Forecasting with jury-based probabilistic argumentation

Probabilistic Argumentation naturally supports the integration of quantitative (probabilistic) reasoning and qualitative argumentation. Meanwhile, Jury-based Probabilistic Argumentation supports the combination of opinions by different reasoners. We show how Jury-based Probabilistic Abstract Argumentation (JPAA) and a form of Jury-based Probabilistic Assumption-based Argumentation (JPABA) can naturally support forecasting, whereby subjective probability estimates are combined to make predictions about future events. The form of JPABA we consider is an instance of JPAA and results from integrating Assumption-Based Argumentation (ABA) and probability spaces expressed by Bayesian networks. We show how JPAA and (the considered form of) JPABA can support forecasting by allowing different forecasters to determine the probability of arguments (and, in JPABA, sentences) with respect to their own probability spaces, while sharing arguments (and, in JPABA, their components). We show in turn how this supports the aggregation of individual forecasts into group forecasts.


Introduction
The benefits resulting from a combination of quantitative (e.g.probabilistic) and qualitative (e.g.logic-based) reasoning are widely acknowledged (e.g.see Domingos et al., 2006;Poole, 2011).Several proposals for this combination exist in the literature, e.g.Poole's Independent Choice Logic (Poole, 2000) and Probabilistic Horn Abduction (Poole, 1993), probabilistic logic programs (e.g.see Ng & Subrahmanian, 1992;Raedt & Kimmig, 2015), probabilistic answer sets by Baral et al. (2009), and probabilistic argumentation (Hunter et al., 2021).In this paper, we focus on the latter approach, and in particular on forms of jury-based probabilistic argumentation (Dung & Thang, 2010).There, multiple 'jurors' (which may be artificial agents or humans) express individual, possibly different probabilistic assessments of future events/arguments depending on these events, and the jurors' opinions are aggregated (e.g. using beyond reasonable doubt and majority voting with preponderance criteria).This has been originally advocated for legal settings, where causal information and norm-based information need to be reasoned with at the same time.In the single-juror case, it has also been advocated for medical settings (Oo et al., 2020), where expert knowledge and Bayesian networks CONTACT Francesca Toni ft@imperial.ac.ukShould Jo, Alex and Pat opt for a tracker mortgage?This depends on how each deems likely that Alex will be jobless soon (amounting to event redundancies) and that the central bank will decrease interest rates (amounting to event interest_increase).They may jointly accept argument J if at least two of them (namely a majority of jurors) accept J with preponderance, i.e. with probability greater than 0.5 (this also covers the case where each accepts J beyond a reasonable doubt, i.e. with probability 1).
This example is representative of a wide range of real-world cases, for instance modelling geopolitical threats such as attacks or pandemics, where 'contingency planning for these threats requires well-calibrated conditional forecasts of the impact of policy interventions' (Lustick & Tetlock, 2021).Conditional forecasting in these real-world cases depends on how different forecasters may assess future events as differently probable and on their acceptance of arguments based on the occurrence of these events.Thus, the scenario in Example 1.1, albeit simple for illustrative purposes, is not simplistic.
The purpose of this paper is to explore how jury-based probabilistic argumentation can shed light on conditional forecasting applications which, like Example 1.1, require both qualitative and probabilistic reasoning.As in Dung and Thang (2010), we will consider two forms of jury-based probabilistic argumentation: Jury-based Probabilistic Abstract Argumentation (JPAA) and Jury-based Probabilistic Assumption-Based Argumentation (JPABA).These integrate probabilistic reasoning and argumentation by using probability spaces in conjunction with, respectively, Abstract Argumentation (AA) by Dung (1995) and Assumption-Based Argumentation (ABA) by Bondarenko et al. (1997), to determine the probability of arguments and, in the case of JPABA, sentences.We will consider JPABA of the general form advocated in Dung and Thang (2010) (whereby probabilistic reasoning is mixed with argumentative reasoning) as well as of a simpler form inspired by Hung (2016a); Oo et al. (2020), whereby probabilistic reasoning is defined by a separate Bayesian network integrated with argumentative reasoning.Rather than restricting attention to grounded extensions as in Dung and Thang (2010), we will also consider sceptically preferred and ideal extensions (Dung et al., 2007).Overall, our contributions are as follows: • We clearly position Jury-based Probabilistic Argumentation as a natural candidate for supporting conditional forecasting computationally.• We point to the possibility of generalising existing forms of (Jury-based) Probabilistic Argumentation to other sceptical semantics, beyond grounded extensions (see Definition 4.5 and Proposition 4.12).This is useful to broaden applicability, given that grounded extensions are the most sceptical and may limit the acceptance of arguments and thus the ability to make forecasts in some cases.
• We position the form of Probabilistic Abstract Argumentation by Dung and Thang (2010) within the broad constellation approach to probabilistic argumentation (see Corollary 4.14), identifying an important property for forecasting that some other constellation approaches may violate.• We define a novel, simple form of (J)PABA (see Definition 5.2) in the spirit of notions in Hung (2016a), Oo et al. (2020).• We firm the relation between (J)PAA and the proposed form of (J)PABA by connecting overall probabilities of arguments in PAA and finer-grained probabilities of sentences in PABA (see Proposition 5.9) that are needed to support forecasting, thus pointing to the need to take a structured argumentation view in this important application area.
The paper is structured as follows.In Section 2 we discuss related work, focusing on computational approaches to forecasting and applications of probabilistic argumentation related to forecasting.In Section 3, we give preliminaries on AA, ABA, probability spaces and Bayesian networks.In Section 4 we give JPAA and its properties.In Section 5 we give a form of JPABA and its properties (including that it is an instance of JPAA, generalising binary Bayesian networks as well as standard ABA).We then conclude in Section 6.

Related work
The integration of argumentation and probabilistic reasoning is the objective of several existing works (see Haenni, 2009;Hunter et al., 2021;Pfeifer & Fermüller, 2023 for overviews of some works in this area).The forms of probabilistic argumentation we consider to support forecasting belong to the so-called constellation approach (Hunter et al., 2021), where uncertainty is expressed by a probability distribution over argumentation frameworks as possible worlds, and standard argumentation semantics are used to determine probabilistic acceptance of arguments or sentences.In the forecasting scenarios, we are focusing on, these possible worlds capture uncertainty about future events, whereas legitimate arguments in these worlds (amounting to sub-graphs of a 'global' argumentation view expressed as a graph) capture epistemic uncertainty.
Several attempts at combining argumentation and Bayesian networks already exist (e.g.see Vreeswijk (2004), Saha and Sen (2005), Williams and Williamson (2006), Timmer et al. (2015) and references therein).These approaches are mostly concerned with argumentative reasoning with Bayesian networks, e.g. to extract probabilistically supported arguments (Timmer et al., 2015) or to explain probabilistic predictions (Williams & Williamson, 2006).In order to support forecasting, we advocate an instance of JPAA which adds an argumentative layer on top of a Bayesian network, keeping the two separate and allowing to build probabilistic arguments supported by assumptions in ABA as well as random variables in a given Bayesian network.Argumentation and Bayesian networks are also kept separate by Saha and Sen (2005) in a multi-agent setting, where argumentation-based negotiation is aided by opponent modelling using Bayesian networks.It would be interesting to see whether JPABA, as a general-purpose argumentation formalism, could be deployed also to integrate opponent modelling within argumentation-based dialogues, generalising the ABA dialogues of Fan and Toni (2014).
Forms of probabilistic argumentation have been applied in several settings.Dung and Thang (2010) use it for legal reasoning, where causal information and norm-based information need to be reasoned with at the same time.In the single-juror case, it has also been advocated for medical reasoning (Oo et al., 2020).Hung et al. (2022) use forms of probabilistic ABA for combining argumentation systems with machine learning components.Usefulness of probabilistic argumentation is generally supported by empirical research (Polberg & Hunter, 2018).
The benefits in adopting human judgement as a mechanism for making predictions, particularly in cases where statistical methods alone are unsuitable, have been touted by many, e.g.Arvan et al. (2019); Baets and Harvey (2020); Petropoulos et al. (2016) and, notably, Tetlock and Gardner (2016).This has led to the field of judgemental forecasting (Lawrence et al., 2006), aiming at making predictions by utilising subjective opinion with probability estimates (see Lawrence et al., 2006;Zellner et al., 2021 for some surveys).Most relevant to this paper, Irwin et al. (2022aIrwin et al. ( , 2022b) ) have explored the use of argumentation for supporting judgemental forecasting, leveraging on argumentation's capability for representing human-like reasoning and for forming consensus via conflict resolution.However, these works commit to the use of quantitative argumentation debate frameworks (Baroni et al., 2015), rather than probabilistic argumentation as we do here.This choice allows us to use structured arguments, which hold distinct advantages in supporting forecasting activities, as we will see.

Preliminaries
An abstract argumentation (AA) framework (as proposed by Dung (1995)) is a pair F = (Ar, Att), where Ar is a set of arguments and Att is a binary relation over Ar representing attacks between arguments, with (α, β) ∈ Att meaning 'α attacks β'.For simplicity, we restrict ourself to AA frameworks with finite sets of arguments.
Let S ⊆ Ar be a set of arguments.S attacks an argument α ∈ Ar if some argument in S attacks α. S attacks a set S ⊆ Ar of arguments if S attacks some argument in S .S is conflict-free iff it does not attack itself.An argument α ∈ Ar is acceptable with respect to S iff S attacks each argument attacking α. S is admissible iff S is conflict-free and each argument in S is acceptable with respect to S. S is a preferred extension iff S is maximally (with respect to set inclusion) admissible.S is sceptically preferred iff it is the intersection of all preferred extensions.We also refer to the sceptical preferred set as the sceptical preferred extension.
The semantics of argumentation can also be characterised by a fixpoint theory of the characteristic function F(S) = {α ∈ Ar | α is acceptable with respect to S} (Dung, 1995).Then S is admissible iff S is conflict-free and S ⊆ F(S).As F is monotonic, it follows that S is a preferred extension iff S is conflict-free and a maximal fixpoint of F. The least fixed point of F gives the grounded extension.
The grounded extension and the sceptically preferred extension are guaranteed to be unique (Dung, 1995) and can be deemed to support sceptical forms of reasoning.An alternative sceptical semantics is given by the ideal extension (proposed by Dung et al. (2007)), namely a maximal (with respect to set inclusion) S ⊆ Ar such that S is admissible and S is contained in every preferred extension.In this paper, we use G, S and I to stand for the grounded, sceptically preferred and ideal extension, respectively, for any given AA framework.Then G ⊆ I ⊆ S for any AA framework (Dung et al., 2007).
An Assumption-based argumentation (ABA) framework (as originally proposed by Bondarenko et al. (1997), but presented here following recent accounts by Dung et al. (2009) and Toni ( 2014)) is a tuple L, R, A, where • L, R is a deductive system, where L is a language and R is a set of (inference) rules of the form • ¯is a total mapping from A into L, where a is the contrary of a, for a ∈ A.
Given a rule s 0 ← s 1 , . . ., s m , s 0 is referred to as the head and s 1 , . . ., s m as the body; if m = 0 then the body is said to be empty.In a flat ABA framework assumptions are not heads of rules (Dung et al., 2006).
Several instances of ABA have been studied by Bondarenko et al. (1997), including logic programming and default logic (flat) and non-monotonic modal logic and circumscription (non-flat).We focus here on flat ABA frameworks.
In ABA, arguments are deductions of claims using rules and supported by assumptions, and attacks are directed at the assumptions in the support of arguments.More formally, following (Dung et al., 2009;Toni, 2014): tree with nodes labelled by sentences in L or by true 1 , the root labelled by s, leaves either true or assumptions in A, and non-leaves s with, as children, the elements of the body of some rule in R with head s .
Given an ABA framework L, R, A, , let Ar be the set of all arguments and Att defined as above.Then (Ar, Att) is an AA framework and standard semantics for the latter can be used to determine e.g.grounded, sceptical preferred and ideal extensions Dung et al. (2007). 2  In this paper, a probability space is a pair = (W, P) where W is a finite set of possible worlds and P is a mapping from W into the interval (0, 1] such that w∈W P(w) = 1.Bayesian networks (e.g.see Pearl, 2009) are a widely used mechanism to specify probability spaces, in terms of joint probability distributions.A Bayesian network consists of an acyclic directed graph, whose nodes are random variables with associated (conditional) probabilities.The probabilities at a node specify the probabilities that the random variable at that node takes a value from its domain given the parents of the node, which are all the nodes with an edge to it.In this paper, we consider binary Bayesian networks, where each random variable can take (either of ) two values, that for simplicity we assume to be either true or false.Without loss of generality, we will use atomic propositions and their classical negation to represent assignments of values to random variables.Given some observations on the state of the world, Bayesian networks can be used to infer the probability of random variables taking some values.The independence assumptions underlying Bayesian networks give that, for any

Jury-based probabilistic abstract argumentation
To motivate our use of probabilistic argumentation for forecasting, let us take a closer look at the arguments in Example 1.1.Jo's argument J is based on the lenders' policy (requiring that some preconditions are satisfied) as well as a personal belief.Alex's argument A is based on the causal relationship between the likelihood of an event and a belief.Finally, Pat's argument P is based on the likelihood of an event.Probabilistic abstract argumentation as defined by Dung and Thang ( 2010) is a suitable representational mechanism in this setting as it allows arguments to combine policies, causal relationships and likelihoods, as follows.(1) F = (Ar, Att) is an AA framework; (2) = (W, P) is a probability space; (3) ⊆ W × Ar specifies which arguments are legitimate in worlds in .
Intuitively, a PAA framework combines a standard AA framework, a probability space and a way to ascertain whether arguments are legitimate in worlds in the probability space.The probability space and the notion of legitimacy of arguments in worlds may be seen as the views of a juror.
Here, intuitively, the four worlds may be determined by combinations of the truth value of two independent events (redundancies and interest_increase), and the worlds' probabilities by the product of the probabilities ascribed to the events by Jo. 3 However, in this section, we are interested in the abstract view of probabilistic argumentation, so we will ignore from now on how jurors may instantiate PAA frameworks (and whether thy instantiate them in the same way).
The choice by a juror as to which arguments are legitimate in a world determines which AA framework the juror can reason with in that world.Basically, F w consists of all the arguments legitimate with respect to w and the attack relation between them.
PAA frameworks can be used to infer the probability of arguments with respect to any sceptical semantics for AA, as follows (naturally extending the original definition of grounded probability of arguments in Dung and Thang (2010) to other sceptical semantics): Basically, the χ (grounded, sceptically preferred and ideal) probability of arguments is obtained by summing up the probabilities of the worlds w in (the AA frameworks F w of ) which the arguments belong to the χ -extensions.Note that the uncertainty captured by this notion of (grounded, sceptically preferred, and ideal) probability of arguments is not 'epistemic', in the sense of the epistemic approach to probabilistic argumentation (Hunter et al., 2021), as it does not convey 'the degree to which arguments are believed'; rather it reflects the uncertainty about which possible worlds hold and what can be (defeasibly) inferred in them.
In the remainder of this section, unless specified otherwise, we assume that χ ranges over G, S, I.
Trivially, if n = 1 then the JPAA framework is a PAA framework.Intuitively, each juror decides which worlds and arguments to consider and assigns probabilities to the worlds.The arguments and attacks between them are all drawn from a 'shared' AA framework.This is particularly suitable for forecasting, whereby it makes sense that forecasters (jurors) make decisions after having shared information.(2) P(w 1 ) = 0.18, P(w 2 ) = 0.72, P(w 3 ) = 0.02, P(w 4 ) = 0.08.
These two worlds may result from Pat's belief that the two events, redundancies and interest_increase, are dependent, and either both occur (w 1 ) or neither occur (w 2 ).
Note that we assume that when aggregating views the jurors 'agree' on which semantics (χ ) to adopt for determining their probabilities.Acceptance by preponderance, as defined here, may only work with an odd number of jurors, and require a tie-breaking mechanism otherwise.Note also that other forms of aggregation may be suitable for forecasting, such as weighted averaging of probabilities, including rules borrowed from the computational social choice literature.
Example 4.10 (Examples 4.2 and 4.8 Continued): J is not accepted beyond reasonable doubt (with respect to the grounded semantics) because Prob G (J) = P(w 4 ) = 0.18 < 1 (from Example 4.6).J is likewise not accepted with preponderance (with respect to the grounded semantics) because for Alex argument J is groundedly accepted only in world w 4 too, which has probability 0.08 < 0.5 (from Example 4.8), so that for majority of the jurors the probability of J is less than half.
As this example shows, forecasters (jurors) generate arguments and ascribe their own probabilities to their own worlds, after which their views can be aggregated to form a forecast.In the running example, whichever the method for aggregation, the forecast amounts to not accepting J (and thus not accepting tracker_m).The forecast could be accompanied by a measure, aggregating the jurors' probabilities in accordance with the aggregation methods deployed: we leave this for future work.
In the remainder of this section, unless specified otherwise, we assume as given a generic JPAA framework (F , 1 , . . ., n , 1 , . . ., n ), with i = (W i , P i ) and Prob i χ standing, respectively, for the probability space and the χ probabilities for any juror i ∈ {1, . . ., n}.
We conclude this section by giving some novel, simple properties of (J)PAA and stressing their usefulness for forecasting.The following result, which follows directly from the definitions, sanctions that the (grounded, sceptically preferred and ideal) probability of an argument is guaranteed to be in the correct interval, and provides a characterisation of when it achieves the boundary values.
Proposition 4.11: For any α ∈ Ar, The first part of this result is a sanity check for the probabilities of arguments.By the second part, if all given arguments are 'non-probabilistic' (i.e.∀w ∈ W i , ∀α ∈ Ar : w α), then the (grounded, sceptically preferred or ideal) probability of an argument amounts to membership in the (grounded, sceptically preferred or ideal) extension of the AA framework component of the given JPAA framework.Therefore, this second part indicates, in the case of conditional forecasting, that, if the uncertainty about future events does not affect whether some argument is being accepted, then the probability of that argument reflects the acceptance status.This is an important property for forecasting, as its violation would lead to arbitrarily drawing conclusions.
The following proposition sanctions formally the relationship between the notions of grounded, sceptically preferred and ideal probabilities, based on known relationships between semantics (see Section 3).Proposition 4.12: For any α ∈ Ar, The choice of which notion, amongst grounded, sceptically preferred and ideal probability, to adopt is in general dictated by applications and the level of scepticism of the jurors, exactly as in standard AA.For forecasting, the possibility of being less sceptical than when adopting grounded extensions may be important, as it may increase the ability to forecast in some circumstances.For example, it may be the case that Prob i G (α) = 0 but Prob i S (α) > 0 and Prob i I (α) > 0. Proposition 4.12 however states, e.g. that when Prob i G (α) = 1 then there is no need to consider other sceptical semantics.
Finally, note that the special case of a single juror (namely of Definition 4.1), amounts to a form of probabilistic argumentation according to the constellation approach (Hunter et al., 2021), namely a probability distribution over the AA frameworks with respect to the given worlds.This is a more restricted form of probabilistic argumentation then in other constellation approaches (e.g. by Hunter (2014)), in that it is guaranteed to satisfy (by definition) the following property, for (F , , ) any PAA framework with F = (Ar, Att) and = (W, P): Proposition 4.13: For every α, β ∈ Ar, for every w ∈ W, if α, β ∈ Ar w then (α, β) ∈ Att w iff (α, β) ∈ Att.
Corollary 4.14: For any world w, any χ w is conflict-free with respect to F. This property is important in the context of forecasting, as it imposes a form of 'coherence' across jurors.It may instead be violated in the constellation approach to probabilistic argumentation of Hunter (2014), if in some worlds attacks between arguments are removed while keeping the arguments.

Jury-based Bayesian probabilistic assumption-based argumentation
In general, the definition of PAA framework (Definition 4.1) puts no restriction on the way a probability space = (W, P) is defined.This could, for example, be presented using statistical, probabilistic theories or calculus.In Dung and Thang (2010), a general form of Jury-based Probabilistic ABA is presented whereby the probability space is given in terms of probabilistic rules and probabilistic assumptions, combined with standard rules and standard assumptions.In this section, we show how it can result from the integration of a Bayesian network within ABA, in the spirit of Oo et al. (2020), so that random variables, whose probability is determined by a Bayesian network, form part of the support of arguments.Before we define the resulting JBPABA formally and show that it is an instance of PAA, let us consider again Example 1.1.
Example 5.1: Arguments J, A and P can be obtained from the following rules in the instance of ABA corresponding to logic programming (see Bondarenko et al., 1997), where assumptions are negation-as-failure literals, of the form not_l, with literals l being their contrary: Specifically, argument J is obtained from rules r 1 , r 2 , r 4 , argument A is obtained from the (probabilistic) assumption redundancies, and argument P is obtained from rule r 3 .These arguments are supported by negation as failure assumptions not_high_interest and not_redundancies as well as assumptions that correspond to (binary) random variables redundancies and interest_increase in a simple Bayesian network consisting simply of two nodes labelled by these variables.Definition 5.2: A Bayesian PABA (BPABA) framework is a triple (F a , BN, RV) where is a flat ABA framework; • BN is a Bayesian network over a set RV of binary random variables; • Let the assignment of a binary random variable ϑ ∈ RV to true or false be denoted by ϑ or ¬ϑ, respectively, both referred to as probabilistic literals.Let L RV be the set of all such probabilistic literals.Then: (1) (1)no probabilistic literal is the head of any rule in R; (2) (2)L RV ∩ A = ∅ and L RV ⊆ L.
Intuitively, BPABA frameworks are obtained by adding a structured argumentation layer on top of standard Bayesian networks, with assignments to random variables (i.e.probabilistic literals) required to be distinguished from assumptions in the underlying ABA framework, while being included in its language.Note that this definition of BPABA is inspired by Definition 5 in Oo et al. (2020), which in turn draws inspiration from Definition 3 in Hung (2016a): whereas those works obtain BPABA by instantiating the general definition of PABA in Dung and Thang (2010), we give a (novel) standalone definition, showing (below) that it is an instance of PAA.The form of BPABA has the same computational advantages as the Bayesian instance of PABA in Hung (2016a), and can be supported, e.g. by the tool therefor from Hung (2016b).Note that, differently from Dung and Thang (2010), probabilistic literals cannot be in the head of rules in a BPABA framework, by definition; also, they are subject to the usual (in)dependence assumptions within the Bayesian network.
We often denote a set of probabilistic literals by S V , where V is the set of random variables appearing in S V .Also, in the remainder of this section, unless specified otherwise, we assume as given a generic BPABA framework (F a , BN, RV) and we refer to it as F B .
Example 5.3: The BPABA framework sketched in Example 5.1 consists, formally, of with 5 (1) R = {r 1 , r 2 , r 3 , r 4 }, (2) A = {not_l | l is an atom without prefix not_ in R}, and (3) not_l = l, for every assumption not_l ∈ A, namely the contrary of every assumption of the form not_l in A is l; • the Bayesian network BN with two nodes holding the random variables in RV and P(interest_increase) = 0.1 and P(redundancies) = 0.8. 6 The deployment of BPABA requires a knowledge representation effort, as concerns in particular which sentences should be assumptions and which probabilistic literals.As a rule of thumb, assumptions express epistemic uncertainty, whereas probabilistic literals express aleatory, quantifiable uncertainty, e.g.drawn from statistical analysis of data.For illustration, in Example 5.3, redundancies may be computed from past data on the company and interest_increase may be drawn from financial and political data; instead, not_high_interest and not_redundancies serve the role of making rules r 1 and r 4 defeasible.As an additional illustration, taking an action needs to be an assumption in the BPABA framework of the agent who needs to decide whether to take the action or not, but can be a probabilistic literal in the BPABA framework of an agent, if it is to be taken by another agent.Both probabilistic literals and standard assumptions can be added to the rules to obtain arguments and attacks (see Definition 5.4 below) and debate arguments' acceptability under the chosen semantics, but the choice of the former contributes to the identification of probability spaces (see Definition 5.5 below) whereas the choice of the latter will lead to extensions.Their combination is responsible for probabilistic reasoning outcomes (see Definition 5.8 below).
The notion of argument in BPABA extends the same notion in ABA by allowing probabilistic literals to serve as support, alongside the assumptions of standard ABA.Attacks between BPABA arguments however are still directed against assumptions.

Definition 5.4:
• A BPABA argument α with claim s ∈ L supported by A ⊆ A and a set S V ⊆ L RV of probabilistic literals (denoted by A, S V s) is a finite tree with nodes labelled by sentences in L or by true, the root labelled by s, leaves either true or assumptions in A or elements in S V , and for non-leaves s the children of s are labelled by the elements of the body of some rule r in R with the head of r labelling s .
BPABA frameworks are instances of PAA frameworks, by using the following definition of probability space: Definition 5.5: The probability space (F B ) of BPABA framework F B is defined as (W, P) such that RV there is exactly one probabilistic literal ϑ or ¬ϑ in S V }. • P is the probability distribution over W defined by the Bayesian network BN.
Thus, the Bayesian network component of a BPABA framework is used exclusively to define a probability space, so that we can see BPABA frameworks as instances of PAA frameworks.Note that, although from a purely theoretical viewpoint the use of Bayesian networks alongside ABA frameworks could be replaced by any other means to formalise a probability space to instantiate PAA, the use of Bayesian networks is crucial for deployment in forecasting (as in healthcare Oo et al., 2020) due to the standard use of these mechanisms for representing uncertainty in forecasting scenarios, e.g. as in Stiber et al. (2004).
Basically, we enforce that possible worlds are maximally consistent sets of probabilistic literals, namely assignments (of true or false, but not both) to all random variables.Note that, according to Definition 5.5, a possible world is a consistent set of probabilistic literals; when identifying legitimate arguments in a given possible world (see third bullet above), we basically take those probabilistic literals as facts : then, in the chosen possible world, the chosen semantics can be used to identify (grounded, sceptically preferred or ideal) extensions which are guaranteed to be conflict-free by definition.Given that probabilistic literals are not heads of rules and that their choice is made up-front consistently, with the given world, no inconsistency amongst probabilistic literals can emerge in any such extension.
In the remainder of this section, (F , , ) with F = (Ar, Att) and = (W, P), will be the PAA framework corresponding to our generic F B .
Trivially, the PAA framework corresponding to F B is a PAA framework, as in Definition 4.1.Thus, we can assess the grounded, sceptically preferred and ideal probability of BPABA arguments, as illustrated next.It is easy to see that w i J for all i = 1, 2, 3, 4. Also, the grounded extensions of the AA frameworks with respect to these worlds are such that: Thus, J ∈ G w i iff i = 4 and Prob G (J) = P(w 4 ) = 0.18.Similarly Prob S (J) = Prob I (J) = P(w 4 ) = 0.18.
In addition to assessing the probability of arguments, borrowed from PAA, assessing the probability of sentences can also be defined in BPABA, as follows, where cl(α) is used to denote the claim of argument α (see Definition 5.4).
In Example 5.7 there is a single argument (J) with the sentence tracker_m as claim, and the probability of the sentence coincides with the probability of the argument supporting it.In general, there may be several arguments supporting the same sentence, and the probability of the sentence and of the arguments are related as follows: Proposition 5.9: For any s ∈ L, for χ any of G, S, I:

Prob χ (α).
Proof: Let χ = G.Let W α , for α ∈ Ar, and W s be defined as Therefore α ∈ G w , and, by definition of W s , we have w ∈ W s .
Thus, Prob G (s) = Prob(W s ) = Prob( α∈Ar and cl(α)=s W α ).Since Informally, this proposition sanctions that the probability of a sentence is upperbounded by the sum of the probabilities of arguments admitting that sentence as their claim.Intuitively, this is because the same sentence may be the claim of several legitimate arguments in the same world.Thus, in the forecasting scenario, where the interest is typically on the probability of a sentence (e.g.whether to choose a tracker mortgage in the running example, or whether there will be flooding in some part of the world, in a geopolitical forecasting setting), the view afforded by (J)PABA is more fine-grained and suitable.
Note that BPABA is a generalisation of both standard ABA and Bayesian networks, trivially.In particular, if the given Bayesian network is empty, or the Bayesian network and the ABA framework are 'disconnected', namely no probabilistic literals occur in the body of rules in the ABA framework, then the grounded/sceptically preferred/ideal probability of an argument/sentence is determined by its membership in the grounded/sceptically preferred/ideal (respectively) extension of the ABA framework, by virtue of Proposition 4.11.
We conclude with the jury-based version of BPABA frameworks, inspired by Dung and Thang (2010), whereby different jurors may hold different Bayesian networks, while sharing rules, assumptions and their contraries and probabilistic literals.Definition 5.10: A Jury-based BPABA (JBPABA) framework with n ≥ 1 jurors is a tuple (F a , BN 1 , . . ., BN n ) such that, for every i ∈ {1, . . ., n}, BN i is a Bayesian network over the same set RV of binary random variables and (F a , BN i , RV) is a BPABA.
It is easy to see that JBPABA frameworks are instances of JPAA frameworks.Aggregation of views of different jurors can be similarly defined, as for JPAA frameworks, but at the level of claims of arguments too, e.g. to establish whether tracker_m can be forecast.

Conclusions
We have explored the use towards supporting judgemental forecasting of an existing framework for Probabilistic Abstract Argumentation (PAA) (Dung & Thang, 2010) as well as an instantiation thereof, combining a structured argumentation framework, Assumption-Based Argumentation, and Bayesian networks.In the latter, arguments may be supported by (assignments to) random variable s in the underlying Bayesian network, and the probabilities of these random variable s determine the probability of possible worlds and arguments that are legitimate in them as well as the probability of claims of these arguments.
There are several tracks of work relevant to ours.In terms of computational feasibility of probabilistic argumentation, Fazzinga et al. (2016) study the computational complexity of determining whether a set of arguments is an extension according to four semantics (of complete, grounded, preferred and ideal extensions) for PAA.Čyras et al. (2021) study the computational complexity of Probabilistic Assumption-Based Argumentation in general.It would be interesting to see how JBPABA instantiates those results.
We have focused on sceptical semantics of JPABA in determining the acceptance of arguments/conclusions under uncertainty.Other approaches, notably the probabilistic answer sets of Baral et al. (2009), develop probabilistic extensions of logic programming, under (credulous) answer sets.It would be interesting to see whether credulous semantics (e.g.stable extensions) could be fruitfully deployed in JBPABA and a forecasting setting specifically.
The existing approach of Irwin et al. (2022aIrwin et al. ( , 2022b) ) to argumentation-based forecasting introduces a notion of rationality to constrain forecasts to those which are deemed rational before an aggregated prediction is produced: it would be interesting to study suitable notions of rationality of forecasters within the setting of probabilistic argumentation.
Future work also includes developing concrete forecasting methodologies and systems based on probabilistic argumentation, in particular of the form we have considered.This will need to be supported by suitable computational machinery for probabilistic argumentation, e.g.integrating existing computational methods for ABA (such as dispute derivations by Dung et al. (2006Dung et al. ( , 2007)); Toni (2013)), or for AA (such as dispute derivations by Thang et al. (2009)), and algorithms for Bayesian networks (Pearl, 2009)), and/or building upon existing systems for probabilistic argumentation (e.g. as in Hung, 2016a;Oo et al., 2020).Forecasting will require the system to accommodate multiple users, information sharing as well as user-friendly interfaces, possibly leveraging on some aspects of Arg&Dec (www.arganddec.com) or the forecasting system of Irwin et al. (2022a).It would also be interesting to experiment with any system for forecasting based on probabilistic argumentation, in particular with expert users in specific domains (e.g.intelligence).
The superforecasting experiment (Tetlock & Gardner, 2016), in which 5000 volunteers took part in geopolitical forecasting tournaments, demonstrated that a specific cognitive-intellectual approach promoted better forecasting (Mellers et al., 2015a(Mellers et al., , 2015b)): we posit that the use of JBPABA may 'unchain' a skilled forecaster's knowledge.Further, Lawrence et al. (2006) have emphasised the importance of synthesising qualitative and quantitative information in judgemental forecasting, which we believe our formalism achieves here.It would be interesting to verify this experimentally.
Definition 4.5 (Adapted from Dung & Thang, 2010): Let (F , , ) be a PAA framework, with F = (Ar, Att) and = (W, P).Let w ∈ W and let the grounded, sceptically preferred and ideal extensions of F w be denoted by G w , S w and I w , respectively.Then, the grounded, sceptically preferred and ideal probabilities of argument α ∈ Ar are obtained as follows, for χ = G, S, I, respectively: Prob χ (α) = w∈W:α∈χ w P(w).
): J is not groundedly accepted in three worlds w 1 , w 2 and w 3 .However, J is groundedly accepted in world w 4 .Hence J belongs to the grounded extension of F w 4 only and the grounded probability of J is Prob G (J) = P(w 4 ) = 0.18.(Similarly for the other semantics.)Definition 4.7 (Adapted from Dung & Thang, 2010): A jury-based probabilistic AA Juror Alex may fully agree with Jo on worlds and legitimacy of arguments, but not on probabilities, as follows: 4

Definition 5. 6 :
The PAA framework corresponding to F B is (F , , ) with• F = (Ar, Att) where Ar is the set of all BPABA arguments and Att the set of all attacks amongst them, as in Definition 5.4.• = (F B ) = (W, P), as in Definition 5.5.• For any w ∈ W and BPABA argument α ∈ Ar, if α is of the form A, S V s, then w α iff S V ⊆ w.

Example 5. 7 :
The PAA framework corresponding to the BPABA framework in Example 5.3 consists of the AA framework with, amongst its arguments, 7 arguments J, A, P sketched in Example 5.1.Formally these arguments are J : {not_high_interest, not_redundancies}, ∅ tracker_m A : ∅, {redundancies} redundancies P : ∅, {interest_increase} high_interest such that A and P attack J.Let R stand for redundancies and I stand for interest_increase.Then, the probability space of the PAA framework corresponding to the BPABA framework in question has possible worlds: W = {R, ¬R} × {I, ¬I}.Let w 1 = {R, I}, w 2 = {R, ¬I}, w 3 = {¬R, I}, w 4 = {¬R, ¬I} Then W = {w 1 , w 2 , w 3 , w 4 } and the probability distribution in the PAA corresponding to the BPABA framework in question is as in Example 4.2.
for χ = G.The proof for χ = S, χ = I is similar.