Decisionalizing the problem of reliance on expert and machine evidence

Abstract

1. Introduction

With the increasing availability of specialized data, more and more decisions are being made that rely not only on direct observation but also on information provided by some intermediary. These intermediaries, which can be machines or humans, or systems consisting of a machine–human team, take in selected measurements, process them, and provide output in the form of a recommendation. The use of such information, which we collectively refer to as “expert output,” has affected all areas of life, including the legal process, where it has reached unprecedented levels of sophistication and pervasiveness.¹ Forensic geneticists, for instance, use probabilistic genotyping systems to analyze complex DNA mixtures to help consumers of expert evidence to deal with questions such as whether or not a particular individual is a contributor to a recovered DNA trace (Buckleton et al. 2019; Coble and Bright 2019).² Forensic fingerprint examiners use computer software to perform computations of the probative value of observed similarities and differences when comparing a fingermark to a reference print of a person of interest (POI) (e.g. Swofford et al. 2018). Similarly, facial image comparison analysts use specialized software to capture facial features, derive comparison scores, and assess the probative value of such scores (e.g. Jacquet and Champod 2020). Besides expert evidence, various stages in the legal process ranging from policing and sentencing to correction and parole have seen the development and introduction of data-driven “risk-assessment” systems to assess the probability of adverse outcomes such as recidivism (e.g. Garrett and Monahan 2019). The use of expert output is likely to further expand in the future. A special section on this topic in this journal, focusing on fingerprints, is therefore both timely and valuable for the discipline.

The use of humans and machines as intermediaries to generate expert output from data raises a number of questions and concerns (e.g. Roth 2016, 2017; Swofford and Champod 2021). The most prominent and lively debates concern observations of certain systems exhibiting varying performance characteristics, such as accuracy rates, in different demographic groups (Castelvecchi 2020). Such considerations underlie ongoing controversies about the suitability of expert systems³ for practical deployment. More broadly, the use of expert systems and their outputs are at the center of regulatory developments, such as the Artificial Intelligence (AI) Act recently approved by the European Union (EU) Council.⁴

Overall, discussions around AI often focus on what, in the context of this article, can be considered as the admissibility of AI systems in the first place. While important, resolving the question of admissibility still leaves us with the practical question of what an individual consumer of expert output ought to do with the output of a given system that has met the requisite legal standards to be given consideration. Upon encountering such expert output, the individual actor faces the question of whether or not to actually rely on that output, that is to take it into account.

While this question is central to the theme addressed by the papers in the special section of this journal, that is envisioning a future in which algorithmic approaches are a part of the practice of fingerprint examination, we will argue in this article that even when thinking broadly about how practical proceedings in such a future might look like, we can hardly dispense with a formal and analytical approach if we intend future practice to involve coherent reasoning and decision-making. This article offers such a formal analysis of reliance on expert and AI output and discusses the argumentative implications of this analysis. We also anticipate that, ultimately, the question of reliance is fundamental in the sense that it applies to all types of expert and AI output, and thus there is nothing inherently special about fingerprint examination that would allow us to exempt it from the fundamental aspects of reliance that we analyze and discuss.

The bottom line is that, contrary to what is sometimes thought or implicitly assumed, there is no need to make a methodological distinction between statistical/AI models for fingerprint analysis on the one hand and discourses on how to use AI output in practice on the other. The conceptual problem of how to use AI output in practice is as amenable to and in need of formal analysis as the process of generating AI output itself. Therefore, these two aspects can indeed be logically combined.

The question of reliance on experts within the common law has been described as a problem of “deference,” that is “whether fact finders are to be educated by or to defer to experts” (Allen and Miller 1993: 1131). This suggests a categorical acceptance of expert evidence that goes beyond the softer form of reliance that we consider here. The purpose of our article is to approach the notion of reliance from an alternative, more analytical perspective. Specifically, we will understand reliance as the incorporation of expert output into a decision-maker’s overall knowledge base, not its dominance. We analyze reliance in this sense from a decision-theoretic perspective. By interpreting reliance as a decision, we expose and formally state the logical underpinnings of reliance decisions, emphasize the inevitability of dealing with uncertainty, and argue for the need to introduce value judgments. In combination, these ingredients reveal the dimensions that standard metrics for characterizing the performance of AI systems do not account for.

This article is organized as follows. Section 2 provides a general statement of the problem of reliance on expert output, using a generic example of information provided by a forensic fingerprint examiner. In subsequent sections, we explain and delimit the scope of our analysis, define the decision-theoretic model, and analyze its properties. The discussion and conclusions are presented in Section 3.

Methodologically, our development is based on normative decision theory (Baron 2008, 2012). At times, we use graphical models, that is influence diagrams (Kjærulff and Madsen 2008; Taroni et al. 2014), to illustrate the structural assumptions of our model and to keep track of formulaic results. The framework presented in this article uses probability as a measure of uncertainty (Lindley 1987) and loss as a measure of the undesirability of decision consequences. Our goal is to develop a rational decision structure that can be justified independently of how individuals naturally make decisions. This is in contrast to the purely observational and descriptive studies of decision-making that are commonly reported in the scientific literature. Although an important area of research, we do not consider intuitive attitudes toward formal approaches to reasoning and decision-making.⁵

2. Decision-analytic account of reliance

2.1 General statement of the problem

Consider a situation where a decision (action) needs to be taken. The decision-maker may need or benefit from assistance in understanding the available information. The decision-maker therefore needs to consider whether to rely on a source that provides him or her with an output, denoted here as E, processing the input information into a more usable form.⁶ The output E can come from a human expert, a machine (including computers, instruments, etc.), or a combination of the two, such as a machine-assisted human expert.⁷ In the most general sense, the output E takes the form of a recommendation about a hypothesis H, called a proposition here, based on some sort of input information or input measurement. The main example of E that we will use in this article is the report of a forensic fingerprint examiner or even a machine that observed similarities and differences between the features of a recovered fingermark of an unknown source and the fingerprint of a POI provide moderate (or some other degree of) support for the proposition that the POI, rather than an unknown person, is the source of the recovered fingermark.

However, E can also be any form of expert or machine output, including:

• a report by a forensic geneticist that the observed correspondence between the DNA profile of a recovered biological stain and that of a POI provides strong support for the proposition that the POI, rather than an unknown person, is the source of the recovered DNA;

• a report by a digital forensic examiner that the digital data support the prosecution’s proposition that the phone of a POI was at the scene rather than the defense’s proposition that the phone was at the home address (propositions adapted from Tart 2020);

• a report by an intelligence analyst according to which two (or more) traces (e.g. shoe marks, facial images, DNA, fibers, digital traces, bullets, etc.) seized on two (or more) distinct instances (or, locations) come from the same source, rather than from two (or more) different sources;

• a report, informed by the use of a risk assessment instrument that assigns a risk score based on certain factors (e.g. Garrett and Monahan 2019, 2020), regarding the potential of a particular individual to re-offend in the future; and

• a report about a diagnostic test, asserting support for the proposition that a given individual has a particular physiological condition rather than the proposition that the individual does not have the condition of interest (e.g. Kaye 1987).

Thus, there is nothing unique or specific to fingermark evidence that distinguishes it from other types of evidence in terms of how factfinders should decide to rely on the evidence. Note, however, that we do not focus here on what Faigman et al. (2014) called “framework evidence,” that is testimony limited to general statements, such as the occurrence of particular features observed on evidential items.

The expert output E is then used by the decision-maker, along with other information, to make an ultimate decision, with the decision to rely on E being made on the way to the ultimate decision. Crucially, the decision-maker using the output either lacks the ability or willingness to process the input information directly into E, even if the input information is available and presented to the decision-maker. Specifically, with regard to the example of fingermark evidence, the decision-maker is someone who is not a fingerprint examiner. The decision-maker will, we may hope, carefully consider the testimony of an examiner, but will not develop the expertise necessary to conduct fingermark examinations themselves. Thus, the decision to rely on the output is made without actually replicating the work, with the expert or machine being to some extent a sort of black box, even though from the expert’s point of view the process used to produce the output may actually be transparent.

In the context of our running example of forensic expert evidence, machine-assisted or not, in which a fingerprint examiner reports on the comparison of a fingermark of an unknown source (e.g. found on a surface of interest) with the reference print of a POI, we focus on the following key question: what does it mean for a recipient of expert output to rely on the expert output? We will argue that this question of reliance on expert output is itself a decision problem, and that the structural features of this decision problem can be formally stated, analyzed, and discussed from a decision-theoretic perspective.

2.2 Defining the scope of analysis

When modeling inference and decision problems, it is important to be precise not only about what exactly we intend to model, but also about what we do not intend to model. We therefore emphasize that our analysis concentrates only on the decision of reliance on the expert output provided in the instant case. The focus is on the reliance on E, not on the ultimate decisions. Deciding to rely upon a particular output supporting a proposition X does not mean or suggest that we take that proposition X to be true, let alone that we make the ultimate decisions. For example, we may decide to take into account an expert report supporting the proposition that the particular defendant left a mark on a murder weapon without deciding that the defendant is the murderer or that the defendant should be convicted. Making decisions about ultimate propositions is a separate decision problem (e.g. Kaplan 1968; Kaye 1999).

Furthermore, we do not focus on the decision to consider a particular expert output as admissible. In this discussion, we assume that the output is already admissible under the applicable criteria. Given that notions of reliability are often incorporated into definitions of admissibility, it may be tempting to conflate admissibility and reliance. However, the two are logically distinct. Just because something is reliable enough to be admitted does not mean that it is reliable enough to be relied upon in making the ultimate decision, which, as we will see, includes other decision factors. Therefore, we focus here on the decision of whether or not to take an expert output into account, once that information is permitted for consideration in the ultimate decision process.

With respect to the question of how decision-makers ought to use the output of an information source to revise their beliefs, we will rely on principles of standard probabilistic inference (see, e.g. Kaye 1987 for an overview of general principles), although we will not address the empirical problem of the (lack of) epistemic competence of human fact finders (e.g. Mnookin 2008). For the sake of argument, we assume that data on the overall performance of the information source are available. However, we stress that this is not a necessary requirement. Our model is flexible and can accommodate modifications described in the existing literature, in particular the endogenous definition of reliability for partially reliable sources of information when validation or accuracy studies in the traditional sense are not available or applicable (e.g. Bovens and Hartmann 2003; Lau and Biedermann 2020). The key point here is that these other modeling approaches treat reliability as a property of the information source, whereas our analysis deals with reliance understood as a decision of the fact finder.

Another aspect, not covered in our analysis, is the evaluation and comparison of the performance of different sources of information, as well as systems and procedures that produce particular (expert) output, such as forensic value-of-evidence computations (e.g. Ramos and Gonzalez-Rodriguez 2013). These are valuable considerations for experts faced with the question of selecting one of the several available expert systems, or legal decision-makers who must decide on the admissibility of a particular proffered system output. Our analysis focuses instead on a more advanced stage in the ultimate decision process, that is, the question of reliance on particular system output that has already been deemed admissible.

2.3 Modeling the problem of reliance as a decision

The problem of reliance has two main aspects. One aspect, called inference here, is about how to use the expert output to inform our view of the competing propositions. Another aspect deals with the question of whether or not to rely on the expert output. This is a question of decision. The former aspect, inference, is extensively covered in existing literature and will only be touched on briefly here. Our focus is on the latter, the decision problem, and how inference and decision can be logically related within a coherent whole. Figure 1a shows the structure of the model described in the remainder of this article. Table 1 summarizes the node definitions. Key to our model is that inference and decision are related through expressions of preferences between accurate and inaccurate decision outcomes (i.e. a so-called loss function). In this model, the notion of accuracy refers to the congruence between the expert output E and the ground truth H. It is possible to represent this aspect, accuracy, in terms of a separate node, as shown in Fig. 1b. Under certain conditions, such a model can be shown to give the same numerical results as the more compact model (a). However, the formulaic development of the second model, (b), is more burdensome. Therefore, for ease of exposition, we continue our analysis using the model shown in Fig. 1a, recognizing that the two possible models are representations of the same decision problem at different levels of resolution.

Figure 1. (a)

Influence diagram for the problem of deciding whether or not to rely on expert output. The definition of the nodes D, E, H, and L is given in Table 1. The bent dashed arc is an informational link that represents the understanding that E is known at the time a decision is made at the node D. The bent dash-dotted arc represents the direction of inference when reasoning about H in light of the evidence E. (b) Alternative model structure, including a node A to monitor the congruence between the expert output E and the ground truth H.

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A defining feature of our model is that expert output is available and is the starting point of our analysis. That is, in our analysis, we seek to assess the “goodness” of reliance decisions in view of what our beliefs about the propositions would be if the expert output were taken into account. This is in contrast to the well-known model for the “Oil Wildcatter” two-decision problem (e.g. Raiffa 1968; Shenoy 1992; Cowell et al. 1999; Kjærulff and Madsen 2008), which focuses on the decision of whether or not to seek information (i.e. conducting a seismic test) prior to making a principal decision (i.e. drilling for oil). See for example, Biedermann et al. (2020), Gittelson et al. (2013), and Taroni et al. (2014) for forensic and legal applications of the latter model. Note, however, that there is nothing to prevent our model from being extended to handle two-decision problems.

2.3.1 Inference

Our analysis begins with the problem of inference, represented by the network fragment $H→E$⁠. The node H has two states, representing the two propositions “The POI is the source of the fingermark” (H₁) and “An unknown person is the source of the fingermark” (H₂). The two propositions H₁ and H₂ thus capture the entirety of what can happen. The decision-maker’s probabilities for the propositions H₁ and H₂ are organized in a node probability table, associated with the node H. The decision-maker’s probabilities are conditioned on the entirety of knowledge and information I available at the time a decision needs to be made about reliance. We write these probabilities, prior to considering evidence E, as $Pr(Hj|I), j={1,2}$⁠, but we do not explicitly model the information I using a separate node. We also omit I from the notation for simplicity.

The node E represents the expert output, modeled as a child variable of the node H. Expert outputs are, generally, recommendations, but come in many forms. In forensic science, such expert outputs vary widely in reporting style. Some experts make direct statements about disputed events, for example, that a particular fingermark came from a particular POI. Others focus on reporting only the value of their observations, that is the extent to which their observations help to discriminate between competing claims about the source of the fingermark. Our analysis here is entirely general and can accommodate any type of reporting format. For the purpose of discussion, we consider three different conclusions: E₁, the findings support H₁ over H₂; E₂, the findings do not support one proposition over the other; E₃, the findings support H₂ over H₁.⁸

Associated with the node E is a table containing conditional probabilities $Pr(E|H)$⁠, with $∑iPr(Ei|H)=1$⁠. For the time being, we will not go into the details of assigning probabilities, we simply introduce their notation, as shown in Table 2, and some general properties.⁹ Note that, in general, in order for an output E_i to have probative value with respect to the propositions $H{1,2}$⁠, it is required that $Pr(Ei|H1)≠Pr(Ei|H2)$⁠. In particular, the greater the difference between $Pr(Ei|H1)$ and $Pr(Ei|H2)$⁠, the greater the probative capacity of E_i as a piece of evidence with respect to $H{1,2}$⁠. Section 2.4.3 presents more detailed considerations.

The bent dash-dotted edge, pointing from E to H, indicates the direction of reasoning. That is, based on learning the expert’s output E, our degrees of belief in the proposition H are revised, using standard methods of probabilistic reasoning. In influence diagrams, this is operated through Bayes’ theorem (e.g. Kjærulff and Madsen 2008), as will be discussed later in Section 2.4.3. Note that the two-node representation $H→E$ is the simplest representation of an inference scheme, chosen here for the sole purpose of limiting the complexity of subsequent formulaic developments. It is possible to extend this representation by introducing additional intermediate considerations, as described elsewhere in the literature (see Section 2.5 for further details).

2.3.2 Accuracy

Expert output ought to be accurate. That is, the output should accord with the actual ground truth and thus give us factually correct guidance. In the context of medical diagnosis, if an individual truly has a particular disease, we expect the diagnostic test to be positive. Similarly, in the context of measuring physical quantities, accuracy refers to the distance between the true value and the value obtained during measurement.¹⁰ In our running example, if the POI truly is the source of the fingermark (H₁), we would want the expert to report E₁, “The findings provide support for H₁ over H₂,” because such a conclusion would be accurate. This is a deductive understanding of the notion of accuracy, to be distinguished from the endogenous definition of reliability for partially reliable sources of information modeled with a different network structure (e.g. Bovens and Hartmann 2003; Lau and Biedermann 2020). That is, in our model, outputs E₁ and E₃ are accurate whenever H₁ and H₂, respectively, are true. Note, however, that the situation is more subtle for the output of type E₂, which does not assert support for one proposition over the other. This type of output, which can be seen as a neutral stance, cannot be accurate or inaccurate in the way that E₁ and E₃ can be (Koehler 2008; Biedermann and Kotsoglou 2021; Swofford et al. 2024).

The model shown in Fig. 1a incorporates the above understanding of accuracy, although not explicitly. As mentioned earlier, the model does not contain a separate node that indicates the probability¹¹ of accuracy as a function of the congruence between E_i and H_j. Nevertheless, accuracy is taken into account indirectly through the node L, which is used to express preferences between the different possible decision consequences, that is accurate and inaccurate decision outcomes and situations where the expert output is E₂. Further properties of the node L are presented in Section 2.3.3.

It is important to emphasize that what is being contemplated here is not domain-wide accuracy, or the accuracy of an expert or machine in general, a notion widely associated with aggregate error rate measurement. Instead, what is being considered here is the accuracy of the particular expert output E that has been provided for the particular ultimate decision problem at hand.

This is a subtle and important distinction. General measures of performance, such as those provided by conventional error rates, may be a relevant preliminary consideration in deciding whether a particular type of evidence, or a particular expert in a particular field of specialization, should be admitted and heard. However, a case-specific consideration of accuracy is still required when deciding whether or not to rely on the particular expert information for the situation at hand. Section 2.3.3 explains how this task is handled by the model shown in Fig. 1a. In summary, accuracy in the aggregate case is a preliminary consideration, but not equivalent to the accuracy of expert output in the instant case. Our model discussed here considers only the latter type of accuracy.

2.3.3 Decision

The part of the model that deals with the problem of deciding reliance needs to be understood in the context of the four nodes E, D, H, and L, which are arranged in a converging connection with E, D, and H as parent variables of node L (see Fig. 1a). Central here is node D, a decision node, with two states, D₁, relying on expert output, and D₂, not relying on expert output. The bent dashed arc pointing from E to D is an informational link, indicating that E is known at the time a decision at node D is made.¹² This accounts for the reality that there is nothing to decide about reliance as long as no expert output E is available. Note that in our model the decision is not whether to accept H₁ or H₂, as this would amount to a decision-theoretic account of hypothesis testing. Our analysis focuses solely on whether or not to rely on expert output regarding proposition H. That is, given the available information, reliance is about whether or not to accept the inferential impact of the information on the relevant propositions at hand.

Generally speaking, making a decision leads to a particular consequence. A consequence is defined as the combination of a decision and a particular state of nature. Decision-making under uncertainty, as understood here, means that we do not know what the state of nature is when we make a decision, and therefore we do not know which decision consequence will be obtained. For example, when a fact finder concludes that a fingermark was left by a particular person, the fact finder does not actually know whether that conclusion is accurate or erroneous. However, the degree to which we believe one proposition to be true rather than another can be expressed in terms of probability and be taken into account. Note that the literature sometimes distinguishes between the concepts of risk and uncertainty. According to these accounts, risk refers to situations where the decision-maker is able to specify probabilities (or chances). In turn, those accounts use the notion of uncertainty for situations in which probabilities cannot be specified and are said to be unknown or unknowable. We do not make this distinction here because probabilities are not a case of being knowable or not (de Finetti 1974; Lindley 1987). By definition, a probability is a measure of our (your, anybody’s) uncertainty about the truth or otherwise of a proposition of interest. The point is that people are more or less willing to articulate their probabilities, and that there are situations in which the specification of probability is felt to be easier (e.g. the toss of a fair coin) than in others (e.g. the outcome of an election) (Lindley 1985). Thus, for the purposes of our analyses, decision-making is considered to be decision-making under uncertainty, with probabilities assumed to be ascertainable in principle.¹³

In our model, a decision about reliance is made given knowledge of the type of expert output E_i under consideration, together with expressions of (1) preferences among the possible consequences of reliance (and non-reliance) on that output and (2) uncertainty about the ground truth state H. In this framework, decisions thus lead to well-defined consequences. More formally, a decision consequence C is defined here as the combination of an expert output E_i, a ground truth state H_j and a decision D_k, denoted $Cijk=(Ei,Hj,Dk)$⁠, for $i={1,2,3}$ and $j,k={1,2}$⁠. We do not model decision consequences explicitly here, using a separate node C.¹⁴ Instead, we model the decision-maker’s preferences among decision consequences directly, using a utility node L. For this discussion, we frame preferences between decision consequences in terms of undesirability, using the concept of loss. Thus, we assign a loss $L(·)$ to each consequence $Cijk=(Ei,Hj,Dk)$⁠. Structurally, this view is expressed by the arcs pointing from nodes E, D, and H to node L.

What remains to be done is to specify the loss function $L(·)$⁠. The specification of loss functions is an intricate topic. Arguments are needed to rationalize and justify particular loss functions. Start by considering what is called a $0−1$ non-negative loss function.¹⁵ This function assigns the loss value 0 to the best consequence(s), and the value 1 to the worst consequence(s). Using the value 0 for the most preferable consequence(s) expresses the idea that nothing is lost by obtaining the consequence of interest, as no better outcome could have been obtained. For our running example, we specify the following losses (see Table 3 for a summary):

• For findings E₁ that assert support of H₁ over H₂:

  • $L(E1,H1,D1)=L(E1,H2,D2)=0$⁠: Reliance (D₁) in the case of accurate output (i.e. E₁ and H₁ hold), and non-reliance (D₂) in the case of non-accurate output (i.e. E₁ and H₂ hold), are the most desirable outcomes, and thus imply zero loss.

  • $L(E1,H2,D1)=ℓ1=1$⁠: Relying on inaccurately incriminating output represents the worst consequence. Therefore, we assign the loss value 1.

  • $L(E1,H1,D2)=ℓ2$⁠, for $0<ℓ2≤1$⁠: Not relying (D₂) on accurate output (i.e. E₁ and H₁ hold) is suboptimal, hence requires us to assign a loss different from zero. The question is whether this consequence—the failure to rely on accurate output—is as undesirable as relying on inaccurately incriminating output, that is (⁠$E1,H2,D1$⁠). For shortness of notation, we will occasionally express the loss for (⁠$E1,H1,D2$⁠) as $ℓ2$⁠.

• For findings E₃ that assert support of H₂ over H₁:

  • $L(E3,H2,D1)=L(E3,H1,D2)=0$⁠: Analogous to the assignments above, we assign a loss of zero to reliance (D₁) in the case of accurate output (i.e. E₃ and H₂ hold), and to non-reliance (D₂) in the case of non-accurate output (i.e. E₃ and H₁ hold), because these are two desirable outcomes.

  • $L(E3,H1,D1)=ℓ3$⁠, for $0<ℓ3≤1$⁠: Relying (D₁) on inaccurately exculpatory expert output (i.e. E₃ and H₁ hold) is suboptimal, that is to some extent undesirable. The question is whether this is as undesirable as failing to rely on accurately exculpatory expert output, (⁠$E3,H2,D2$⁠), to which the loss value 1 is assigned. Again, for shortness of notation, we will designate the loss for (⁠$E3,H1,D1$⁠) by $ℓ3$⁠.

  • $L(E3,H2,D2)=ℓ4=1$⁠: Not relying (D₂) on accurately exculpatory expert output (i.e. E₃ and H₂ hold) is considered a highly undesirable result. This is analogous to relying on inaccurately incriminating output, defined above. A loss value of 1 is therefore assigned.

• For findings E₂ that do not assert support of one proposition over the other: Recall from Section 2.3.2 that output of type E₂ cannot be accurate or inaccurate in the sense that E₁ and E₃ can be. Yet, the question the decision-maker needs to ask is how the two types of losses $L(E2,Hj,D1)$ and $L(E2,Hj,D2)$⁠, for j = 1, 2, compare to one another. Recall that, ultimately, we do not alter our belief about H in the case of an output of type E₂: if we decide D₂, not to rely on E₂, this means that we discard E₂ altogether; if we decide D₁, to rely on E₂, we do not alter our beliefs either because E₂ does not assert support for one proposition over the other.¹⁶ However, in our view, the latter course of action would represent a greater loss, because we have the burden (or cost) of reliance on an item of information from which we derive no inferential guidance. So let $L(E2,Hj,D1)>L(E2,Hj,D2)$⁠, j = 1, 2. For the purpose of further analysis, we set $L(E2,Hj,D2)=0$ and $L(E2,Hj,D1)=γ$⁠, for $0<γ≤1$ and j = 1, 2.

We reiterate that the loss values specified above can be broadly understood as expressions of regret related to the consequences of reliance decisions and the congruence between expert output and ground truth. We reiterate that they should not be confused with losses resulting from ultimate decisions, as ultimate decision-making may take into account other factors unrelated to belief in the facts of the situation at hand.

2.4 Analysis of model properties

The model described in Section 2.3 provides a static representation of the problem of reliance. The model clarifies the variables that are thought to capture the essential elements of the problem at hand, along with the relationships that are assumed to hold between the variables. Note that the presence of an arc between a pair of nodes, representing a direct influence of one variable on another, is as informative as the absence of an arc. For example, besides an informational link (Section 2.3.3), there is no straight, plain edge pointing from node E to node D. This means that expert output does not predicate or otherwise directly influence a decision about reliance.

However, this descriptive account does not tell us yet how to choose between the rival decisions D₁ and D₂, for any given type of expert output E. We need to look deeper into this formal framework to uncover its logical implications for deciding about reliance. In the next two sections, we consider different decision criteria and examine their properties.

2.4.1 Deciding without probabilities?

While it is uncontroversial that we ought to rely on accurate expert output, this is usually easier said than done. We cannot know for sure whether expert output is accurate or not, so we cannot readily determine which decision, D₁ or D₂, will minimize actual loss. An exception, however, is the case of expert output E₂, which affirms no support for one proposition over its alternative. As noted in Section 2.3.2, the notion of accuracy in the traditional sense is not applicable to such output. From the assigned loss function in Table 3, we can directly determine D₂ as the decision that minimizes loss, as $L(D2,E2)<L(D1,E2)$⁠, regardless of the ground truth state H. Thus, there is no decisional problem of reliance with output of type E₂. In the remainder of this article, we will not consider this type of output any further.

For output of type E₁ and E₃, as mentioned in Section 2.3.2, accuracy depends on whether H₁ or H₂ is true. Thus, by definition, in deciding whether to rely on these types of outputs, we inevitably need to deal with probabilities. Yet, many view probabilities with skepticism. Some might even claim that, in their decision-making, they dispense with probabilities altogether. This raises the question of whether it is reasonably possible to ignore the inevitability of probability and conceive of a decision criterion that does not involve probability.¹⁷ As we shall see, such a deterministic approach does not result in quality decision-making.

A well-known example of a probability-agnostic decision procedure is the minimax rule for individual decision-making. Minimax is a concept used more widely by game theorists to study situations where the decision-maker faces an active adversary. However, the problem of reliance does not involve a game in the traditional sense of game theory. Instead, our hypothetical decision-maker faces possible states of nature that define the accuracy of expert output.

Applying the minimax rule to individual decision-making involves the following steps. For each course of action (decision), identify the worst consequence (i.e. the one with the highest loss). Then choose the decision that minimizes the maximum loss over the different states of nature. Let us see where this leads in our running example for outputs of type E₁ and E₃.

To begin our analysis, consider a special case of the loss function $L(·)$⁠, known as a symmetric loss function, by assigning $ℓ2=1$ and $ℓ3=1$⁠. In words, this assignment means that, in the case of expert output E₁, reliance D₁ when H₂ is true (i.e. the output is non-accurate) is taken to be as undesirable as non-reliance D₂ when H₁ is true (i.e. the output is accurate). Similarly, in the case of expert output E₃, we assume that reliance D₁ when H₁ is true (i.e. the output is non-accurate) is as undesirable as non-reliance D₂ when H₂ is true (i.e. the output is accurate). See also Table 3 for a summary. With such a loss function, we immediately see that for each decision $D{1,2}$⁠, and each type of expert output $E{1,3}$⁠, we have exactly one optimal outcome (with loss 0), and one worst consequence (with loss 1). However, since the adverse consequence for both courses of action implies the same loss, here the maximum value 1, the minimax method does not help us in selecting one of the two rival decisions. In terms of the minimax criterion, the two rival courses of action appear equally suitable when a symmetric loss function is assumed.

With these loss assignments in mind, we can now apply the minimax procedure as follows:

• For expert output of type E₁: If we decide D₁, the worst that can happen is that H₂ is true and, hence, the output is inaccurate, $L(E1,H2,D1)=1$⁠. The worst that can happen under D₂ is not to rely on the output even though it is accurate, because H₁ is true, $L(E1,H1,D2)=ℓ2$⁠. Since $ℓ2<1$⁠, the minimax decision thus is D₂. Stated otherwise, the loss incurred with D₂, in the case of an adverse outcome, is smaller than the loss incurred with D₁ in the case of an unfavorable outcome.

• For expert output of type E₃: If we decide D₁, the worst that can happen is to rely on inaccurate output (i.e. H₁ is true), $L(E3,H1,D1)=ℓ3$⁠. If we decide D₂, then the worst consequence is not to rely on accurate exculpatory output, $L(E3,H2,D2)=1$⁠. Thus, for $ℓ3<1$⁠, the minimax decision is to select D₁, to rely on output E₃. Stated otherwise, the loss incurred with D₁, in the event of an adverse outcome, is smaller than with D₂ in the event of an adverse outcome.

Thus, under the chosen loss function, the minimax criterion tells us to always rely on the expert output of type E₃, and to never rely on the expert output of type E₁. Clearly, this looks unsatisfactory and may be perceived as unbalanced because it amounts to selectively depriving ourselves of one type of expert output (E₁), while systematically relying on another type of expert output (E₃). This would amount to systematically relying on information that asserts support for one viewpoint (or party), and never relying on information that asserts support for the other. So, where lies the problem?

While the above result is due to the use of an asymmetric loss function, the problem does not lie in the choice of the actual numbers (assigned loss values). What is important is the ranking of the consequences. Specifically, for each type of expert output E₁ and E₃, the loss function defines two optimal consequences, a worst consequence and an intermediate (i.e. less than worst) consequence.

Rather, the problem lies in the minimax procedure, in particular the fact that it focuses solely on preferences among decision consequences. The driving consideration is the avoidance of maximum loss. By assuming that the worst will occur, the procedure advises to select the decision with the smallest loss in the worst case. This is a paranoid attitude to decision-making, which is appropriate in cases where there is an opponent who is able to force the maximum loss for any decision but which may not be applicable in many of the practical situations involving the use of expert output.

Indeed, while avoiding excessive losses can be seen as a laudable goal, it is important to recognize that this goal comes at a price. The price is never to rely on a type of expert output, here E₁, regardless of the probability of leading to an adverse outcome.¹⁸ This brings us back to the consideration of probability that is explicitly avoided in the minimax account. What we can see is that if we want to maintain the decision to rely on output E₁, rather than systematically rejecting it in order to avoid any possibility of incurring maximum loss, we have to somehow accept the fact that incurring maximum loss is a possibility.¹⁹ The question, then, is how to conceive of a decision criterion that allows us to rationalize reliance (in case of E₁), rather than non-reliance, despite the potential of reliance to lead to the worst consequence.²⁰ We return to this topic in the next section.

2.4.2 Probabilistic decision criterion: Scoring rival decisions through expected loss

The conclusion at the end of the previous section highlights the observation that, at times, we make certain decisions, such as reliance, despite their potential to lead to maximum loss, provided that the probability of a non-adverse outcome is (sufficiently) high. How high this probability ought to be in order to warrant a particular decision thus becomes a key question. In the remainder of this section, we will elaborate on this point by focusing only on the expert output of type E₁, although the logic of the analysis applies equally to the other expert outputs.²¹

Recall that the “problem” of deciding between reliance, D₁, and non-reliance, D₂, is due to the fact that the ground truth (variable H) is not known, and therefore the accuracy status of the expert output is affected by uncertainty. Indeed, if we knew whether the expert output E₁ was accurate, we would know that decision D₁ would lead to the best consequence. Conversely, if we knew that E₁ was not accurate, we would know that D₂, not to rely on the expert output, was the best decision. In other words, in situations of certainty about the relevant state of nature (variable H), we could minimize the actual loss.

Thus, the expected loss of the reliance decision D₁ is equal to the probability that proposition H₂ is true, that is the probability that expert output E₁ is inaccurate. This is a fairly intuitive property: the expected loss of the decision to rely on expert output E₁ is a function of the probability that the ground truth state asserted by the expert output is true. More explicitly, the higher the probability of H₁ given E₁, the smaller the expected loss of relying on expert output E₁. This is because $Pr(H2|E1)=1−Pr(H1|E1)$⁠.

In a sense, the expected loss $EL(Dk|E1)$ of a decision can be thought of as a measure of what to expect in terms of loss when making decision D_k, given knowledge of E₁. We insist that this is a case-based assessment. In particular, we make no claim about the average loss of making decisions of type D_k in a series of unrelated cases involving distinct expert outputs of type E₁.²²

The question remains as to how the notion of $EL$ can help in decision-making about reliance on expert output E_i. The point is that the notion of $EL$ provides a criterion that allows us to compare rival decisions about reliance in order to guide decision-making about reasonable reliance. Being able to compare rival decisions allows us to rank them: the decision-maker can consider whether reliance is, so to speak, “better” than nonreliance, in terms of expected loss.²³

To examine this expression, we start by considering—as in Section 2.4.1—a symmetric loss function with $ℓ2=1$⁠. This particular loss assignment means that we consider the consequences $L(E1,H2,D1)$⁠, reliance in the case of nonaccurate output, and $L(E1,H1,D2)$⁠, non-reliance in the case of accurate output, to be equally undesirable (see also Table 3). Under this assumption, Expression (2) says that decision D₁ has the smaller expected loss than (and is therefore preferable to) decision D₂ if the posterior probability of the first proposition, $Pr(H1|E1)$⁠, is greater than the posterior probability of the alternative proposition, $Pr(H2|E1)$⁠. However, since the variable H is binary, the decision criterion, Expression (2), comes down to inquiring whether the probability $Pr(H1|E1)$ is greater than 0.5.²⁴

The result of our analysis here makes perfect sense because if the probability that the expert output E₁ is accurate (i.e. H₁ is true) is greater than the probability that E₁ is not accurate (i.e. H₂ is true), the assumption of a symmetric loss function implies that the decision D₁ offers the smaller probability of incurring an undesirable outcome than D₂. More generally, the greater the probability $Pr(H1|E1)$⁠, the more advantageous it becomes to decide D₁ rather than D₂. Stated otherwise, deciding D₂, not to rely on expert output, would be suboptimal as it would have a higher probability for an undesirable outcome, than deciding D₁. Note that this is a first advantage over the minimax procedure described in Section 2.4.1, which cannot provide guidance in the case of a symmetric loss function.

Nevertheless, the result of our analysis is rather unspectacular. The above result comes down to the recommendation that, when faced with two courses of reliance, where the best outcomes are as desirable as the worst outcomes are undesirable, we should make the decision of reliance based on the greater (smaller) probability of obtaining a desirable (undesirable) outcome. Intuitively, we already use this rationale in our daily lives.²⁵

Expression (3) states that in the case of expert output E₁, the expected loss of decision D₁, reliance on E₁, is smaller than the expected loss of decision D₂, non-reliance on E₁, whenever the odds in favor of proposition H₁ over H₂, given expert output E₁, exceed the ratio of losses associated with adverse consequences of decisions D₁ and D₂, respectively.²⁶

Expression (3) is well suited to help examine the argumentative implications of assuming an asymmetric loss function, $ℓ1≠ℓ2$⁠. In particular, we can now see that the more the losses $ℓ1$ and $ℓ2$ differ, the higher the required odds in favor of H₁ over H₂ in order to satisfy Expression (2).²⁷ Stated otherwise, Expression (3) asks us to compare—or so-to-say “weigh”—relative beliefs against relative losses (of adverse decision consequences).

Let us look at some numbers to get a feel for this relationship. Let us start with a short digression back to a symmetric loss function, assuming $ℓ1=ℓ2$⁠. We can immediately see from Expression (3) that the required odds in favor of H₁ over H₂ must be greater than evens. This is in agreement with the result $Pr(H1|E1)>0.5$ found in the above discussion of Expression (2) and the condition under which D₁ is the optimal decision when assuming a symmetric loss function. Figure 2a provides a graphical illustration of this result: the lines representing the expected loss of decisions D₁ (solid line) and D₂ (dashed line) intersect at $Pr(H1|E1)=0.5$ (vertical dotted line).²⁸ That is, for $Pr(H1|E1)<0.5$⁠, decision D₂ has the smaller expected loss, whereas for $Pr(H1|E1)>0.5$⁠, decision D₁ has the smaller expected loss. As an example, consider a case where $Pr(H1|E1)=0.4$⁠, highlighted in Fig. 2a with a vertical dotted line: here, $EL(D1|E1)=0.6$ and $EL(D2|E1)=0.4$⁠, so D₂ is the decision that minimizes the expected loss.

Figure 2. (a)

Expected loss of decisions D₁ (relying on expert output E₁; solid line) and D₂ (nonreliance; dashed line) as a function of $Pr(H1|E1)$ using a $0−1{·}$ loss function with l₁=1 (loss of relying on inaccurate output E₁) and $l2={0.1,0.25,0.5,0.75,1}$ (loss of nonreliance on accurate output E₁). The bold line highlights the optimal decision, as a function of $Pr(H1|E1)$⁠, for a case in which $ℓ1$ is twice as great as $ℓ2$⁠. (b) Minimum probability $Pr(H1|E1)$ necessary for the expected loss of decision D₁ to be smaller than the expected loss of decision D₂, as a function of the loss ratio $ℓ1/ℓ2$⁠. The gray-shaded area shows pairs of values for the loss ratio and probability $Pr(H1|E1)$ for which D₁ minimizes expected loss.

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Now suppose that $ℓ1$ is not equal to $ℓ2$⁠, but greater, say ten (a hundred, a thousand, etc.) times greater. This means that reliance on nonaccurate output is taken to be worse than nonreliance on accurate output. A common proxy to investigate this assumption is Blackstone’s formulation: “It is better that ten guilty persons escape than that one innocent suffer” (Blackstone 1769: 352). We call this a proxy here because Blackstone’s formulation tends to refer to error rates across multiple cases (Dekay 1996; Kaye 1999) and also to ultimate decision-making about guilt, whereas our focus here is on the relative losses involved in reliance decisions and the relationship of one person with a particular crime scene artifact.²⁹

Following this line of reasoning, suppose that $ℓ1$ is 10 times greater than $ℓ2$⁠, that is the loss ratio is $ℓ1/ℓ2=1/0.1=10$⁠. Expression (3) now states that D₁ is optimal, that is has a smaller expected loss than decision D₂, for odds $Pr(H1|E1)/Pr(H2|E1)>10$⁠. This is equivalent to requiring that $Pr(H1|E1)$ is greater than $10/11=0.91$ (rounded result). Figure 2a shows this result graphically: the lines representing the expected loss of decisions D₁ and D₂ intersect at 0.91.³⁰

The same result is shown in Fig. 2b. More generally, Fig. 2b shows the minimum probability $Pr(H1|E1)$ required for decision D₁ to have the smaller expected loss than decision D₂, as a function of the loss ratio $ℓ1/ℓ2$⁠. As can be seen, for loss ratios of the order of one hundred and above, the threshold probability tends to 1 to an extent that in practice becomes increasingly difficult to conceptualize and articulate in entirely nonnumerical terms. This does not render this result useless. Quite to the contrary, it retains its practical relevance. While we can easily think of loss ratios in terms of orders of magnitude (e.g. tens, hundreds, etc.), the guidance for practical thinking can be expressed in the following terms: “The larger your ratio of losses associated with adverse decision consequences, the more certain you should be that H₁ is true rather than H₂, given E₁, for D₁ to be the better decision than D₂.” Furthermore, our analysis shows how reliance decisions can reveal the nature of the loss functions themselves. This allows decisions of reliance to expose the extent to which other factors may govern the ultimate decision-making as well as to compare reliance decisions over multiple decisions.

However, it would go beyond the scope of our analysis if we interpret our decision-theoretic result in a prescriptive way (Biedermann et al. 2020), not least because our development is based on a model that captures only some but not all dimensions of the decision problem at hand. Nevertheless, the aspects we do cover—such as accuracy³¹ and (un-)desirability of decision consequences—are fundamental. Furthermore, Fig. 1a represents the relationships between the different variables in a way that corresponds to our understanding of the decision problem at hand. For example, the converging connection at node L represents our understanding that we are focused on the relative value of decision consequences, given an observed value for E and different possible ground truth states H.

Our development, based on expected loss, thus provides one way of quantifying the “goodness” of a decision, taking into account a selected number of variables (Biedermann et al. 2018). We recognize and accept that, beyond the limited number of aspects covered by our formal development, there may be yet other considerations that govern practical decision-making. What we emphasize here is that for decision-makers who are concerned about the losses associated with uncertain decision consequences, there is a way of formally expressing relevant considerations in terms of “weighing” the losses associated with adverse decision consequences against the probability of their occurrence.

2.4.3 Probabilities for competing propositions

Our analysis so far shows that a decision to rely on expert output depends critically on our degree of belief in the truth state of the main proposition H. The degree of belief we have in the truth or otherwise of H is conditioned on the type of expert output (i.e. state of the variable E). More generally, as noted in Section 2.3.1, the degree of belief that we have in the truth or falsity of H is based on the totality of our knowledge and information available at the time that a decision about reliance needs to be made. However, in our account, we have been rather agnostic about particular values for these probabilities. Rather, we have made general statements about the constraints on the orders of magnitude of beliefs compared to the relative losses of adverse outcomes, in order to warrant particular decisions. Below we explore some further technicalities.

Recall the graphical specification of our model (Fig. 1a). The graphical structure implies two aspects that are essential for the probabilities of target propositions. The first is the temporal ordering between the node E, representing the expert output, and the decision node D. As noted in Section 2.3.3, this ordering reflects the understanding that some expert output E must be available prior to a decision at node D.³² The second aspect concerns the relationship between the variables E and H, since expert output E is information that is potentially³³ pertinent to the variable H. Thus, since a decision analysis requires some expert output E as a starting point, the model structure implies that the probability of H, which is used in the computation of expected loss, is conditioned on E, that is $Pr(H|E)$⁠. This amounts to a subtle point in hypothetical reasoning: when we make a decision about whether to rely on some expert output E, we should do so in the light of our degree of belief in the proposition H conditioned on the expert output under consideration. This may seem circular, but it is needed in order to properly compute expected loss. After all, in order to properly score rival decisions regarding reliance, the computations of expected loss of these rival decisions are necessarily based on what our beliefs would be if we took into account the expert output at hand.

The question that remains is how to arrive at $Pr(H|E)$⁠. Conceptually, assigning a value to $Pr(H|E)$ is itself a decision, and the constraints on that decision can be stated in decision-theoretic terms, using notions such as scoring rules, originally developed in the context of evaluating and comparing the performance of forecasters (Brier 1950).³⁴ From a Bayesian epistemological point of view, $Pr(H|E)$ is the result of probabilistic belief updating based on a partially reliable source of information (e.g. Bovens and Hartmann 2003). In practice, most, if not all, sources of information are only partially reliable to some extent. That is, the information—here expert output E—occurs not only when a given proposition of interest is true, but also when that proposition is not true. A fully reliable source of information would, among other characteristics, never provide output that asserts support for a proposition that is not true, or in other words, has a false-positive rate of zero. In practice, this is almost never the case. We will therefore use less hypothetical values.

Thus, the value of information of E₁ with respect to H₁ and H₂ is the likelihood ratio m/p, and takes values greater than 1.³⁵ We do not consider cases where m < p, because this would mean to take the expert output E₁, asserted support H₁ over H₂, as weakening our preexisting belief in H₁. To make this clearer, this would be equivalent to a case where a weather forecaster’s announcement of rain would make us less persuaded of rain, even when we think that it would rain based on what we can see from the window.

For the remainder of this section, our discussion will focus only on expert output of type E₁. Readers interested in evaluating the expert output of type E₂ will need to assign values to n and q. The remaining probabilities, $Pr(E3|H1)$ and $Pr(E3|H2)$⁠, which characterize the probative value of expert output of type E₃, are set once we define values for m and p, and n and q, respectively (see also Table 2).

The term on the left is the likelihood ratio of E₁ with respect to H₁ and H₂. The term on the right is the (prior) odds of H₁ against H₂. Expression (4) says that whatever our odds on H₁ versus H₂, learning that evidence E₁ is available will multiply those odds by the factor given by the likelihood ratio.

It is important at this point to include some comments about the likelihood ratio m/p. The ratio m/p is the value that the decision-maker assigns to expert output E₁, indicative of the importance of E₁ to the overall decision-making if E₁ were to be relied upon. This ratio m/p must be distinguished from any statement about the probability of H₁ and H₂, including likelihood ratios, reported by the expert. Recall that, following the definition given in Section 2.3.1, expert output E₁ can refer to either a likelihood ratio with respect to H₁ and H₂, or a direct and categorical assertion regarding the truth or falsity of H₁. Conversely, the likelihoods m and p refer to the occurrence of a report of type E₁ given H₁ and H₂, respectively, regardless of the actual strength of support asserted by E₁. This means that even if an expert reports that the findings support H₁ over H₂ in terms of a likelihood ratio of, say, a million, or makes a categorical assertion that the POI is the source of a given fingermark, the decision about reliance requires asking “what is the probability that this expert would assert E₁in this particular case if $H{1,2}$ were true?”. This can barely lead to a likelihood ratio of a million or more as this would require the assumption that, for example, the sensitivity m is 1 and the probability of a false-positive report p is as small as one in a million.³⁶ Few if any forensic disciplines or individual scientists can provide empirical support for such figures.³⁷

However, even if we had such relevant data, it would still not be sufficient to obtain values for m and p. The reason for this is that data from accuracy or validation studies only provide information about the examiner’s diagnostic performance in the aggregate case, which should not be confused with, and is not informative about, the selectivity of the features actually observed in the case at hand. By definition, a rate, proportion, or relative frequency of cases (outcomes) of a certain type is not equivalent—though related—to the probability of a single event (Lindley 2006), a distinction that is often misunderstood in the forensic science literature.³⁸ For decisions of reliance in routine situations, where consequences are minor, we may equate m and p with relative frequencies. For more serious decisions, such as those involving forensic science, the use of m and p from standard validation studies is at best a crude proxy that may need to be adjusted according to the circumstances of the particular case.

Notwithstanding these intricacies, data from accuracy or validation studies, including in particular proficiency testing of examiners, provide a relevant starting point for decision-makers. They can use it to provide an anchor, as argued by Koehler (2008). Champod et al. (2020) provide an illustrative example of how proficiency test data can be obtained at the individual examiner level and used to pragmatically assess the trustworthiness of a fingerprint examiner. However, when using data from validation studies, it is recommended to ensure that the data collected relate to test items and experimental conditions that reflect casework conditions, for example, in terms of the level of difficulty imposed by the quality of the items submitted for examination (i.e. degradation, pollution, etc.). Imwinkelried has referred to this as “the general notion of range of validation” (Imwinkelried 2020). For a similar viewpoint in the context of forensic voice comparison and the results of likelihood ratio computation procedures, see also Morrison et al. (2021).

However, one type of data (source) alone cannot fully resolve the assessment of expert or method performance. There are a number of ways in which relevant information can be obtained. For example, in the field of proficiency testing, at least two types of tests should be distinguished (Koehler 2017). One, more general, focuses on the ability of examiners to follow standard examination procedures. The other, more focused on operational aspects, is concerned with performance under varying casework conditions. Another dimension that decision-makers should consider is whether or not the testing was conducted in a blind manner (Mejia et al. 2020). The purpose of blind testing is to prevent examiners from approaching proficiency testing in a different, potentially more cautious or prudent, manner than they would approach regular casework, thereby compromising the validity of the resulting data. More generally, since all currently available data are imperfect in some way, it is recommended that data be used with caution.

It is worth noting that our model is not prescriptive about the data to be used to inform probabilities, nor about the methods of assigning probabilities. Furthermore, the model is flexible in the sense that it allows, but does not require, for example, a reported likelihood ratio, expert output E₁, to be equated with the decision-maker’s likelihood ratio m/p. However, this can be problematic, especially when the reported likelihood ratios are in the billions, trillions, and beyond. Indeed, what many such likelihood ratios take into account is only the assessed rarity of occurrence of analytical traits. However, the target is not the abstract occurrence of analytical traits in examined items, but the real-world event of a process which has at its core an expert (or machine) providing particular output.³⁹ This again emphasizes the importance of information about the performance of the individual examiner and/or method in case-relevant circumstances, as noted above.

The $log 10$ transformation makes the combination of likelihood ratios and odds additive. Readers may notice that working with the logarithm introduces symmetry. For example, the odds of H₁ versus H₂ of 10 (to 1), that is a $log 10$ of 1, correspond to a segment on a number line between 0 and 1. Conversely, odds on H₂ against H₁ of 10 (to 1), that is a $log 10$ of –1, correspond to a segment on a number line of the same length, but going from –1 to 0.

The reformulated decision criterion in Expression (5) states that the decision-maker ought to compare the entirety of evidence available at the time a decision needs to be made—expert output E₁ combined with any previous knowledge and data—with the magnitude of the ratio of losses associated with adverse decision consequences. Table 4 provides a few examples. Suppose the decision-maker receives expert output E₁ to which he assigns a likelihood ratio of 1000, that is a $log 10$ of 3. Suppose further that the prior odds are even $(1:1)$⁠, corresponding to a $log 10$ of 0. The combination of likelihood ratio and prior odds in terms of their $log 10$ gives 1000, that is a $log 10$ of 3. This is the limiting value that, according to Expression (5), the ratio of losses $ℓ1/ℓ2$ must not exceed for decision D₁ (reliance on expert output E₁) to be preferable to decision D₂ (not relying on expert output E₁). For $log 10$ prior odds other than zero, for example, $−2,−1,1,2$⁠, the limiting loss ratio changes accordingly (see Table 4).

As noted previously, we are not asking the reader to pin down specific numbers but to think in terms of general orders of magnitude. The broader conclusion here is that if $ℓ1$⁠, the loss of the adverse outcome of reliance, is, say, x times greater than $ℓ2$⁠, the loss of the adverse outcome of non-reliance, then coherence requires that we ought to be at least x times more certain that H₁ is true than H₂, given the entirety of available knowledge and data, including expert output E₁.

To some extent, this finding runs counter to common understandings in debates about expert evidence. Mainstream arguments regarding reliance are often narrowly framed in terms of aggregate performance measures, such as rates of false positives and false negatives in suitably designed⁴⁰ validation studies, which ought to be satisfied. However, there is no agreed, specific performance characteristic or threshold that must be met for a particular expert output to be considered reliable. This is rather unsurprising, since attempting to define a generic criterion for reliance on expert output would mean to dissociate expert evidence from the instant case, and thus from considerations of the inconvenience of adverse consequences of reliance in the instant case. Our account avoids this impasse by embedding probative value in a coherent weighing procedure along with two other essential ingredients: prior information and relative losses associated with the decisions of reliance and nonreliance on expert output.

Figure 3 provides a graphical illustration. It shows plots of examples of likelihood ratio values $m/p={1,10,100,1000,10000}$ that indicate the prior probability $Pr(H1)$ that must be exceeded in order to result—when combined with expert output E₁—in posterior odds that exceed the loss ratio $ℓ1/ℓ2$ and thus make decision D₁ (reliance on expert output E₁) preferable to decision D₂ (not relying on E₁), in the sense defined by Expression (3). For example, suppose the decision-maker’s loss ratio $ℓ1/ℓ2$ is 10⁴ and there is expert output E₁ to which a likelihood ratio m/p of 10³ is assigned. In such a case, decision D₁, that is relying on expert output E₁, is optimal only if the prior probability $Pr(H1)$ is greater than 0.91, that is the prior odds $Pr(H1)/Pr(H2)$ are greater than $10:1$⁠. If the decision-maker’s prior probability is lower, then decision D₁ is preferable to D₂—that is optimal in the sense of Expression (3)—only if the loss ratio is lowered accordingly. Conversely, if the assigned likelihood ratio m/p is 10⁴ a prior probability greater than 0.5 is sufficient to ensure that D₁ is the optimal decision. Figure 3 illustrates these two cases with the horizontal dashed lines at 0.91 and 0.5, respectively.

Figure 3.

Plot of curves representing examples of likelihood ratio values m/p for expert output E₁, indicating the prior probability $Pr(H1)$ (y-axis) that must be exceeded in order for the posterior odds $Pr(H1|E1)/Pr(H2|E1)$ to be greater than the loss ratio $ℓ1/ℓ2$ (x-axis) according to Expression (3), making decision D₁ preferable to decision D₂. The dashed lines highlight specific cases discussed in the text.

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Note that this account does not exclude the limiting case of reliance on expert output E₁ to which the decision-maker assigns no value, that is a likelihood ratio m/p of 1. This is expert output for which we take the sensitivity to be equal to the probability of a false positive. In other words, the decision-maker does not consider the expert output to add any value over and above the other evidence already considered. In such a case, the prior $Pr(H1)$ is unaffected by the expert output and Expression (3) amounts to a direct comparison of the prior odds against the loss ratio $ℓ1/ℓ2$⁠. Figure 3 illustrates this for a case where the loss ratio is 10 (vertical dashed line). Here, prior odds greater than $10:1$⁠, that is $Pr(H1)>0.91$⁠, are required for decision D₁ to be preferable to decision D₂. It may seem strange that D₁ can be optimal at all, even though E₁ appears to be considered uninformative.

There are two ways to rationalize this. First, it is important to remember that the comparison of rival decisions in terms of Expression (3) is based on a loss function for outcomes defined in terms of the variable H and the type of expert output E. However, the procedure is flexible regarding the decision-maker’s assigned likelihood ratio m/p. Stated otherwise, while the question of whether or not to rely on a given output E₁ is a function of the ground truth H, there is nothing in the procedure that prevents the decision-maker from considering the output to have no probative value. Second, and related to the previous point, the probability of H₁, that is the proposition toward which the expert output tends, may already be sufficiently high (e.g. $Pr(H1)=0.91$ as noted above) based on other available evidence, thus warranting decision D₁ even though the specific expert’s output is assigned no value. In such a case, reliance D₁ would mean to acknowledge the direction of inference conveyed by the expert even though we consider the expert in fact adds no information to what is already known. Incidentally, this is what distinguishes $(m/p)=1$ assigned to the assertive expert output E₁ from the expert output E₂ which asserts no support for one proposition over the other.

2.4.4 Illustration by means of an influence diagram

Consider the results presented in the previous sections in the form of an influence diagram, as shown in Fig. 4. This influence diagram has been created in Python using aGrUM/pyAgrum (version 0.20.2).⁴¹ The model has the structure shown in Fig. 1a, although the positions of nodes E and H—as automatically arranged by the program—are slightly different. The influence diagram here shows a situation where expert output of type E₁ is available. This is conveyed by setting the state of node E to E₁. The node H shows posterior probabilities $Pr(H1|E1)$ and $Pr(H2|E1)$ of 0.818 and 0.182, respectively, obtained by updating a uniform prior⁴² with a likelihood ratio m/p of $0.9/0.2=4.5$⁠. The probabilities $Pr(Hj|Ei)$ are one ingredient in the computation of the expected loss as defined in Equation (1). Another is the loss function. For the purposes of illustration, we assume $ℓ1=(E1,H2,D1)=1$ and $ℓ2=(E1,H1,D2)=0.1$⁠, that is a ratio of losses from adverse outcomes of 10. The remaining (optimal) outcomes are assigned a loss of zero. We can now obtain the expected loss for decision D₁, given E₁, using Equation (1), which reduces to $EL(D1|E1)=ℓ1×Pr(H2|E1)$⁠. Following the assignments above, this gives $1×0.182=0.182$ and is displayed next to the state D₁ of node D. Similarly, we obtain the expected loss for decision D₂, given E₁, as $ℓ2×Pr(H1|E1)$⁠. The result is $0.1×0.818=0.0818$ and is displayed next to state D₂ of node D. Thus, in our example here, $EL(D2|E1)<EL(D1|E1)$⁠, so D₂ is the better decision in terms of expected loss.

Figure 4.

Illustration of the influence diagram shown in Fig. 1a, constructed in Python using aGrUM/pyAgrum (version 0.20.2). Node definitions are as given in Table 1, and quantitative assessments are as given in Section 2.4.4. The diagram illustrates a case where there is an output of type E₁ and the optimal decision is D₂ because its expected loss is smaller than that of decision D₁.

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To put this result in context, consider Fig. 2b. This figure shows that for a loss ratio $ℓ1/ℓ2$ of 10, a posterior probability $Pr(H1|E1)$ greater than 0.91 is required for decision D₁ to be the optimal decision. In other words, the posterior odds $Pr(H1|E1)/Pr(H2|E1)$ obtained here do not “outweigh” the loss ratio in the sense defined by Expression (3). As another way of looking at this result, consider Fig. 2a. This figure shows that for a posterior probability $Pr(H1|E1)$ of 0.818, the y-axis value of the $EL$ function of D₂ (dashed line) is smaller than that of D₁.

2.5 Modularity and flexibility

The modeling framework invoked in this article is both modular and flexible. It is modular in the sense that different aspects, in particular inference and decision, are captured by different but related sub-models. The framework is flexible in the sense that each sub-model can be refined as needed. To illustrate this feature, consider again the aspect of inference. In our model, Fig. 1a, we have chosen a minimal representation that includes only the nodes E (for expert output) and H (for the main propositions of interest). Other accounts in the literature have further dissected this relationship. For example, when dealing with human witnesses, a cascaded chain of inference can be constructed, relating the variable representing a witness’s assertion about the occurrence of a given event to the variable representing that event through a series of intermediate nodes. These intermediate nodes represent propositions such as the witness’s senses registering the target event and the witness actually believing that the event occurred (e.g. Taroni et al. 2014). Similarly, in the context of forensic DNA evidence, Thompson et al. (2003) distinguish between an expert’s report of the occurrence of a correspondence between two compared DNA profiles, on the one hand, and the actual (but unobserved) event of corresponding DNA profiles, on the other. The latter includes considerations of the rarity of the features in the relevant population, referred to as diagnosticity (Koehler 2008), whereas the former accounts for the probability of (human) error, that is reliability (Koehler 2008). See, for example, Taroni et al. (2004) for a representation of this analytical view using a Bayesian network. It could be used in the models shown in Fig. 1 to replace the $H→E$ fragment, thus illustrating the notions of modularity and flexibility. For further accounts of cascaded inference based on human sources of information, see also Schum (1994) and Thompson (2016).

3. Discussion and conclusions

The special paper section in this journal asks the question of what a future with practically implemented statistics-driven evaluation methods for forensic results, especially in fingerprint examination, might look like. We argue that this future will involve, more or less explicitly, some form of reliance. In this article, we have chosen to examine the notion of reliance through the lens of formal analysis. We derive a number of insights, which we summarize below.

Overall, our decision-analytic account of reliance on the output of an information source in the instant case differs from mainstream accounts of general performance assessment based on response rates in two-category classification problems, as largely described in the AI (machine learning) literature (e.g. Murphy 2012; Shalev-Shwartz and Ben-David 2014; Russell and Norvig 2016) and similarly adopted by black box studies in forensic comparison disciplines. Aggregate performance metrics, such as those derived from a confusion matrix, can be useful in method development, evaluation, and comparison, and can inform general discourses about the admissibility of particular methods. However, these metrics cannot be used directly to capture specific expert output in a particular case. They have also been shown not to be conducive to understanding (Burnell et al. 2023). Our account clarifies why exactly aggregate performance metrics fall short of the needs in case-specific decision-making about reliance.

First, general performance metrics derived from test cases under controlled conditions are, by design, entirely data-based, making them seemingly objective and human independent. However, this falls short of the needs of decision-making in the individual case. What we mean by this is that reliance decisions, by definition, imply consequences of varying degrees of (un-)desirability, especially adverse consequences. In our context here, adverse consequences are reliance on inaccurate system output and nonreliance on accurate system output. What is particularly intricate is that the maker of the decision of reliance is not necessarily the party who has to bear the consequences of the decision, a consideration that is completely absent in controlled studies of forensic comparison disciplines such as fingermarks. Recognizing and quantifying the relative undesirability of decisional outcomes thus becomes a key feature of discourses over case-based decisions about reliance. This highlights the need to introduce value judgments, but data-centric performance metrics provide no help with this. The deeper insight here is that what characterizes the “goodness” of a method in general cannot serve as a criterion for characterizing the “goodness” of individual decisions (Biedermann et al. 2018). Thus, with respect to the future of machine-supported forensic feature comparison practice, our analysis suggests that (test) data alone, while useful for some purposes, is inherently insufficient to solve the problem of reliance in the instant case. Instead, as discussed below, there is an inherent judgmental component that ultimately relies on human input, even if that input was somehow programmed into a computational procedure.

Second, in our model, the expected loss associated with the decision of reliance in the individual case depends crucially on the probability of—in our context here—the accuracy of the expert output. It is important to note that we are talking about the accuracy of the individual output, not the accuracy of the method in general. Accuracy in a particular case necessarily depends on what evidence is available other than the expert output at hand, that is evidence that informs the decision-maker’s beliefs about the relevant ground truth state. While the introduction of probability into decision-making procedures might be perceived as retrograde and to be avoided, we have shown its necessity. In particular, if we try to design a decision process based on avoiding excessive losses using a deterministic approach, this leads to paralysis. For example, in the case of system output that asserts support for a particular proposition rather than the relevant alternative, the use of an asymmetric loss function, reflecting a reasonable attitude, leads to the recommendation that one should never decide to rely on a particular type of expert output. Thus, if we want to allow for reliance decisions in the latter type of case, we cannot do so without accepting a nonzero probability of incurring the overall worst decision consequence. Our account makes this understanding formally precise. Qualitatively, the decision-theoretic criterion we have derived states that the higher the ratio of losses associated with the two ways in which reliance and nonreliance can lead to adverse outcomes, the more we shall be sure that the target ground truth state asserted by the system output is true, rather than the relevant alternative.

Ultimately, our account is agnostic about the type of expert output. It makes no difference whether the information comes from a traditional human expert, a machine, or a combination of the two. For this reason, our account alleviates common concerns about the potential epistemological challenges raised by the use of AI output, of which forensic fingerprint expertise is just one example. In terms of the logic of reliance decisions, our account emphasizes that there are essentially two components that need to be coherently aggregated: an expression of uncertainty about the relevant conditioning ground truth states, and an expression or judgment about the value (or, undesirability) of decision consequences. Neither of these can reasonably be externalized because, as noted above, they require a personal stance on the part of the decision-maker. We argue that these considerations should be at the heart of current and future discourses about the use of machine-generated or -assisted evidence, particularly fingerprint examination. However, our analysis in this article shows that even the seemingly simple question of reliance on information has no easy answers if a rationally rigorous procedure is to be devised.

The above result also provides a perspective on the broader question of whether decision-making can (justifiably) be delegated to AI systems. Critical readers may argue that this is a nonissue, as there are already many operating systems that are entrusted with autonomous decision-making, such as one-to-one comparisons in biometric verification tasks. However, these applications typically operate with high-quality input information and are based on decision thresholds defined by performance metrics in the aggregate case, which are considered acceptable by the designers of such systems and by those who implement them. This is different from applications that make “one-to-many” comparisons, with possibly sub-optimal input information (i.e. a fingermark of poor quality), and that involve case-based value judgments and individualized assessments of uncertainty. In addition, the former applications (of type “one-to-one”) implicitly have a built-in value judgment that the errors have tolerable consequences. As our article helps to illustrate, value judgments in one-to-many comparisons, possibly leading to associations between a POI (defendant) and a crime scene fingermark, obviously reflect completely different stakes (Kotsoglou and Biedermann 2022). In this sense, genuine delegation of decision-making tasks would require either the encoding of these core features of decision-making in the application, or the acceptance of a procedure based on a proxy for these features.

Footnotes

Acknowledgements

The authors thank Pierre-Henri Wuillemin from Sorbonne University, Paris, for his precious advice on the aGrUM/pyAgrum framework. The authors are indebted to the Guest Editors at Law, Probability, and Risk, the anonymous reviewers and Clare Lau of the Johns Hopkins University Applied Physics Laboratory for their valuable comments on the manuscript.