2. Three types of analogical inference

In this section, we briefly discuss the traditional characterization and distinguish three types of analogical inference. This will provide the basis for the discussion of analogical inference in the Bayesian framework in subsequent sections. A standard form of an argument by analogy goes as follows (cf. Walton Reference Walton2005, 96):

Similarity Premise: Generally, case ${s}^{\prime } $ is similar to case s.

Base Premise: ${Q}^{\prime } $ holds in ${s}^{\prime } $ .

Conclusion: Q, which is similar to ${Q}^{\prime } $ , holds in s.

Similarly, Bartha (Reference Bartha2010, 1 and 13) states that “an analogical argument is an explicit representation of analogical reasoning that cites accepted similarities between two systems in support of the conclusion that some further similarity exists,” which amounts to the following schema:

1. (1) ${s}^{\prime } $ is similar to s in certain respects ${P}_i^{\prime } $ and ${P}_i $ (where ${P}_i^{\prime } $ and ${P}_i $ with $1\le i\le \kern1em n $ are similar).

2. (2) ${s}^{\prime } $ has some additional feature ${Q}^{\prime } $ .

3. (3) Therefore, s has feature Q (where Q and ${Q}^{\prime } $ are similar).

In her seminal monograph on analogical reasoning, Hesse (Reference Hesse1966, 59f) suggests to represent analogies by help of a tabular representing features of a source and target system:Footnote ¹

Note that Hesse (Reference Hesse1966, 59f; see also Hesse, Reference Hesse1974) already distinguished between horizontal and vertical relations between the target and the source system’s features. The horizontal relations between echoing and reflection, loudness and brightness, etc., consist in similarity or identity, while the vertical relations between echoing, loudness, etc., and propagation in air consist in causal dependence. Explicitly describing these relations constitutes an argument by analogy and brings the tabular notion in line with the schemata above. In accordance with Bartha (Reference Bartha2010, 24), one can further distinguish between empirically established relations and merely possible (or conjectured) relations. The similarity of s and ${s}^{\prime } $ with respect to ${P}_i $ and ${P}_i^{\prime } $ (with $1\le i\le 4 $ ), for example, was a de facto established relation back at the time of ether theories. Hence, the similarities between ${P}_i $ and ${P}_i^{\prime } $ were considered to be horizontal relations (simpliciter). However, the similarity between Q and ${Q}^{\prime } $ was not an established relation, but also not excluded back then. It was, hence, considered to be a possible horizontal relation. It is interesting to note that Bartha (Reference Bartha2010) also applies this more subtle distinction to vertical relations. The relation between echoing and propagation in air, for example, was an already established causal relation and, hence, considered to be a vertical relation (simpliciter). The similarity between reflection and propagation in ether, on the other hand, was not established on empirical grounds; it was considered to be a possible vertical relation. By extending the approach of Bartha (Reference Bartha2010, 24), who considers possible horizontal and vertical relations not independent of each other, but as mutually constrained, we can distinguish two types of analogical inference, namely the type of inferring a (possible) horizontal relation and the type of inferring a (possible) vertical relation. The schemata of these two types of inference are provided in Figure 1.

Figure 1. Horizontal and vertical relations in analogical reasoning.

For an analogical inference of a (possible) horizontal relation we have already provided an example above: the ether example. A prime example of an analogical inference of a (possible) vertical relation would be the famous violinist case presented by Thomson (Reference Thomson1971). Here it is argued that similarly to the case where a violinist’s right to live does not establish a claim on someone else’s body (whose kidney is used to keep the violinist alive), a baby’s right to live does not establish a claim on the mother’s body. Put into the argumentation schema above, this amounts to:

1. (1) Generally, the violinist case ${s}^{\prime } $ is similar to the case of unwanted pregnancy s, i.e., ${P}_1^{\prime },{P}_1 $ and ${Q}^{\prime },Q $ are similar.

2. (2) A violinist’s right to live ${P}_1^{\prime}\left({s}^{\prime}\right) $ does not establish a right to use someone else’s body ${Q}^{\prime }(s) $ .

3. (3) Therefore, also a baby’s right to live ${P}_1(s) $ does not establish a right to use the mother’s body $Q(s) $ .

Let us simplify this structure by considering only single hypotheses (conclusions) about and evidence (premises) for features of the source and target systems. Let H stand for such a hypothesis about and E for such evidence for features of the target system s. Likewise, let ${H}^{\prime } $ and ${E}^{\prime } $ stand for such a hypothesis about and such evidence for features of the source system ${s}^{\prime } $ . Furthermore, let us, in accordance with Hesse (Reference Hesse1964) and Bartha (Reference Bartha2010), assume that the possible horizontal and vertical relations are confirmational relations. Then characterizing the first kind of analogical inference (type I)—which is based on a possible horizontal relation between $Q(s) $ and ${Q}^{\prime}\left({s}^{\prime}\right) $ —amounts to the task of evaluating the confirmational impact of ${E}^{\prime } $ on E. Note that, speaking in terms of the traditional terminology, in this type of confirmation the possible horizontal relation is established on the basis of established positive analogies (similarities); what role these positive analogies play will become more clear when we present Dardasthi et al.’s (Reference Dardashti, Hartmann, Thébault and Winsberg2019) Bayesian approach in the next section. Characterizing the second kind of analogical inference (type II)—establishing a possible vertical relation between $Q(s) $ and ${P}_1(s),\dots, {P}_n(s) $ —amounts to the task of evaluating the confirmational impact of E on H on the basis of ${E}^{\prime } $ ’s confirmational impact on ${H}^{\prime } $ .

Note that these two types are not the only possibilities of how evidence about features of the source systems could be used for analogical reasoning. It is easy to see that there is also a third kind of analogical inference (type III) which consists in evaluating the confirmational impact of evidence ${E}^{\prime } $ (source system) on hypothesis H (target system). This third type is not explicitly discussed in the traditional literature on analogical reasoning. However, as we will see in the next section, it is the central kind of inference discussed in the literature on so-called analogue simulation. In the traditional framework, this type of analogical reasoning could be described as an inference along the lines of a possible diagonal relation. The three different types of inference by analogy are depicted in Figure 2.

Figure 2. Three types of analogical inference. Thick arrows represent inferred possible relations, dotted arrows established relations, and dashed lines similarity relations.

3. Confirmation by analogy Bayesian style

In this section, we present Dardasthi et al.’s (Reference Dardashti, Hartmann, Thébault and Winsberg2019) Bayesian approach to confirmation based on analogical reasoning and show that it can cover type I and III analogical inferences. We will use a toy example for illustration throughout the paper: assume we are interested in the efficacy of a new antiviral compound on humans who suffer from a certain viral disease. Experts think that this new antiviral compound is incredibly promising. To avoid possible negative side effects, however, physicians decide to first test the antiviral compound on nonhuman model organisms such as rats. Only after several successful trials with rats would the physicians consider cautiously starting to investigate the efficacy of the antiviral compound with human test subjects. This procedure seems to suggest that they make some kind of inference from the success of the treatment on rats to the future success of the treatment on humans. In other words: they seem to assume that a successful treatment of the model organism can be employed to confirm a corresponding hypothesis about humans. Physicians might justify this by pointing to the fact that the immune system of rats works analogously to the immune system of humans in relevant respects.

Let us now describe the toy example introduced above in a more precise way. (For an illustration, see Figure 3.) Let s stand for human test persons and ${s}^{\prime } $ for rats. Let H be a hypothesis about the human immune system and ${H}^{\prime } $ a hypothesis about the immune system of rats. If H is true, then humans suffering from the viral disease who have been treated with the antiviral compound will recover without severe negative side effects. Likewise, if ${H}^{\prime } $ is true, then rats treated with the antiviral compound will recover without noteworthy negative side effects. Finally, let E describe whether infected humans treated with the antiviral compound recover without severe negative side effects and ${E}^{\prime } $ for whether infected rats do.

Figure 3. Schema of analogical reasoning.

What goes on in the above toy example seems to be the following: at the beginning, the physicians are quite confident that H (claims about the human immune system) holds, but they still consider it as too risky to perform human trials in order to directly confirm H. So they start to test the antiviral compound on rats instead. They also think that a hypothesis ${H}^{\prime } $ similar to H holds for the immune system of rats. So the experts are quite confident that the immune systems of humans and rats (s and ${s}^{\prime } $ , respectively) are adequately modeled by H and ${H}^{\prime } $ , respectively, and that the immune systems of humans (s) and rats ( ${s}^{\prime } $ ) share important structural features (i.e., are in many relevant respects analogous). They then collect evidence ${E}^{\prime } $ (treatment success with rats), which directly confirms ${H}^{\prime } $ . Because the immune systems of humans and rats are assumed to share relevant structural features, physicians conclude that the results of the rat study can somehow be used to at least weakly confirm H.

So far the rough schema. The problem is that it is still not well understood how the confirmation involved here works. Here is a rational reconstruction of how the confirmation would work in Bayesian terms according to Dardashti et al. (Reference Dardashti, Hartmann, Thébault and Winsberg2019). They would stress the fact that the inference seems to crucially employ assumptions about the structure shared by the immune systems of humans and rats.Footnote ² Consequently, they would propose to model this shared structure with a variable X that is a common ancestor of H and ${H}^{\prime } $ in a Bayesian network. Bayesian networks are especially well suited for Dardashti et al.’s endeavor because they allow for modeling and graphically representing the paths over which probabilistic information spreads between variables. A Bayesian network consists of a set V of random variables ${X}_1,\dots, {X}_n $ , a set E of directed edges ( $\to $ ) connecting some of these variables, and a probability distribution P over V. A triple $\left\langle \mathbf{V},\mathbf{E}, P\right\rangle $ is a Bayesian network if and only if it conforms to the Markov factorization (Pearl Reference Pearl2000, 16):

(1) $$ P\left({X}_1,\dots, {X}_n\right)=\prod \limits_{i=1}^nP\left({X}_i|\mathbf{Par}\left({X}_i\right)\right), $$

where $\mathbf{Par}\left({X}_i\right) $ is the set of ${X}_i $ ’s parents in the Bayesian network, i.e., all ${X}_j\in \mathbf{V} $ with ${X}_j\to {X}_i $ . If P factors according to equation 1, then one can read off certain independencies from the graph $\left\langle \mathbf{V},\mathbf{E}\right\rangle $ . In particular, every ${X}_i\in \mathbf{V} $ has to be independent of every ${X}_j $ (where $ j \neq i $) that is not connected to ${X}_i $ over a path ${X}_i\to \dots \to {X}_j $ conditional on $\mathbf{Par}\left({X}_i\right) $ .

Now, Dardashti et al. (Reference Dardashti, Hartmann, Thébault and Winsberg2019) would analyze the kind of inference used in our toy example above by means of the Bayesian network in Figure 4: this Bayesian network allows for probability flow from ${E}^{\prime } $ to H over the path $H\leftarrow X\to {H}^{\prime}\to {E}^{\prime } $ . In particular, observing ${E}^{\prime } $ confirms ${H}^{\prime } $ . This increases the probability of X which, in turn, increases the probability of H. Thus, ${E}^{\prime } $ increases the probability of H which, according to Dardashti et al., shows that ${E}^{\prime } $ indirectly confirms H. So, according to their understanding, confirmation based on analogical reasoning coincides with Bayesian confirmation of H given ${E}^{\prime } $ . They claim that their approach provides a general schema for indirect confirmation by analogy, where by “indirect confirmation” we mean confirmation citing evidence ${E}^{\prime } $ from the source system, but not evidence E from the target system.

Figure 4. Bayesian network of analogical reasoning type III.

Before we go on to generalize Dardashti et al.’s (Reference Dardashti, Hartmann, Thébault and Winsberg2019) approach in such a way that it can also cover causal scenarios (which play a central role in the traditional literature on analogical inference; see, e.g., Hesse Reference Hesse1964; Bartha Reference Bartha2010) and discuss several possible problems, some remarks are in order. First, note that the role of X (the shared structure) is to mediate probabilistic influence between the source system ${s}^{\prime } $ described by ${H}^{\prime } $ and the target system s modeled by H. How exactly X must be specified is not entirely clear yet. We will come back to this issue and provide a general theoretical recipe in subsection 4.b. Second, the interpretation of the directed edges in the underlying Bayesian network is more or less flexible. The arrow ${H}^{\prime}\to {E}^{\prime } $ stands for some kind of direct confirmation relation, while the arrows $X\to H $ and $X\to {H}^{\prime } $ express that X describes a common feature of both systems relevant for H and ${H}^{\prime } $ . We will argue in section 4 that X could also be understood as a common cause of H and ${H}^{\prime } $ . Third, note that Dardashti et al. do not speak of hypotheses H and ${H}^{\prime } $ , but rather of models that are assumed to adequately represent certain systems. This is due to their interest of explicating what is called an analogue simulation in the modeling literature (Winsberg Reference Winsberg2009). We have decided to speak of hypotheses instead because this is much more in line with the traditional philosophical debate on analogies and confirmation (see, e.g., Carnap Reference Carnap1962, appendix D; Hesse, Reference Hesse1964). However, it does not make that much of a difference whether one speaks of hypotheses or of models, because it seems quite straightforward to think of models as complex hypotheses about certain systems. Fourth, note that ${E}^{\prime } $ can increase the probability of H only under certain circumstances. Dardashti et al. identify the following four conditions that have to be assumed in addition to the structure of the Bayesian network depicted in Figure 4:

1. (i) $0<P(X)<1 $

2. (ii) $P\left(H|X\right)>P\left(H|\neg X\right) $

3. (iii) $P\left({H}^{\prime }|X\right)>P\left({H}^{\prime }|\neg X\right) $

4. (iv) $P\left({E}^{\prime }|{H}^{\prime}\right)>P\left({E}^{\prime }|\neg {H}^{\prime}\right) $

Condition (i) states that the probability distribution over X is not extreme. Conditions (ii) and (iii) say that H and ${H}^{\prime } $ must become more plausible in the light of X. Finally, condition (iv) is a necessary condition for ${E}^{\prime } $ to be considered evidence for ${H}^{\prime } $ . It can be proven that $P\left(H|{E}^{\prime}\right)>P(H) $ holds if conditions (i) through (iv) are satisfied and the probability distribution $P $ over $\mathbf{V}=\left\{X,H,{H}^{\prime },{E}^{\prime}\right\} $ factors according to equation 1 (Dardashti et al. Reference Dardashti, Hartmann, Thébault and Winsberg2019, Theorem 1). Dardashti et al. see this as support for the view that evidence ${E}^{\prime } $ of the source system can actually confirm hypothesis H about the target system. As a measure for indirect confirmation by analogy, one could, according to this approach, simply use the ordinary Bayesian difference measure

(2) $$ Bconf\left(H|{E}^{\prime}\right)=P\left(H|{E}^{\prime}\right)-P(H). $$

In this paper, we use this particular measure as a proxy for all kinds of Bayesian confirmation measures. (For an overview, see Fitelson Reference Fitelson1999.)

Note that the Bayesian approach proposed by Dardashti et al. (Reference Dardashti, Hartmann, Thébault and Winsberg2019) amounts to a model of what we labeled type III analogical inference in section 2. It aims at confirming H on the basis of ${E}^{\prime } $ —where the confirmational impact can be expressed by $Bconf\left(H|{E}^{\prime}\right) $ —which corresponds to the confirmatory relation indicated by the thick diagonal arrow in Figure 2. Their approach can be easily expanded to a model that can also cover type I analogical inference by adding a variable E standing for (unknown) direct evidence for H to one’s model and by assuming the additional condition (v) $P\left(E|H\right)>P\left(E|\neg H\right) $ . It follows that $P\left(E|{E}^{\prime}\right)>P(E) $ holds and, hence, that E can be Bayes confirmed by ${E}^{\prime } $ , which corresponds to the confirmatory relation expressed by the thick horizontal arrow in the type I pattern in Figure2. The confirmational impact can, again, be expressed by $Bconf\left(E|{E}^{\prime}\right) $ .Footnote ³

4. Shared structures, common causes, and problems with indirect confirmation

In this section, we will do two things: First, we will generalize Dardashti et al.’s (Reference Dardashti, Hartmann, Thébault and Winsberg2019) Bayesian approach to indirect confirmation by analogy in such a way that it can also cover causal scenarios. Since analogies involving causal dependencies are particularly appreciated in the traditional literature on analogies (Bartha, Reference Bartha2010), supplementing their approach with such an interpretation seems promising. To this end, we will start this section by introducing the basics required for the causal interpretation of Bayesian networks. Second, we will highlight possible problems this generalized approach has to face. These problems do not depend on causal setups and, hence, might pose a threat also to Dardashti et al.’s original non-causal approach. However, they are not intended to deny the rationale of confirmation by analogy type I and III in general. By putting them forward we rather want to stress that inferences by analogy type II (which are not exposed to these problems) and other strategies for using evidence of the source system for confirming a hypothesis about the target system are significant as well.

The causal interpretation of Bayesian networks was developed by Spirtes, Glymour, and Scheines (Reference Spirtes, Glymour and Scheines1993) and later by Pearl (Reference Pearl2000). In this interpretation, the arrows of a Bayesian network are interpreted as direct causal dependencies: If ${X}_i\to {X}_j $ , then ${X}_i $ is a direct cause of ${X}_j $ . If there is a path of the form ${X}_i\to \dots \to {X}_j $ , then ${X}_i $ is called a (direct or indirect) cause of ${X}_j $ . If a variable ${X}_k $ lies on a path of the form ${X}_i\to \dots \to {X}_k\to \dots \to {X}_j $ , then ${X}_k $ is called an intermediate cause. A variable ${X}_k $ lying on a path ${X}_i\leftarrow \dots \leftarrow {X}_k\to \dots \to {X}_j $ is called a common cause of ${X}_i $ and ${X}_j $ , and a variable ${X}_k $ lying on a path ${X}_i\to \dots \to {X}_k\leftarrow \dots \leftarrow {X}_j $ is called a common effect of ${X}_i $ and ${X}_j $ .

Under the causal interpretation of Bayesian networks, the Markov factorization (equation 1) implies (among other things) that conditionalizing on intermediate causes or common causes ${X}_k $ on a path ${X}_i\to \dots \to {X}_k\to \dots \to {X}_j $ or ${X}_i\leftarrow \dots \leftarrow {X}_k\to \dots \to {X}_j $ , respectively, screens ${X}_i $ and ${X}_j $ off each other (provided ${X}_i $ and ${X}_j $ are not connected via other paths), and that variables ${X}_i $ and ${X}_j $ , connected only via a common effect path ${X}_i\to \dots \to {X}_k\leftarrow \dots \leftarrow {X}_j $ , are independent but might become dependent after conditionalizing on their common effect ${X}_k $ or on one of its effects.

Note that from a formal point of view, all that a Bayesian network does is provide a graphical representation from which certain independencies in an associated probability distribution can be read off. A common direct ancestor structure such as $H\leftarrow X\to {H}^{\prime } $ , for example, produces the same screening off effect regardless of whether the directed edges are causally interpreted or not. Whatever the true nature of the relation between X and the hypotheses H and ${H}^{\prime } $ might be, as long as it has the right formal properties, it can be represented by the structure $H\leftarrow X\to {H}^{\prime } $ .Footnote ⁴ If dependence due to shared structure is adequately represented by $H\leftarrow X\to {H}^{\prime } $ , then it seems to be straightforward to assume that if Dardashti et al.’s (Reference Dardashti, Hartmann, Thébault and Winsberg2019) story about indirect confirmation via a path $H\leftarrow X\to {H}^{\prime } $ through a variable X representing relevant shared structural properties is correct, then ${E}^{\prime } $ should also confirm H if X represents a common cause of H and ${H}^{\prime } $ instead of a shared structural feature as long as the conditions (i) through (iv) from section 3 are satisfied. Indirect confirmation via common cause paths would work exactly like indirect confirmation by analogy in Dardashti et al.’s approach does. Thus, Dardashti et al.’s Bayesian approach to indirect confirmation by analogy type III (and also type I) can be straightforwardly generalized in such a way that it also covers causal structures.

Let us now highlight several possible problems with this generalized Bayesian approach to indirect confirmation. The first two problems (subsections 4.a and 4.b) can be solved. We will argue that the other problems (subsection 4.c) should raise doubt with respect to the kind of cross-system confirmation employed by type I and III analogical inferences. Our diagnosis will be that sometimes confirmation should be restricted to system-internal confirmation and, hence, to analogies of type II (to be discussed in section 5). We will also suggest an alternative procedure for how one can confirm H on the basis of collecting evidence ${E}^{\prime } $ that can avoid these problems.

4.a Paths between hypotheses

The first possible objection one might raise is that it is not clear how to draw the arrows between the variables X, H, ${H}^{\prime } $ , and ${E}^{\prime } $ . It is quite uncontroversial that hypothesis ${H}^{\prime } $ and evidence ${E}^{\prime } $ are correctly represented by ${H}^{\prime}\to {E}^{\prime } $ . (The reason for this is simply that further evidence for ${H}^{\prime } $ is typically assumed to be independent of ${E}^{\prime } $ conditional on ${H}^{\prime } $ .) But what about the connection of X to H and ${H}^{\prime } $ ? Recall Dardashti et al. (Reference Dardashti, Hartmann, Thébault and Winsberg2019) originally introduced X to mediate probabilistic information between H and ${H}^{\prime } $ . There are basically four possible structures that would allow X to play this role (see Figure 5). Since X represents something the systems described by H and ${H}^{\prime } $ have in common, it seems plausible to eliminate possibilities (a) and (b). But why should (c) and not (d) be the correct structure? In the common cause scenario it is quite trivial that (c) is the right representation. But what about the shared structure interpretation?

Figure 5. Four possibilities of connecting H and H' via X.

We think that we can come up with the following argumentation for why (c) is also the correct representation in the shared structure case: (c) seems, contrary to (d), to be able to capture just the right dependencies and independencies. Assume that the probability distribution over X is not extreme (i.e., satisfies condition (i) from section 3). Now H and ${H}^{\prime } $ can be expected to be dependent. When we learn, for example, ${H}^{\prime } $ , then this might increase the probability of H simply because H and ${H}^{\prime } $ partially share the same structure (described by X). But once we conditionalize on X, H and ${H}^{\prime } $ will become independent. When fixing X’s value, the general structural features (represented by X) of the systems described by H and ${H}^{\prime } $ cannot change anymore. So all changes in H or ${H}^{\prime } $ must be independent of these structural features. But since probability flow between H and ${H}^{\prime } $ is only established via the shared structure described by X, learning about other features of the system described by ${H}^{\prime } $ would give us no additional probabilistic information for H and vice versa. The dependence and independence features we have described are exactly the ones that come with structure (c); they are, on the other hand, incompatible with structure (d). This will become even clearer in the next subsection.

4.b Choosing X and the Markov factorization

Here is another possible problem that is somehow related to the first one. To establish probability flow between the target and the source system, not only conditions 1 through 4 are required, but one must also be justified in assuming that $H\leftarrow X\to {H}^{\prime}\to {E}^{\prime } $ is a Bayesian network—one must be justified in assuming that one’s probability distribution factors according to equation 1. In particular, X does not only have to mediate between the source and the target system, but also to screen the two systems off each other if conditioned on. A necessary condition for the latter is that

(3) $$ P\left(H|\neg X,{H}^{\prime}\right)=P\left(H|\neg X\right) $$

holds. But why exactly should X satisfy this screening off condition? The problem is that X cannot be chosen just in any way sufficient for establishing probability flow between the target and the source system. Assume, for example, X would have the two possible values $H\wedge {H}^{\prime } $ and $\neg H\vee \neg {H}^{\prime } $ .Footnote ⁵ We would then get $P\left(H|\neg X,{H}^{\prime}\right)=P\left(H|\neg H\vee \neg {H}^{\prime },{H}^{\prime}\right)=0 $ and $P\left(H|\neg X\right)=P\left(H|\neg H\vee \neg {H}^{\prime}\right)>0 $ and, thus, equation 3 would be violated. So, again, are there general rules how X must be specified for analogical inference, and how can introducing an X that satisfies these rules be justified?

First of all, note that finding and specifying X has two aspects: a theoretical and an empirical aspect. The theoretical task consists in two subtasks, in identifying general constraints for X that guarantee that X will have the right formal properties—it must mediate probabilistic influence between the target and the source system and screen both systems off each other if conditioned on—and in finding a general method for inferring the existence of such an X. Let us say a few words on the latter first. One possibility to infer the existence of a suitable X is creative abduction (Schurz Reference Schurz2008). The basic idea is that a stable strict correlation between two systems requires an explanation. According to Reichenbach’s (Reference Reichenbach1956) principle of the common cause, there must be a common cause if these systems are not directly causally connected. One key feature of common causes is that they screen off their effects, which guarantees that the inferred X has the right formal properties. This basic idea can be generalized to nonstrict correlations and noncausal structures as well (Feldbacher-Escamilla and Gebharter Reference Feldbacher-Escamilla and Gebharter2019). In that case, one infers dispositional or shared structural features of the correlated systems instead of a common cause (see also Glymour Reference Glymour2019).

Though creative abduction can be used for inferring the existence of a suitable X, this does not yet tell us more about X than that it must have the right formal properties. So, how could X generally be specified? With this question, we have arrived at the other part of the theoretical aspect. A possible answer to it comes again from the formal similarity between common causes and shared structural features. Common causes do screen off if they are maximal, i.e., if all common causes are subsumed under X. The same holds for shared components or constitutents (see, e.g., Gebharter Reference Gebharter2017b) and should also hold for shared structural features. If probability flow between the target and the source system is due to shared structural features, then X must, to fully explain this correlation and to guarantee the screening off property, cover all the features shared by both systems. One can, for example, think in terms of INUS conditions (Insufficient but Necessary parts of a condition which is itself Unnecessary but Sufficient; cf. Mackie Reference Mackie1980) about these matters. Assume C is an effect to be explained and A and B are causally relevant factors for C. Their logical connection to C can then be described via

(4) $$ AU\vee BV\vee W\leftrightarrow C, $$

where U, V, and W stand for further possibly unknown explanatory relevant factors. Note that each causally relevant factor (such as A and B) is an INUS condition for C, i.e., an insufficient but non-redundant part of an unnecessary but sufficient condition for C. Similar representations have been used for constitutively relevant parts (Couch Reference Couch2011; Harbecke Reference Harbecke2010) and might, in the same way, be used for characterizing structural relevance as well. Note that for establishing probability flow between H and ${H}^{\prime } $ , two descriptions such as equation 4 having at least one INUS condition in common are required. These descriptions could, for example, be

(5) $$ AU\vee BV\vee W\leftrightarrow H $$

and

(6) $$ A{U}^{\prime}\vee B{V}^{\prime}\vee {W}^{\prime}\leftrightarrow {H}^{\prime } $$

which share the structural features A and B. Now, for X to screen H and ${H}^{\prime } $ off each other, it is required that all INUS conditions appearing in both descriptions are featured in X. In particular, $X $ ’s values must be the possible conjunctions of INUS conditions (and their negations) appearing in both descriptions; it is in this sense that X must be maximal. According to this idea, X would, in our example, have the four values $A\wedge B $ (both structural features are present), $A\wedge \neg B $ (only $A $ is present), $\neg A\wedge B $ (only $B $ is present), and $\neg A\wedge \neg B $ (neither $A $ nor $B $ are present).

We think that there is not much more that can be said about X from a theoretical perspective. We pointed to a method for inferring the existence of a suitable X and tried to provide a general logical pattern for how X could be specified in order to have the right formal properties. Inferring and specifying X in concrete cases (such as the dynamics of black holes studied by Dardashti et al. Reference Dardashti, Hartmann, Thébault and Winsberg2019, Dardashti et al. Reference Dardashti, Thébault and Winsberg2015, or our rat case example) cannot be done a priori. Filling in these details is the empirical task mentioned before that might require different methods depending on the specific domain of application and the specific science (or sciences) concerned.

4.c Possible problems with cross system confirmation

In what follows, we put forward another type of problems whose brief discussion aims at a further clarification of the model(s) on confirmation based on analogical reasoning. As we will argue, confirmation across systems (type I and type III) might undermine intuitions about confirmation by licensing inflation, scattering, employment of distrusted evidence, and outweighing of direct evidence by indirect evidence.

Confirmation inflationism. This problem arises for the shared structure as well as for the common cause interpretation. Let us have a look at the common cause interpretation first. Physicists are quite confident (but not absolutely certain) about the existence of the Big Bang. Let us say that X describes the Big Bang. Since the Big Bang is a common cause of everything going on in the universe at any later point in time, we can formulate hypotheses H and ${H}^{\prime } $ about any two systems s and ${s}^{\prime } $ in the world. Given the generalization of Dardashti et al.’s (Reference Dardashti, Hartmann, Thébault and Winsberg2019) story about indirect confirmation—given evidence ${E}^{\prime } $ can actually confirm a hypothesis H if there is some mediator X between H and ${H}^{\prime } $ —then evidence ${E}^{\prime } $ for any hypothesis ${H}^{\prime } $ indirectly confirms hypothesis H about (almost) any other system simply because all current systems have the Big Bang as a common cause. Everything required for indirect confirmation in a Bayesian framework is that conditions (i) through (iv) from section 3 are satisfied, which seems to be very plausible in the case of the Big Bang.

A similar point can be made for the shared structure interpretation. It seems that for almost all real world systems there are some common structural features that could be described by X. Note that conditions (i) through (iv) from section 3 may already be satisfied if there are very weak dependencies of hypotheses H and ${H}^{\prime } $ on X. This means that if all relations of confirmation by analogy would be modelled by analogical confirmation type I and III only, then almost everything would at least very weakly confirm almost everything else. We would have to face some kind of full blown confirmation inflationism.

Confirmation of scattered hypotheses. Here is another possible problem. It is related to the last one. If the generalized Bayesian approach to indirect confirmation is correct, then confirmation just consists in Bayesian update, meaning that everything needed for a hypothesis H to be confirmed by some ${E}^{\prime } $ is that $P\left(H|{E}^{\prime}\right)>P(H) $ holds. But then variables that are quite scattered across a Bayesian network adequately representing a scenario of interest can, in principle, be used for confirmation. $P\left(H|{E}^{\prime}\right)>P(H) $ might already hold in some circumstances if there is some (maybe complex) path connecting ${E}^{\prime } $ with H in such a way that it can propagate probabilistic influence from ${E}^{\prime } $ to H. This leads to cases of confirmation that seem, at least at first glance, quite odd. Let us illustrate this by means of the exemplary Bayesian network whose structure is depicted in Figure 6: the ALARM network which was originally developed as a tool for monitoring medical systems of a certain kind (Beinlich, Suermondt, Chavez, and Cooper Reference Beinlich, Suermondt, Chavez and Cooper1989).

Figure 6. The ALARM belief network (reconstruction based on Spirtes et al., Reference Spirtes, Glymour and Scheines1993, p. 12).

The details of the ALARM network and the medical system modeled are not that important for the point we want to make. Let us rather focus on the following two variables of the ALARM network: $\boxed{19} $ and $\boxed{23} $ . $\boxed{19} $ could represent the hypothesis that an anaphylaxis occurs, and $\boxed{23} $ the hypothesis that the ventilation tube is kinked. Now $\boxed{19} $ and $\boxed{23} $ are not directly connected to each other in the ALARM network; they are rather scattered. They are, however, connected by several quite complex paths. The path consisting of the bold arrows in Figure 6, for example, can transport probabilistic dependence between $\boxed{19} $ and $\boxed{23} $ conditional on $\boxed{9} $ (which stands for heart rate obtained from oximeter). Now assume that we observe some evidence ${E}^{\prime } $ for $\boxed{23} $ . Since conditionalizing on $\boxed{23} $ increases the probability of $\boxed{19} $ conditional on $\boxed{9} $ , it might well be the case that $P\left(\boxed{19}|{E}^{\prime },\boxed{9}\right)>P\left(\boxed{19}|\boxed{9}\right) $ also holds. But in the generalized Bayesian approach to indirect confirmation, this just means that ${E}^{\prime } $ confirms $\boxed{19} $ in the context of $\boxed{9} $ . However, one might hesitate to speak of confirmation in this situation simply because the hypotheses $\boxed{19} $ and $\boxed{23} $ are too scattered. It is not clear why ${E}^{\prime } $ should count as evidence for $\boxed{19} $ .

Distrust in indirect evidence. The third possible problem is that the generalized Bayesian approach to indirect confirmation by analogy type I and III does not seem to be in line with how many scientists understand and use the concept of confirmation as well as with the common sense understanding of that concept. Let us illustrate this by means of the rat study example introduced in section 3. Assume that Jones was affected with a certain virus. She goes to a physician who offers her two possible treatments: medical compound A or B. A has been tested on humans and it turned out that 80% of humans affected with the virus recovered after taking A. B, on the other hand, has only been tested on rats so far. Let us assume that 95% of rats affected with the virus recovered after taking B. Let us further assume that physicians have used Dardashti et al.’s (Reference Dardashti, Hartmann, Thébault and Winsberg2019) method of indirect confirmation. They chose the priors of X and everything else required for indirect confirmation to the best of their knowledge and are quite certain that no mistakes occurred. They might then come to the conclusion that 80% of humans affected with the virus should recover after taking compound B. So which compound should Jones choose? Our guess is that almost every patient (and physician) would prefer compound A over B, simply because A has been tested directly on humans. The rat study investigating B clearly gives us probabilistic information for whether B will work when given to humans and makes it equally plausible that B will lead to recovery when given to affected humans. Still, we distrust B to some extent. Maybe the best explanation for this is that we hesitate to accept results from the rat study as evidence for hypotheses about B’s effect on human recovery. If this diagnosis is correct, then at least sometimes confirmation has to be constrained by structural rules (system internal confirmation versus confirmation across systems).

Irrelevance of direct evidence. Another possible problem for considering the generalized Bayesian approach to confirmation of a hypothesis H by analogy type III is that direct evidence E for the hypothesis H about the target system does not play any role for confirming H in that approach. All the confirmatory work is actually done by evidence ${E}^{\prime } $ for the hypothesis ${H}^{\prime } $ about the source system and the connection between H and ${H}^{\prime } $ established by the common cause or shared structure X. However, E often plays an important role in confirming H in scientific practice, even when scientists reason on the basis of analogies. If the generalized Bayesian approach to confirmation by analogy type III were the only mode of indirect confirmation of a hypothesis H about the target system, then it is not clear why, in cases such as the rat study example, scientists are looking for direct evidence E though they have already confirmed hypothesis H about the target system indirectly. Even more problematic, it is easy to find scenarios in which indirect evidence ${E}^{\prime } $ has more confirmatory impact on H than direct evidence E has. Here is an exemplary probability distribution:

From this distribution the impact of indirect evidence ${E}^{\prime } $ on H can be computed as $Bconf\left(H|{E}^{\prime}\right)=.053 $ , which overshoots the impact of direct evidence E on H which can be computed as $Bconf\left(H|E\right)=.048 $ .

Here is our diagnosis of the possible problems highlighted above. We think that the culprit is an unconstrained application of Bayesian confirmation. In particular, it seems to be problematic to only rely on probabilistic information $P\left(H|{E}^{\prime}\right)>P(H) $ when doing confirmation, especially if there is also structural information available. The situation seems to be similar to what was historically going on in several other philosophical debates such as debates about explanation, causation, decision theory, etc. According to Hempel’s (Reference Hempel1965, 338) classical inductive-statistical model of explanation, for example, an explanation of a phenomenon described by E consists of a statistical law of the form $P\left(E|{C}_1,\dots, {C}_n\right)\approx 1 $ and statements ${C}_1,\dots, {C}_n $ about the occurrence of certain events. If $P\left(E|{C}_1,\dots, {C}_n\right)\approx 1 $ and ${C}_1,\dots, {C}_n $ together make E very likely, then $P\left(E|{C}_1,\dots, {C}_n\right)\approx 1 $ and ${C}_1,\dots, {C}_n $ explain E. After striking counterexamples (see, e.g., Salmon Reference Salmon1984) it turned out that structural information was missing in Hempel’s approach: for a successful explanation it is also required that the events described by ${C}_1,\dots, {C}_n $ are causally relevant for the event described by E. Naïve probabilistic theories of causation also share this problem. They claim that C is a cause of E if and only if $P\left(E|C\right)>P(E) $ (this is, admittedly, an oversimplified version of a probabilistic theory of causation and more sophisticated probabilistic theories typically make additional assumptions; they might, for example, require that the cause always occurs before the effect, etc.). It turned out that a purely probabilistic criterion for causation is not enough. Here, one also needs additional information about structure. Modern accounts (e.g., Cartwright Reference Cartwright1979) rather characterize causation by referring to additional causal (i.e., structural) information. Finally, a similar move can also be found in the philosophical literature on decision theory. It is well known that the purely probabilistic account (i.e., evidential decision theory) has to face several problems. Some of them can be solved by endorsing causal decision theory, which already comes with some structural constraints. The approaches proposed by Meek and Glymour (Reference Meek and Glymour1994) and Hitchcock (Reference Hitchcock2016) consider richer causal structural information, which allows them to handle even more problems. We think that structural restrictions might also help to overcome problems that a generalized Bayesian approach to analogical inference has to face.

In this section, we discussed three types of possible problems for analogical inference Bayesian style. We think that the first two (subsection 4.a and 4.b) can be solved. The third type of problems (subsection 4.c) concerns confirmation across systems and it seems that these problems force one to make a choice depending on whether one finds them alarming. If one does, one might want to subscribe, as indicated above, to a structural constraint for confirmation:

• (*). Only evidence E accessible via paths within a particular system s can be used for confirmation of a hypothesis H about s.

We do not want to commit ourselves to one of the two options mentioned. Rather, we think that whatever path one takes, it still stresses the relevance of confirmation by analogy type II. (For details, see section 5.) However, it is clear that if one subscribes to (*), then cross-system confirmation of type I and III is excluded and type II is the only classical inference pattern (of the three introduced in section 2) left. But even if one considers indirect confirmation of type I and III to be perfectly fine, it seems that type I and III only work within their proper bounds: when considering confirmation on the quantitative scale, for example via $Bconf $ , then the problems of confirmation inflationism and confirmation of scattered hypotheses might vanish simply because the degree of the general background confirmation or that of confirmation of scattered hypotheses might be so low that it can be considered negligible. However, these strategies will not work for a purely qualitative model of confirmation.

In the next section, we will discuss alternatives that conform to (*) and, thus, are not plagued by the cross-system problems discussed in subsection 4.c. If evidence E of the target system is accessible, one can go for the last remaining classical inference pattern: type II analogical inference. For the case that E is inaccessible, we will propose an alternative to cross-system confirmation of type I and III that adheres to the structural constraint (*).

5. Confirmation by analogy: Bayes meets Jeffrey

Let us start this section with type II analogical inference. The basic idea is to use information about confirmation within the source system ${s}^{\prime } $ for making confirmational claims about the target system s. Schematically, the problem is to define $Bconf\left(H|E\right) $ on the basis of $Bconf\left({H}^{\prime }|{E}^{\prime}\right) $ . This corresponds to establishing the confirmational relation indicated by the vertical thick arrow in Figure 2. To this end, we first need to add a variable for evidence about the target system (E) to our model (see Figure 7). Next, we have to think about how to establish the vertical confirmatory relation within the target system on the basis of what we can learn about the source system. Note that we cannot simply equate $Bconf\left(H|E\right) $ with $Bconf\left({H}^{\prime }|{E}^{\prime}\right) $ . According to equation 2, $Bconf\left(H|E\right) $ is defined on the basis of $P\left(H|E\right) $ and $P(H) $ and, hence, $Bconf\left(H|E\right) $ cannot be determined independently of the Bayesian network’s parameters a, c, and e. (Likewise, $Bconf\left({H}^{\prime }|{E}^{\prime}\right) $ is defined on the basis of $P\left({H}^{\prime }|{E}^{\prime}\right) $ and $P\left({H}^{\prime}\right) $ and $Bconf\left({H}^{\prime }|{E}^{\prime}\right) $ cannot be specified independently of the parameters b, d, and e.) The problem is that the nonparameter probabilities within a Bayesian network are fully determined by and cannot be varied independently of the Bayesian network’s parameters a through e, where $a=\left\langle P\left(E|H\right),P\left(E|\neg H\right)\right\rangle $ , $b=\left\langle P\left({E}^{\prime }|{H}^{\prime}\right),P\left({E}^{\prime }|\neg {H}^{\prime}\right)\right\rangle $ , $c=\left\langle P\left(H|X\right),P\left(H|\neg X\right)\right\rangle $ , $d=\left\langle P\left({H}^{\prime }|X\right),P\left({H}^{\prime }|\neg X\right)\right\rangle $ and $e=P(X) $ . Since parameters b through e are already fixed, specifying a is the only way to influence the probabilities $P\left(H|E\right) $ and $P(E) $ relevant for determining $Bconf\left(H|E\right) $ . Because of this, we suggest implementing type II analogical inference via equating parameter a of the target system with the respective parameter b of the source system: $a=b $ . This move amounts to inferring a similar vertical relation for the target system as for the source system on the basis of the similarity between these systems expressed via $H\leftarrow X\to {H}^{\prime } $ .

Figure 7. Parameters in the Bayesian network for analogical reasoning type II.

Note that one could say a little bit more about the similarity relations between features of the source and the target system in terms of the parameters c and d. That the first element of $c=\left\langle P\left(H|X\right),P\left(H|\neg X\right)\right\rangle $ is significantly greater than the first element of $d=\left\langle P\left({H}^{\prime }|X\right),P\left({H}^{\prime }|\neg X\right)\right\rangle $ could, for example, represent the fact that the structural features modeled by X have more impact on the target system than on the source system. Equal parameters ( $c=d $ ), on the other hand, could model perfect similarity between the impact of these features on s and ${s}^{\prime } $ . In case of such a perfect similarity, making a type II analogical inference, i.e., identifying a with b, implies that $E $ ’s confirmational impact on H equals ${E}^{\prime } $ ’s confirmational impact on ${H}^{\prime } $ :

$$ \mathbf{Observation}: If\;c=d\kern.6em and\kern.6em a=b,\ then\kern0.17em Bconf\left(H|E\right)= Bconf\left({H}^{\prime }|{E}^{\prime}\right). $$

Note that type II analogical inference adheres to the structural constraint (*) from section 4: confirmation does not happen over cross-system pathways. Thus, type II analogical inference does not have to face the problems of cross-system confirmation discussed in subsection 4.c. However, type II analogical inference clearly requires that evidence E of the target system is accessible. This requirement will often not be satisfied. This was what made type I and III analogical inference so tempting in the first place. So, what if E is inaccessible? Can we still somehow use E for confirming H given the similarity between the source and the target system while, at the same time, respecting the structural constraint (*)? This question will be our focus in the remainder of this section. It is especially relevant if we find the problems discussed in subsection 4.c alarming and are somewhat skeptical regarding type I and III analogical inference.

Here comes our alternative to cross-system confirmation of type I and III: we propose a two-step approach: in the first step, one updates the probability distribution over the target system by conditionalizing on evidence ${E}^{\prime } $ of the source system. So the first step consists in Bayesian updating; no confirmation is involved here and, thus, (*) is respected. Confirmation happens system-internal in the second step: here we use the updated probability of evidence E to confirm H. Of course, E is not observed (in the strict sense); in the first step we have just reduced our uncertainty about E to some degree. For this reason, we have to use an update method in step 2 that can handle uncertain evidence. A standard method for this is Jeffrey conditionalization (Jeffrey Reference Jeffrey1983).

If one is ready to subscribe to (*), then one still needs to make sense of scientific practice as, for example, illustrated in the rat study case. But if one finds cross-system confirmation type I and III problematic, then one will not find the reconstruction of the rat study case provided in section 3 very tempting. In the following, we will use this example again for two purposes: First, to show how it can be rationally reconstructed in terms of our two-step approach if one endorses (*), and second to illustrate the two-step approach. In the rat case example physicians were, for moral reasons, not able to collect evidence E about how humans react to the antiviral compound under investigation. So they were not able to directly confirm the hypothesis H about the human immune system. For this reason, they studied the effects of the antiviral compound on a model organism such as rats. Let us assume that the physicians were able to gather evidence ${E}^{\prime } $ for a similar hypothesis ${H}^{\prime } $ about the immune system of rats. Since it has already been empirically established that the source system and the target system share relevant structural features, they infer, from observing ${E}^{\prime } $ , that if they were to start testing the antiviral compound on humans, they would very likely find evidence E. This is the first step in our two-step approach based on ordinary Bayesian updating; no confirmation is involved here. They then use this inferred decreased uncertainty about E as evidence for H, meaning that they infer that it is sufficiently likely that infected humans will recover without severe side effects if treated with the antiviral compound. This is the second step in our two-step approach; it is based on Jeffrey conditionalization and marks the point where (system-internal) confirmation happens. Having confirmed H that way will be sufficient for physicians to start tests with humans to further confirm H (directly).

Here is our story again, this time with all the technical details: we assume that the rat study example can be adequately represented by a Bayesian network with the structure depicted in Figure 7. To confirm H, one needs access to direct evidence E. But, in our example, E cannot be observed directly. However, what the experts have available is their (prior) estimation of the probability that direct evidence E would occur if the antiviral compound were administered to affected humans. What they can do now is employ the information they gathered from their investigation of the model organism to update the probability $P(E) $ Bayesian style to ${P}^{\ast }(E) $ via

(7) $$ {P}^{\ast }(E)=P\left(E|{E}^{\prime}\right). $$

Since ${E}^{\prime } $ is ordinarily gathered evidence—the physicians just observe ${E}^{\prime } $ —this move seems to be straightforward. In this way, evidence ${E}^{\prime } $ about the source system is used to achieve an empirically informed estimation of the probability of the direct evidence E. Now this updated probability ${P}^{\ast }(E) $ can be used to confirm H directly. Because E is still uncertain—not observed in the strict sense—we use Jeffrey conditionalization for this last step. We say that uncertain evidence E confirms H by analogy if

(8) $$ P\left(H|E\right)\cdot {P}^{\ast }(E)+P\left(H|\neg E\right)\cdot {P}^{\ast}\left(\neg E\right)>P(H) $$

holds. The left part of this inequality expresses the impact of direct but uncertain evidence E on hypothesis H Jeffrey style (i.e., the impact on H given the updated probability of E). We have to use the prior conditional probabilities $P\left(H|E\right) $ and $P\left(H|\neg E\right) $ from the original distribution P here. Using the posterior conditional probabilities ${P}^{\ast}\left(H|E\right) $ and ${P}^{\ast}\left(H|\neg E\right) $ would amount to computing the impact of ${E}^{\prime } $ on H by simple Bayesian conditionalization and, thus, we would face at least the problems of distrust in indirect evidence and irrelevance of direct evidence which we highlighted in subsection 4.c. The structural constraint (*) which only allows for confirmation within a system impedes to directly use external evidence for confirmation. However, such evidence is not completely irrelevant: even if ${E}^{\prime } $ cannot be used to confirm H, observing ${E}^{\prime } $ can still (cross systemwise) decrease uncertainty about our expectations of direct evidence E. We can then use this decreased uncertainty to directly confirm (system-internal) H.

Before we go on to define a measure for this kind of confirmation, note that Jeffrey conditionalization (but also Bayesian conditionalization) requires a rigidity condition to hold: one has to use the initial conditional probabilities in calculating the unconditional ones. Since the conditional probabilities remain unchanged, our two-step approach adheres to this rigidity condition. Next to the uncertainty of evidence E of the target system, this is another important respect in which the two-step approach differs from type II inference by analogy in form of parameter mapping: parameter mapping changes the conditional probabilities and, hence, is a form of nonrigid updating.

Based on the two-step approach to confirmation outlined above, we can define the simple incremental confirmation measure

(9) $$ {BJconf}_{E^{\prime }}\left(H|E\right)=\left[P\left(H|E\right)\cdot {P}^{\ast }(E)+P\left(H|\neg E\right)\cdot {P}^{\ast}\left(\neg E\right)\right]-P(H). $$

BJ stands for the two steps: B for the first step using Bayesian update of the probability of E in the light of ${E}^{\prime } $ and J for the second step using Jeffrey conditionalization for confirmation.

A similar result as the one Dardashti et al. (Reference Dardashti, Hartmann, Thébault and Winsberg2019) proved for $Bconf $ can be proven for $BJconf $ if

1. $$(v)\kern0.36em P\left(E|H\right)>P\left(E|\neg H\right) $$

is added as a fifth condition to (i) through (iv) from section 3. Condition (v) is analogue to (iv). It guarantees that E can serve as evidence for H. If all five conditions are satisfied and the probability distribution P over $\mathbf{V}=\left\{X,H,{H}^{\prime },E,{E}^{\prime}\right\} $ factors according to equation 1, where the parents are determined by the structure of the Bayesian network in Figure 7, then E confirms H on the basis of observing ${E}^{\prime } $ , even if E cannot be accessed directly.

Before we go on, let us briefly compare the measures $Bconf $ and $BJconf $ . It can be shown that under conditions (i) through (v) $BJconf $ is always weaker than or equal to $Bconf $ .

The confirmational impact according to the two measures can be illustrated by means of Figure 8: the formula in the squared brackets of equation 9 is a weighted average $wa $ with maximum $P\left(H|E\right) $ and minimum $P\left(H|\neg E\right) $ and weights ${P}^{\ast }(E) $ and ${P}^{\ast}\left(\neg E\right) $ , respectively. According to Theorem 1, $wa $ lies above $P(H) $ and the distance between $wa $ and $P(H) $ corresponds to the degree of confirmation according to $BJconf $ . When replacing $P\left(H|E\right) $ with ${P}^{\ast}\left(H|E\right) $ and $P\left(H|\neg E\right) $ with ${P}^{\ast}\left(H|\neg E\right) $ in equation 9, one gets Dardashti et al.’s (Reference Dardashti, Hartmann, Thébault and Winsberg2019) confirmation measure $Bconf $ . According to Theorem 2, ${wa}^{\ast } $ of the posterior probabilities is equal to or lies above $wa $ used in our measure $BJconf $ simply because the former maximum and minimum are equal or lie above the latter. Again, the distance between ${wa}^{\ast } $ and $P(H) $ represents the degree of confirmation according to $Bconf $ .

Figure 8. Illustration of confirmational impact of Bconf and BJconf.

Recall from section 4 that, according to Dardashti et al.’s (Reference Dardashti, Hartmann, Thébault and Winsberg2019) approach, indirect evidence ${E}^{\prime } $ sometimes provides more confirmation of a hypothesis H than direct evidence E for H, which contradicts in relevant cases our intuitions and scientific practice. The next theorem shows that the two-step approach does not share this problem with the generalized Bayesian approach to confirmation by analogy type III.

This result fits nicely with our intuitions and scientific practice. It shows that our approach allows for an explanation of why, for example, the scientists in the rat study scenario still collect direct evidence for the efficacy of the antiviral compound on humans though they have already tested it on rats: in this case, direct evidence has an impact at least as strong as evidence from different but structurally analogous systems; even more, direct evidence will almost always have more impact. So, we think that in the case of the rat study, for example, the Bayes-Jeffrey measure for confirmation provides a more adequate model of the confirmational relation in question than the Bayes measure for type III confirmation.