Refining a Heuristic for Constructing Bayesian
Networks from Structured Arguments
Remi Wieten1
1, Floris Bex1 , Linda C. van der Gaag1 , Henry Prakken1,2 , andSilja Renooij1

Department of Information and Computing Sciences, Utrecht University, NL
2Faculty of Law, University of Groningen, NL
Corresponding author: g.m.wieten@uu.nl

Abstract. Recently, a heuristic was proposed for constructing Bayesian
networks (BNs) from structured arguments. This heuristic helps domain
experts who are accustomed to argumentation to transform their reasoning into a BN and subsequently weigh their case evidence in a probabilistic manner. While the underlying undirected graph of the BN is automatically constructed by following the heuristic, the arc directions are to be
set manually by a BN engineer in consultation with the domain expert.
As the knowledge elicitation involved is known to be time-consuming, it
is of value to (partly) automate this step. We propose a refinement of
the heuristic to this end, which specifies the directions in which arcs areto be set given specific conditions on structured arguments.
Keywords: Bayesian networks · Structured argumentation

1

Introduction

In recent years, efforts have been made to gain a better understanding of therelation between different normative frameworks for evidential reasoning, such asargumentative and probabilistic approaches [9]. Argumentative approaches are
particularly suited for adversarial settings, where arguments for and against a
specific conclusion are constructed from evidence. The inferences which are used
to draw conclusions from evidence are generally defeasible, in that the conclusionof an argument does not universally hold given the evidence. Arguments can be
attacked by other arguments; it can then be established which arguments are
accepted and which are rejected. In current argumentative approaches, however,there is no emphasis on incorporating graded uncertainty.
In contrast, probabilistic approaches are well suited for handling graded uncertainty. In particular, Bayesian networks (BNs) [2, 3] are powerful tools tothis end. BNs are compact graphical models of joint probability distributions,which allow for evidence evaluation by calculating the probability of the truthof a proposition of interest. However, BNs are generally difficult to construct;
in fact, they are often constructed by modelers with the relevant mathematicalbackground, called BN engineers, in consultation with a domain expert.

Structured
arguments(Sect. 2.1)

Bayesian
networks(Sect. 2.2)

Support graph method [6](Sect. 3.1)
Heuristic of Bex and Renooij [1](Sect. 3.2)

Fig. 1. Outline of Sects. 2 and 3 of this paper.

Recently, a heuristic for constructing BNs from structured arguments was
proposed by Bex and Renooij [1]; in this paper, the heuristic will be referred
to as the BR heuristic. The heuristic helps domain experts who are more accustomed to argumentation to transform their reasoning into a BN (cf. Fig. 1)
and subsequently weigh their case evidence in a probabilistic manner. The focusof the BR heuristic lies on obtaining the graphical structure of the BN, called
the BN graph, which captures the independence relations between the domain
variables. While the underlying undirected graph, or skeleton, of the BN graphcan be automatically constructed by following the BR heuristic, the heuristicprescribes that the arc directions should be set manually by a BN engineer in
consultation with a domain expert. Although the heuristic further suggests thatthe commonly used notion of causality be taken as a guiding principle [3], theresulting graph still has to be verified and refined in terms of the independence
relations it represents. This type of knowledge elicitation is known to be timeconsuming [7], however, and moreover needs to be repeated for every adjustment
to the original arguments. As a consequence, letting arc directions be set by a BN
engineer is practically infeasible in investigative contexts such as police investigations, where evidence changes dynamically. It is, therefore, of value to investigate
whether the process of setting arc directions can be (partly) automated.
Accordingly, in this paper we propose a refinement of the BR heuristic, which
specifies the directions in which the arcs should be set in a BN graph under
specific conditions on structured arguments. These conditions are identified byapplying a method called the support graph method [6]. This method essentiallyworks in the opposite direction of the BR heuristic, in that structured arguments
are constructed from BNs (cf. Fig. 1). By applying the support graph method
to BN graphs obtained with the BR heuristic, it is determined whether andunder which conditions the original arguments are re-obtained. If the original
arguments are not re-obtained from the thus constructed BN graph, it may
be concluded that this graph represents the original arguments in a different,
possibly incorrect, way. Our refinement of the BR heuristic now ensures thatBN graphs from which the original arguments are not returned by the support
graph method are not constructed.

The paper is structured as follows. Sections 2 and 3 provide some preliminaries on structured argumentation, BNs, the support graph method and the BR
heuristic. In Sect. 4, our refinement to the BR heuristic is proposed, based on
observations from applying the support graph method. In Sect. 5, our findings
are summarized and possible directions for future research are discussed.

2

Preliminaries

In this section, structured argumentation and BNs are briefly reviewed.
2.1

Structured Argumentation

A simplified version of the ASPIC+ framework for structured argumentation [4]is assumed throughout this paper. Let L be a non-empty propositional literallanguage with the unary negation symbol ¬. Informally, L contains the basic elements which can be argued about. Given a knowledge base K ⊆ L of premises,arguments are constructed by chaining inference rules. These rules are definedover L and are defeasible, in that the conclusion of a defeasible rule does not universally hold given the premises, in contrast with the strict inferences of classicallogic. Let R be a set of defeasible inference rules of the form d : φ1 , . . . , φn ⇒ φ,where φ1 , . . . , φn and φ are meta-variables ranging over well-formed formulasin L. An argument A is then either: (1) φ if φ ∈ K, where the conclusion of
the argument A, denoted by Conc(A), is equal to φ; or (2) A1 , . . . , An ⇒ φwith φ ∈ L \ K, where A1 , . . . , An are arguments such that there exists a rule
Conc(A1 ), . . . , Conc(An ) ⇒ φ in R. In the first case, Conc(A) is an elementfrom the knowledge base, while in the second case, Conc(A) follows by applying
a defeasible rule to the conclusion(s) of arguments A1 , . . . , An , which are called
the immediate sub-arguments of A. Generally, a sub-argument of an argument
A is either A itself or an argument that is (iteratively) used to construct A. The
smallest set of finite arguments which can be constructed from L, K and R isdenoted by A. An argument graph of A then graphically displays the arguments
in A and their sub-arguments. Fig. 3a shows an example of an argument graph.
The general ASPIC+ framework further includes the notion of attack. Informally, an argument in A is attacked on one of its non-premise sub-arguments byanother argument in A with the opposite conclusion of that sub-argument. Dueto space limitations, the focus of the current paper lies on argument structures
without attack relations.

2.2

Bayesian Networks

BNs [3] are graphical probabilistic models which are being applied in manydifferent fields, including medicine and law [2]. A BN is a compact representationof a joint probability distribution Pr(V) over a finite set of discrete randomvariables V. The random variables are represented as nodes in a directed acyclic
graph G, where each node1 can take one of a number of mutually exclusive andexhaustive values; in this paper, we assume all nodes to be Boolean. A node A
is a parent of another node B, called the child, in G if G contains an arc from A
to B. The BN further includes, for each node, a conditional probability table, or
CPT, given its parents; this table specifies the probabilities of the values of thenode itself conditioned on the possible joint value combinations of its parents.
A node is called instantiated iff it is fixed in a specific value. Given a set of
instantiated nodes, conditional probability distributions over the other nodes in
the network can be computed using probability calculus [3].
The BN graph captures the independence relations between its variables.
Let a chain be defined as a simple path in the underlying undirected graph, orskeleton, of a 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 is blocked iff it includes a node V
such that (1) V is an uninstantiated head-to-head node on c without instantiated descendants; (2) V is not a head-to-head node on c and is instantiated. In
addition, instantiated end-points of the chain c, that is, instantiated nodes withat most one incoming or outgoing arc on c, serve to block the chain [5]. A chain
is inactive if it is blocked; otherwise it is called active. Two nodes A 6= B are
called d-separated by a set of nodes Z if no active chains exist between A and
B given instantiations of nodes in Z. If two nodes are d-separated by Z, thenthey are considered conditionally independent given Z. We note that conditional
independence thereby depends on the set of instantiated nodes [8].
An immorality in a BN graph is defined as a triple of nodes (A, B, C), where
A and C are parents of B that are not directly connected by an arc. Two BNs are
said to be Markov equivalent iff they share the same skeleton and immoralities.
Markov equivalent networks constitute an equivalence class, for which Vermaand Pearl [10] proved that any two elements represent the same independence
relations over the variables involved. Arcs between nodes that are not involvedin an immorality can thus be reversed without changing the represented independence relations as long as no new immoralities arise. Immoralities derive
their importance from providing for intercausal reasoning [11]. Specifically, if
the head-to-head node involved in an immorality is instantiated, an active chain
arises between the parents of the node. These parents can be seen as different
causes of the same effect modeled by the head-to-head node. If one of the causes
is now observed, then the probability of the other cause being present as wellcan either increase, decrease or stay the same upon updating, depending on theprobabilities in the CPT of the head-to-head node.
1

The terms ‘node’ and ‘variable’ are used interchangeably.

3

Two Methods for Translating between Structured
Arguments and Bayesian Networks

In this section, the support graph method [6] and the BR heuristic [1] are reviewed; the support graph method is used to build structured arguments from
BNs, while the BR heuristic is used to construct BN graphs from structured
arguments.
3.1

The Support Graph Method

The support graph method, proposed by Timmer and colleagues [6], is a twophase method for constructing argument structures from BNs. The method allows domain experts who are not familiar with BNs but are accustomed toargumentation to understand the knowledge and reasoning patterns captured
by a BN. To this end, the method summarizes all reasoning chains from a set ofevidence to a conclusion in a given BN.
In the first phase of the method, a directed graph called the support graph(SG) is constructed from a BN given a variable of interest V ∗ ; in this SG,
all reasoning chains in the BN ending in V ∗ are captured. The SG does notdepend on specific instantiations, and can thus be re-used to build argument
structures for different evidence. An SG is iteratively constructed, starting witha graph containing only V ∗ . New parents are added to existing nodes in the
SG as new inference steps are identified in the BN. Three types of inference
step are distinguished: (1) an inference step along an arc from a parent to achild; (2) an inference step along an arc from a child to a parent; and (3) an
inference step between two parents in an immorality. The last type directly
accommodates intercausal reasoning steps which occur between the parents ofan immorality, and summarizes the inference from one parent of an immorality
to another parent via the common child. In the constructed SG, V ∗ is the only
node without children; every other node in the SG is an ancestor of V ∗ .
In the second phase of the support graph method, arguments are constructedfrom the SG for a given set of node instantiations. Given this evidence, the SG ispruned such that only paths remain that start in an instantiated node. From thethus pruned graph, arguments are constructed as follows. The logical language
L is taken to consist of all literals which correspond to the values of the nodes
in the BN; two literals φ, ψ ∈ L negate each other iff φ and ψ correspond withthe different values of the same node. Given the evidence, the knowledge base K
consists of those literals in L that correspond with the values of the instantiatednodes. The defeasible rules in R are of the form (N1 , o1 ), . . . , (Nk , ok ) ⇒ (N, o),where N1 , . . . , Nk are parents of the node N in the pruned SG and o1 , . . . , ok , o
are values of these nodes. From L, K, and R, a set of arguments A is thenconstructed.

CrimeMotivePsych
report

Crime

Twin

DNA
match
(a)

MotiveTwin

DNA match

Psych report(b)Fig. 2. A BN graph (a) and the corresponding SG for the variable of interest Crime(b); Twin is pruned from the SG as only Psych report and DNA match are instantiated.

Example 1. An example by Timmer and colleagues [6] from the legal domain
is reviewed to demonstrate the support graph method. In the example, the BNgraph from Fig. 2a2 is constructed for a criminal case, in which we are interested
in whether the suspect committed the crime, that is, whether Crime = true.
Evidence for this possible conclusion would be the existence of a motive, which
may be mentioned in a psychological report. A match between the suspect’s
DNA and DNA found at the crime scene would further support the proposition
that the suspect committed the crime. This finding might also be explained,
however, if the suspect had an identical twin. For the variable of interest Crime,
the SG of Fig. 2b is obtained; the node Twin is directly added as a parent of
Crime, as the triplet (Crime, DNA match, Twin) is an immorality in the BNgraph. The literals in L are the possible values of all nodes in the BN graph, that
is, L contains crime, ¬crime, motive, ¬motive, . . . . Now, if we assume that Psych
report and DNA match are instantiated with the value true conform available
evidence, and Twin is not instantiated, then the path starting at the node Twinis pruned from the SG. The knowledge base K then consists of psych reportand dna match. Among the defeasible rules extracted from the pruned SG are
d1 : psych report ⇒ motive and d2 : dna match, motive ⇒ crime. The arguments
A1 : psych report, A2 : dna match, A3 : A1 ⇒ motive, and A4 : A2 , A3 ⇒ crime
can then be constructed. Also the rules d3 : psych report ⇒ ¬motive and d4 : dna
match, ¬motive ⇒ ¬crime are extracted from the SG, from which arguments
A5 : A1 ⇒ ¬motive and A6 : A2 , A5 ⇒ ¬crime are constructed. These arguments
have opposite conclusions of A3 and A4 .

It should be noted that, when using the support graph method, the reasons pro
and con a given conclusion are not distributed over separate arguments, as is
usual in argumentation, but are instead encapsulated in a single argument. Thatis, all literals that are relevant for a specific proposition are taken as the premises
2

In figures in this paper, circles are used in BN graphs, rectangles are used in argument graphs and rounded rectangles are used in SGs. Nodes and propositions
corresponding to evidence are shaded. Capital letters are used for the nodes in BNgraphs and SGs, and lowercase letters are used for propositions.

of an argument for that proposition, which reflects the way in which Bayesian
networks internally weigh all evidence.
For every argument that is returned from a BN by the support graph method,
the method also returns an argument with the same ‘structure’ but with the
opposite conclusion. Timmer and colleagues [6] employ a quantitative step tofilter the set of arguments returned. As in the current paper the focus lies on
the graphical structures of BNs and not on the modeled probability distribution,
this quantitative step is not further discussed here.
3.2

The BR Heuristic for Constructing Bayesian Networks from
Structured Arguments

Bex and Renooij [1] have proposed the BR heuristic for constructing BN graphsfrom structured arguments. This heuristic allows domain experts who are accustomed to argumentation to translate their reasoning expressed as arguments
into a BN graph. This graph is then supplemented with CPTs to arrive at a fully
specified BN for probabilistic inference over the original arguments. Focusing onargument structures in which no attack relations are present, from a given setof arguments A constructed from a logical language L, knowledge base K, anda set of defeasible rules R, BN graphs are constructed as follows:
1. For every proposition φ ∈ L used in A, the BN graph includes a single node
V such that V = true corresponds to φ and V = false corresponds to ¬φ. For
every e ∈ K, the corresponding node is instantiated at the observed value.
2. For every defeasible rule d : φ1 , . . . , φn ⇒ φ ∈ R used in A, a set of undirectededges between the node associated with φ and each of the nodes associatedwith φ1 , . . . , φn is created for inclusion in the BN graph.
3. The direction of the edges from the previous step is decided upon by a BN
engineer in consultation with the domain expert, where a causal direction ischosen if possible, and an arbitrary direction otherwise. The resulting arcs
are inserted in the BN graph.
4. The BN engineer verifies that the graph is acyclic and that all chains thatshould be active in the graph indeed are; if the graph does not yet exhibit
these properties, appropriate arcs are removed or reversed, once more in
consultation with the domain expert.
Example 2. A simple example is introduced to demonstrate the BR heuristic.
The logical language, knowledge base and defeasible rules involved are L ={p, ¬p, q, ¬q, r, ¬r}, K = {p} and R = {p ⇒ q; q ⇒ r}. The constructed arguments are A = {A1 : p; A2 : A1 ⇒ q; A3 : A2 ⇒ r}; the argument graph of A is
depicted in Fig. 3a. Following steps 1 and 2 of the BR heuristic, the skeleton of
the BN graph corresponding to this argument structure consists of nodes P , Q
and R, with undirected edges between P and Q and between Q and R. Following
step 3, one of the BN graphs of Fig. 3b-e is obtained, depending on how the arc
directions are set.


r

P

Q

R

P

(b)

Q

R

(c)

q
Q

p

(a)

P

P

R

(d)

R

Q

(e)

Fig. 3. An argument graph with arguments from p to r via q (a); the four corresponding
BN graphs which can be constructed by following the BR heuristic (b-e).

For a given set of arguments A, the skeleton of the BN graph is automaticallyconstructed by following the first two steps of the BR heuristic. Step 3 thenprescribes that the directions of the arcs should be set manually by a BN engineer in consultation with the domain expert, using the notion of causality as a
guiding principle (see also [3]). For example, if the domain expert indicates fora defeasible rule d : p ⇒ q that p is a typical cause of q, then the arc is set fromnode P to node Q. Since immoralities can result from following this guiding principle, the independence relations in the constructed BN graph should be verified
manually, as prescribed by step 4 of the BR heuristic. This type of knowledge
elicitation and verification is known to be a time-consuming and error-prone process in general [7]. Especially for larger or more densely connected BN graphs,
it quickly becomes infeasible to verify all independence relations manually, as
all possible chains for all possible combinations of instantiated variables need to
be investigated. Moreover, the elicitation and verification needs to be repeatedfor every adjustment to the original argument graph. As this step is practicallyinfeasible in investigative contexts, such as police investigations, in which theevidence for a case changes dynamically, the arc directions are preferably set(semi-)automatically.

4

Refining the BR Heuristic

We propose a refinement of step 3 of the BR heuristic, which specifies the directions in which arcs should be set in a BN graph under specific conditions on
structured arguments. These conditions are identified from applying the support
graph method. To this end, the arguments to which the BR heuristic is applied
are compared to the arguments returned by the support graph method whenapplied to a BN graph constructed by steps 1-3 of the BR heuristic. In order toapply the support graph method, a variable of interest has to be chosen. In thispaper, we assume that there is a single ultimate conclusion in the input argument graph, that is, a single argument that is not an immediate sub-argument ofanother argument. The node corresponding to this ultimate conclusion is taken

as the node of interest. We further assume that the input arguments for the BR
heuristic are linked, in the sense that all premises relevant for a conclusion areencapsulated in a single argument; Fig. 5a shows an example of an argument
graph with linked arguments only. Linked argument graphs are similar to thetype of argument graphs that are returned by the support graph method.
When applying the support graph method to a BN graph constructed bysteps 1-3 of the BR heuristic, a set of arguments is returned. This set may bedifferent from the set of arguments that was used as input for the heuristic. As
measures for the differences found, we distinguish between recall and precision,
which for a given BN graph respectively measure the proportion of original arguments returned and the proportion of additional arguments returned. Formally,
let A be the set of input arguments for the BR heuristic, let B be a BN graphconstructed from A by steps 1-3 of the heuristic, and let A0 be the set of arguments returned from B by the support graph method. We define the recall andprecision of B as follows:- Recall (B)= |A ∩ A0 |/|A|- Precision(B) = |A ∩ A0 |/|A0 |where B has maximum recall and precision if these fractions are equal to 1.
In Sect. 4.1, we propose a refinement of the third step of the BR heuristic,
which serves to increase the recall of constructed BN graphs. In Sect. 4.2, weaddress precision. As argued before, Timmer and colleagues [6] propose a quantitative step for filtering the set of arguments returned by the support graphmethod, which suggests that for improving the precision of constructed BNs,
the CPTs need to be taken into account. As in this paper, the focus lies on thegraphical structure of a BN, we propose a further refinement of the third stepof the BR heuristic based on graphical considerations only.
4.1

Refining the BR Heuristic to Improve Recall

To illustrate how the BR heuristic can be refined such that BN graphs withhigher recall are constructed, we revisit Example 2 from Sect. 3.2. By applying
steps 1-3 of the heuristic to the argument graph of Fig. 3a, four possible BNgraphs over the nodes P , Q and R were constructed, as shown in Figs. 3b-e.
These graphs fall into two Markov equivalence classes; the first class consists of
the BN graphs of Figs. 3b-d, and the second class consists of the graph of Fig. 3e.
Timmer and colleagues [6] proved that for two Markov equivalent BNs and thesame node of interest, the same SG is obtained. By applying the support graphmethod for the node of interest R, we now show that the recall of the original
arguments from the BN graph in the second equivalence class is lower than thatof the BN graphs in the first class. Since the logical language and knowledge baseof the argument structure returned by the support graph method are derived
from the BN skeleton, L03 = {p, ¬p, q, ¬q, r, ¬r} and K0 = {p} are the same3

The prime symbol is used to denote objects which result from applying the support
graph method.

R

Q

R

P

Q

P
(a)

(b)

Fig. 4. The (pruned) SG obtained from the BN graphs of Figs. 3b-d (a), and the SG
obtained from the BN graph of Fig. 3e (b), where Q is pruned as only P is instantiated.

for all four BN graphs. For the graphs in the first equivalence class, the SG
of Fig. 4a is obtained. The defeasible rules of the returned argument structure
correspond to the arcs of this SG, that is, R0 = {p ⇒ q; p ⇒ ¬q; ¬p ⇒ q; ¬p ⇒¬q; q ⇒ r; q ⇒ ¬r; ¬q ⇒ r; ¬q ⇒ ¬r}. As L ⊆ L0 , K = K0 and R ⊆ R0 , alloriginal arguments A1 , A2 , A3 ∈ A are re-obtained from the SG. Therefore, theBN graphs of Figs. 3b-d have maximal recall.
For the BN graph in the second equivalence class, the SG of Fig. 4b is constructed. In this SG, node P is a direct parent of R and not of Q, as (P, Q, R)
is an immorality. We recall that an SG is meant for constructing arguments fordifferent sets of evidence. In the example, where just P is instantiated, node Q
is pruned from the SG. The defeasible rules corresponding to this pruned SG
are R0 = {p ⇒ r; p ⇒ ¬r; ¬p ⇒ r; ¬p ⇒ ¬r} and the arguments which can beconstructed are A1 : p, A02 : A1 ⇒ r and A002 : A1 ⇒ ¬r. Timmer and colleagues[6] employ a quantitative step using the CPTs from the original BN to filter theset of constructed arguments; by this step, arguments A02 and A002 are filteredout, as P and R are independent given that Q is not instantiated. The original
arguments A2 are A3 are not returned by the support graph method. The recallof the BN graph from Fig. 3e is 13 , which is lower than that of the BN graphs in
the first equivalence class. It therefore seems desirable to prohibit constructionof this BN graph when using the BR heuristic.
Generalizing from the example, let A1 , . . . , An ∈ A, where Ai is an immediate
sub-argument of Ai+1 for all i ∈ {1, . . . , n − 1}, let Conc(Ai ) = pi , p1 ∈ K,
and let pn be the ultimate conclusion of the argument graph of A. Further
assume that no immorality (Pi−1 , Pi , Pi+1 ) is formed for i ∈ {2, . . . , n − 1} bysteps 1-3 of the BR heuristic. As no immoralities (Pi−1 , Pi , Pi+1 ) are present fori ∈ {2, . . . , n − 1}, upon constructing the SG for the node of interest Pn parents
are added iteratively, that is, Pn−1 is added as a parent of Pn , . . . , P1 is added asa parent of P2 . As P1 corresponds to an instantiated variable, the path startingin P1 is not pruned from the SG. The support graph method, therefore, returns
the arguments A1 , . . . , An , and the recall is maximal. On the other hand, if for agiven i ∈ {2, . . . , n − 1} an immorality (Pi−1 , Pi , Pi+1 ) would be formed by steps
1-3 of the BR heuristic, then an SG would result in which Pi+1 is an ancestor of

Pn . As Pi−1 is directly added as a parent of Pi+1 , the argument Ai would not
be returned, and the recall would not be maximal.
Based on the above observations, the following refinement of step 3 of the
BR heuristic is proposed:
30 . Let A1 , . . . , An ∈ A, where Ai is an immediate sub-argument of Ai+1 forany i ∈ {1, . . . , n − 1} and where Conc(Ai ) = pi . Then, the directions of
the arcs are set such that no immoralities (Pi−1 , Pi , Pi+1 ) are formed for anyi ∈ {2, . . . , n − 1}. Taking this constraint into account, the directions of the(remaining) arcs are set by a BN engineer in consultation with the domain
expert, where a causal direction is chosen if possible.
4.2

A Further Refinement of the BR Heuristic

While in the previous section, simple chains in an argument structure were shown
to be best translated in the BN graph by a chain without any immoralities, we
now focus on argument structures that do enforce immoralities in the BN graphand propose a further refinement of the refined third step of the heuristic.
Example 3. We consider the linked argument graph of Fig. 5a. The logical language, knowledge base and defeasible rules involved are L = {p, ¬p, q, ¬q, r, ¬r,s, ¬s, t, ¬t}, K = {p, q}, and R = {p ⇒ r; p, q ⇒ s; r, s ⇒ t}; the constructed arguments are A = {A1 : p; A2 : A1 ⇒ r; A3 : q; A4 : A1 , A3 ⇒ s; A5 : A2 , A4 ⇒ t}.

t

r

T

s

R

p

q

S

P

(a)

(b)

T

T

R

S

P

(c)

Q

R

Q

S

P

Q
(d)

Fig. 5. An argument graph (a) and the corresponding BN skeleton that is constructedby the BR heuristic (b); a corresponding BN graph with the immorality (R, P, S) (c)and a BN graph with the immorality (R, T, S) (d).

Steps 1 and 2 of the BR heuristic result in the BN skeleton of Fig. 5b. In orderto obtain an acyclic directed graph from this skeleton, at least one immorality
has to be created in the subgraph induced by the nodes P , R, S and T .
According to the refined third step of the BR heuristic, an immorality (T, R, P )should not be formed, as A1 : p is an immediate sub-argument of A2 : A1 ⇒ r,which in turn is an immediate sub-argument of A5 : A2 , A4 ⇒ t. Similarly, theimmorality (T, S, P ) should not be formed. Now, the equivalence class of BNgraphs is considered which includes just the immorality (R, P, S); the BN graphdepicted in Fig. 5c is an element of this class. With T as the node of interest,
the SG of Fig. 6 is obtained from this graph. The logical language and knowledge base corresponding to this SG are L0 = {p, ¬p, q, ¬q, r, ¬r, s, ¬s, t, ¬t} andK0 = {p, q}, matching those of the original argument graph. The set of defeasible rules R corresponding to the SG includes the rules p, q ⇒ s; p, s ⇒ r;
p ⇒ r; p, q, r ⇒ s and r, s ⇒ t. Among the arguments which can be constructedfrom the SG are A1 : p, A2 : A1 ⇒ r, A3 : q, A4 : A1 , A3 ⇒ s, A5 : A2 , A4 ⇒ t,
A02 : A1 , A4 ⇒ r, A04 : A1 , A2 , A3 ⇒ s, and A05 : A02 , A04 ⇒ t. While the recall of
the BN graphs from Fig. 5c is maximal, the precision is not; more specifically,
the returned arguments A02 , A04 and A05 were not in the original argument set A.
Now, the equivalence class of BN graphs with just the immorality (R, T, S)
is addressed; the BN graph depicted in Fig. 5d is an element of this class. From
this BN graph, again the SG of Fig. 6 is constructed for the node of interest
T , and thus the same arguments as above are returned. While the precision of
the BN graph of Fig. 5d is equal to that of the BN graph from Fig. 5c, we notethat the nodes R and S are conditionally independent given the evidence for
Z = {P, Q} in the former graph, that is, in the BN graph with just the immorality (R, T, S). The immediate sub-argument A4 of A02 and the immediate
sub-argument A2 of A04 , therefore, appear to be irrelevant, as the associatedreasoning is non-existent in this BN graph. As noted before, Timmer and col-

T

S

R

A1

A1
P

A'5 A'2 A4 A1

Q

A3

Q

P

P

S

R

A3
P

A1 A2 A'4

Fig. 6. The SG corresponding to the BN graph of Figs. 5c and 5d, with T as the nodeof interest; the SG is annotated with some of the possible arguments which can beextracted from it.

leagues [6] employ a quantitative step to filter the set of arguments returned by
the support graph method; specifically, as the nodes R and S are conditionallyindependent given the evidence in the BN graph in Fig. 5d, A4 and A2 are filtered out as immediate sub-arguments of A02 and A04 respectively. Building on theconditional independence relations that can be inferred from the BN graph given
the set of instantiated nodes, however, irrelevance of A4 and A2 as immediate
sub-arguments of A02 and A04 can be decided upon by graphical considerations
only, without involving the CPTs of the nodes.

Based on the above example, we propose to set the directions of arcs in a BN
skeleton such that no instantiated head-to-head nodes or head-to-head nodes
with instantiated descendants are formed, as such head-to-head nodes may introduce unwarranted dependence relations. More specifically, the following refinement of step 30 of the BR heuristic is proposed, which fully specifies the
directions of the arcs in a BN graph corresponding to a set of arguments A:
300 . The directions of the arcs in a BN graph are set in the same direction as thearcs in the argument graph, that is, if A is an immediate sub-argument ofB, then an arc should be drawn from the node corresponding to Conc(A)to the node corresponding to Conc(B).
We note that step 300 is a further refinement of step 30 , as none of the immoralities(Pi−1 , Pi , Pi+1 ) mentioned in that step are formed if arcs are set in the same
direction as in the argument graph. By step 300 , arcs are guaranteed to be set
such that head-to-head nodes are not instantiated and do not have instantiateddescendants, as the premise arguments in the argument graph, and hence theinstantiated nodes in the BN graph, only have outgoing arcs. Finally, we notethat step 300 is not a strict specification of the directions of the arcs in a BNgraph; directions can possibly be reversed, given that an element from the same
Markov equivalence class as specified by step 300 is obtained.

5

Conclusion and Future Research

In this paper, we have proposed a refinement of the heuristic of Bex and Renooij[1] for constructing BN graphs from structured arguments. This heuristic is
aimed at aiding domain experts who are accustomed to argumentation to transform their reasoning into BNs and subsequently weigh their case evidence in aprobabilistic manner. Our refinement consists of fully specifying the directions
in which arcs should be set in a BN graph for a given argument structure without attack relations; more specifically, when employing the refined heuristic fora set of arguments A, the directions of the arcs in the BN graph are set in thesame direction as the arcs in the original argument graph of A. By our refinedheuristic, BN graphs with maximal recall are constructed, that is, the original arguments are returned by applying the support graph method to the constructed
BN graphs. Furthermore, our refined heuristic prevents the creation of directintercausal dependence relations between variables in the BN graph that did notexist between the corresponding propositions in the original argument graph. In
the near future, we will evaluate the heuristic in practice by establishing, forexample, the extent to which the automatically derived arc directions match theperceived real-world causality or the judgments of domain experts.
In this paper, we focused on improving the recall of BN graphs constructedby the BR heuristic. In our future research, we will address the construction of
BN graphs with increased precision. Furthermore, we will extend our research
to a more general framework of argumentation [4], not restricting ourselves tolinked argument graphs without attack relations.

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