allen and Pardo, 2019).The other way maybe the experts' subjective opinion, as indicated in the guideline of ENFSI (ENFSI, 2015): "In my opinion, the findings provide moderately strong support for the proposition that…" It just is the subject opinion.

Box 3 | An example of the extended probability paradigm
The probability of a certain kind of feature-matching degree of the same individual in the population is the in-class matching probability, and the probability of the matching degree of this kind of feature between different individuals in the population is the random matching probability RMP (probability assumption); A scene feature with a certain degree of matching with the suspect's feature was observed; the likelihood ratio is LR, supporting the scene feature and suspect feature originate from the same individual.*The likelihood ratio is "the probability odds of a matching degree between the scene feature and the suspect's feature under the same source condition, and the different source condition." Box 4 | Logical reasoning structure of our framework Major Premise: The relative identification value of evidence is >0, it supports the common source hypothesis and the degree of support is proportional to the relative identification value; Minor premise: The feature-matching relative identification value of evidence A is >0; Conclusion: The feature-matching relative identification value of evidence A supports the common source hypothesis and the degree of support is proportional to the relative identification value.
Further, Bayes theorem is not a natural tool for causal inference.As Jaynes said, "Bayes theorem expresses nothing more than that Aristotelian logic is commutative" (Jaynes, 1988), which means "(A and B) is true" and "(B and A) is true" tell the same thing.We use P(A,B) to denote the probability of event "(A and B) is true", then we can have P(A,B) = P(B,A) according to the Bayes' rule.When we analyze the two events, "hypothesis B" and "feature matching A", of that have a clear causal relationship instead of only an association relationship, the "probability of feature matching under the condition that a given hypothesis is true" is easy to understand and express definite causality.And it is a causal reasoning process, which can also be obtained through observation, statistics, etc.; whereas "the probability that a hypothesis is true under the condition of feature matching" represents inductive reasoning.We can get P(A|B)P(B) = P(B|A) P(A) from P(A,B) = P(B,A).From a logical point of view, this relationship means that the probability of "(A and B) is true" or "(B and A) is true" can be obtained through causal reasoning or inductive reasoning.Whether this argument is true or not may be a huge controversy, where is the place of long-term controversy in the fields of philosophy and logic (Popper, 2002).
If the principle of source identification is based on this commutative property used in causality, we call in question on the reliability of this theory.Bayes theorem does not stipulate the relationship between two events, and does no guarantee that there can be a causal relationship between events.If two causal events are applied to Bayes theorem in the source identification, we need to be very careful about the logic in them.
The purpose of this study.Although Bayes methods have been defended by considerable scholars (Evett, 2015;Evett, 1996;Taroni et al., 1998;Aitken and Taroni, 2004;Foreman et al., 2003;Jackson et al., 2013;Cook et al., 1998b), and particularly by Fenton (Fenton et al., 2016) with Bayes Network, it is still worth to find feature comparison methods that do not rely on Bayes.
In this work, we try to establish such a feature comparison principle, and unify the three paradigms.The basic idea comes from the following two aspects: the first is the feature comparison paradigm can rest on the basis without connection to the Bayes method, and thus the second is that the resistance within the legal profession to the Bayes method should not affect the acceptance of feature comparison method.
In the rest of the paper, we highlight the key theoretical concepts: feature-matching value and feature-matching identification value in the second section.The meaning and logic of the concepts are analyzed.As a result, a scientific and unified principal framework for the source identification is established based on the concepts.The third section gives a detailly discussion about the framework in premise assumption, feature matching and conclusion statement.The conclusion of the paper is presented in the last section.
The proposed principle framework of the source identification In the introduction section, we have discussed the existing paradigms and challenges of source identification.This section illustrates our proposed principles.

Symbols and instructions.
In order to facilitate further discussion, the following basic concepts are presented firstly.
Common sense C. Common sense, also known as experience, is the basis how we understand the world.When expertize is lack, our cognition depends primarily on common sense.Judges rely on experience and logic in their trials.For identification, experience and logic are also the most important basis.
Proposition H (Hypothesis).Proposition is a cognition of facts.And proposition is called hypothesis until it is confirmed.In the course of evidentiary reasoning, it is usually expressed in hypothetical terms, and denote the claims or affirms of the prosecution and the defense.A simple example is "The defendant is the source of the DNA at the crime scene", the opposite hypothesis is "The defendant is not the source of the DNA at the crime scene" or "The DNA at the crime scene is from random source".This hypothesis and its opposite together constitute a hypothesis space.Although in some cases, the hypothesis space can be divided into multiple mutually exclusive and exhaustive hypotheses, we only consider two cases of the source identification problem.Hypotheses (including opposite hypothesis H) are the facts that we want to know, and they are unknown to the forensic scientists.
Features matching E. Feature matching is a general statement of the relationship between the features of the evidence to be compared.It can be considered as the matching relationship between features and the measurement of the matching relationship.If the test material and the sample have a similar set of features, it can be called matched features.For example, if the height features of two people are analyzed, the difference in height can be used as the matching degree of height features.Feature-matching metric (feature-matching degree) describes a more general situation of feature association.Feature-matching links common sense with factual statements about source identification.In the identification process, forensic scientists mainly judge H (or H) based on C and E.
Feature-matching likelihood: P(E|C), and feature-matching likelihood under hypothesis H: P(E|H,C).A critical point to source identification is the understanding of feature matching.Regardless of what kind of evidence, it is necessary to extract the features of the evidence for further comparison and analysis, so we discuss the feature matching firstly.
For the macro-level understanding of feature matching, the main point is the possibility of feature matching.In this paper we focus on two situations.
First is the general understanding of feature matching based on common sense, which can be expressed in the form of conditional probability P(E|C), which is the probability of feature matching appearing under common-sense conditions.It is called featurematching likelihood.For example, we find that the sun shines and we can take the light as the feature of the sun.The sun shines yesterday and today is a feature matching.Then according to common sense, the feature-matching likelihood that the sun will shine is P(E|C) = 1, which means the sun shining must appear according to common sense.Moreover, when tossing a coin with uniform weight, the feature-matching likelihood of the coin landing on heads is Secondly is about the common sense and hypothesis, which can be expressed in the form of a conditional probability P(E|H,C) and named as feature-matching likelihood under hypothesis condition.It means the probability of feature matching appears under common sense and hypothesis conditions.For example, this formula can express the probability that the two DNA types match under the conditions of common sense and the hypothesis that the DNA at crime scene and the suspect's DNA came from the common source.In this case, the DNA at crime scene, the suspect's DNA, and the DNA profile matching are all common sense.There is a similar definition PðEj H; CÞ of the opposite hypothesis against H.In general, P(E|H,C) is the probability distribution of feature matching under a given hypothesis.
However, for source identification applications, we focus on its significance rather than its concrete distribution forms.
Before further discussion, it should be noted that: (I) Using probability here indicates that the basic law of probability is accepted, such as the likelihood value of feature matching is between 0 and 1 showing the possibility of feature matching.(II) This possibility does not imply a prediction of the likelihood of future events, but a generalization and speculation of some of the possibilities involved in feature-matching analysis in past events as Evett presented (Evett, 1996).(III) Common sense appears in the understanding of feature matching, indicating that any cognition is based on certain common-sense experiences.It is a prerequisite for understanding, and plays an important role in the process of understanding.
Definition.Based on the aforementioned concepts, several key models are constructed below.We start with the feature-matching value model.
Feature-matching value.For the feature matching E, we introduce the definition of feature-matching value based on information theory.Shannon's pioneering work discussed information entropy, which reflected the idea that an event with greater uncertainty (smaller probability) contains more information (Shannon, 1948).In the paper, the logarithm of reciprocal probability is expressed as the surprise rate.We believe the surprise rate of feature matching reflects the feature matching value in feature comparison.The feature-matching value is defined as: This definition is based on the following proposition: Proposition 1: The feature-matching value Ln 1 PðEjCÞ represents the amount of information beyond common sense obtained through feature matching.
Proof: The feature-matching value Ln 1 PðEjCÞ is a measurement of the uncertainty (Shannon, 1948) of feature matching under common-sense conditions.After the feature matching is observed, Ln 1 PðEjCÞ is the amount of uncertainty, which is reduced.These reduced uncertainties are the amount of new information in addition to common sense obtained through feature matching because common sense is known information.
This definition is a quantization of the intuition about feature matching, which indicates that the rarer a feature matched, the more unknown information can be brought.And this information is more likely to support new understandings.The greater the probability of feature matching, the less information is provided.When the probability is 1, feature matching is guaranteed based on common senses.This means that no new information is provided than experience and no new understanding can be formed.For example, in the process of fact-finding, investigators search for abnormal events based on experience; in the process of features comparison, people look for rare features (such as random, and individual features) for matching; and in scientific research, scientists study novel things and phenomena.The above analysis suggests that when an unusual (less likely) event in everyday experience occurs, it may result in more information.This is the key information given in the above definition.We defined this empirical observation as the inverted probability of the new information's measurement.We name it the feature-matching value (the value of feature matching).
According to the definition, the feature-matching value is greater than or equal to 0 and approaches 0 when the featurematching likelihood approaches 1. Obviously, feature-matching value seems to be required greater than or equal to 0. According to the definition, when a feature-matching likelihood is 0, the feature-matching value provides infinite information, which means that the observed phenomenon cannot be explained by common sense at all.For instance, bright stars were observed in the sky for dozens of days during the daytime in the year of 1054.Since this phenomenon may have been observed for the first time in that era, the feature matching (between bright star appearing in the daytime and supernovae) likelihood is 0, which means that common sense at that time cannot explain and understand the phenomenon.However since then, it has been recorded, studied, and defined as supernovae.And thus, under today's common sense, its feature-matching likelihood approaches to 1.
For multiple independent feature matching, the total value is defined as the sum of the matching value of a single feature: This definition is the most important foundation in our framework, which indicates different feature-matching values provide varying degrees of support for perceptions other than common sense.Based on this, we can build the entire framework related to source identification.
The identification value of feature matching.Under certain hypothesis conditions, the value of feature matching in supporting the hypothesis is in the following form: The identification value of feature matching for hypothesis is the logarithm of reciprocal for feature-matching likelihood under opposing hypothesis conditions.The larger the identification value is, the greater the value of feature matching support for the hypothesis is.This is also a quantitative representation of the intuitive experience and logic of feature-matching identification value, which is an extension of the definition of feature-matching value.This definition contains two aspects of understanding: meaning and logic.
(1) The definition first indicates the basic meaning of featurematching value, which means the rarer the featurematching appears, the greater the value is.For simplicity, the former in the above formula can be referred to as the prosecution identification value (the identification value of the prosecution hypothesis) and the other as the defense identification value (the identification value of the defense hypothesis).The meaning of common sense in this definition is directly inherited from definition of featurematching value, thus also indicating that the identification value of feature matching is the support for cognition other than common sense.
(2) In this definition, the feature-matching identification value that supports the hypothesis is the feature matching under the condition of the opposite hypothesis.Although it may seem counterintuitive, it is actually logical and also the core supporting the identification reasoning by the definition of the identification value.Following is an example with detailed analysis of feature-matching identification value with P(E|H,C).The meaning of the likelihood is that with common sense as the background, the probability (possibility) of a feature matching occurs under the condition that the hypothesis H is true.This is all the information given by the feature-matching likelihood condition with hypothesis.
If the value of P(E|H,C) is close to 1, it means that under the condition that the hypothesis H occurs, the feature matching must appear.However, this does not mean that if the feature matching appears, the hypothesis H must occur or occur at a certain probability.Regardless of the value of the P(E|H,C), if only based on that likelihood, we cannot judge whether the hypothesis occurred after observing the feature matching.Therefore, the new information contained in Ln 1 PðEjH;CÞ does not directly support the recognition about the hypothesis H. So, what does this new information support?In our opinion, the answer is to support the opposite hypothesis, which is logically reserved for the opposite hypothesis.To explain this point more clearly, we use the following extreme case as an example.If the probability of the feature-matching condition with hypothesis is close to 0 (i.e., feature matching is almost impossible under hypothesis conditions), then the identification value Ln 1 PðEjH;CÞ ! 1, the information provided by it can neither directly support the occurrence of hypothesis nor directly support its not occurrence.Therefore, it can only indicate that the appearance of feature matching must supported by the reason outside the hypothesis, which is the occurrence of opposite hypothesis caused (in this case, an implicit relationship is, for unknown space other than common sense, divided into hypothesis and opposite hypothesis, which will be explained in the discussion section).If the probability of the feature-matching condition with hypothesis is 1, the value of identification Ln 1 PðEjH;CÞ ¼ 0. This means that the value of identification provides the least information (no information) and the least support for the opposite hypothesis.Therefore, if the probability of the feature-matching condition with hypothesis occurs between 0 and 1, the identification value Ln 1 PðEjH;CÞ supports the opposite hypothesis H to some extent.
The reasoning process belongs to deductive reasoning other than induction or abduction reasoning.Inman consider "Inductive inference is a primary tool for evaluating physical evidence gathered from a criminal event" (Inman and Rudin, 2001).However, we believe that deductive reasoning is a more solid basis for evidence identification.This can be illustrated by specific examples such as DNA evidence.If the hypothesis is "common source" and the opposite hypothesis is "random match", a feature (DNA profile) matching is observed.And if the feature-matching condition of "same source" appears to be small, its identification value Ln 1 PðEjH;CÞ is supporting the opposite hypothesis "random match".Conversely, if the feature-matching conditional on "random matching" appear to be small, the identification value Ln 1 PðEj H;CÞ is supporting "common source".
Principle framework.A unified framework for the evidence source identification can be concluded based on previous proposed definitions.
A key step in the identification process is to evaluate the identification value of the feature matching.We evaluate it by measuring this value condition with two hypotheses, which is the basic meaning of feature comparison in evidence identification without ambiguity.The simplest and most intuitive way is to compare the two values as the evaluation results: where VS is the relative identification value (RIV) of feature matching.If the result is >0, it means that the feature matching is more supportive of the hypothesis, otherwise it means that the feature matching is more supportive of the opposite hypothesis.Furthermore, this equation can reveal more meanings for evidence identification.
(1) We have derived a deductive reasoning result based on the value of feature matching.Our logical structure is (Box 4): Based on our feature matching value definition, this logical structure can cover the three identification paradigms mentioned in the introduction section.
The first paradigm is a special case of our principle.When the identification value of feature matching for hypothesis is infinity and the identification value of feature matching for opposite hypothesis is 0, our principle is equivalent to the unique assumption in the first paradigm.At this point, the feature matching RIV is infinite, thus obtains a definitive conclusion.Take fingerprint identification as an example, with "the scene fingerprint and the suspect's sample fingerprint is not the same source" as a condition, the two features cannot match, meaning the identification value for the condition "the scene fingerprint and the suspect's sample fingerprint are the same source" is infinite.Then a definitive conclusion can be got as "the scene fingerprint and the suspect's sample fingerprint are the same source".
the relationship between the relative identification value and the likelihood ratio is shown clearly.Besides, the hypothesis is supported if the likelihood ratio is >1.Thus, the existing paradigm with a likelihood ratio as the core can be explained by our principles.
The third paradigm can also be described by our principles.
In the concept established by our principle, feature matching is a general interpretation that includes traditional feature matching, as well as measurement for feature matching.In the principle, by evaluating the value of feature matching measurement, the relative identification value of feature matching is obtained.And thus, the third paradigm can naturally be reflected in our principles.Thus, a new unified principle framework of source identification is constructed through this simple logical reasoning structure combined with the relevant definition of feature matching value.(2) Our framework provides a way of using deduction logic style to discuss about how to direct support hypothesis.(I) We do not need Bayesian theory as the principle, which give the meaning of the likelihood ratio in forensic evidence identification through an interpretation about belief update.Although this interpretation is the mainstream interpretation of momentum, and regarded as one of the core bases supporting the likelihood paradigm method.It is also accepted by many scholars such as Sullivan (Sullivan, 2019), Fenton (Fenton et al., 2016), Aitken (Aitken et al., 2010), and some forensic scientific institutions such as ENFSI (ENFSI, 2015), while we believe that Bayes theory can be used as a tool to support the calculation of the likelihood ratio, other than the principle candidates of the evidence identification or the basis of that principle.Furthermore, even the definition of entropy is closer to the principle of evidence identification than the Bayes theory.(II) In our framework, we don't need induction reasoning to support the interpretation of the likelihood ratio, although some scholars think that the evidence reasoning can be summed up in a kind of induction reasoning.(III) One recent work has applied The Law of Likelihood to solve court evidence reasoning, which is also a better tool for source identification than Bayes framework interpretation (Sullivan, 2019).However, our theory is more concise and we don't have to rely on a theorem.If a theorem appears in the basic principles of a professional field, it will inevitably increase the difficulty of application and interpretation of the principle.(3) Through our principle, there is no need to formulate such an incomprehensible conclusion as "the odds of the probability of occurrence under the conditions of hypothesis and the probability of occurrence under the conditions of opposite hypothesis is LR".When the RIV is >0, we can directly draw the conclusion: "The feature matching RIV of evidence A supports the common source hypothesis and the degree of support is proportional to the value".

Analysis and discussion
As a fundamental principle in a certain field, it should be enough simple and robust.The core problem in the field of forensic science lies in the identification of evidence, which the principle is only based on comparing the identification value of feature matching.We believe that the principle presented in this paper is simple enough, while whether it is strong enough still needs to be explored.In the previous arguments, the logical structure has been analyzed, and the following will focus on the discussion of premise assumption, feature matching and conclusion.
The premise assumption.Our premise assumption is "If the feature-matching relative identification value of evidence is >0, it supports the common source hypothesis and the degree of support is positively related to the identification value".This assumption focuses on the fundamental relationship in the identification, which is the relationship between the comparison result of the feature-matching identification value and the supporting to the hypothesis.This relationship is the soul of the forensic science comparison, so our premise assumption is more basic and general.Furthermore, current existing paradigm assumptions such as Kirk's uniqueness assumption and random match probability assumption can be included in this assumption as different special cases.Two extreme cases of this assumption have been analyzed in the previous section.Here, we review them again: (1) if the identification value of feature matching is infinite, the uncertainty of the hypothesis is minimized; and (2) if the identification value of feature matching infinitely approaches to zero, the uncertainty of the hypothesis remains unchanged.In the middle of these two extreme cases, the identification value of feature matching is monotonically decreasing functioned with the uncertainty of the hypothesis.This monotonic relationship stems from the definition of feature-matching values: the greater the feature-matching value, the smaller the chance of evidence appearance; the greater the degree of surprise, the more information provided about unknown understanding.
In the definition of feature-matching value proposed in this study, the concept of common sense is introduced, which indicates that there is a great unknown space in our understanding of the world.Under the guidance of this idea from Epistemology of Philosophy, it is possible to define the featurematching identification value smoothly.If we remove the common-sense item in the definition, the resulting logical vacuum can be found: where P(E|C) degenerates to P(E), i.e., a prior probability, and Ln 1 PðEjCÞ degenerates to Ln 1 PðEÞ .At this point, if we directly define it as feature-matching value, the meaning of Ln 1 PðEÞ ¼ 0 and Ln 1 PðEÞ ! 1 is difficult to explain, not to mention what can be supported by the feature-matching value.Only after introducing the common-sense items, it can be clearly explained that the feature-matching value supports the understanding other than common sense.We use Fig. 1 to illustrate this definition more intuitively, where the ellipse represents the composition of known and unknown area in the knowledge space in the source identification.The left side represents the common-sense part and the right shadow represents the unknown part.The feature-matching value represents the support for the unknown part.Its support level is positively proportional to the feature-matching value and the arrow represents the support relationship.
This definition provides the basis for feature-matching identification value, in which Ln 1 PðEjH;CÞ ¼ k and Ln 1 PðEj H;CÞ ¼ s.The situation of reducing the uncertainty is shown in Fig. 2, which indicates that the feature-matching identification value evaluation is based on common sense and logic.
In the introduction section, we query the validity of using Bayes theory as a principle of source identification.For those who adhere to Bayes' argument, they may argue that the definition and derivation proposed here has a similar form of the Bayes formula, and is nothing more a distortion of Bayes' theory.Now we discuss it in more detail and reveal the difficulties and potential problems with using Bayes formula as a principle.
Judea Pearl used two expressions PðHjEÞ ¼ PðEjHÞPðHÞ PðEÞ (form I) and PðHjEÞ ¼ PðH;EÞ PðEÞ (form II) to explain the Bayesian theorem (Pearl and Mackenzie, 2018) (as he did in his earlier work (Pearl, 2009), both of which contain the item 1 PðEÞ and form II is a concise version of form I. Since in form I, 1 PðEÞ is regarded as a normalized constant, the Bayes formula is used as an update rule of belief in assumptions after the emergence of evidence, which does not provide meaningful support for the principle of source identification.The form II expresses that "among other things, that the belief a person attributes to H after discovering E is never lower than that attributed to H ∧ E before discovering E." (Pearl and Mackenzie, 2018).He also noted: Bayes theorem implies that the more the degree of surprise of the evidence is, the smaller P(E), the more people should believe that the cause of evidence exists or occurs.
If feature matching is used as an item of evidence in the theorem, Judea Pearl's expression means that the smaller P(E) (that is, the larger 1 PðEÞ ), the more the feature matching supports its cause.This is clearly a plausible statement for source identification and does not clearly indicate what feature matching should support, since feature matching corresponds to two causes (hypothesis and opposite hypothesis).So 1 PðEÞ doesn't have a clear meaning in form II.
According to the Bayes theorem form pointed out by Jaynes (Jaynes, 2003), form II includes a data item.However, many scholars such as Finkelstein, Fenton, and Aitken, have omitted this item (Finkelstein and Fairley, 1970;Fenton et al., 2016;Aitken et al., 2010).When common-sense items have been introduced into form II, it can be viewed as this data item.Moreover, form II becomes ...However, it still can't explain 1 PðEjCÞ clearly.Next, we turn to the third form, i.e., the Bayesian odds PðHjEÞ Pð HjEÞ ¼ PðEjHÞ PðEj HÞ PðHÞ Pð HÞ .This has been used in many source identification discussions such as Fenton, Nordgard, and USFJC (Fenton et al., 2013;Nordgaard and Rasmusson, 2012;Fenton et al., 2016;The US Federal Judicial Center, 2011).Inman (Inman and Rudin, 2001) and Evett (Evett, 2015) where I represents environmental evidence and can also be considered as common sense.
The first item on the right side of the equation called the likelihood ratio gives the meaning of the multiple relationship between the posterior ratio and the prior ratio, which plays a role for changing the belief in the hypothesis after observing the evidence.In other words, it seems that it needs to be placed in the context of the prior ratio and the posterior ratio to reflect its meaning.However, these contexts are the subjects that forensic scientists try to avoid when using the probability paradigm, since their analytical results can be expressed using only the likelihood ratio term without considering the prior (or prior ratio) (Inman and Rudin, 2001;Fenton et al., 2016).Furthermore, the challenge to the subjective probability in prior probability always exist (Tribe, 1971;Allen, 2017).Moreover, the posterior ratio is the result that the judge needs.Moreover, it seems that the meaning of likelihood ratio is only the meaning by its odds form (probability of feature matching under hypothesis conditions divided by the probability of feature matching under opposite hypothesis), and cannot draw a simple conclusion from the meaning of likelihood ratio.Although there is clear relationship between the feature-matching identification value and the likelihood ratio, but Bayes theory does not give the meaning of the feature-matching relative identification value.However, our principle assumption has revealed the deep meaning under the veil of the likelihood ratio, and we do not need to put it in the context of the prior and posterior ratio.Therefore, our assumption can explain the likelihood ratio, and the likelihood ratio supported by the Bayes theorem does not effectively explain the meaning of the assumption in our principle.
The reason that Bayes theorem cannot give the meaning of feature-matching value is decided by the logical meaning of this theorem.As the statement of Jaynes says: "Bayes theorem expresses nothing more than that Aristotelian logic is commutative" (Jaynes, 1988).The Bayes theorem (form I and form II) actually considers two events (feature matching and hypothesis), while three events (feature matching, hypothesis, and opposite hypothesis) are involved in the process of source identification.In the form III, as shown in Eq. ( 3), it is actually the ratio of two Bayes posterior probability values, which seems to involve three events.However, the item in this form that really reflects the role of feature matching is the posterior ratio with the feature matching as a condition, which appears on the isolated side of the Eq. ( 3) (left side of the equation), supported by likelihood ratio and prior ratio, and this is the result belong to the fact-finder.Furthermore, 1 PðEjCÞ , which reflects the value of the feature matching, is eliminated by the division of the two Bayes posterior probabilities.This important item has been neglected by many forensic scientists or been considered as a meaningless normalization factor (Fenton et al., 2016).So, the forensic scientists can only retreat to the second, using the likelihood ratio to indirectly reflect the role of feature matching.
Inspired by the definition of entropy and supported by deductive logic, we propose the definition of feature-matching value and feature-matching identification value in this study, which also comes from the general experience in many fields.Compared with the probability paradigm supported by the Bayes framework, our assumptions not only explain the likelihood ratio, but also support the identification more reasonably.As a result, our assumptions are scientific and robust enough.
Feature matching.From the previous discussion, feature matching can be regarded as the key element of the principle.As far as we are concerned, the core position of the principle is feature matching other than feature itself.Thus, the evaluation method of feature-matching identification value is our principle.
In addition, feature matching is the technical basis of physical evidence source identification.Usually, different evidences will adopt different feature-matching methods.However, in terms of feature-matching theory itself, a more general view of feature matching may be more easily adapted to the changes in forensic science.For instance, the study of facial comparison methods carried out by Rodriguez has fully reflected this point of view (Rodriguez et al., 2020).According to the principle theory proposed in this research, we use feature-matching degree as an extension of feature matching.The degree refers to the result derived by measuring the feature matching under some predefined distance function.For example, in the fingerprint or DNA feature matching, the simplest case is a binary result, i.e., matching or mismatching; but for the latest development in human facial image feature matching, the result may be expressed as a continuous metric, which is a score with the value between 0 and 1 (Jacquet and Champod, 2019;Rodriguez et al., 2020).Using this concept, more types of feature comparison methods can be incorporated, especially with new technologies such as artificial intelligence (AI) and deep learning approaches.The following figures show the distribution of different feature-matching metrics.As shown in Fig. 3, if the probability of random matching is 0, the graph can describe the traditional uniqueness assumption.Otherwise, when the probability of random matching is not 0, the figure describes the feature-matching distribution of the probability paradigm.Figure 4 represents a general featurematching metric in which probabilities vary with different scores and the metric is used to compatible with the extended probability paradigm.

Conclusion statement. Conclusion expression is an extension of
the principle of identification and the embodiment of the thought in principle.There has long been a heated debate about how to express the results of the feature-comparison identification, especially the attribution of the source.On the one hand, this debate stems from the dispute over the principle.On the other hand, it comes from the forensic scientific practical needs.
The common source conclusions obtained by the traditional paradigm come from Kirk's uniqueness assumption.Thus, the criticism of this conclusion naturally transfers to the criticism of uniqueness, such as Saks, and Cole (Saks and Koehler, 2005;Cole, 2009).However, some scholars who support this assumption, such as Kaye, claimed that the uniqueness assumption can provide usable results in the forensic scientific practice (Kaye, 2010).
As for the probability paradigm, many scholars insist that this is a paradigm that forensic science should have, but the conclusion derived by the probability paradigm is still criticized by other scholars.For example, in the ENFSI's guidance document, the recommendation of the likely ratio of conclusion states that "in my opinion, the finding is in the order of 400 times more likely if Mr. J was the person who handled the bag rather than someone else handled the bag and Mr J's DNA transferred via Officer P." (ENFSI, 2015) Although this conclusion is not a simple source identification conclusion, it is still a conclusion of the probability paradigm.However, Allen believes that the recommendations of the likely ratio of conclusions is misleading because the meaning of the likelihood ratio in the paradigm is not clear, which makes it difficult to understand in court (Allen, 2017).Moreover, the ultimate reason of this criticism is that the Bayes theorem should not appear in the principle of featurecomparison identification.
However, the conclusion expression in our proposed source identification principle comes from our premise assumption, which is supported by the deductive logic structure.So there will be no logic problem.Therefore, what needs to be carefully discussed here is whether and to what extent our statement of conclusions can be used in the forensic scientific practice.The conclusion of the paradigm proposed in this work is expressed as "The feature-matching RIV of evidence A supports the common source hypothesis and the degree of support is proportional to the value".That means the goal is to support the hypothesis of the source based on the feature-matching test of evidence for identification.One feature matching has a RIV, and these RIVs are closely related to the degree of uncertainty of the conclusion, i.e., the greater the RIV, the less uncertainty.
The interpretation about the identification conclusion.The interpretation of the identification conclusion draws the attention of the fact-finders.So we analyze the interpretation of the likelihood ratio value and its impact on the identification conclusion.Many scholars have debated on the numerical meaning of the likelihood ratio.Allen argues that the number of 400 times mentioned in ENFSI's guidance document (ENFSI, 2015) means that "in a forensic context is precisely that the expert would expect to find this distribution of DNA in about one in 400 similar cases involving similar facts."(Allen, 2017).If the likelihood ratio is interpreted by this interpretation, another inferred problem occurs.Which is, the identification process evaluates the source of a particular evidence based on feature matching, other than the statistics on the frequency of errors in the identification resulted by "similar cases involving similar facts".
Sullivan (Sullivan, 2019) uses the following method to explain the meaning of the likelihood ratio.i.e., the probability of the event, that the likelihood ratio is greater than or equal to k under the condition that the opposite hypothesis is true, is not greater than 1 k : This explanation is undoubtedly novel for forensic science, but it does not give a simple and clear explanation about the meaning of the likelihood ratio.
Compared with the interpretation of the likelihood ratio, the interpretation of the RIV may be more intuitive.In our principle, the identification value of feature matching reflects the amount of information provided to the observer about hypothesis (or opposite hypothesis) after observing feature matching, i.e., the amount of uncertainty reduction about hypothesis (or opposite hypothesis).This is the process similar to human cognition of the objective world: the subject gets more information about the object, and the uncertainty of the object's understanding decreases.Relative identification value (numerically equaling to the logarithm of a likelihood ratio) represents a relative decrease in uncertainty about hypothesis and opposite hypothesis after observing a feature matching.A positive relative identification value means the reduction of uncertainty about hypothesis is more than opposite hypothesis, i.e., more support for hypothesis than for opposite hypothesis.Compared with the meaning of likelihood ratio, the relative identification value may pass information to the fact-finder more accurately.For example, the ENFSI guideline provides a reference conclusion statement about shoe evidence (ENFSI, 2015): "The observations made on the right shoe, item RT/4 and on the mark in S/10, recovered from Scene 3, are, in my view, far more likely if the first proposition were true rather than the alternative.By far more likely I indicate that the results are in the order of 2000 times more probable given the proposition that 'Mr Suspect's left shoe is the source of the mark', than Explicit reference is made to the propositions at hand." (PP.78 in the guideline) With the new principle, it can be described by: "The observations made on the right shoe, item RT/4 and on the mark in S/10, recovered from Scene 3, are, support the first  proposition were true rather than the alternative, with the degree 3.3.The first proposition: 'Mr Suspect's left shoe is the source of the mark'; the alternative: 'Explicit reference is made to the propositions at hand'." Further, the Eq. ( 2) indicates the relationship between the RIV and likelihood ratio.The logarithm likelihood ratio (log-LR) form, which numerically equal to the RIV in the equation, is adopted by Good (Good, 1991), Aitken (Aitken and Taroni, 1998).Nordgaard et al. demonstrated how an ordinal scale of conclusions for likelihood ratios can be constructed and used (Nordgaard et al., 2012).This scale is now in use at the Swedish National Laboratory of Forensic Science (SKL).It is close to a logarithmic scale in some interval.The RIV provide a clearer meaning to the log-LR.
Understand the "false positive" or "false negative" in the source identification.Although the forensic scientists instinctively refuse to talk about errors in the source identification, the errors do exist (Saks and Koehler, 2005).For a real different sources case, if the same source is supported from the perspective of feature-matching evaluation, it is a "false positive" error.In contrast, for real common source cases, if the different sources are supported from the perspective of feature match evaluation, it is a "false negative" error.
Here, we do not intend to pursue an identification principle to avoid these kinds of errors.As viewed from a philosophical point, identification is a kind of cognition process of facts in which errors are inevitable.However, what we can try to do is to understand the source of the error, avoiding the risks caused by the error, and establish a reference for error control.Errors in the traditional paradigm led people to question the unique assumptions in the paradigm, then question the principle.The probability paradigm and the extended probability paradigm avoid this situation by acknowledging the possibility of error.They avoid the risk to principle at the expense of abandoning the determinism conclusion.However, from the point of view in this research, the essence of source identification is to measure and evaluate the identification value of feature matching.Thus, these errors are not related to our source identification principle, but come from the feature itself.As the differences for the featurematching identification values of different physical evidences and different features of the same physical evidence do exist objectively, the principle of identification aims to provide a unified framework to reflect and measure these differences.Moreover, a unified framework that can reasonably reflect the differences of feature-matching identification values is more beneficial to control errors.
Each conclusion from source identification is based only on the evaluation results of some specific feature-matching identification values.Thus, the errors appeared in conclusions are also different.The final judgment on whether the evidences to be common source or not seems to be left to the judge after a comprehensive analysis on the evaluation of the multi-category feature matching and other evidences.The goal of the source identification principle is to provide a scientific and reasonable way to evaluate the feature-matching identification value.

Conclusions
Forensic science has been developed for nearly a hundred years, and the call for the shift of forensic science paradigm has been raised many times (Saks and Koehler, 2005;Cole, 2009).In this paper, a unified principle framework for the source identification is established based on two newly proposed concepts, namely the feature-matching value and the feature-matching identification value.This principle framework has solid scientific foundation and covers the current three prevalent paradigms by unifying the logical reasoning structures of different paradigms into deductive logical reasoning.Our principle of source identification does not rely on such advanced mathematical theorems as Bayes theory and the likelihood theorem, but on more basic logic and common sense, which come from the fundamental theory of philosophical anthology.More importantly, our principle is simple enough to provide an effective basis to support the court's evidence reasoning.According to our principle, the conclusion and its meaning of source identification maybe easier to interpret and passed on to judges, juries, and defense lawyers.Thus, we try to bridge, in a simple way, the gaps as Fenton called "the cultural barriers that exist between science and law" (Fenton, 2011).
As Kirk has pointed out: "All sciences rest on simple principles."(Kirk, 1963) For forensic scientific source identification, our work has established such a simple principle.However, a new principle, before it is accepted by forensic scientists and factfinders, needs interpretation, discussion, training, application, and empirical data to back it.
As a result, in this study, we only discussed a framework of source identification principles, and did not involve the detailed identification process of specific types and forms of evidence, like face images, soil evidence, shooting residue identification, etc.In addition, as Cook proposed, the hypothesis can be divided into different levels (Cook et al., 1998a).According to our definition, our premise assumption is not limited to the types of hypotheses.Moreover, the most basic source-level hypothesis has been discussed in this paper, but how to apply it to sub-source-level, activity-level, and offense-level hypothesis is not discussed.These issues warrant further study under this framework.

Fig. 1
Fig. 1 Feature-matching value.It supports the composition beyond the common source.

Fig. 2
Fig. 2 Feature-matching identification value.The feature-matching identification value for hypothesis is defined by the feature-matching likelihood of opposite hypothesis.Ln 1 PðEjH;CÞ reduce the uncertainty of H, and Ln 1 PðEj H;CÞ reduce the uncertainty of H. H and H are unknown for the forensic scientists.

Fig. 4
Fig. 4 Feature-matching measurements.The measurements of featurematching distribute in a range.
use a slightly different form: