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Constructing Bayesian Network Graphs from Labeled Arguments
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Constructing Bayesian Network Graphsfrom Labeled Arguments
Remi Wieten1(B) , Floris Bex1,2 , Henry Prakken1,3 , and Silja Renooij1
1

Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands{g.m.wieten,f.j.bex,h.prakken,s.renooij}@uu.nl
2
Institute for Law, Technology and Society, Tilburg University, Tilburg, The Netherlands
3
Faculty of Law, University of Groningen, Groningen, The Netherlands
Abstract. Bayesian networks (BNs) are powerful tools that are wellsuited for reasoning about the uncertain consequences that can be inferred from evidence. Domain experts, however, typically do not have theexpertise to construct BNs and instead resort to using other tools suchas argument diagrams and mind maps. Recently, we proposed a structured approach to construct a BN graph from arguments annotated withcausality information. As argumentative inferences may not be causal,
we generalize this approach to include other types of inferences in thispaper. Moreover, we prove a number of formal properties of the generalized approach and identify assumptions under which the constructionof an initial BN graph can be fully automated.
Keywords: Bayesian networks · Argumentation · Inference · Reasoning.

1

Introduction

Bayesian networks (BNs) [11] are compact graphical models of joint probability
distributions that have found applications in many different fields where uncertainty plays a role, including medicine, forensics and law [6]. BNs are well-suitedfor reasoning about the uncertain consequences that can be inferred from evidence. However, especially in data-poor domains, their construction needs tobe done mostly manually, which is a difficult, time-consuming and error-prone
process [7], and domain experts typically resort to using other tools such asargument diagrams, mind maps and ontologies [4, 8]. Hence, we believe BN construction can be facilitated by automatically extracting information relevant fora BN from such tools. More specifically, in this paper we study how information
expressed as structured arguments [2] about the domain can inform the designof a BN graph, a directed acyclic graph (DAG) which captures the independence
relation among variables.
In previous research, Bex and Renooij [3] identified constraints on BNs given
structured arguments, but these only suffice for constructing an undirected skeleton of a BN graph. Recently, we were able to derive a directed graph [18], but
only by assuming that all inferences in the initial structured arguments are explicitly labeled with causality information [1,14]. Arcs in the BN graph are thenset in the causal direction, following the heuristic typically used in the manual
construction of BN graphs [11]. However, in [18] it is assumed that all inferencesare labeled with causality information, which precludes the use of other typesof inferences, such as mere statistical correlations and definitions. Furthermore,formal properties of the proposed proposals were not studied in both [3] and [18].
Accordingly, in this paper we present an approach that generalizes our previously proposed construction approach [18] to other types of inference. In addition, we formally prove that BN graphs constructed by our approach allow
reasoning patterns similar to the inferences represented in the original structured arguments. Moreover, we identify assumptions under which the fully automatically constructed initial graph is guaranteed to be a DAG, and we identify
bounds on the complexity of inference in BNs constructed by our approach.
The paper is structured as follows. Section 2 provides preliminaries on argumentation and BNs. In Sect. 3, we present our generalized approach for constructing BN graphs from inferences. In Sect. 4, we prove a number of formal
properties of the approach. In Sect. 5, we discuss related research and conclude.

2

Preliminaries

2.1

Argumentation

Throughout this paper, we assume that the domain experts’ analysis is captured
in an argument graph (AG), in which claims are substantiated by chaining inferences from the observed evidence; an example is depicted in Fig. 1a. AGs
are closely related to argument diagrams and mind maps [4], familiar to many
domain experts. Formally, an AG is a directed graph GA = (P, AA ), where P
is a set of nodes representing propositions from a literal language with ordinary
negation symbol ¬, and AA is a set of directed (hyper)arcs. We write p = −q
in case p = ¬q or q = ¬p. Nodes Ep ⊆ P corresponding to the (observed) evidence are root nodes in GA . We assume that for every p ∈ Ep it holds that¬p ∈/ P. AA is comprised of three pairwise disjoint sets S, R and U, which aresets of support arcs, rebuttal arcs and undercutter arcs, respectively. A support
arc is a (hyper)arc s : {p1 , . . . , pn } → p ∈ S, indicating an inference step from
{p1 , . . . , pn } ⊆ P (called the tails of s, denoted by Tails(s)) to a single proposition p ∈ P (called the head of s, denoted by head(s)). Here, curly brackets
are omitted in case |Tails(s)| = 1. Support arcs s1 , . . . , sm form a support chain(s1 , . . . , sm ) iff head(si ) ∈ Tails(si+1 ) for 1 ≤ i < m.
There are two types of attack arcs. A rebuttal arc r ∈ R is a bidirectionalarc r : p ←→ ¬p in GA that exists for every pair p, ¬p ∈ P. An undercutter arc
u ∈ U is a hyperarc u : p → (s), where p ∈ P undercuts s ∈ S. Informally, a
rebuttal is an attack on a proposition, while an undercutter attacks an inference
by providing exceptional circumstances under which the inference may not beapplicable. In figures in this paper, nodes in GA corresponding to elements of
Ep are shaded. Support arcs are denoted by solid (hyper)arcs and rebuttal arcs
and undercutter arcs are denoted by dashed (hyper)arcs.
In reasoning about evidence, a distinction can be made between causal andevidential inferences [1,14]. Causal inferences are of the form “c is a cause for e”(e.g. fire causes smoke), whereas evidential inferences are of the form “e is causedby c” (e.g. smoke is caused by fire). Inferences may also be neither causal nor

evidential. For instance, definitions, or abstractions [5], allow for reasoning at different levels of abstraction, such as stating that guns can generally be considered
deadly weapons. Another example of a different type of inference is an inference
representing a mere statistical correlation, such as a correlation between homelessness and criminality. While there may be one or more confounding factorsthat cause both homelessness and criminality (e.g. unemployment), a domain
expert may be unaware of these factors or may wish to refrain from capturing
them in the AG. For our current purposes, we assume that support arcs in S are
either annotated with a causal “c” label, an evidential “e” label, or are labeled“o” for all other types of inferences. S then divides into three disjoint sets Sc ,Se and So of causal, evidential and other types of support arcs, respectively. In
figures in this paper, “o” labels are omitted.
In this paper, some further assumptions are made. We assume that support
chains are non-repetitive in that there does not exist a support chain (s, . . . , s) in
AG. We assume that for every support chain (s1 , . . . , sn ) the heads of s1 , . . . , sn
are consistent in that @i, j ∈ {1, . . . , n}, i 6= j such that head(si ) = −head(sj ).
Furthermore, we assume that AGs do not include causal cycles in that there do
not exist two support chains (s1 , . . . , sn ) and (s01 , . . . , s0m ) in AG with s1 , . . . , sn ∈Sc , s01 , . . . , s0m ∈ Se , Tails(s1 ) ∩ Tails(s01 ) 6= ∅ and head(sn ) = head(s0m ) orhead(sn ) = −head(s0m ). Informally, this assumption says that for every p, q ∈ P,
if p is a cause of q, then q (or −q) cannot be a cause of p (see also [1]).
As noted by Pearl [14], the chaining of a causal inference and an evidential
inference can lead to undesirable results. Consider the example in which a causal
inference states that a smoke machine causes smoke and an evidential inference
states that smoke is evidence for fire. Chaining these inferences would make us
conclude there is a fire when seeing a smoke machine, which is clearly undesirable.
We therefore assume that an AG does not include a support chain (s1 , s2 ) wheres1 ∈ Sc , s2 ∈ Se , and refer to this assumption as Pearl’s C-E constraint.
For those familiar with argumentation, we note that, although we use theterm “argument graph”, the graph only represents inferences and attacks between propositions by means of arcs; actual arguments are not represented in
the graph. Preferences over arguments, as well as their status, are thus not taken
into account in our formalism, since they are not needed for our current purposes. Our formalism can be straightforwardly mapped to ASPIC+ (cf. [2]) if all
inferences are considered to be defeasible.
2.2

Bayesian Networks

A BN [11] compactly represents a joint probability distribution Pr(V) over afinite set of discrete random variables V; in this paper we assume all variables
to be Boolean. The variables are represented as nodes in a DAG GB = (V, AB ),where AB ⊆ V × V is a set of directed arcs Vi → Vj from parent Vi to child Vj .
The BN further includes, for each node, a conditional probability table (CPT)
specifying the probabilities of the values of the node conditioned on the possible
joint value combinations of its parents. A node is called instantiated iff it is set
to a specific value. Given a set of instantiations, or evidence, for nodes EV ⊆ V,
the probability distributions over the other nodes in the network can be updatedthrough probabilistic inference [11]. An example of a BN graph is depicted inFig. 1b, where ovals represent nodes and instantiated nodes are shaded.
The BN graph GB captures the independence relation among its variables.
Let a chain be defined as a sequence of distinct nodes and arcs in the BN graph.
A node V is called a head-to-head node on a chain c if it has two incoming
arcs on c. A chain c between nodes V1 and V2 is blocked iff it includes a nodeV ∈/ {V1 , V2 } such that (1) V is an uninstantiated head-to-head node on cwithout instantiated descendants; or (2) V is instantiated and has at most one
incoming arc on c. A chain that is not blocked is called active. If no active
chains exist between V1 and V2 given instantiations of Z ⊆ V, then they are
considered conditionally independent given Z. In case a head-to-head node or
one of its descendants is instantiated, an active chain is induced between its
parents, allowing for interparental interactions. If one of the parents is now true,
then the probability of another parent being true as well may change, dependingon the specific synergistic effect modeled in the CPT for the head-to-head node.
BN construction is typically an iterative process. After constructing an initial
BN graph, we should verify that this graph is acyclic and that it correctly captures the (conditional) independencies. If the graph does not yet exhibit theseproperties, arcs should be reversed, added or removed by the BN modeler in
consultation with the domain expert. We call this the “graph validation step”.

3

Constructing BN Graphs from Argument Graphs

To facilitate the BN construction process, we previously proposed a stepwise approach for constructing an initial BN graph from domain knowledge represented
in AGs with support arcs in Sc ∪ Se only [18]. In this section, we generalize thisapproach to include inferences in So .
Upon using an AG to inform BN construction, we have to consider theirdifference in semantics. An AG, by means of its support chains, describes theiterative inference steps that can be made from the observed evidence towards
the conclusions. In comparison, a BN describes a joint probability distribution
which does not model such directionality. Only when probabilistic inference isperformed is available evidence propagated through the network using the existing active chains. To mimic the inferences described by an AG in a BN, we
will focus on ensuring that the (chains of) support arcs in the AG, originatingfrom evidence Ep ⊆ P, are captured in the BN graph by means of active chainsfor propagating instantiations of EV ⊆ V (see also [18]). Note that since thenotion of an active chain is a symmetrical concept, a BN graph will also capture
reasoning patterns in the direction opposite of the support chains present in the
AG. In Sect. 4, we formally prove that all support chains in an AG indeed havecorresponding active chains in the BN when following our generalized approach.
In the manual construction of BN graphs, arcs are typically directed using
the notion of causality as a guiding principle [11]. By following this heuristic, twocompeting causes form a head-to-head connection in the node corresponding to
the common effect, allowing synergistic effects between the causes to be directly
captured in the CPT for this node. Hence, we propose to use the same heuristic

in automatically directing arcs, where we exploit causality information explicitly
expressed in an AG by means of “c” and “e” labels.
Undercutters attack inferences in support chains by providing exceptions to
the inference. For instance, if an inference is in the evidential direction, thenan undercutter suggests an alternative cause for the same effect. Accordingly,
we propose to enable capturing such interactions between an undercutter and a
support arc in the CPT of a head-to-head node formed in the BN graph.
3.1

The Generalized Approach

In this subsection, we present and explain the steps of the generalized approach.
Let var : P → V be an operator mapping every proposition p or ¬p ∈ P in an
AG to a BN variable var(p) = var(¬p) ∈ V describing values p and ¬p. For an
AG GA = (P, AA ), a BN graph GB = (V, AB ) is constructed as follows:1) ∀p, ¬p ∈ P, include var(p) in V; if p or ¬p ∈ Ep , also include var(p) in EV .
2) For every support arc s : {p1 , . . . , pn } → p:2a) If s ∈ Se , include var(p) → var(pi ), i = 1, . . . , n in AB .
2b) If s ∈ Sc , include var(pi ) → var(p), i = 1, . . . , n in AB .
2c) If s ∈ So and @s1 ∈ Se such that (s, s1 ) form a support chain, include
var(pi ) → var(p), i = 1, . . . , n in AB .
2d) If s ∈ So and ∃s1 , . . . , sm ∈ Se such that (s1 , . . . , sm ) is a maximal
chain of evidential support arcs in AG following s, include var(pi ) →var(head(sm )), i = 1, . . . , n in AB .
3) For every undercutter arc u : p → (s) ∈ U with s : {q1 , . . . , qn } → q:3a) If s ∈ Se , include var(p) → var(qi ), i = 1, . . . , n in AB .
3b) If s ∈/ Se , include var(p) → var(q) in AB .
4) Verify the properties of the constructed graph GB :4a) Break cycles in GB introduced by so-called evidential shortcuts resulting
from the combination of steps 2a and 2d (see Sect. 3.3 for further details).
4b) Apply the standard graph validation step (see Sect. 2.2).
While our approach exploits the domain knowledge captured in the AG in constructing a BN graph, the AG may lack information needed to prevent cycles
and unwarranted (in)dependencies in the obtained BN graph; hence the manualvalidation step (step 4b above), which is standard in BN construction.
The first step is to capture every proposition in GA and its negation as twovalues of a random variable in GB . By the same step, two propositions involvedin a rebuttal are captured as two mutually exclusive values of the same node.
The steps pertaining to s ∈ Sc ∪ Se are analogous to those proposed previouslyin [18]. These steps formalize the approach of setting arcs using the notion of
causality as a guiding principle [11]. For further details, the reader is referredto [18]. In Sects. 3.2 and 3.3, we motivate and explain the steps pertaining tos ∈ So with several examples.
3.2

Explanation and Motivation of Steps 2c and 3b

Consider Fig. 1, illustrating steps 1 − 2c and 3b of the generalized approach
for a forensic example. A dead body was found and we are interested in theangularmallethammer

stone

e
tes1

e
tes2

Mallete
autopsy

AngularHammer

Stone

Tes1

Tes2

Autopsy(a)(b)
Fig. 1. An AG including support arcs in So (a); the corresponding BN graph constructed by steps 1 − 2c and 3b of the generalized approach (b).

cause of death of this person. According to witness testimony (tes1 ), the person
was hit with a hammer (hammer ); however, according to another testimony(tes2 ), the person was hit with a stone (stone). We conclude that the person
was hit with an angular object (angular ), as hammers and stones can generally
be considered to be angular. Note that the relation between hammer (stone)
and angular is neither causal nor evidential; instead, the support arcs between
these propositions express that, at a higher level of abstraction, both hammers
and stones can generally be considered angular objects. A mallet was found at
the crime scene (mallet), which undercuts inference hammer → angular sincea mallet is an exceptional type of hammer that is not angular but instead has alarge cylindrical head. Finally, an autopsy report (autopsy) further supports theclaim that the person was hit with an angular object. By following steps 1 − 3of the generalized approach, the BN graph of Fig. 1b is constructed from the
AG in Fig. 1a. By steps 2c and 3b, variables Hammer and Stone and variables
Mallet and Hammer respectively form head-to-head connections in Angular.
In general, by step 2c head-to-head nodes are formed in the nodes corresponding to the heads of support arcs in So . Specifically, let p1 , . . . , pn be tails of one
or more si ∈ So with head(si ) = p. Then AB includes arcs var(pj ) → var(p),j = 1, . . . , n by step 2c; head-to-head nodes are, therefore, formed in var(p). By
setting arcs as per step 2c, we thus allow for including synergistic effects, if any,of the tails on the probability of p in the CPT for the head-to-head node.
Similarly, by step 3b head-to-head nodes are formed in the nodes corresponding to the heads of undercut support arcs in Sc ∪ So . Specifically, letu : p → (s) ∈ U be an undercutter of s : {q1 , . . . , qn } → q ∈ Sc ∪So . Then by step
3b, head-to-head nodes are formed in var(q) as AB includes arc var(p) → var(q).
Again, this allows for modeling possible interactions between p and qi , and hence
between var(p) and var(qi ), directly in the CPT for var(q). Bex and Renooij [3]previously noted that the presence of an undercutter should decrease the probability that the conclusion of the undercut inference is true. By setting arcs as per
step 3b, this interaction can be directly captured by the following constraints on
the CPT for var(q): Pr(q | p, qi ) < Pr(q | ¬p, qi ) for i = 1, . . . , n.
3.3

Explanation and Motivation of Steps 2d and 4a

Next, consider Figs. 2a and 2b, illustrating step 2d of the generalized approach
for a medical example (taken from [7]). After performing a CT scan (scan) on
a patient who has severe difficulty swallowing, it is established that a tumor isreflux

distalSs

Refluxecylindricale
Cylindrical
Distal

r

R

¬r

e

e

q1

q2

Q2

Q1

e

e
scan

Scan

p

P

(a)

(b)

(c)

(d)

Fig. 2. An AG (a) and the corresponding BN graph (b) illustrating step 2d of thegeneralized approach; an AG (c) and the corresponding BN graph (d), illustrating theconditions under which a cycle is introduced in step 2d.

present in the lower (distal ) part of his esophagus. Clinical studies indicate astrong correlation between the location of an esophageal tumor and its cell type;however, neither can be considered a cause of the other. Distal tumors generally
consist of cylindrical cells (cylindrical ), often formed as a result of frequent
gastric reflux (reflux ). The BN graph constructed by steps 1 − 2a and 2d ofthe generalized approach from the AG in Fig. 2a is depicted in Fig. 2b. As arcs
Distal → Reflux and Reflux → Cylindrical are included in AB and the involved
nodes are not instantiated, active chains exist between Distal and Reflux andDistal and Cylindrical. Note that we do not wish to set arcs as per step 2c, as
in this case a head-to-head node would instead be formed in Cylindrical which
would block the chain between Distal and Reflux.
Under specific conditions, cycles are introduced in step 2d of the generalized
approach, namely when a so-called evidential shortcut exists in the AG, i.e. if in
addition to the conditions of step 2d, also ∃s01 , . . . , s0k ∈ Se such that (s01 , . . . , s0k )form a support chain, Tails(s) ∩ Tails(s01 ) 6= ∅ and head(s0k ) = head(sj ) orhead(s0k ) = −head(sj ) for a j ∈ {1, . . . , m}. An example is depicted in Fig.
2c. In this example, s : p → q1 ∈ So is followed by a chain of support arcs
s1 : q1 → r, s2 : r → s ∈ Se , where there also exists a chain of support arcs
s01 : p → q2 , s02 : q2 → ¬r ∈ Se . By step 2a, arcs S → R, R → Q2 and Q2 → P
are included in AB . By step 2d, arc P → S is also included, introducing a cycle
in GB . We note that this arc can safely be removed, as an active chain alreadyexists between P and S via Q2 and R. In general, cycles are broken in step 4aby removing arcs var(pl ) → var(head(sm )) from AB ∀pl ∈ Tails(s) ∩ Tails(s01 ).

4

Properties of the Generalized Approach

In this section, we prove a number of formal properties of the generalized approach. The first property states that for every support chain in a given AG
there indeed exists a corresponding active chain in the BN graph.
Proposition 1. Let GA = (P, AA ) be an AG with root nodes Ep , and let GB =(V, AB ) be the corresponding BN graph constructed according to steps 1 − 4aof the generalized approach. Let (s1 , . . . , sn ) be any support chain in GA , where
Tails(s1 ) = {p1 , . . . , pm } and head(sn ) = q. Then there exist active chains
between var(pi ) and var(q) in GB given EV for every i ∈ {1, . . . , m}.

Proof (sketch). The following cases are distinguished:− If sk ∈ Sc ∪ Se ∀k ∈ {1, . . . , n}, then when following steps 2a and 2b a headto-head node can only be formed in var(head(sj )) for an arbitrary sj , j ∈{1, . . . , n − 1} if sj ∈ Sc , sj+1 ∈ Se ; however, this construction is prohibitedas it violates Pearl’s C-E constraint (see Sect. 2.1). Furthermore, since headsof support arcs are not propositions in Ep , corresponding nodes in GB are notinstantiated. Chains between var(pi ) and var(q) are thus never blocked.
− If (s1 , . . . , sn ) includes support arcs in So and none of these arcs is followedby an s ∈ Se , then arcs in AB are set similarly as for s ∈ Sc by step 2c. As
per the above proof, chains are not blocked.
− Let an sj ∈ So , 1 ≤ j < n be followed by a chain of support arcs in Se ,and let (sj+1 , . . . , sj+l ) be a maximal such chain. If j + l ≤ n, then step 2d
introduces direct arcs, and therefore active chains, between nodes in {var(p) |p ∈ Tails(sj )} and var(head(sj+l )). If j + l > n, then AB in addition includes
a directed path from var(head(sj+l )) to var(head(sn )) by step 2a; therefore,
chains between nodes in {var(p) | p ∈ Tails(sj )} and var(head(sn )) via
var(head(sj+l )) are active, as var(head(sj+l )) is not a head-to-head node. In
step 4a, a subset of the arcs introduced in step 2d is removed (see Sect. 3.3)
iff an evidential shortcut and a corresponding active chain already exist.
Finally, AB is only extended for undercutter arcs in step 3; active chains formed
between var(pi ) and var(q) in step 2 are, therefore, not affected by this step. 
In Proposition 2, we prove that under specific conditions on AGs an acyclic
graph is automatically obtained when following steps 1 − 4a of the approach,which simplifies the manual verification involved in step 4b. Conditions a) andb) concern the existence of undercutter arcs within and between connected subgraphs of AGs. Condition c) is a generalization of our assumption that no causal
cycles exist in AGs (see Sect. 2.1) to support arcs in So .
Proposition 2. Let GA = (P, AA ), and let G∗A = (P, A∗A ) be the subgraph of
GA with A∗A = AA \ U. Let an AG component of GA be defined as a connectedcomponent of G∗A . Assume the following conditions are satisfied:a) For any AG component C = (P0 , A0A ) of GA with P0 ⊆ P, A0A ⊆ A∗A , there
does not exist a u : p → (s) ∈ U with p ∈ P0 , s ∈ A0A .
b) For every pair of AG components C1 = (P0 , A0A ) and C2 = (P00 , A00A ) of GA
with P0 , P00 ⊆ P, A0A , A00A ⊆ A∗A , there does not exist both a u1 : p1 → (s1 ) ∈U with p1 ∈ P0 , s1 ∈ A00A and a u2 : p2 → (s2 ) ∈ U with p2 ∈ P00 , s2 ∈ A0A .
c) There do not exist two support chains (s1 , . . . , sn ) and (s01 , . . . , s0m ) with
s1 , . . . , sn ∈ Sc ∪ So , s01 , . . . , s0m ∈ Se , Tails(s1 ) ∩ Tails(s01 ) 6= ∅, andhead(sn ) = head(s0m ) or head(sn ) = −head(s0m ).
Let GB = (V, AB ) be the graph constructed from GA according to steps 1 − 4aof the generalized approach. Then GB is a DAG.
Proof (sketch). The following cases are distinguished:− In steps 2a and 2b, no cycles are introduced. Specifically, our non-repetitivenessassumption and our consistency assumption (see Sect. 2.1) jointly assume that

¬r

r

q1

q2
p

R
Q1

r1

r2

R1

R2

Q2

q1

q2

Q1

Q2

P

p1

p2

P1

P2

(a)
(b)
(c)
(d)
Fig. 3. Examples of AGs (a, c) for which a cyclic graph is constructed by steps 1 − 4aof the generalized approach (b, d).

for every p ∈ P, p or −p cannot be inferred via a chain of support arcs. Therefore, no chain of arcs exists in AB from a node P to itself. The only other casein which cycles can be introduced is when a causal cycle exists in GA , whichis also prohibited by assumption (see Sect. 2.1).
− No cycles are introduced in step 2c if condition c) is satisfied. Cycles are only
introduced in step 2d if an evidential shortcut exists; however, these cycles
are broken again in step 4a as described in Sect. 3.3.
− After step 2, there is a correspondence between AG components and theconnected components of the underlying undirected graph S of the thus far
constructed BN graph. Under condition a), no cycles are introduced withina connected component of S when including additional arcs in AB for everyu ∈ U in step 3. Furthermore, for every pair of AG components C1 and C2of GA with corresponding connected components C10 and C20 of S, no cycles
are introduced between components C10 and C20 in step 3 under condition b).Figures 3a and 3c depict examples of AGs that do not satisfy conditions a) andb) of Proposition 2, respectively. In the validation step that follows the initialconstruction of these BN graphs, arcs can be reversed or removed to make thesegraphs acyclic. The choice of arc to reverse or remove will depend on its effecton active chains, including those between nodes not directly incident on the arc.
We note that this type of manual verification is standard in BN construction,
especially in data-poor domains. While the domain knowledge expressed in theoriginal AG has been exploited to construct an initial BN graph, additional
domain knowledge may need to be elicited to obtain a valid graph.
Proposition 3 gives an upper-bound on the number of parents introduced by
the approach for each node var(p) in a BN graph, which bounds both the size of
the CPTs and the complexity of inference in the BN. This bound captures thenumber of support arcs and undercutters that involve either proposition p or ¬p.
The proof of this result is straightforward and omitted due to space limitations.
Proposition 3. Let GA = (P, AA ) be an AG, and let GB = (V, AB ) be theBN graph constructed according to steps 1 − 4a of the generalized approach. For
every p ∈ P, let Parp = {pi | pi ∈ Tails(s), s ∈ Sc ∪ So , head(s) = p}
and let Par0p = {pi | pi ∈ Tails(s), s ∈ So , s is followed by maximal chain
s1 , . . . , sm ∈ Se with head(sm ) = p or head(sm ) = ¬p}. Let Sep be a subset of
Se , where s ∈ Sep iff p ∈ Tails(s). Let Uep ⊆ U be the subset of undercutter arcs
directed to an s ∈ Sep or s ∈ Se¬p . Similarly, let Ucp , Uop ⊆ U be the subsets ofundercutter arcs directed to an s ∈ Sc respectively So for which head(s) = p orhead(s) = ¬p. Then an upper-bound for the number of parents of var(p) is:
1) |Parp | + |Par¬p | + |Par0p | + |Ucp | + |Uop | if Sep = Se¬p = ∅;2) |Parp | + |Se¬p | + |Par0p | + |Uep | + |Ucp | + |Uop | if Sep = ∅ and Se¬p 6= ∅;3) |Par¬p | + |Sep | + |Par0p | + |Uep | + |Ucp | + |Uop | if Sep 6= ∅ and Se¬p = ∅;4) |Sep | + |Se¬p | + |Uep | + |Uop | if Sep 6= ∅ and Se¬p 6= ∅.

5Conclusion

In this paper, we have studied how domain knowledge expressed as labeled arguments can be exploited to construct a BN graph. Firstly, we have generalized ourpreviously proposed approach [18] by allowing inference types that are neither
causal nor evidential. Moreover, we have formally proven that, as intended, ourapproach captures all support chains in an AG in the form of active chains in
the BN graph. We have also identified conditions on AGs under which a DAG is
automatically constructed by the approach, simplifying the manual verification
step. Lastly, we have identified bounds on the size of the CPTs and the complexity of inference in BNs constructed by our approach. All properties also holdfor the limited case considered in [18] but were not proven in that paper.
The generalized approach allows us to construct an initial BN graph froma domain expert’s initial argument-based analysis, capturing similar reasoning
patterns as their original AG; it thereby simplifies the BN elicitation process.
We note that BN construction is an iterative process in which both the domain
expert and BN modeler should stay involved; this also holds when applying ourapproach, as the provided AG may be incomplete or incorrect. To aid in thisiterative process, approaches were proposed in related work which allow experts
to use argumentation to argue about the BN under construction instead of about
the domain [13, 19]. In other related work, approaches for explaining the reasoning patterns captured in BNs in terms of argumentation were proposed [12, 17],
which allow domain experts more accustomed to argumentation to understandthe probabilistic reasoning captured in a BN. Compared to the present paper,
this work is in the reverse direction, namely from BNs to arguments.
Recently, there has been much other work on probabilistic argumentation.
However, most approaches concern abstract argumentation (see e.g. [10] for an
overview) while we need structured arguments. Rienstra [16] considers probabilistic structured argumentation; however, he takes what Hunter [9] calls theconstellations approach to probabilistic argumentation by considering uncertainty in the existence of arguments. Instead, we take what Hunter calls the
epistemic approach to probabilistic argumentation by considering probabilities
to express uncertainty concerning the reliability of an argument’s inferences.
There is some work on the epistemic approach to probabilistic structured argumentation (e.g. [9, 15]). In future work, this may become relevant for deriving
probabilistic constraints on BNs.

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