1 Introduction

Relationships in a dataset can be expressed in terms of inductive rules, \(E \rightarrow H\), meaning that the knowledge of evidence/premise E corroborates hypothesis/conclusion H. Such rules can be supported by data with different intensities, which often need to be estimated to rank the rules, for example with respect to their reliability: the more reliable the rules, the higher the interest. Many interestingness measures have been conceived and selected for data mining and knowledge discovery (see, e.g., Chakhar et al. 2020; Greco et al. 2016; Lenca et al. 2008; McGarry 2005; Tan et al. 2004).

Bayesian Confirmation Measures (BCMs) are interestingness measures aimed at evaluating the degree to which an evidence E supports or contradicts the conclusion H, using prior probability \(Pr(H)\), posterior probability \(Pr(H|E)\) and \(Pr(E)\), the probability of evidence E. The emerging of evidence E may indeed change the knowledge about the occurrence of H, since conclusion H may be confirmed when \(Pr(H|E)>Pr(H)\), or disconfirmed when \(Pr(H|E)<Pr(H)\).

Many confirmation measures were defined and used in different frameworks (some of them are recalled in Sect. 2), many of them were even reinvented and renamed by different researchers, being actually the same measure. A question that arises is whether there are BCMs that perform better than other ones, given specific assessment criteria and contexts of application.

To highlight in which way BCMs are similar or not, geometric and visual approaches have been proved to be particularly useful, in fact they allow to uncover some properties of BCMs and to suggest how to select the more apt to be used for a specific purpose (see, e.g., Brzezinski et al. 2018; Celotto 2016; Fürnkranz and Flach 2005; Susmaga and Szczȩch 2015, 2016). Nevertheless, it is the study of the analytical properties of BCMs that allows to definitely master the differences among measures (Fitelson 2007; Greco et al. 2012b; Susmaga and Szczȩch 2013).

Symmetries are, not surprisingly, among the properties that can be better captured by a visual approach and they also have been deeply examined from an analytical point of view. The first symmetry properties that were defined by Carnap (1950), concern the confirmation values attained by the measure when negation of evidence is substituted to evidence E, negation of conclusion is substituted to conclusion H, or when the roles of evidence and conclusion are inverted. Crupi et al. (2007) completed the set of symmetries proposed by Carnap considering all their possible compositions.

A topic discussed by the literature concerns desirability of BCMs’ symmetry properties, namely, only some symmetries should be considered as desirable properties for a BCM and some as undesirable (see Crupi et al. 2007; Eells and Fitelson 2002; Glass 2013; Greco et al. 2012a, b, 2016). Therefore it becomes relevant to discuss whether a BCM satisfies a desirable combination of symmetry properties, as in Susmaga and Szczȩch (2016). Only some combinations of symmetry properties can be concurrently satisfied, as it was recently pointed out by means of group theory arguments by Susmaga and Szczȩch (2016): they consider the group of symmetries of \(2\times 2\) contingency tables entries, its isomorphism with the dihedral group \(D_8\) and its subgroups structure, allowing to detect the symmetries that can coexist.

In Sects. 3 and 4 we propose a different group theoretic approach to discuss BCMs symmetry properties. Following Crupi et al. (2007), we define a set of symmetry functions acting on couples (H, E) of hypothesis and evidence sentences, and then we endow the set with the composition of symmetries obtaining a group structure, which, also in this case, turns out to be isomorphic to the dihedral group \(D_8\).

By exploiting the group structure within the new framework, in Sect. 5 we can partition the BCMs into 10 equivalence classes based on the set of symmetry properties they satisfy. In this way we can determine the subsets of symmetries that can coexist, thus obtaining results in line with those of Susmaga and Szczȩch (2016).

The subgroup structure inherited from the dihedral group plays another important role as it allows to propose an algorithm that simplifies the checking of symmetry properties.

A relevant issue we tackle in Sect. 6 is to find a representative element for each of the 10 equivalence classes: at this aim, we make use of 8 known BCM and define two new BCMs to conclude that all equivalence classes have at least one representative.

Section 7 concerns desirability of symmetry properties. This problem was addressed in the contingency tables context in Susmaga and Szczȩch (2016), we face it here in the framework of conclusion-evidence pairs. Different BCMs can be this way considered as more or less well crafted, depending on which equivalence classes they belong to.

But, are all BCMs belonging to the same class equally desirable? In other terms, if a BCM does not meet a specific symmetry requirement, how far is it from satisfying it? Different BCMs that do not satisfy the same symmetry property may indeed exhibit different degrees of asymmetry, this way resulting more (less) desirable, provided a degree of asymmetry is defined, of course. We define a family of asymmetry measures based on p-norms and illustrate their possible use computing the 1-norm and the 2-norm asymmetry measures for each of the 10 class representatives of feasible symmetry combinations (Sect. 9).

The numerical results suggest several possible uses of asymmetry measures. For example, providing a feasible way to appraise the asymmetry degrees, they allow to rank BCMs with respect to a fixed symmetry property. Moreover, an asymmetry measure that vanishes, suggests that the BCM owns the corresponding symmetry property and that a further investigation may prove it analytically.

Concluding remarks and future work hints are reported in Sect. 10.

2 A sample of Bayesian Confirmation Measures

In this Section we propose a selection of Bayesian Confirmation Measures, some of them are well known BCMs and are often used, appreciated or, even, criticised in the literature, depending on the properties they comply with. They will be used in Sect. 6 to illustrate symmetry properties they do possess and, if they do not possess some symmetry properties, to assess their degree of asymmetry (Sect. 9).

We recall that a Bayesian Confirmation Measure is a function c(H, E), that provides an evaluation of the change of probability of conclusion H due to the knowledge variation associated with an additional evidence E. A widely accepted definition is (see, e.g., Crupi et al. 2010; Geng and Hamilton 2006):

Definition 1

A function c of conclusion H and evidence E, is a Bayesian Confirmation Measure (BCM) when

  • \(c(H,E)>0\,\,\) if \(Pr(H|E)>Pr(H) \qquad\) (confirmation case)

  • \(c(H,E)=0\,\,\) if \(Pr(H|E)=Pr(H) \qquad\) (neutrality case)

  • \(c(H,E)<0\,\,\) if \(Pr(H|E)<Pr(H) \qquad\) (disconfirmation case).

Till now, a lot of BCMs have been defined in many different ways, with different names and meanings, with different analytical formalisations, depending on the reference framework. Nevertheless, the main idea of a BCMs can be traced back to some elementary kinds of evaluations, like the difference \(Pr(H|E)-Pr(H)\) [which is Carnap’s difference measure d, see Carnap (1950)], the proportional difference \((Pr(H|E)-Pr(H))/ Pr(H)\) [which was considered in Finch (1960)] or the idea of probabilistic independence \(Pr(H\cap E)-Pr(H)Pr(E)\) (as in Carnap’s Relevance measure b in Carnap (1950)). In this paper we will refer to the well known BCMs which are collected in Table 1 along with their definitions.

To provide examples of BCMs that satisfy some possible combinations of symmetry properties (see Sect. 6) we need to complete the sample with two BCMs that, at least to the best of our knowledge, were never defined before in the literature.

Table 1 A set of well known Bayesian Confirmation Measures
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We will call Predictive minus false over False confirmation measure (\(c_{PF}\)) the function

$$\begin{aligned} c_{PF}(H,E) = \frac{(Pr(H|E)+Pr(\lnot H|\lnot E))-(Pr(\lnot E|H) + Pr(E|\lnot H))}{Pr(\lnot E|H)+ Pr(E|\lnot H)}. \end{aligned}$$
(1)

This function is a BCM since it satisfies Definition 1. In fact, if we consider, for example, the confirmation case, from the condition \(Pr(H|E)>Pr(H)\) we obtain the inequalities

$$\begin{aligned} Pr(\lnot H|\lnot E)>Pr(\lnot H) \quad Pr(\lnot E|H)<Pr(\lnot E) \quad Pr(E| \lnot H)<Pr(E) \end{aligned}$$

and therefore,

$$\begin{aligned} c_{PF}(H,E) = \frac{Pr(H|E)+Pr(\lnot H|\lnot E)}{Pr(\lnot E|H)+ Pr(E|\lnot H)} -1 > \frac{Pr(H)+Pr(\lnot H)}{Pr(\lnot E)+ Pr(E)}-1=0. \end{aligned}$$

That is, if \(Pr(H|E)>Pr(H)\) then \(c_{PF}(H,E)>0\).

The name Predictive minus false over False confirmation measure is inspired by the fact that the measure can be recast in a meaningful way using the language of diagnostic tests. In fact, \(Pr(H|E)+Pr(\lnot H|\lnot E)\) can be rewritten as \(PPV+NPV\) where PPV (Positive Predictive Value) denotes the percentage of correct positive predictions of a statistical test, and NPV (Negative Predictive Value) indicates the percentage of correct negative predictions of the test. Moreover, \(Pr(\lnot E|H)+ Pr(E|\lnot H)\) can be written as \(FNR+FPR\) with FNR (False Negative Rate) and FPR (False Positive Rate) denoting the percentages of positive conditions which yield negative test outcomes and, respectively, of negative conditions which yield positive test outcomes. This means that we can write

$$\begin{aligned} c_{PF}(H,E) = \frac{(PPV+NPV)-(FNR+FPR)}{FNR+FPR} . \end{aligned}$$
(2)

Equation (1) can be rewritten as

$$\begin{aligned} c_{PF}(H,E) = \frac{Pr(H|E)+Pr(\lnot H|\lnot E)}{Pr(\lnot E|H)+ Pr(E|\lnot H)} -1 \end{aligned}$$
(3)

or, in other terms,

$$\begin{aligned} c_{PF}(H,E)= \frac{PPV+NPV}{FNR+FPR} -1 . \end{aligned}$$
(4)

Related to (4) is the definition of the following function that we will call the Logarithm of Predictive over False confirmation measure (\(c_{LPF}\)):

$$\begin{aligned} c_{LPF}(H,E) = \log \frac{PPV+NPV}{FNR+FPR} \ \end{aligned}$$
(5)

that is,

$$\begin{aligned} c_{LPF}(H,E)=\log \frac{Pr(H|E)+Pr(\lnot H|\lnot E)}{Pr(\lnot E|H)+ Pr(E|\lnot H)} . \end{aligned}$$
(6)

In the same way, also \(c_{LPF}\) is a BCM as it can be easily checked.

The two confirmation measures \(c_{PF}\) and \(c_{LPF}\) are ordinally equivalent since \(c_{LPF}(H,E)=\log (c_{PF}(H,E)+1)\). Nevertheless, as we will see in Sect. 6, the sets of symmetries satisfied by the two BCMs are different.

3 The group of symmetry functions defined on (Hypotesis, Evidence) pairs

With the aim to set up a comprehensive scheme of symmetry properties of BCMs, we follow Crupi et al. (2007) that, as a first step, consider symmetry functions, i.e., maps defined on couples (H, E) of sentences H, called the Hypothesis, and E, called the Evidence. Sentences H and E are assumed to belong to the set of sentences \(\Gamma\), a set closed under negation and conjunction. The couple \((H,E) \in \Gamma \times \Gamma\) is mapped by a symmetry function into another couple of sentences in \(\Gamma \times \Gamma\).

Following Crupi et al. (2007), we first consider the symmetry functions \(e\), \(h\) and \(i\) defined as

  • Negation of Evidence: \(e(H,E)=(H,\lnot E)\);

  • Negation of Hypothesis: \(h(H,E)=(\lnot H,E)\);

  • Inversion of Evidence and Hypothesis: \(i(H,E)=(E,H)\).

In other words, when we consider these symmetry functions acting on the rule \(E \rightarrow H\), the rule is changed into \(\lnot E \rightarrow H\) by \(e\), into \(E \rightarrow \lnot H\) by \(h\) and into \(H \rightarrow E\) by \(i\).

Further, four different symmetry functions, proposed in Crupi et al. (2007) as well, can be easily obtained by composition of \(e\), \(h\) and \(i\):

  • Negation of Evidence and Hypothesis

  •                               \(eh(H,E) = (\lnot H,\lnot E);\)

  • Inversion after negation of Evidence

  •                               \(\, ei(H,E) = (\lnot E,H);\)

  • Inversion after negation of Hypothesis

  •                               \(\, hi(H,E) = (E,\lnot H);\)

  • Inversion after negation of Evidence and Hypothesis

  •                               \(ehi(H,E) = (\lnot E,\lnot H)\).

To the seven above recalled symmetry functions we add the Identity function \({\sigma }_0\) (i.e., \({\sigma }_0(H,E)=(H,E)\) for all (H, E)) and consider the set of symmetry functions

$$\begin{aligned} \Sigma = \{{\sigma }_0, e, h, i, eh, ei, hi, ehi\}. \end{aligned}$$

The set \(\Sigma\) can be endowed with the composition operation \(\otimes\) defined by the Cayley Table reported in Table 2, where entry (k, j) corresponds to the symmetry function \({\sigma }_k \otimes {\sigma }_j\) defined as \({\sigma }_k \otimes {\sigma }_j (H,E) = {\sigma }_j ({\sigma }_k (H,E))\).

Table 2 Cayley table of \((\Sigma , \otimes )\) (symmetry in the first column operates first)
Full size table

Table 2 reveals that \((\Sigma ,\otimes )\) is a non-commutative (observe, e.g., that \(e\otimes i=ei\) does not coincide with \(i\otimes e=hi\)) group that can be related to \(D_8\), the dihedral group of order 8, i.e., the group of symmetries of the square.Footnote 1

The group \(D_8\) contains eight elements, the counter-clockwise rotation by \(\pi /2\) (denoted by \(\rho\)), the reflection (or flip) (denoted by \(\varphi\)), the identity (denoted by \(1\)) and their compositions defined by the Cayley’s table reported on Table 3, i.e. all the possible symmetries of the square. To relate \((\Sigma , \otimes )\) with the dihedral group \(D_8\), we consider the map from \(D_8\) to \((\Sigma ,\otimes )\) defined by Table 4, which is an isomorphism, as is easily proved by comparing the Cayley tables of the two groups. Since the two groups \((\Sigma , \otimes )\) and \(D_8\) are isomorphic, the classical lattice representation of the structure of the dihedral group \(D_8\) can be used to represent the symmetry functions subgroups in \((\Sigma ,\otimes )\) as in Fig. 1 (see, e.g., subwiki 2018). The structure of the subgroup lattice will be used in Sect. 6 to suggest an apt way to explore symmetry properties of a BCM.

Table 3 Cayley table of \(D_8\) (element in the first column operates first)
Full size table
Table 4 Isomomorphism of \(D_8\) into \(\Sigma\)
Full size table
Fig. 1
figure 1

The lattice of subgroups of \((\Sigma ,\otimes )\)

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4 Symmetries of Bayesian Confirmation Measures

In Susmaga and Szczȩch (2016), a group theoretic approach has been suggested to consider symmetries as permutations of the elements of \(2\times 2\) contingency tables, and compositions of symmetries as compositions of permutations.

Changing the point of view, we focus on the inductive rule \(E \rightarrow H\), applying each symmetry function \(\sigma \in \Sigma\) to the evidence-hypothesis couple (H, E). We will see that this way the group structure of \((\Sigma ,\otimes )\) can be used to detect in a straightforward manner which symmetry properties of a BCM can coexist (see Sect. 5).

Since we want to exploit the group structure of \((\Sigma ,\otimes )\) to have an in-depth look on symmetry properties of BCMs, we observe, first, that each symmetry function \(\sigma \in (\Sigma ,\otimes )\) can be used to define a corresponding symmetry property defined on BCMs. For example, a BCM c(H, E) is said to satisfy evidence symmetry if \(c(H,E)=-c(H,\lnot E)\) (see Crupi et al. 2007): alternatively, we can consider the negation of evidence symmetry function \(e\) and say that c(H, E) satisfies evidence symmetry if \(c(H,E)=-c(e(H,E))\). Likewise, using the negation of hypothesis symmetry function \(h\), it is possible to restate the definition of hypothesis symmetry property (see Crupi et al. 2007) of a BCM as \(c(H,E)=-c(h(H,E))=-c(\lnot H,E)\).

More in general, taking into account the sign requirements of a BCM (see Definition 1), it is possible to define symmetries for BCMs as follows.

Definition 2

Given a symmetry function \(\sigma \in {\Sigma }^*=\Sigma {\setminus } \{{\sigma }_0\}\), a BCM c(H, E) satisfies the symmetry property \(\sigma\) when

$$\begin{aligned} c(H,E)=sign(\sigma ) \cdot c(\sigma (H,E)) \end{aligned}$$

where \(sign (\sigma )=(-1)^k\) and k is the number of negations (\(\lnot\)) contained in the definition of \(\sigma\).

Observe that in the above definition we do not consider the identity \({\sigma }_0\), thus obtaining the following symmetries that are studied in Crupi et al. (2007).

Definition 3

A confirmation measure c satisfies

Evidence Symmetry (ES) when:
$$\begin{aligned} c(H,E)=-c(e(H,E))=-c(H,\lnot E); \end{aligned}$$
Hypothesis Symmetry (HS) when:
$$\begin{aligned} c(H,E)=-c(h(H,E))=-c(\lnot H,E); \end{aligned}$$
Inversion Symmetry (IS) when:
$$\begin{aligned} c(H,E)=c(i(H,E))=c(E,H); \end{aligned}$$
Evidence Hypothesis Symmetry (EHS) when:
$$\begin{aligned} c(H,E)=c(eh(H,E))=c(\lnot H,\lnot E); \end{aligned}$$
Evidence Inversion Symmetry (EIS) when:
$$\begin{aligned} c(H,E)=-c(ei(H,E))=-c(\lnot E,H); \end{aligned}$$
Hypothesis Inversion Symmetry (HIS) when:
$$\begin{aligned} c(H,E)=-c(hi(H,E))=-c(E,\lnot H); \end{aligned}$$
Evidence Hypothesis Inversion Symmetry (EHIS) when:
$$\begin{aligned} c(H,E)=c(ehi(H,E))=c(\lnot E,\lnot H). \end{aligned}$$

Note that in Eells and Fitelson (2002) and Greco et al. (2004) Inversion Symmetry IS and Evidence Hypothesis Symmetry EHS are called Commutativity Symmetry CS and Total Symmetry TS, respectively.

The meaning and interpretation of the symmetries defined for BCMs is simple. For example, ES means that evidence E confirms/disconfirms hypothesis H to the same extent that \(\lnot E\) does disconfirm/confirm the same hypothesis H.

The literature has already observed some links between these symmetry properties and these relationships can be proved by referring to the algebraic properties of symmetries, as highlighted in Susmaga and Szczȩch (2016) in the context of contingency tables where symmetries are considered as permutations (and compositions of permutations). In the next section we will reformulate the symmetry properties using directly the group structure of the symmetry functions \((\Sigma ,\otimes )\), this allows to easily identify the possible coexistence of properties and to formulate a quick way to analyze which symmetry properties are satisfied by a BCM.

5 Exploiting the structure of the symmetry functions group

The group structure of \((\Sigma ,\otimes )\) can be used to precisely detect all the combinations of symmetry properties that a BCM can simultaneously satisfy. In fact, while in Sects. 3 and 4 our new perspective allowed to easily detect the isomorphism between \((\Sigma , \otimes )\) and \(D_8\), in the current section we exploit the lattice structure of the subgroups of \(D_8\) to explicitly determine the number of symmetries that can coexist (Proposition 2) and, more precisely, the subsets of symmetries that can coexist (Proposition 3). This way, Propositions 2 and 3 generalize some results proposed in Susmaga and Szczȩch (2016) concerning the fact that some subsets of symmetries may be not closed under symmetry composition (in Susmaga and Szczȩch (2016) this phenomenon is called incompleteness).

Our different perspective of fully exploiting the lattice structure of the subgroups of \(D_8\) is key for partitioning the BCMs into equivalence classes, which allows to design an algorithm suitable for a rapid examination of the symmetry properties.

Let’s first focus on how symmetry properties of a BCM are inherited from other symmetries.

Proposition 1

Given a confirmation measure c satisfying two symmetry properties, say \(S_1\), \(S_2\), with corresponding symmetry functions \({\sigma }_1\), \({\sigma }_2 \in {\Sigma }^*\) and signs \((-1)^{k_1}\), \((-1)^{k_2}\), then necessarily c satisfies other two properties, say \(S_3\), \(S_4\), corresponding to symmetry functions \({\sigma }_3={\sigma }_1 \otimes {\sigma }_2\) and \({\sigma }_4={\sigma }_2 \otimes {\sigma }_1\), with sign \((-1)^{k_1+k_2}\).

Proof

Observe that if \({\sigma }_3\) or \({\sigma }_4\) coincide with the identity \({\sigma }_0\), then the new properties \(S_3\) or \(S_4\) are obviously satisfied by c.

Given that c is supposed to satisfy symmetry properties \(S_1\) and \(S_2\), it is \(c(H,E)=(-1)^{k_1}c({\sigma }_1(H,E))\) and \(c(H,E)=(-1)^{k_2}c({\sigma }_2(H,E))\) where \({\sigma }_1\) and \({\sigma }_2\) are the corresponding symmetry functions, and \(k_1, k_2\) denote the number of negations \(\lnot\) contained in the symmetry function definitions.

By applying \({\sigma }_1\) to the couple (H, E) and then \({\sigma }_2\) to the couple \({\sigma }_1(H,E)\), we obtain \(c(H,E)=(-1)^{k_1}c({\sigma }_1(H,E))=(-1)^{k_1}(-1)^{k_2}c({\sigma }_2({\sigma }_1(H,E))\).

In the same way, \(c(H,E)=(-1)^{k_2}c({\sigma }_2(H,E))=(-1)^{k_2}(-1)^{k_1}c({\sigma }_1({\sigma }_2(H,E))\), so that \(c(H,E)=(-1)^{k_1+k_2}c({\sigma }_1({\sigma }_2(H,E))=(-1)^{k_1+k_2}c({\sigma }_2({\sigma }_1(H,E))).\) \(\square\)

A first consequence of the group structure of \((\Sigma ,\otimes )\) can now be given in the next Proposition.

Proposition 2

A Bayesian Confirmation Measure satisfies either 1, 3 or 7 symmetry properties, or none.

Proof

From Proposition 1 it follows that if \(S_1\) and \(S_2\) hold, with corresponding symmetry functions \({\sigma }_1\), \({\sigma }_2 \in \Sigma\), then necessarily also \(S_3\), with corresponding symmetry \({\sigma }_3={\sigma }_1 \otimes {\sigma }_2\), and \(S_4\), with corresponding symmetry \({\sigma }_4={\sigma }_2 \otimes {\sigma }_1\), must hold, where \({\sigma }_1\), \({\sigma }_2\), \({\sigma }_3\) and \({\sigma }_4\) obviously belong to the same subgroup of \((\Sigma ,\otimes )\). Proposition 1 allows therefore to state that each non-empty subset of symmetries of a BCM is closed under operation \(\otimes\) and, so, it constitutes a subgroup of \((\Sigma ,\otimes )\).

By Lagrange’s Theorem (see, e.g., Beardon 2005) we know that in finite groups the order of a subgroup divides the order of the group and \((\Sigma ,\otimes )\) has order 8. Therefore, subgroups have either 1, 2, 4 or 8 elements, identity included. The result follows excluding from each possible subgroup the identity symmetry function since, of course, it does not correspond to any BCM symmetry. \(\square\)

The link between symmetry functions and the symmetries of BCMs allows to precisely deduce the symmetry properties that can be concurrently satisfied by a BCM, a result that follows by considering all the possible subgroups of the group of symmetry functions \((\Sigma ,\otimes )\) and the corresponding symmetry properties.

Proposition 3

For a Bayesian Confirmation Measure c, exactly one of the following statements is true:

  1. (i)

    c satisfies no symmetry property;

  2. (ii)

    c satisfies exactly one of the symmetries IS, EHIS, EHS, ES, HS;

  3. (iii)

    c satisfies all the symmetries in one of the sets \(\{IS,EHS,EHIS\}\), \(\{EHS,EIS,HIS\}\), \(\{ES,HS,EHS\}\);

  4. (iv)

    c satisfies all the symmetry properties.

Proof

Proposition 2 establishes the cardinality of each subset of concurrent symmetry properties; given the isomorphism of \(\Sigma\) into \(D_8\) it is possible to explicitly write all the possible subgroups of symmetry functions of \((\Sigma , \otimes )\) and therefore the possible corresponding symmetry properties that can be jointly satisfied. The possible subgroups are (see Fig. 1):

order 1::

\(\{\sigma _{0} \}\),

order 2::

\(\{\sigma _{0}, i\}\), \(\{\sigma _{0}, ehi\}\), \(\{\sigma _{0}, eh\}\), \(\{\sigma _{0}, e\}\), \(\{\sigma _{0}, h \}\),

order 4::

\(\{\sigma _{0}, i, eh, ehi\}\), \(\{\sigma _{0}, eh, ei, hi\}\), \(\{\sigma _{0}, e, h, eh\}\),

order 8::

the whole group \(\{ \sigma _{0} ,e,h,i,eh,ei,hi,ehi\}\).

It turns out that the symmetry properties that a BCM can simultaneously satisfy are those stated in the Proposition. \(\square\)

Given the isomorphism of \((\Sigma , \otimes )\) with \(D_8\), Proposition 3 identifies thus a partition of the set of BCMs into 10 equivalence classes, each class being defined by the set of satisfied symmetries.

Moreover, the lattice structure of subgroups, reported in Fig. 1, suggests quick ways to check the fulfillment of concurrent symmetry properties.

For example, the fact that \(\{\sigma _0, eh\}\) is the Frattini subgroup of \((\Sigma , \otimes )\), i.e. the intersection of all its maximal subgroups (see, e.g., Rose 2009), suggests to explore symmetry properties of a BCM starting from symmetry EHS, which corresponds to the element \(eh\) of the Frattini subgroup. Indeed, if the BCM does not satisfy EHS, then the set of properties it satisfies cannot correspond to any subgroup of order higher than 2 (i.e., the BCM cannot exhibit more than one symmetry property). Thus, we are allowed to check only symmetry properties corresponding to subgroups of dimension 2, so that we can check only symmetries IS, EHIS, ES and HS to possibly conclude that the BCM does not possess any symmetry property. Using analogous considerations, it is possible to define a procedure for an easy check of concurrent symmetry properties of a BCM, as in the algorithm shown in Fig. 2.

Fig. 2
figure 2

Symmetry Test algorithm for checking concurrent symmetries of the Bayesian Confirmation Measure c

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6 Examples of BCMs that concurrently satisfy feasible combinations of symmetries

Let us now focus our attention to the existence of BCMs satisfying the admissible combinations of symmetries specified by Proposition 3: we are wondering if, for each possible subgroup of \((\Sigma ,\otimes )\), there exists at least one confirmation measure c that satisfies all the corresponding symmetry properties (and only them) specified by Proposition 3. In other terms, given the 10 equivalence classes identified by Proposition 3 in the set of BCMs, we want to understand if some of them are empty. In order to do that, we make use of the BCMs sample defined in Sect. 2.

We can now state that for any feasible combination of symmetries there exist one BCM satisfying exactly those properties. More precisely:

Proposition 4

For each of the following alternative statements, there is at least one BCM c that satisfies it:

  1. (i)

    c satisfies no symmetry property;

  2. (ii)

    c satisfies exactly one of the symmetries IS, EHIS, EHS, ES, HS;

  3. (iii)

    c satisfies all the symmetries in one of the sets \(\{IS,EHS,EHIS\}\), \(\{EHS,EIS,HIS\}\), \(\{ES,HS,EHS\}\);

  4. (iv)

    c satisfies all the symmetry properties.

Proof

(i) The confirmation measure \(c_{GSS}(H,E)\) proposed by Greco et al. doesn’t satisfy any symmetry property (see Greco et al. 2016).

(ii) To prove that \(c_{PF}\), the BCM defined in Sect. 2 as

$$\begin{aligned} c_{PF}(H,E) = \frac{Pr(H|E)+Pr(\lnot H|\lnot E)}{Pr(\lnot E|H)+ Pr(E|\lnot H)} -1 \end{aligned}$$
(7)

satisfies only EHS symmetry, we follow the steps prescribed by the Symmetry Test algorithm presented in Fig. 2.

  1. (a)

    \(c_{PF}\) satisfies EHS, in fact

    $$\begin{aligned} c_{PF}(\lnot H,\lnot E) = \frac{Pr(\lnot H|\lnot E)+Pr(\lnot (\lnot H)|\lnot (\lnot E))}{Pr(\lnot (\lnot E)|\lnot H)+ Pr(\lnot E|\lnot (\lnot H))} -1 = c_{PF}(H,E) \end{aligned}$$
  2. (b)

    to prove that \(c_{PF}\) does not satisfy EHIS, EIS and ES, we consider the case in which

    $$\begin{aligned} Pr(H \cap E)=0.2; \ Pr(H \cap \lnot E)=0.1; \ Pr(\lnot H \cap E)=0.6; \ Pr(\lnot H \cap \lnot E)=0.1. \end{aligned}$$

    We find that \(c_{PF}(H,E)=-0.37\) while

    1. (b1)

      \(c_{PF}(\lnot E,\lnot H)\cong -0.3524\), so EHIS fails;

    2. (b2)

      \(-c_{PF}(\lnot E,H)\cong -0.5873\), so \(c_{PF}\) does not satisfy EIS;

    3. (b3)

      \(-c_{PF}(H,\lnot E)\cong -0.5441\), then ES is not satisfied.

Thus, \(c_{PF}\) is a BCM satisfying only EHS.

In a similar way, it is possible to prove that Finch’s R(H, E), Rips’s G(H, E), Mortimer’s M(H, E) and Carnap’s d(H, E) satisfy only IS, EHIS, ES and HS, respectively. Detailed proofs are omitted, see Celotto (2016), Eells and Fitelson (2002) and Greco et al. (2016).

(iii) Considering the subsets of 3 symmetries, now we prove that \(c_{LPF}\), the BCM defined by (6) and proposed in Sect. 2, satisfies only EHS, EIS and HIS. We follow again the steps suggested by the algorithm.

  1. (a)

    \(c_{LPF}\) satisfies EHS since

    $$\begin{aligned} c_{LPF}(\lnot H,\lnot E) = \log \frac{Pr(\lnot H|\lnot E)+Pr(\lnot (\lnot H)|\lnot (\lnot E))}{Pr(\lnot (\lnot E)|\lnot H)+ Pr(\lnot E|\lnot (\lnot H))} = c_{LPF}(H,E) \end{aligned}$$
  2. (b)

    \(c_{LPF}\) doesn not satisfy EHIS since considering the same numerical example used for \(c_{PF}\), one has \(c_{LPF}(H,E) \cong -0.2007\) while \(c_{LPF}(\lnot E,\lnot H)\cong -0.1887\).

  3. (c)

    \(c_{LPF}\) satisfies EIS since

    $$\begin{aligned} - c_{LPF}(\lnot E,H) = - \log \frac{Pr(\lnot E|H)+Pr(\lnot (\lnot E)|\lnot H)}{Pr(\lnot H|\lnot E)+ Pr(H|\lnot (\lnot E))} = c_{LPF}(H,E). \end{aligned}$$

Therefore, \(c_{LPF}\) meets exactly properties EHS, EIS and (necessarily) HIS.

In a similar way, following the same algorithm, it is possible to prove that Cohen’s K(H, E), Nozick’s N(H, E) are confirmation measures for which only \(\{IS, EHS, EHIS\}\), and \(\{ES, HS, EHS\}\), respectively, are met. Proofs are omitted (see again Celotto 2016; Eells and Fitelson 2002; Greco et al. 2016).

(iv) Finally, Carnap’s b(H, E) has all symmetry properties because it satisfies ES, HS, EHS and IS (see Eells and Fitelson 2002) and Proposition 2 ensures validity of all other symmetry properties. \(\square\)

Fig. 3
figure 3

BCMs that satisfy the feasible combinations of symmetries, embedded in the subgroup lattice structure of \((\Sigma ,\otimes )\)

Full size image

The 10 BCMs used to prove Proposition 4 can eventually be considered to complete Fig. 2 by associating to each possibile combination of symmetries a BCM that satisfies all of them, like in Fig. 3.

7 Group structure and the debate on desirability of symmetry properties

The fact that some combinations of symmetry properties are possible while others are not is connected with the debate on the desirability of the symmetry properties of a Bayesian confirmation measure (Crupi et al. 2007; Greco et al. 2016) and is related to the concept of inconsistency in preferential assessments of symmetries (see Susmaga and Szczȩch 2016).

Several authors have analyzed the desirability and, conversely, the undesirability of particular symmetry properties of confirmation measures.

Among them, we recall Eells and Fitelson (2002) and Glass (2013); they argued that an adequate BCM should not exhibit symmetries ES, EHS and IS, they proposed HS as the only compelling desideratum while they did not consider EHIS, EIS and HIS symmetries at all. Along with Proposition 3, undesirability of EHS necessarily implies that BCMs satisfying the concurrent symmetries in the sets \(\{IS, EHS, EHIS\}\), \(\{EHS, EIS, HIS\}\), and \(\{ES, EHS, HS\}\) should not be considered. At the same time, any BCM for which either just one symmetry among IS, EHIS, EHS and ES is met, or, even, if it satisfies all the seven symmetries, should be considered as not adequate. Remark that those authors, implicitly assert that only BCMs that belong to the equivalence class satisfying solely HS hold desirable symmetry properties. Carnap’s d is an example of adequate confirmation measure in Eells and Fitelson’s framework. Crupi et al. (2007) widened the set of the symmetries that an adequate BCM should (or should not) satisfy; in their discussion, they distinguished the cases of confirmation and disconfirmation. In case of confirmation, they argued that only HS, HIS and EHIS can be considered as desirable properties, while all other symmetries should be considered undesirable. In case of disconfirmation, they have proposed HS, EIS and IS as properties that an adequate measure of confirmation should possess, while the other symmetry properties were considered as not desirable. Again, only HS was considered a desirable symmetry in both situations, confirming among the BCMs considered in Sect. 2, only Carnap’s d as an adequate BCM.

In the field of rule interestingness, Greco et al. (2013) considered ES, EHS and HS as desirable properties, all the other symmetry properties were classified as undesirable: it is interesting to recognise in their suggestion one of the possible coexistent combinations of symmetries in the subgroup lattice structure of \((\Sigma ,\otimes )\), i.e. the Klein subgroup \(\{ \sigma _{0} ,e,h,eh\}\). Considering the whole list of desirable/undesirable symmetry properties, a good measure appears to be, for example, Nozick’s N (see Fig. 2), as Greco et al. (2013) actually suggest; but, taking their suggestion in a less restrictive way, one could consider adequate also a BCM that satisfies just one of the symmetries \(\{ES\}\) (like Mortimer’s M), \(\{EHS\}\) (as the above defined \(c_{PF}\)) or \(\{HS\}\) (e.g., Carnap’s d), hence assuming implicitly the existence of a quite larger group of adequate BCMs.

More in general, given different contexts and opinions on which symmetry properties should be satisfied by a confirmation measure, Proposition 3, i.e., the lattice of subgroups of \((\Sigma ,\otimes )\), allows to determine quite easily whether (and which) symmetry requirements may coexist or not.

8 Lack of symmetry and degree of asymmetry

We focus now on symmetry properties that a confirmation measure does not meet: how far is the measure from satisfying them?

For example, Kohen’s K and Finch’s R do not satisfy HS, a symmetry property acknolewdged as desirable by the literature: how far are the two measures from satisfying it? We are interested in measuring the lack of symmetry, or, in other terms, we want to define a degree of asymmetry of a confirmation measure, with respect to a given symmetry. For this purpose, we will consider a more spatial point of view on BCMs and on symmetry properties, similar approaches can be found, for example, in Celotto (2016), Celotto et al. (2016) and Susmaga and Szczȩch (2015).

In particular, in the so-called Confirmation Space defined as a two dimensional space with dimensions \(x=Pr(H|E)\) and \(y=Pr(H)\), Celotto in Celotto (2016) visualised in a vivid way the IFPD confirmation measuresFootnote 2 allowing a geometric interpretation of their properties.

Here we consider the Extended Confirmation Space (see Celotto et al. 2018), i.e. the three dimensional space where

$$\begin{aligned} x=Pr(H|E) \quad y=Pr(H) \quad z=Pr(E) \end{aligned}$$

thus allowing to consider also BCMs that are not necessarily IFPD.Footnote 3

Definition 4

The extended confirmation space for the evidence-hypothesis couple (H, E) is the three-dimensional space of points (x, y, z), where \(x = Pr(H|E)\), \(0< x < 1\), \(y = Pr(H)\), \(0< y < 1\), \(z = Pr(E)\), \(0< z < 1\).

A confirmation measure c(x, y, z) can be seen, therefore, as a real-valued function with domain \((0,1)\times (0,1)\times (0,1)\) and studied with usual calculus. Note that restricting x, y and z to the open interval (0, 1) is necessary to prevent Bayesian Confirmation Measures from being not defined. Since probabilities x, y and z must satisfy the Total Probability Theorem, i.e. \(Pr(E \cap H)+Pr(\lnot E \cap H)+Pr(E\cap \lnot H)+Pr(\lnot E \cap \lnot H)=1\) and \(x=Pr(H|E) = Pr(E \cap H)/[Pr(E \cap H)+Pr(E \cap \lnot H)]\), \(y=Pr(H) = Pr(E \cap H)+Pr(\lnot E \cap H)\), \(z=Pr(E) = Pr(E \cap H)+ Pr(E \cap \lnot H)\), all components in the ordered triples (x, y, z) must satisfy also the double constraint \(zx \le y \le zx + 1-z\).

Table 5 reports the definitions of our BCMs sample in terms of the variables x, y and z.

Table 5 Representative BCMs expressed in the Extended Confirmation Space
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The symmetry properties, written in Definition 3 in terms of Hypothesis H, Evidence E and their negations \(\lnot H\) and \(\lnot E\), can also be formulated in terms of the probabilities which are involved in the definition of the confirmation measure (see e.g. Celotto 2016; Celotto et al. 2016), i.e. in terms of x, y and z.Footnote 4

For example, the symmetry HS, i.e.,

$$\begin{aligned} c(H,E)=-c(\lnot H,E) \end{aligned}$$

can be written in terms of probabilities as

$$\begin{aligned} c(Pr(H|E),Pr(H),Pr(E))=-c(Pr(\lnot H|E),Pr(\lnot H),Pr(E)) \end{aligned}$$

or,

$$\begin{aligned} c(Pr(H|E),Pr(H),Pr(E))=-c(Pr(1-Pr(H|E),1-Pr(H),Pr(E)) \end{aligned}$$

that is, in the Extended Confirmation Space,

$$\begin{aligned} c(x,y,z)=-c(1-x,1-y,z). \end{aligned}$$

In a similar way, we can express each symmetry property in terms of x, y and z (see Table 6).

Accordingly, we can say that a confirmation measure c satisfies a symmetry property \(\sigma\), or that c is \(\sigma\)-symmetric, if

$$\begin{aligned} c(x,y,z)={c}^{\sigma }(x,y,z) \end{aligned}$$
(8)

where \({c}^{\sigma }\) is the righthandside function in the last column of Table 6.

Table 6 Symmetries in the Extended Confirmation Space
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When condition (8) is not satisfied, the measure c will be called \(\sigma\)-asymmetric (or, asymmetric with respect to \(\sigma\)). For example (see Sect. 6) Kohen’s K is ES, HS, EIS, \(HIS-\)asymmetric, while Finch’s R is asymmetric with respect to ES, HS, EHS, EIS, HIS and EHIS.

Since both Kohen’s K and Finch’s R are HS-asymmetric measures, is one of the two more asymmetric than the other one? Is it possible to measure the magnitude of their HS-asymmetry? To define the degree of asymmetry of a BCM, our approach recalls some studies on the degree of exchangeability of continuous identically distributed random variables (see Celotto et al. 2018; Klement and Mesiar 2006; Siburg and Stoimenov 2011; Siburg et al. 2016).

To evaluate the degree of asymmetry of a BCM, that is, how far it is from being symmetric, we first introduce an order of asymmetry: this definition is necessary to avoid the possibility of facing two BCMs which are inversely ordered by different asymmetry measures (Siburg et al. 2016).

Definition 5

In the Extended Confirmation Space, a BCM \(c_{1}\) is less \(\sigma\)-asymmetric than a BCM \(c_2\), written \(c_{1} {\prec }_{\sigma a} c_2\), when

$$\begin{aligned} \frac{|c_{1}(x,y,z)-{c}^{\sigma }_1(x,y,z)|}{|c_{1}(x,y,z)|} \le \frac{|c_2(x,y,z)-{c}^{\sigma }_2(x,y,z)|}{|c_2(x,y,z)|} \end{aligned}$$

for all \((x,y,z) \in (0,1)\times (0,1)\times (0,1)\) s.t. \(zx \le y \le zx + 1-z\), with \(x \not =y\).Footnote 5

Observe that, by Definition 1, \(c(x,y,z)=c^{\sigma }(x,y,z)=0\) if and only if \(x=y\), i.e., if and only if H and E are independent. The points (x, y, z) with \(x=y\) are excluded in Definition 5, since they cannot affect the lack of symmetry properties of the BCM, at the same time preventing denominators from vanishing.

For each symmetry \(\sigma\), the relation \({\prec }_{\sigma a}\) is a preorder (i.e. reflexive and transitive); moreover, it is not antisymmetric because for each symmetry \(\sigma\) it is possible to find a couple of confirmation measures \(c_1\) and \(c_2\) both \(\sigma\)-symmetric, for which clearly

$$\begin{aligned} c_{1}(x,y,z)-{c}^{\sigma }_1(x,y,z)= c_{2}(x,y,z)-{c}^{\sigma }_2(x,y,z)=0. \end{aligned}$$

For example, Nozick’s N and Carnap’s d are distinct confirmation measures both satisfying Hypothesis Symmetry.

Finally, \({\prec }_{\sigma a}\) is not a total order for each symmetry \(\sigma\), since not all BCMs can be ordered by \({\prec }_{\sigma a}\): if we consider for example Carnap’s d and Finch’s R that are both not \(EIS-\)symmetric, when \((x,y,z)=(1/4,3/10,4/5)\) it is

$$\begin{aligned} \frac{|d(x,y,z)-{{d}}^{EIS}(x,y,z)|}{|d(x,y,z)|} < \frac{|R(x,y,z)-{{R}^{EIS}}(x,y,z)|}{|R(x,y,z)|} \end{aligned}$$

while the inequality is reversed in \((x,y,z)=(2/5,1/3,1/2)\).

Finally, we propose a general definition of asymmetry measure which is compatible with the partial order \({\prec }_{\sigma a}\):

Definition 6

A measure of \(\sigma\)-asymmetry is a function \(\mu _{\sigma a}\) that satisfies the following conditions:

  1. 1.

    \(\mu _{\sigma a}(c)=0\) if and only if the BCM c is \(\sigma\)-symmetric;

  2. 2.

    given the BCMs \(c_{1}\) and \(c_{2}\), if \(c_{1} {\prec }_{\sigma a} c_2\) then \(\mu _{\sigma a}(c_{1}) \le \mu _{\sigma a}(c_2)\).

In the following, we consider the class of \(\sigma\)-asymmetry measures defined as:

$$\begin{aligned} \mu _{\sigma a}^{p}(c)=\frac{\left\| c-{c}^{\sigma }\right\| _p}{\left\| c\right\| _p} \end{aligned}$$
(9)

where \(\Vert \cdot \Vert _p\) denotes the \(L^p\)-norm, for each \(p \in [1,\infty ]\). Dividing \(\left\| c-{c}^{\sigma }\right\| _p\) by \(\left\| c\right\| _p\) allows to compare asymmetry measures \(\mu _{\sigma a}^{p}\) of different BCM functions. Observe that the denominator in Eq. (9) cannot vanish as a BCM c cannot be identically equal to zero. The proposed measures \(\mu _{\sigma a}^{p}\) result to be compatible with the partial order \({\prec }_{\sigma a}\) as required in Definition 6. Furthermore, by choosing different values of p, it is possible to emphasize different types of \(\sigma\)-asymmetries: for example, when \(p=1\) attention is put on the average degree of asymmetry, while \(p=\infty\) drives the focus on the maximum degree of asymmetry that can be attained at single points. Moreover, observe that for a given BCM, the asymmetry order and the asymmetry measures defined above may depend on the chosen variables x, y and z, even if the symmetry properties do not.

9 Asymmetry degree evaluations

In this section we present some asymmetry degree computations performed with Wolfram’s software Mathematica (Wolfram 2021). More precisely, we used the numerical integration procedure NIntegrateFootnote 6 computing the \(L^p\) norms of functions \(c-{c}^{\sigma }\) and c on the feasible set

$$\begin{aligned} \{(x,y,z) \in (0,1)^3: zx \le y \le zx + 1-z\}. \end{aligned}$$

The asymmetry measures with two \(L^p\) norms, \(L^1\) and \(L^2\), of all the confirmation measures recalled in Table 5 are presented in Tables 7 and 8, respectively.

Table 7 Degree of asymmetry of some BCMs (\(L^1\) norm and \(\mu _{\sigma a}^{1}\))
Full size table
Table 8 Degree of asymmetry of some BCMs (\(L^2\) norm and \(\mu _{\sigma a}^{2}\))
Full size table

Let us now highlight some features of the BCMs that can be deduced from this kind of calculations. For example, by observing the results concerning symmetry ES, the confirmation measures M and N appear to be ES symmetric, as suggested by the vanishing of the corresponding asymmetry measures: this fact can be proved analytically with some tricky algebraic manipulations, but it is easily suggested by the numerical results.

The asymmetry degree \(\mu _{\sigma a}\) provides even more interesting information when a confirmation measure c does not satisfy a symmetry property, i.e., when \(\mu _{\sigma a}(c)>0\). If, for example, we consider the \(L^1\)-norm, and Cohen’s K and Finch’s R, the proposed asymmetry measures highlight the lack of HS (see Table 7), but what is really interesting is being able to compare their degrees of asymmetry. Indeed, since

$$\begin{aligned} \mu _{HSa}^1(K)=0.33206841 \quad \hbox {and} \quad \mu _{HSa}^1(R)=1.07571540 \end{aligned}$$

we can assert that K is less HS-asymmetric than R: given that HS is considered a desirable property (e.g., by Eells and Fitelson 2002; Glass 2013), K could be considered preferable to R. Instead with respect to ES which is often considered an undesirable symmetry, R performs better than K as (see Table 7)

$$\begin{aligned} \mu _{ESa}^1(K)=0.33206841 \qquad \hbox {and} \qquad \mu _{ESa}^1(R)=0.80840290 \end{aligned}$$

from which the different suggestion to choose R, because it is more ES-asymmetric than K.

Note that with respect to HS and ES symmetries, the ranking induced by the \(L^1\)-norm on the considered BCMs does not change if we consider the \(L^2\)-norm (see Table 8), but the rankings induced by different norms are in general not necessarily equivalent. As an example, consider the IS property and the confirmation measures \(c_{PF}\) and \(c_{GSS}\): in the case of \(L^1\)-norm,

$$\begin{aligned} \mu _{ISa}^1(c_{PF})=0.30837210 \quad \hbox {and} \quad \mu _{ISa}^1(c_{GSS})=0.51024700 \end{aligned}$$

or, in other words, \(c_{PF}\) is less IS-asymmetric than \(c_{GSS}\), while, using \(L^2\)-norm, it is

$$\begin{aligned} \mu _{ISa}^2(c_{PF})=0.71194341 \quad \hbox {and} \quad \mu _{ISa}^2(c_{GSS})=0.62243166 \end{aligned}$$

so that \(c_{PF}\) turns out to be more IS-asymmetric than \(c_{GSS}\).

Indeed, different choices of \(L^p\)-norms allow to consider different characteristics of the BCMs with respect to asymmetry properties, for example, \(L^1\)-norm reacts more softly than the \(L^2\)-norm to extreme values in the difference \(c-c^{\sigma }\).

Moreover, the intensity with which a confirmation measure does not satisfy a symmetry property can highlight some unexpected features. For example, let’s focus on what happens with Cohen’s K: in this particular case, all non-zero asymmetry measures are exactly the same (0.33206841 with \(L^1\) and 0.4243428 with \(L^2\)). What does this result suggest? If we consider the \({K}^{\sigma }(x,y,z)\) functions corresponding to the symmetries ES, HS, EI and HIS, we can easily verify that the function \({K}^{\sigma }\) is the same, that is

$$\begin{aligned} {K}^{\sigma }(x,y,z)=\frac{2(y-x)z}{1-y-z+2yz} \end{aligned}$$

a result not immediate to predict at a first glance.

10 Conclusions

The large number of available Bayesian Confirmation Measures makes the choice of the measure to be used a rather tricky task. Geometric, visual and analytical ways to compare the measures have already been suggested (see Brzezinski et al. 2018; Celotto 2016; Susmaga and Szczȩch 2015, 2016), among them, the use of symmetry properties appears to be particularly effective and has been largely discussed (see, e.g., Celotto et al. 2016; Crupi et al. 2007; Eells and Fitelson 2002; Greco et al. 2016; Koscholke 2018).

In this paper, the adoption of a group theoretical point of view allows a clear insight on symmetry properties by considering the isomorphism between the group of symmetry functions on sentences pairs and the dihedral group \(D_8\). Another similar group theoretic approach was already proposed by Susmaga and Szczȩch (2016) but referring to contingency tables and relating symmetries to the permutation group \(S_4\) of order 24. Our approach contributes a clear overview on BCMs symmetries and thus, a simple way to infer their properties. Making use of the subgroup lattice of the group of symmetry functions, it is possible to partition the BCMs into 10 equivalence classes depending on the set of symmetry properties they satisfy.

Seeking for a set of representative BCMs, one for each of the 10 possible sets of symmetries that can coexist, we did’nt find known BCMs satisfying exactly the three symmetries EHS, EIS and HIS, or only the symmetry EHS. Therefore, we defined two new BCMs. The first, Predictive minus false over False confirmation measure (\(c_{PF}\)) satisfies exactly the three symmetries EHS, EIS and HIS, and the second, Logarithm of Predictive over False confirmation measure (\(c_{LPF}\)) satisfies only the symmetry EHS. These BCMs, which are strictly related to the concepts of Positive and Negative Predictive Values and of False Positive and Negative Rates, may deserve further investigation, especially for what concernes other properties, like different kinds of monotonicity, and their practical use.

The subgroup lattice structure allows also to suggest an algorithm to neatly detect which symmetry properties a BCM satisfies. The algorithm was used to simplify the proof of the existence of BCMs satisfying the admissible combinations of symmetries (Proposition 4).

In addition, the group structure can be exploited to clearly assess and understand the simultaneous satisfaction of other properties of BCMs. For example, consider the monotonicity properties of BCMs and their compatibility with symmetry properties. One of the results proposed by Greco et al. (2016) states that there is no confirmation measure strictly monotonic in the Bayesian perspective satisfying EHS: Proposition 4 in Sect. 6 automatically rules out the possibility that in the Bayesian perspective a strictly monotonic confirmation measure can meet more than one symmetry at a time (see Fig. 5).

Moreover, we propose the definition of an asymmetry measure that easily suggests whether a BCM satisfies or not a symmetry property and, if not, how far it is from satisfying it.

Numerical results show how by using an asymmetry measure it is possible to easily highlight features of a BCM which are not obvious ex ante and, sometimes, to discover even unexpected properties, thus paving the way to their interpretation and analytical proof (see Sect. 9).

The choice of an asymmetry measure is not unique. In our proposal, different \(L^p\) norms provide an evaluation of the degree of asymmetry with different sensitivities on extreme values. To illustrate this fact, we computed the degree of asymmetry using norm \(L^1\) (Table  7) and norm \(L^2\) (Table 8), obtaining different rankings of the 10 representative BCMs.

Our approach to the study of symmetry properties could be extended to other symmetry approaches, as in Koscholke (2018), and it is also not limited to BCMs but can be extended to other interestingness measures, like already partially done for Fuzzy Confirmation Measures (FCMs) (see e.g. Celotto et al. 2019; Glass 2008; Wang et al. 2012) and for Distorted Bayesian Confirmation Measures (Ellero and Ferretti 2020).