From Arguments to Constraints on a
Bayesian Networka

Floris BEX a , Silja RENOOIJ a
Information and Computing Sciences, Utrecht University, The Netherlands
Abstract. In this paper, we propose a way to derive constraints for a Bayesian Network from structured arguments. Argumentation and Bayesian networks can bothbe considered decision support techniques, but are typically used by experts withdifferent backgrounds. Bayesian network experts have the mathematical skills tounderstand and construct such networks, but lack expertise in the application domain; domain experts may feel more comfortable with argumentation approaches.
Our proposed method allows us to check Bayesian networks given arguments constructed for the same problem, and also allows for transforming arguments into a
Bayesian network structure, thereby facilitating Bayesian network construction.

1. Introduction
Bayesian networks, graphical representations of probability distributions, are very well
suited to epistemic reasoning because they capture the probabilistic (in)dependencies between variables in the domain of discourse. They have found a number of applications indomains such as medicine, forensics and the law [1]. However, constructing a Bayesian
network and understanding the modelled influences between variables requires knowledge of the Bayesian network formalism, which means that domain experts (doctors,
lawyers) can often only construct a network with the help of a modeller with the relevant
mathematical background. In contrast, argumentation approaches can be said to moreclosely follow the reasoning of the domain experts, especially in legal or organizational
contexts which are less mathematically inclined [2, 3]. Our aim is to bridge the communication gap between domain experts and Bayesian network analysts by developing abetter understanding of the relation between the two kinds of reasoning.
The relations between arguments and Bayesian networks (BNs) can be consideredfrom two directions. For the first direction, BNs are transformed into arguments or argument diagrams [4–6], which allows the knowledge captured in the BN to be understood
more easily by domain experts accustomed to argumentative reasoning. This does notdirectly help domain experts to construct a BN based on argumentative reasoning, forwhich we need to transform in the other direction, that is, from arguments to BNs. A
classic example is [3], who use Wigmore graphs, which are very similar to evidential arguments [7], as the basis for BNs. Schum and Kadane, however, do not provide a formal
definition of their transformation. Such a definition is given in [8], in which Carneades
argument evaluation structures are transformed into BNs, thus allowing existing BNs to
be extended with the information contained in the arguments. The main focus in [8],

however, is on simulating the Carneades method of evaluating arguments through BNs.
In contrast, the main aim of this article is to explore how domain knowledge expressed
as argument structures can inform the construction of BNs.
One of the difficulties of interpreting generic argument structures as a BN is thattheoretical assumptions have to be made about, for example, the causal direction [9] andstrength of the inference rules [10, 11]. From a practical point of view, it is also an issue
that different BNs can represent the same reasoning. Consider for example a case wherea forensic specialist constructs a BN for a part of the case, while a judge constructs anargument about the same part of the case. If we directly translate this argument, we might
not end up with the exact same network as the forensic specialist, but this does not meanthat the judge and the forensic specialist disagree, as different BNs can represent thesame probability distribution. Because a set of evidential arguments therefore do not –and cannot – uniquely define a BN about the same part of the case, it makes sense totransform the arguments to a set of constraints that can be met by multiple possible BNs.
We hence distinguish two cases for transforming arguments to (constraints on) BNs.
In the first case, a network already exists – hand-crafted by experts or learned from data
– for the same case as the arguments and we want to check if the BN complies withthe arguments. Here, the arguments indicate a number of basic constraints a BN has toadhere to. In the second case, we use the identified constraints to construct a new BN
on the basis of the available arguments. Because the constraints are not exhaustive (i.e.,
given a set of constraints based on an argument there will be multiple BNs that adhere tothese constraints), we propose a general heuristic for transforming arguments into BNs.
Sections 2 and 3 briefly discuss the formal preliminaries: a structured argumentation
framework and Bayesian networks. In Section 4.1, we then discuss possible constraints
on the graph of a BN given structured arguments, and in Section 4.2 we propose a heuristic for transforming arguments to a BN. Note that initially, the focus is on constraints on
the structure (i.e. the graph) of the Bayesian network. Deriving constraints on the (conditional) probabilities of a network from structured arguments is more difficult, because it
depends on probabilistic interpretations of argumentation which are more contentious. In
Section 5 we briefly discuss some possible constraints on the (conditional) probabilities
in the Bayesian network, and how they can be used to expand our heuristic.

2. Structured Argumentation
In this section, we define a simple propositional system for argumentation based on the
ASPIC+ framework [12], which captures the basic elements of structured argumentation.
Because the idea is that domain experts have to be able to relatively easily construct arguments, we want to impose as few formal constraints as possible on these arguments. As
has been shown [7], ASPIC+ allows for arguments that are very similar to the Wigmore
graphs [3] with which many lawyers and judges are familiar.
Definition 2.1 [Argumentation systems] An argumentation system is a triple AS =(L, R, n) where:• L is a logical language with contrariness function − : L → 2L .
• Rs and Rd are two disjoint sets of strict (Rs ) and defeasible (Rd ) inference rulesof the form ϕ1 , . . . , ϕn → ϕ and ϕ1 , . . . , ϕn ⇒ ϕ respectively (where ϕi , ϕ aremeta-variables ranging over well-formed formulas in L).

• n is a naming convention for defeasible rules, which to each rule r in Rd assigns
a well-formed formula ϕ from L (written as n(r) = ϕ).
Here, L is a propositional language, where ϕ is a contrary of ψ if ϕ ∈ ψ and ψ 6∈ ϕ(i.e. asymmetrical conflict) and ϕ is a contradictory of ψ, denoted by ‘ϕ = −ψ’, if ϕ ∈ ψand ψ ∈ ϕ (i.e. symmetrical conflict). Furthermore, Rs contains the inference rulesof classical logic. Whereas in evidential reasoning we can distinguish between causal
defeasible rules (fire causes smoke) and evidential defeasible rules (smoke is evidence
for fire) [9], for now we make no assumptions as to the type of rule used. Finally, notethat informally, n(r) is a wff in L which says that rule r ∈ R is applicable.
Definition 2.2 [Knowledge bases] A knowledge base in an AS = (L, R, n) is a set K ⊆L consisting of two disjoint subsets Ke (the evidence) and Kp (the ordinary premises).
The evidence in Ke is similar to axiom premises [12], which cannot be denied or attacked. If, for example, a witness testimony is presented as evidence, then the existenceof the testimony cannot be denied; of course we can still question its veracity. Ordinary
premises can be undermined by other arguments (see Definition 2.4).
Arguments can be constructed from knowledge bases by chaining inference rules
into trees. Here, for any argument A, the function Sub returns all sub-arguments of A;
Prop returns all the formulas in A; Prem returns all the formulas of K (called premises)
used to build A, Conc returns A’s conclusion, Rules returns all inference rules in A and
TopRule returns the last inference rule used in A.
Definition 2.3 [Arguments] An argument A on the basis of a knowledge base K in anargumentation system AS = (L, R, n) is:
1. ϕ if ϕ ∈ K with: Prem(A) = {ϕ}; Conc(A) = ϕ; TopRule(A) = undefined;
Prop(A) = {ϕ}; Sub(A) = {ϕ}.
2. A1 , . . . , An →/⇒ ψ if A1 , . . . , An are arguments such that there exists a strict/defeasible
rule Conc(A1 ), . . . , Conc(An ) →/⇒ ψ in Rs /Rd , with:
Prem(A) = Prem(A1 ) ∪ . . . ∪ Prem(An );Conc(A) = ψ;TopRule(A) = Conc(A1 ), . . . , Conc(An ) →/⇒ ψ;
Prop(A) = Prop(A1 ) ∪ . . . ∪ Prop(An ) ∪ {ψ},Sub(A) = Sub(A1 ) ∪ . . . ∪ Sub(An ) ∪ {A};Rules(A) = Rules(A1 ) ∪ . . . ∪ Rules(An ) ∪ {Conc(A1 ), . . . , Conc(An ) →/⇒ ψ}
Arguments can be attacked in essentially two ways: on their conclusion (rebutting
attack) or on a defeasible inference step (undercutting attack).
Definition 2.4 [Attack] A attacks B iff A undercuts or rebuts, where:• A undercuts argument B (on r) iff Conc(A) ∈ n(r) for some B 0 ∈ Sub(B) suchthat TopRule(B) = r and r is defeasible.
• A rebuts argument B (on ϕ) iff Conc(A) ∈ ϕ and Conc(B 0 ) = ϕ for some B 0 ∈Sub(B) where either TopRule(B) is defeasible or ϕ ∈ Kp .

Argumentation systems plus knowledge bases form argumentation theories, which
induce structured argumentation frameworks. Note that we do not include an orderingon the arguments, as this is not needed for current purposes. Furthermore, because weonly use the structure of arguments and not take the evaluation of arguments into account for our constraints on Bayesian networks, we will not discuss any of the possible
argumentation semantics for ASPIC+ as given in [12].
Definition 2.5 [Structured Argumentation Frameworks] Let AT be an argumentation theory (AS, K). A structured argumentation framework (SAF) defined by AT , isa pair hA, Ci where A is the set of all finite arguments constructed from K in AS and(X, Y ) ∈ C iff X attacks Y .
Example 2.6 As an example of an argument, suppose that a burglary has taken place
and that we are interested in whether some suspect is guilty of committing the burglary(Bur). Forensic analysis (For) shows a match between a pair of shoes owned by thesuspect and footprints (Ftpr) found near the crime scene. However, there is also evidence that there may have been a mix up at the forensic lab (Mix): the exact history of
the footprints from crime-scene to lab has not been properly documented. This suspect
had a motive (Mot) to commit this burglary, which is confirmed by at least one reliable testimony (Tes1). Furthermore, it is argued that the suspect also had the opportunity (Opp) to commit this burglary, but this is denied (-Opp) by a further testimony of
the suspect himself (Tes2). We can now build arguments based on the evidence, where
Ke = {For, Mix, Tes1, Tes2} and Kp = {Opp}. Furthermore, Mix ∈ rf or , whererf or : For ⇒ Ftpr is the rule applied in A01 .
A1 : For
A01 : A1 ⇒ Ftpr
A2 : Opp

A3 : Tes1
A03 : A3 ⇒ Mot
A4 : A01 , A2 , A03 ⇒ Bur

B1 : Tes2
B10 : B1 ⇒ -Opp

C1 : Mix

Figure 1 shows the arguments, where inferences are modelled as dashed arrows (allthe inferences in this example are defeasible) and attacks as thick black arrows. Grey
propositions are in Ke .
Bur
Ftpr

Mot

For

Tes1

Opp

-Opp

Mix
Tes2

Figure 1. An argument in the example case

3. Bayesian Networks
In this section we briefly review Bayesian networks which combine a graph with conditional probability tables (CPTs) to compactly represent a joint probability distribution
over a set of stochastic variables [13].

Definition 3.1 [Bayesian network] A BN is a triple B = (V, A, P) where:• G = (V, A) is an acyclic directed graph with nodes V and arcs A ⊂ V × V (arc(Vi , Vj ) is directed from Vi to Vj );• P = {PrV | V ∈ V} where each PrV is a set of (conditional) distributions
Pr(V | par(V )) over variables V ∈ V, one for each combination of values forthe parents par(V ) of V in graph G; these distributions are typically represented
as tables (CPTs).
Note from the above definition that in a BN there is a one-to-one correspondence between
nodes and stochastic variables. Moreover, a BN allows for defining a joint probability
distribution that respects the independences portrayed by its digraph (see below).
Proposition 3.2 The BN B = (V, A, P) uniquely defines the following joint probability
distribution Pr(V):Y
Pr(V) =
Pr(V | par(V ))V ∈V

A BN thus allows for computing any probability of interest over its variables; a typicalquery of interest is the probability Pr(h | e) of some hypothesis h given a combinationof observations e for a set of observed variables E. The computation of probabilities
from the network specification is called inference and the Bayesian network framework
includes various algorithms to this end.
Although the directed arcs in a BN graph may suggest that the represented relations
are causal, the arcs in fact only have meaning in combination with other arcs: together
they capture the independences among the represented variables by means of the graphical d-separation criterion [13].
Definition 3.3 [d-separation] Consider three sets of nodes X, Y, and Z in graph G. In
addition, consider a simple chain s in G.
• Chain s is said to be blocked, or inactive, given Z if the chain contains a node with
two incoming arcs on the chain (a head-to-head node) which is not in Z and has
no descendants in Z, or it contains a node in Z that has at most one incoming arc
on the chain; a chain that is not blocked by Z is said to be active given Z.
• X and Y are said to be d-separated by Z if all possible chains s between nodes in
X and Y are inactive given Z.
• If X and Y are d-separated by Z, then the two corresponding (sets of) variables X
and Y are probabilistically independent given the third set Z.
Note from the latter statement that the values of the variables in Z are assumed to be actually observed and therefore known. We assume that two nodes that are directly connectedby an arc are not d-separated.
Example 3.4 Figure 2 shows the graph of a Bayesian network, where observed variables
have been shaded; the conditional probabilities are for now left implicit. Note that, compared to the argument graph in Figure 1 there is an additional variable, Rel, representing
the reliability of the witness that gave testimony 1. Note that, for example, given evidence

Tes1

Rel

Bur

Mot

For

Ftpr

Mix

Figure 2. An Bayesian network graph for the example casefor Tes1, nodes Rel and Bur are not d-separated: given the witness testimony, theirreliability will influence the probability of the suspect having committed the burglary.
On the other hand, Mix and Ftpr are d-separated due to the observation for For.

4. From Argumentation to (Constraints on) Bayesian Network Graphs
A structured argumentation framework in ASPIC+ can be used as the basis for (constraints on) the structure of a Bayesian network. First, we will discuss a number ofgeneric constraints that can be put on a Bayesian network given an argumentation framework. After this, we will provide a heuristic for building a Bayesian network graph giventhese constraints based on arguments.
4.1. Constraints on the Structure of a Bayesian Network
First, we need to interpret all the elements of an argumentation framework in terms of
the structure of a BN graph. The graph of a BN conveys the (in)dependencies between
variables represented by the nodes in the graph. With structural constraints we denote
both properties of nodes (which variables must be represented, and what are their values) and chains of arcs. In this paper we assume that all variables V are binary-valued,
with possible values Val(V ) = {true, false}. Since (in)dependencies are dynamic andcan change depending on the observed variables, we will also include observations forvariables in the structural constraints.
Constraint 4.1 [Nodes and values] Given a SAF = hA, Ci, the following constraints
can be put on the nodes V of a Bayesian network.
• For every atomic proposition v or −v in Prop(A), where A ∈ A, there exists a
single node V ∈ V such that v ≡ V = true and −v ≡ V = false;• Every atomic proposition ei or −ei in Ke is taken to represent the observed valueof node Ei from the set of observed variables E ⊂ V.
We say that propositions v and −v are associated with node V .
We now address the (chains of) arcs A that capture the (in)dependencies between
variables in a BN. Whenever a rule is applied to infer some conclusion from a set ofpremises, we should be able to reason from each node associated with the premises tothe node associated with the consequence in the BN. That is, given the context of theevidence upon which an argument is built, there should exist active chains between thenodes associated with the applied rules.

Constraint 4.2 [Inference chains] Given a SAF = hA, Ci, the following constraints
based on inferences can be put on the chains of arcs A of a Bayesian network.
• For every rule r : ϕ1 , . . . , ϕn → ψ or r : ϕ1 , . . . , ϕn ⇒ ψ such thatr = Rules(A) and A ∈ A, there exist active chains between each of the nodes
associated with ϕ1 , . . . , ϕn and the node associated with ψ, given the observed
nodes E associated with evidence in Ke .
An argumentation framework also includes a set of attack relations C. As for inference, it makes sense to assume that whenever two propositions in arguments are in conflict (i.e. contrary or contradictory), that there is an influence between the values representing these propositions in the BN. Because contradictory propositions ϕ and −ϕ are
captured as two values of a single variable (see Constraint 4.1), not all of the propositions
will translate to separate nodes in the BN, and hence the influences between the values
associated with ϕ and −ϕ will be captured in the (conditional) probabilities involving
these values. In the case of ϕ being contrary to ψ – which occurs in rebutting attacks– the influence should be captured as an active chain between the two associated nodes
given the evidence. Similarly, if a proposition undercuts the inference from one proposition to another then, given the evidence, there should be active chains between the nodes
associated with this undercutting attack.
Constraint 4.3 [Attack chains] Given a SAF = hA, Ci, the following constraints basedon attacks can be put on the chains of arcs A of a Bayesian network. For every attack
relation (A, B) ∈ C:
• if Conc(A) = ϕ is a contrary of Conc(B) = ψ, then there exists an active chain
between the node associated with ϕ and the node associated with ψ, given theobserved nodes E associated with evidence in Ke .
• if A undercuts B on r, where Conc(A) = χ and r : ϕ1 , . . . , ϕn ⇒ ψ, then there
exist active chains between the node associated with χ and the nodes associatedwith ϕ1 , . . . , ϕn , ψ, given the observed nodes E associated with evidence in Ke .
Example 4.4 In our case, a Bayesian network expert with limited domain knowledge
builds the network in Figure 2, and a judge builds the argument in Figure 1. The question
is now whether, given the evidence, the network conforms to the constraints imposed by
the arguments. In the argumentation framework of the case, there is an argument about
the suspect having the opportunity to commit the burglary (Opp), and a counterargument
based on the suspect’s testimony (Tes2). The BN, however, does not include variables
representing Opp or Tes2 and it thus violates some of the node constraints posed by theargumentation framework. Note that the BN also includes additional information that is
not captured in the argumentation framework: there is a variable representing the reliability of a witness testimony (Rel) whereas none of the arguments includes a proposition about witness reliability. Remember that our current aim is to use the argumentation
framework to put constraints on the BN and not vice versa, so any additional knowledge
or reasoning in the BN is fine as long as it does not lead to violation of the constraints.
In order to determine the active chains in the BN of Figure 2, we have to consider
the set of observations E based on the evidence in Ke and the nodes in the BN, so
E = {For = true, Mix = true, Tes1 = true}. With respect to inference chains,we now see that there are active chains from Ftpr to For (inference in A01 ), Tes1 to
Mot (inference in A03 ), Ftpr to Bur and Mot to Bur (both part of the inference in A04 ).
Because the node representing Opp is not in the network, there is no chain from Opp to
Bur as per the constraint based on the top rule application in A4 . Furthermore, because
Tes2 is not a node in the network, there is no active chain from Tes2 to Opp, whichis required by both the inference chain constraint and the first attack chain constraints.
Finally, the second attack chain constraint says that, because Mix undercuts the application of rf or : For ⇒ Ftpr in A01 , there should be active chains from Mix to For andfrom Mix to Ftpr. While these chains are both in the BN, the latter one, from Mix to
Ftpr is blocked by the observation For = true, so the constraint is violated.
4.2. Transforming Arguments into Bayesian Network Graphs
In this section we propose a heuristic for the construction of a BN from arguments. Our
BN graph construction heuristic starts by building an undirected skeleton in which allnodes that should be connected by active chains, according to the constraints from section 4.1, are connected directly by an edge. The resulting undirected graph will typicallybe densely connected and represent only few independencies. After this, it has to be decided which connections to retain and how to direct the edges. Recall that arguments are
typically constructed by performing successive inference steps based on evidence. The
rules behind these inferences can be either causal (fire causes smoke) or evidential rules(smoke is evidence for fire) [9]. In BNs – though the graph is just a representation of
an independence relation – people tend to attach a causal interpretation to the arcs, andthe notion of causality is typically used as a heuristic to guide the construction of thegraph with the help of domain experts [13]. Hence, we propose to use the same heuristic
for choosing the direction of the arcs. Furthermore, undercutting can be seen as a formof intercausal reasoning called explaining away (see Section 5). In BNs head-to-headnodes (see Definition 3.3) are explicitly used to model induced intercausal dependencies, such as exploited in explaining away, and we therefore propose to use head-to-headconnections among the nodes involved in an undercutting attack. Our heuristic is then as
follows:
BN graph construction heuristic
1) construct nodes according to Constraint 4.1;
2) each active chain dictated by Constraints 4.2 and 4.3 is implemented as a direct,
undirected edge;
3) for all undercutting attacks: if a rule r : ϕ1 , . . . , ϕn ⇒ ψ is undercut by χ ∈ n(r)then remove edge (χ, ψ) (or (ψ, χ)) and turn the other edges involved into thefollowing directed ones: (χ, ϕi ) and (ψ, ϕi ) for all ϕi ;4) for all remaining undirected edges, choose the causal direction if a causal interpretation is possible, and an arbitrary direction, otherwise;5) verify that the graph is acyclic and that all chains that should be active indeed are;otherwise remove or reverse appropriate arcs in consultation with a domain expert.
With respect to directing the edges we remark that BN graphs which share the same
skeleton (undirected edges) and the same immoralities (head-to-head nodes for which
the parents are not directly connected) are said to be Markov equivalent and capture

the same independence relation. Arc reversal therefore does not change the modelled
independence relation as long as the resulting graph is Markov equivalent.
Example 4.5 Using the heuristic described above on the arguments depicted in Figure 1
we can first construct an undirected graph G0 (Figure 3). We then apply the third stepof the heuristic: Mix – which in the argumentation framework is modelled as an undercutter of the rule For ⇒ Ftpr – is now captured in the BN as a cause of For, together with Ftpr. The observation of head-to-head node For makes the chain between
Mix and Ftpr active, so evidence for a mix-up will now change the probability of the
footprints being the cause of the forensic results – further probabilistic constraints (seeSection 5) could ensure that this intercausal effect is indeed “explaining away”. For the
fourth step of the heuristic, we can quite sensibly interpret the edges causally: having a
motive and opportunity cause the suspect to commit the burglary, which in turn causes
the footprints to be found. Furthermore, the evidential observations are caused by theevents that led to them. Now it can be verified that all chains that should be active given
E = {Mix, For, Tes1, Tes2} indeed are, and we end up with graph G in Figure 3.
graph
G’

Bur

Mot

Opp

Ftpr

graph
G

Bur

Mot

Opp

Ftpr

Mix
Tes1

Tes2

For

Mix
Tes1

Tes2

For

Figure 3. Undirected and directed graphs based on the argument in Figure1

The structure of the final graph G is quite close to the structure of the original arguments in Figure 1. This structural similarity can also be seen in [3]: in a Wigmore
diagram, the arrows denote evidential influences, which are then reversed in the causal
direction to form a BN graph. A difference between the informal heuristic of [3] and ours
is that we do not assume all influences in the argument to be evidential.

5. Constraints on the Probabilities of a Bayesian Network
Since a BN represents a probability distribution with its independence relation, an important part of the BN is formed by the (conditional) probabilities over the variables in a
network. Putting constraints on these probabilities based on the evidential arguments ina case requires a probabilistic interpretation of such arguments, which is not straightforward (see, a.o., [8, 10, 11, 14]). A full exploration of probabilistic constraints is therefore
beyond the scope of this paper. Rather, we will show a few possibilities of transforming
argument structures into probabilistic constraints, and briefly discuss some of the issues
concerning a probabilistic interpretation of structured arguments.
As for structural constraints, conditional probability constraints can be derived fromthe inferences and attacks in an argumentation framework. The interpretation of a strictrule, for example, is fairly straightforward: the conclusion is necessarily true given thepremises. However, with respect to defeasible rules different ideas exist on how theyshould be modelled probabilistically.
Constraint 5.1 [Inference probabilities] Given a SAF = hA, Ci, the following constraints can be put on the distribution Pr defined by a Bayesian network.
• For every strict rule r : ϕ1 , . . . , ϕn → ψ such that r = Rules(A), A ∈ A, wehave that Pr(ψ | ϕ1 , . . . , ϕn ) = 1;
• For every defeasible rule r : ϕ1 , . . . , ϕn ⇒ ψ such that r = Rules(A), A ∈ A,
we have that Pr(ψ | ϕ1 , . . . , ϕn ) > 0.
Note that all probabilities are taken to hold in the context of the available evidence,that is, each probability should also be conditioned on the subset of evidence for thenodes not associated with the rule. The above constraint for defeasible rules, which has
been adapted from [11], is fairly weak. It can, for example, be argued that the premiseshould make it more likely than not that the conclusion is true (Pr(ψ | ϕ) > 0.5), or thatthe premises are only a relevant reason for the conclusion if they increase the belief in the
conclusion (e.g. Pr(ψ | ϕ) > Pr(ψ) [10]). The point is that there are many measures of
support [14], and choosing exactly which ones to use to interpret inference is not trivial.
As for attacks, it makes sense to interpret rebuttal using classical negation, as rebutting propositions ϕ and −ϕ are values of one variable. Undercutting, where the application of a rule is attacked, can be seen as a form of explaining away, where the influenceof a certain observation should be removed, or at least weakened, given an additional
observation [13] (for more on the relations between undercutting evidential arguments
and alternative causal explanations also see [9]). In BNs, explaining away is considered
to be a type of intercausal reasoning often found between two causes (say ψ and χ) of acommon effect (say ϕ): if the effect is observed, both causes become more likely; however, if we subsequently know which cause actually occurred (e.g. χ), the probability ofthe other cause (ψ) being present as well decreases.
Constraint 5.2 [Attack probabilities] Given a SAF = hA, Ci, the following constraints
based on attacks can be put on the conditional probabilities of a BN. For every attack
relation (A, B) ∈ C:
• if Conc(A) = ϕ and Conc(B) = ψ and ϕ ∈ ψ, then Pr(ψ | ϕ) = 0.
• if Conc(A) = χ ∈ n(r) and r : ϕ1 , . . . , ϕn ⇒ ψ such that r = Rules(B), thenPr(ψ | ϕ1 , . . . , ϕn , χ) < Pr(ψ | ϕ1 , . . . , ϕn ).
A stronger version of the second constraint says that Pr(ψ | ϕ, χ) = 0 [11], whichfits with the idea that undercutting arguments always defeat the argument they attack[12]. Similarly, in [8] an argument is not applicable given an exception.
Example 5.3 Again consider the arguments in Figure 1 and the BN in Figure 2. Consider the two defeasible rules rf or : For ⇒ Ftpr and rmot : Mot ⇒ Bur, and theundercutting attack (C1 , A01 ). In the context of the relevant evidence e0 , our choice of
probabilistic interpretation now gives the following constraints:
Pr(Ftpr = true | For = true, e0 ) > 0

Pr(Ftpr = true | For = true, Mix = true, e0 ) < Pr(Ftpr = true | For =true, e0 )From Figure 2 we see that given For, the effect of a mix-up on footprints is blocked,
which means that the last constraint is violated since due to the blocking influence of
Mix it holds that Pr(Ftpr = true | For = true, Mix = true, e0 ) = Pr(Ftpr =true | For = true, e0 ).
In addition to rules and attacks, there are also other aspects of argumentation frameworks that can be interpreted probabilistically. For example, ASPIC+ includes the option
to define preferences over ordinary premises in Kp and defeasible rules in Rd , whichtranslate more or less directly to constraints on probabilities: if premise ϕ is preferredover premise ψ, then Pr(ϕ) > Pr(ψ). Furthermore, it is also possible to capture theevaluation of arguments probabilistically, as in [8], where the (conditional) probabilities
depend on the status of propositions in the Carneades argument evaluation structure.
Once we have a probabilistic interpretation of an argumentation framework, we canuse the probability constraints in an elicitation procedure for obtaining the required local
probability distributions [15]. However, the above discussion shows that there are different ways to interpret arguments probabilistically, and that different aspects of argumentation frameworks can be incorporated as constraints on probabilities. The difficulty here
lies in the fact that the exact probabilistic interpretation of arguments and evidence, andhence the various types of constraints on a BN, is a contentious issue. In fact, one way
to deal with discussions surrounding constraints on probabilities is to allow arguments
about the various probabilistic constraints in a BN [16].

6. Conclusion
In this paper we proposed a method for establishing constraints on Bayesian networks
from structured arguments. Using our method, the type of arguments typically constructed by, for example, legal experts can be used to check BNs and in particular theknowledge contained in them. Observed differences between the arguments and a BN
may carry useful information, as they point to possible differences in the reasoning of a
Bayesian network expert and the domain expert. Hence, the communication gap between
these two types of experts can be bridged.
In addition to deriving constraints for BNs from a set of arguments, we have also
designed a heuristic for constructing a BN graph based on these constraints. In this way,
the constraints can help with the construction of a BN: domain experts can build evidential arguments, which are then transformed into BNs or BN skeletons, which canin turn be interpreted and further extended by Bayesian network experts in conjunction
with the domain experts. The proposed heuristic thus expands the toolbox for building a
BN relatively easily, and can be used together with existing techniques for building BNs
such as idioms, recurrent BN structures based on standard arguments such as inductive
arguments or arguments based on evidence [1].
One of the key issues when transforming arguments into BNs is the probabilistic
interpretation of the elements of argumentation. It turns out that there are numerous possible interpretations of evidential support and argument strength [10, 11, 14], which obviously lead to different constraints on the probability distribution represented by a BN.

It is at present not clear if and how the choice of constraints influences the outcomes of
the reasoning in BNs. Furthermore, the choice of interpretation also depends on the (implicit) assumptions regarding evidential argument strength the domain experts that build
structured arguments have. We would like to address these questions in future research.
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