=Paper=
{{Paper
|id=Vol-3086/paper3
|storemode=property
|title=Forging Argumentative Explanations from Causal Models
|pdfUrl=https://ceur-ws.org/Vol-3086/paper3.pdf
|volume=Vol-3086
|authors=Antonio Rago,Fabrizio Russo,Emanuele Albini,Pietro Baroni,Francesca Toni
|dblpUrl=https://dblp.org/rec/conf/aiia/0001RABT21
}}
==Forging Argumentative Explanations from Causal Models==
https://ceur-ws.org/Vol-3086/paper3.pdf
Forging Argumentative Explanations from Causal Models
Antonio Ragoa , Fabrizio Russoa , Emanuele Albinia , Pietro Baronib and
Francesca Tonia
a
Department of Computing, Imperial College London, UK
b
Dipartimento di Ingegneria dell’Informazione, Università degli Studi di Brescia, Italy
Abstract
We introduce a conceptualisation for generating argumentation frameworks (AFs) from causal
models for the purpose of forging explanations for models’ outputs. The conceptualisation is
based on reinterpreting properties of semantics of AFs as explanation moulds, which are means
for characterising argumentative relations. We demonstrate our methodology by reinterpreting
the property of bi-variate reinforcement in bipolar AFs, showing how the extracted bipolar
AFs may be used as relation-based explanations for the outputs of causal models.
Keywords
Explainable AI, Argumentation frameworks, Causal models
1. Introduction
The field of explainable AI (XAI) has in recent years become a major focal point of the
efforts of researchers, with a wide variety of models for explanation being proposed (see
[1] for an overview). More recently, incorporating a causal perspective into explanations
has been explored by some, e.g. [2, 3, 4]. The link between causes and explanations has
long been studied [5]; indeed, the two have even been equated (under a broad sense of the
concept of “cause”) [6]. Causal reasoning is, in fact, how humans explain to one another
[7], and so mimicking such a trend lends credence to the hypothesis that machines should
do likewise. Further, research from the social sciences [8] has indicated the value of
causal links, particularly in the form of counterfactual reasoning, within explanations,
and that the importance of such information surpasses that of probabilities or statistical
relationships for users.
Despite these findings, many of the approaches for generating explanations for AI
models have, nevertheless, neglected causality as a potential drive for explainability. Some
of the most popular methods are heuristic and model-agnostic [9, 10], and, although
5th Workshop on Advances In Argumentation In Artificial Intelligence (AI3 2021)
Envelope-Open antonio@imperial.ac.uk (A. Rago); fabrizio@imperial.ac.uk (F. Russo); emanuele@imperial.ac.uk
(E. Albini); pietro.baroni@unibs.it (P. Baroni); f.toni@imperial.ac.uk (F. Toni)
GLOBE https://www.doc.ic.ac.uk/~afr114/ (A. Rago); https://briziorusso.github.io/ (F. Russo);
https://www.emanuelealbini.com/ (E. Albini); https://pietro-baroni.unibs.it/ (P. Baroni);
https://www.imperial.ac.uk/people/f.toni (F. Toni)
Orcid 0000-0001-5323-7739 (A. Rago); 0000-0003-2964-4638 (E. Albini); 0000-0001-5439-9561 (P. Baroni);
0000-0001-8194-1459 (F. Toni)
© 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution
4.0 International (CC BY 4.0).
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073
CEUR Workshop Proceedings (CEUR-WS.org)
they are useful, particularly with regards to their wide-ranging applicability, they neglect
how models are determining their outputs and therefore the underlying causes therein.
This has arguably left a chasm between how explanations are provided by models at the
forefront of XAI technology and what users actually require from explanations [11].
Meanwhile, computational argumentation (see [12, 13] for recent overviews) has received
increasing interest in recent years as a means for providing explanations of the outputs of
a number of AI models, e.g. recommender systems [14], classifiers [15], Bayesian networks
[16] and PageRank [17]. Argumentative explanations have also been advocated in the
social sciences [18, 8], and several works focus on the power of argumentation to provide
a bridge between explained models and users, validated by user studies [19, 20]. While
argumentative explanations are wide-ranging in their application (see [21, 22] for recent
surveys), the links between causal models and argumentative explanations have remained
largely unexplored to date.
In this paper, we introduce a conceptualisation for generating argumentation frame-
works (AFs) with any number of dialectical relations as envisaged in [23, 24], from causal
models for the purpose of forging explanations for the models’ outputs. Like [25], we focus
not on explaining by features, but instead by relations, hence the use of argumentation as
the underpinning explanatory mechanism. After giving the necessary background (§2), we
show how properties of argumentation semantics from the literature can be reinterpreted
to serve as explanation moulds, i.e. means for characterising argumentative relations (§3).
In (§4) we propose a way to define explanation moulds based on inverting properties
of argumentation semantics. Briefly, the idea is to detect, inside a causal model, the
satisfaction of the conditions specified by some semantics property: if these conditions
are satisfied by some influence in the causal model, then the influence can be assigned an
explanatory role by casting it as a dialectical relation, whose type is in correspondence
with the detected property. The identified dialectical relations compose, altogether,
an argumentation framework. We demonstrate our methodology by reinterpreting the
property of bi-variate reinforcement [26] from bipolar AFs [27] and then showing in (§5)
how the extracted bipolar AFs may be used as counterfactual explanations for the outputs
of causal models representing different classification methods. Finally, we discuss related
work (§6) before concluding, indicating potentially fruitful future work (§7).
Overall, we make the following main contributions:
• We propose a novel concept for defining relation-based explanations for causal
models by inverting properties of argumentation semantics.
• We use this concept to define a novel form of reinforcement explanation (RX) for
causal models.
• We show deployability of RXs with two machine-learning models, from which causal
models are drawn.
2. Background
Our method relies upon causal models and some notions from computational argumenta-
tion. We provide core background for both.
Causal models. A causal model [28] is a triple ⟨𝑈 , 𝑉 , 𝐸⟩, where:
• 𝑈 is a (finite) set of exogenous variables, i.e. variables whose values are determined
by external factors (outside the causal model);
• 𝑉 is a (finite) set of endogenous variables, i.e. variables whose values are determined
by internal factors, namely by (the values of some of the) variables in 𝑈 ∪ 𝑉;
• each variable may take any values in its associated domain; we refer to the domain
of 𝑊𝑖 ∈ 𝑈 ∪ 𝑉 as 𝒟 (𝑊𝑖 );
• 𝐸 is a (finite) set of structural equations that, for each endogenous variable 𝑉𝑖 ∈ 𝑉,
define 𝑉𝑖 ’s values as a function 𝑓𝑉𝑖 of the values of 𝑉𝑖 ’ parents 𝑃𝐴(𝑉𝑖 ) ⊆ 𝑈 ∪ 𝑉 ⧵ {𝑉𝑖 }.
Example 1. Let us consider a simple causal model ⟨𝑈 , 𝑉 , 𝐸⟩ comprising 𝑈 = {𝑈1 , 𝑈2 },
𝑉 = {𝑉1 , 𝑉2 } and for all 𝑊𝑖 ∈ 𝑈 ∪ 𝑉, 𝒟 (𝑊𝑖 ) = {⊤, ⊥}. Figure 1i (we ignore Figure 1ii for the
moment: this will be discussed later in §4) visualises the variables’ parents, and Table 1
gives the combinations of values for the variables resulting from the structural equations
𝐸. This may represent a group’s decision on whether or not to enter a restaurant, with
variables 𝑈1 : “margherita” is spelt correctly on the menu, not like the drink; 𝑈2 : there is
pineapple on the pizzas; 𝑉1 : the pizzeria seems to be legitimately Italian; and 𝑉2 : the
group chooses to enter the pizzeria.
Figure 1: (i) Variables and parents for Example 1, with parents indicated by dashed arrows (for
example {𝑈1 , 𝑈2 } = 𝑃𝐴(𝑉1 ), i.e. 𝑈1 and 𝑈2 are the parents of 𝑉1 ). (ii) SAF explanation (see §3) for
the assignment to exogenous variables u ∈ 𝒰 such that 𝑓𝑈1 [u] = ⊤ and 𝑓𝑈2 [u] = ⊤.
Given a causal model ⟨𝑈 , 𝑉 , 𝐸⟩ where 𝑈 = {𝑈1 , … , 𝑈𝑖 }, we denote with 𝒰 = 𝒟 (𝑈1 )×…×𝒟 (𝑈𝑖 )
the a set of all possible combinations of values of the exogenous variables (realisations).
With an abuse of notation, we refer to the value of any variable 𝑊𝑖 ∈ 𝑈 ∪ 𝑉 given u ∈ 𝒰
as 𝑓𝑊𝑖 [u]: if 𝑊𝑖 is an exogenous variable, 𝑓𝑊𝑖 [u] will be its assigned value in u; if 𝑊𝑖 is
𝑈1 𝑈2 𝑉1 𝑉2
⊤ margherita ⊤ pineapple ⊥ ∼Italian ⊥ ∼enter
⊤ margherita ⊥ ∼pineapple ⊤ Italian ⊤ enter
⊥ margarita ⊤ pineapple ⊥ ∼Italian ⊥ ∼enter
⊥ margarita ⊥ ∼pineapple ⊥ ∼Italian ⊥ ∼enter
Table 1
Combinations of values (⊤ or ⊥) resulting from the structural equations for Example 1. Here
we also indicate the intuitive reading of the assignment of values to variables according to the
illustration in Example 1 (for example, the assignment of ⊤ to 𝑈1 may be read as “margherita” is
spelt correctly on the menu – simply given as ‘margherita’ in the table, and the assignment of 𝑈2
to ⊥ may be read as there is no pineapple on the pizzas – simply given as ‘∼pineapple’ in the
table).
an endogenous variable, it will be the value dictated by the structural equations in the
causal model.
We use the do operator [29] to indicate interventions, i.e., for any variable 𝑉𝑖 ∈ 𝑉 and
value thereof 𝑣𝑖 ∈ 𝒟 (𝑉𝑖 ), 𝑑𝑜(𝑉 = 𝑣𝑖 ) implies that the function 𝑓𝑉𝑖 is replaced by the constant
function 𝑣𝑖 , and for any variable 𝑈𝑖 ∈ 𝑈 and value thereof 𝑢𝑖 ∈ 𝒟 (𝑈𝑖 ), 𝑑𝑜(𝑈𝑖 = 𝑢𝑖 ) implies
that 𝑈𝑖 is assigned 𝑢𝑖 .
Argumentation. In general, an argumentation framework (AF) is any tuple ⟨𝒜 , ℛ1 , … , ℛ𝑙 ⟩,
with 𝒜 a set (of arguments), 𝑙 > 0 and ℛ𝑖 ⊆ 𝒜 × 𝒜, for 𝑖 ∈ {1, … , 𝑙}, (binary and directed)
dialectical relations between arguments [23, 24]. In the abstract argumentation [30]
tradition, arguments in these AFs are unspecified abstract entities that can be instan-
tiated differently to suit different settings of deployment. Several specific choices of
dialectical relations can be made, giving rise to specific AFs instantiating the above
general definition, including abstract AFs (AAFs) [30], with 𝑙 = 1 (and ℛ1 a dialectical
relation of attack, referred to later as ℛ− ), support AFs (SAFs) [31], with 𝑙 = 1 (and
ℛ1 a dialectical relation of support, referred to later as ℛ+ ), and bipolar AFs (BAFs)
[27], with 𝑙 = 2 (and ℛ1 and ℛ2 dialectical relations of attack and support, respectively,
referred to later as ℛ− and ℛ+ ).
The meaning of AFs (including the intended dialectical role of the relations) may be
given in terms of gradual semantics (e.g. see [24, 32] for BAFs), defined, for AFs with
arguments 𝒜, by means of mappings 𝜎 ∶ 𝒜 → 𝕍, with 𝕍 a given set of values of interest
for evaluating arguments.
The choice of gradual semantics for AFs may be guided by properties that the mappings
𝜎 should satisfy (e.g. as in [26, 32]). We will utilise, in §4, a variant of the property of
bi-variate reinforcement for BAFs from [26].
3. From Causal Models to Explanation Moulds and
Argumentative Explanations
In this section we see the task of obtaining explanations for causal models’ assignments
of values to variables as a two-step process: first we define moulds characterising the core
ingredients of explanations; then we use these moulds to obtain, automatically, (instances
of) AFs as argumentative explanations. Moulds and explanations are defined in terms
of influences between variables in the causal model, focusing on those from parents to
children given by the causal structure underpinning the model, as follows.
Definition 1. Let 𝑀 = ⟨𝑈 , 𝑉 , 𝐸⟩ be a causal model. The influence graph corresponding to
𝑀 is the pair ⟨𝒱 , ℐ ⟩ with:
• 𝒱 = 𝑈 ∪ 𝑉 is the set of all (exogenous and endogenous) variables;
• ℐ ⊆ 𝒱 × 𝒱 is defined as ℐ = {(𝑊1 , 𝑊2 )|𝑊1 ∈ 𝑃𝐴(𝑊2 )} (referred to as the set of
influences).
Note that, while straightforward, the concept of influence graph (closely related to the
notion of causal diagram [33]) is useful as it underpins much of what follows.
Next, the idea underlying explanation moulds is that, typically, inside the causal model,
some variables affect others in a way that may not be directly understandable or even
cognitively manageable by a user. The influence graph synthetically expresses which
variables affect which others but does not give an account of how the influences actually
occur in the context (namely, the values given to the exogenous variables) that a user
may be interested in. Thus, the perspective we take is that each influence can be assigned
an explanatory role, indicating how that influence is actually working in that context.
The explanatory roles ascribable to influences can be regarded as a form of explanatory
knowledge which is user specific: different users may be willing (and/or able) to accept
explanations built using different sets of explanatory roles as they correspond to their
understanding of how variables may affect each other. We assume that each explanatory
role is specified by a relation characterisation, i.e. a Boolean logical requirement, which
can be used to mould the explanations to be presented to the users by indicating which
relations play a role in the explanations.
Definition 2. Given a causal model ⟨𝑈 , 𝑉 , 𝐸⟩ and its corresponding influence graph ⟨𝒱 , ℐ ⟩,
an explanation mould is a non-empty set:
{𝑐1 , … , 𝑐𝑚 }
where for all 𝑖 ∈ {1, … , 𝑚}, 𝑐𝑖 ∶ 𝒰 × ℐ → {⊤, ⊥} is a relation characterisation, in the form
of a Boolean condition expressed in some formal language. Given some u ∈ 𝒰 and
(𝑊1 , 𝑊2 ) ∈ ℐ, if 𝑐𝑖 (u, (𝑊1 , 𝑊2 )) = ⊤ we say that the influence (𝑊1 , 𝑊2 ) satisfies 𝑐𝑖 for u.
Note that we are not prescribing any formal language for specifying relation character-
isations, as several such languages may be suitable.
Given an assignment u to the exogenous variables, based on an explanation mould, we
can obtain an AF including, as (different) dialectical relations, the influences satisfying
the (different) relation characterisations for the given u. Thus, the choice of relation
characterisations is to a large extent dictated by the specific form of AF the intended
users expect. Before defining argumentative explanations formally, we give an illustration.
Example 1 (Cont.). Let us imagine a situation where one would like to explain the
behaviour of the causal model from Figure 1i and Table 1 with a SAF (see §2). We thus
require one single form of relation (i.e. support) to be extracted from the corresponding
influence graph ⟨{𝑈1 , 𝑈2 , 𝑉1 , 𝑉2 }, {(𝑈1 , 𝑉1 ), (𝑈2 , 𝑉1 ), (𝑉1 , 𝑉2 )}⟩. In order to define the explana-
tion mould for such a situation, we note that the behaviour defining this relation could
be characterised as changing the state of rejected arguments that it supports to accepted
when the supporting argument’s state is accepted. In our simple causal model, accepted
arguments may amount to variables assigned to value ⊤ and rejected arguments may
amount to variables assigned to value ⊥. Thus, the intended behaviour can be captured
by a relation characterisation 𝑐𝑠 such that, given u ∈ 𝒰 and (𝑊1 , 𝑊2 ) ∈ ℐ:
𝑐𝑠 (u, (𝑊1 , 𝑊2 )) = ⊤ iff
(𝑓𝑊1 [u] = ⊤∧𝑓𝑊2 [u] = ⊤ ∧𝑓𝑊2 [u, 𝑑𝑜(𝑊1 = ⊥)] = ⊥)∨
(𝑓𝑊1 [u] = ⊥ ∧𝑓𝑊2 [u] = ⊥ ∧ 𝑓𝑊2 [u, 𝑑𝑜(𝑊1 = ⊤)] = ⊤).
Then, for the assignment to exogenous variables u ∈ 𝒰 such that 𝑓𝑈1 [u] = ⊤ and 𝑓𝑈2 [u] = ⊥,
we may obtain the SAF in Figure 1ii (visualised as a graph with nodes as arguments and
edges indicating elements of the support relation). For illustration, consider (𝑈1 , 𝑉1 ) ∈ ℐ
for this u. We can see from Table 1 that 𝑓𝑉1 [u] = ⊤ and also that 𝑓𝑉1 [u, 𝑑𝑜(𝑈1 = ⊥)] = ⊥
and thus from the above it is clear that 𝑐𝑠 (u, (𝑈1 , 𝑉1 )) = ⊤ and thus the influence is of the
type of support that 𝑐𝑠 characterises. Meanwhile, consider (𝑈2 , 𝑉1 ) ∈ ℐ for the same u: the
fact that 𝑓𝑈2 [u] = ⊥ and 𝑓𝑉1 [u] = ⊤ means that 𝑐𝑠 (u, (𝑈2 , 𝑉1 )) = ⊥ and thus the influence is
not cast as a support. Indeed, if we consider the first and second rows of Table 1, we
can see that 𝑈2 being true actually causes 𝑉1 to be false, thus it is no surprise that the
influence is not cast as a support and plays no role in the resulting SAF. If we wanted for
this influence to play a role, we could, for example, choose to incorporate an additional
relation of attack into the explanation mould, to generate instead BAFs (see §2) as
argumentative explanations. This example thus shows how explanation moulds must be
designed to fit causal models depending on external explanatory requirements dictated
by users. It should be noted also that some explanation moulds may be unsuitable to
some causal models, e.g. the explanation mould with the earlier 𝑐𝑠 would not be directly
applicable to causal models with variables with non-binary or continuous domains.
In general, AFs serving as argumentative explanations can be generated as follows.
Definition 3. Given a causal model ⟨𝑈 , 𝑉 , 𝐸⟩, its corresponding influence graph ⟨𝒱 , ℐ ⟩,
some u ∈ 𝒰 and an explanation mould {𝑐1 , … , 𝑐𝑚 }, an argumentative explanation is an AF
⟨𝒜 , ℛ1 , … ℛ𝑚 ⟩, where
• 𝒜 ⊆ 𝒱, and
• ℛ1 , … , ℛ𝑚 ⊆ ℐ ∩ (𝒜 × 𝒜 ) such that, for any 𝑖 = 1 … 𝑚, ℛ𝑖 = {(𝑊1 , 𝑊2 ) ∈ ℐ ∩ (𝒜 ×
𝒜 )|𝑐𝑖 (u, (𝑊1 , 𝑊2 )) = ⊤}.
Note that we have left open the choice of 𝒜 (as a generic, possibly non-strict subset of
𝒱). In practice, 𝒜 may be the full 𝒱, but we envisage that users may prefer to restrict
attention to some variables of interest (for example, excluding variables not “involved” in
any influence satisfying the relation characterisations).
Example 1 (Cont.). The behaviour of the causal model from Figure 1i and Table 1 for u
such that 𝑓𝑈1 [u] = ⊤ and 𝑓𝑈2 [u] = ⊤, using the explanation mould {𝑐𝑠 } given earlier, can
be captured by either of the two SAFs (argumentative explanations) below, depending
on the choice of 𝒜:
• the SAF in Figure 1ii, where every variable is an argument;
• the SAF with the same support relation but 𝑈2 excluded from 𝒜, as not “involved”
and thus not contributing to the explanation.
Both SAFs explain that 𝑓𝑉1 [u] = ⊤ is supported by 𝑓𝑈1 [u] = ⊤, in turn supporting
𝑓𝑉2 [u] = ⊤ . Namely, the causal model recommends that the group should enter the
pizzeria because the pizzeria seems legitimately Italian, given that “margherita” is spelt
correctly on the menu. Note that the pineapple not being on the pizza could also be seen
as a support towards the pizzeria being legitimately Italian, the inclusion of which could
be achieved with a slightly more complex explanation mould.
4. Inverting Properties of Argumentation Semantics:
Reinforcement Explanations
The choice (number and form) of relation characterisations in explanation moulds is
crucial for the generation of explanations concerning the value assignments to endoge-
nous variables in the causal models. Even after having decided which argumentative
relations to include in the AF/argumentative explanation, the definition of the relation
characterisations is non-trivial, in general. In this section we demonstrate a novel con-
cept for utilising properties of gradual semantics for AFs for the definition of relation
characterisations and the consequent extraction of argumentative explanations.
The common usage of these properties in computational argumentation can be roughly
equated to: if a semantics, given an AF, satisfies some desirable properties, then the
semantics is itself desirable (for the intended context, where those properties matter).
We propose a form of inversion of this notion for use in our XAI setting, namely: if some
desirable properties are identified for the gradual semantics of (still unspecified) AFs,
then these properties can guide the definition of the dialectical relations underpinning
the AFs. For this inversion to work, we need to identify first and foremost a suitable
notion of gradual semantics for the AFs we extract from causal models. Given that, with
our AFs, we are trying to explain the results obtained from underlying causal models, we
cannot impose just any gradual semantics from the literature, but need to make sure that
we capture, with the chosen semantics, the behaviour of the causal model itself. This is
similar, in spirit, to recent work to extract (weighted) BAFs from multi-layer perceptrons
(MLPs) [34], using the underlying computation of the MLPs as a gradual semantics, and
to the proposals to explain recommender systems (RSs) via tripolar AFs [35] or BAFs
[20], using the underlying predicted ratings by the RSs as a gradual semantics.
A natural semantic choice for causal models, since we are trying to explain why
endogenous variables are assigned specific values in their domains given assignments to
the exogenous variables, is to use the assignments themselves as a gradual semantics.
Then, the idea of inverting properties of semantics to obtain dialectical relations in AFs
can be recast to obtain relation characterisations in explanation moulds as follows: given
an influence graph and a selected value assignment to exogenous variables, if an influence
satisfies a given, desirable property, then the influence can be cast as part of a dialectical
relation in the resulting AF.
Naturally, for this inversion to be useful, we need to identify useful properties from
an explanatory viewpoint. We will illustrate this concept with the property of bi-
variate reinforcement for BAFs [26], which we posit is generally intuitive in the realm of
explanations. Bi-variate reinforcement is defined when the set of values 𝕍 for evaluating
arguments is equipped with a pre-order <. Intuitively, bi-variate reinforcement states that1
strengthening an attacker (a supporter) cannot strengthen (cannot weaken, respectively)
an argument it attacks (supports, respectively), where strengthening an argument amounts
to increasing its value from 𝑣1 ∈ 𝕍 to 𝑣2 ∈ 𝕍 such that 𝑣2 > 𝑣1 (whereas weakening an
argument amounts to decreasing its value from such 𝑣2 to 𝑣1 ). In our formulation of
this property, we require that increasing the value of variables represented as attackers
(supporters) can only decrease (increase, respectively) the values of variables they attack
(support, respectively).
Property 1. Given a causal model ⟨𝑈 , 𝑉 , 𝐸⟩ such that, for each 𝑊𝑖 ∈ 𝑈 ∪𝑉, the domain 𝒟 (𝑊𝑖 )
is equipped with a pre-order <,2 and given its corresponding influence graph ⟨𝒱 , ℐ ⟩, an
argumentative explanation ⟨𝒜 , ℛ− , ℛ+ ⟩ for u ∈ 𝒰 satisfies causal reinforcement iff for
any (𝑊1 , 𝑊2 ) ∈ ℐ where 𝑤1 = 𝑓𝑊1 [u], for any 𝑤− ∈ 𝒟 (𝑊1 ) such that 𝑤− < 𝑤1 , and for any
𝑤+ ∈ 𝒟 (𝑊1 ) such that 𝑤+ > 𝑤1 :
• if (𝑊1 , 𝑊2 ) ∈ ℛ− , then 𝑓𝑊2 [u, 𝑑𝑜(𝑊1 = 𝑤+ )] ≤ 𝑓𝑊2 [u] and 𝑓𝑊2 [u, 𝑑𝑜(𝑊1 = 𝑤− )] ≥
𝑓𝑊2 [u];
• if (𝑊1 , 𝑊2 ) ∈ ℛ+ , then 𝑓𝑊2 [u, 𝑑𝑜(𝑊1 = 𝑤+ )] ≥ 𝑓𝑊2 [u] and 𝑓𝑊2 [u, 𝑑𝑜(𝑊1 = 𝑤− )] ≤
𝑓𝑊2 [u].
We can then invert this property to obtain an explanation mould. In doing so, we
introduce slightly stricter conditions to ensure that influencing variables that have no effect
on influenced variables do not constitute both an attack and a support, a phenomenon
which we believe would be counter-intuitive from an explanation viewpoint.
1
Here, we ignore the intrinsic basic strength of arguments used in the formal definition in [26].
2
With an abuse of notation we use the same symbol for all pre-orders.
Definition 4. Given a causal model ⟨𝑈 , 𝑉 , 𝐸⟩ such that, for each 𝑊𝑖 ∈ 𝑈 ∪ 𝑉, the domain
𝒟 (𝑊𝑖 ) is equipped with a pre-order <, and given its corresponding influence graph ⟨𝒱 , ℐ ⟩,
a reinforcement explanation mould is an explanation mould {𝑐− , 𝑐+ } such that, given some
u ∈ 𝒰 and (𝑊1 , 𝑊2 ) ∈ ℐ, letting 𝑤1 = 𝑓𝑊1 [u]:
• 𝑐− (u, (𝑊1 , 𝑊2 )) = ⊤ iff:
1. ∀𝑤+ ∈ 𝒟 (𝑊1 ) such that 𝑤+ > 𝑤1 , it holds that 𝑓𝑊2 [u, 𝑑𝑜(𝑊1 = 𝑤+ )] ≤ 𝑓𝑊2 [u];
2. ∀𝑤− ∈ 𝒟 (𝑊1 ) such that 𝑤− < 𝑤1 , it holds that 𝑓𝑊2 [u, 𝑑𝑜(𝑊1 = 𝑤− )] ≥ 𝑓𝑊2 [u];
3. ∃≥1 𝑤+ ∈ 𝒟 (𝑊1 ) or ∃≥1 𝑤− ∈ 𝒟 (𝑊1 ) satisfying strictly the inequality conditions
in points 1 and 2 above.
• 𝑐+ (u, (𝑊1 , 𝑊2 )) = ⊤ iff:
1. ∀𝑤+ ∈ 𝒟 (𝑊1 ) such that 𝑤+ > 𝑤1 , it holds that 𝑓𝑊2 [u, 𝑑𝑜(𝑊1 = 𝑤+ )] ≥ 𝑓𝑊2 [u];
2. ∀𝑤− ∈ 𝒟 (𝑊1 ) such that 𝑤− < 𝑤1 , it holds that 𝑓𝑊2 [u, 𝑑𝑜(𝑊1 = 𝑤− )] ≤ 𝑓𝑊2 [u];
3. ∃≥1 𝑤+ ∈ 𝒟 (𝑊1 ) or ∃≥1 𝑤− ∈ 𝒟 (𝑊1 ) satisfying strictly the inequality conditions
in points 1 and 2 above.
We call any argumentative explanation resulting from the explanation mould {𝑐− , 𝑐+ } a
reinforcement explanation (RX).
Note that, as for generic argumentative explanations, we do not commit in general to
any choice of 𝒜 in RXs.
Proposition 1. Any RX satisfies causal reinforcement.
Proof. Follows directly from the definition of Property 1 and Definition 4.
The satisfaction of the property of causal reinforcement indicates how RXs could
be used counterfactually, given that the results of changes to the variables’ values on
influenced variables are guaranteed. For example, if a user is looking to increase an
influenced variable’s value, supporters (attackers) indicate variables whose values should
be increased (decreased, respectively). In the following sections, we will explore the
potential of this capability when causal models provide abstractions of classifiers whose
output needs explaining.
5. Reinforcement Explanations for Classification
In this section, we instantiate causal models for two different AI models commonly used
for classification in the literature and preliminarily discuss a potential use of RXs in this
context.
The two classification models that we use to instantiate causal models are Bayesian
network classifiers (BCs) and classifiers built from feed-forward neural networks (NNs).
Given some assignments to input variables I (from the variables’ domains), these classifiers
can be seen as determining the most likely value for classification variables, which, in
Figure 2: A schematic view of classification by BCs and NNs. We assume C = {𝐶1 , … , 𝐶𝑘 }, for
𝑘 ≥ 1, with each 𝐶𝑖 a binary classification variable, with values 𝑐𝑖1 and 𝑐𝑖0 , such that 𝑃(𝐶𝑖 = 𝑐𝑖0 ) =
1 − 𝑃(𝐶𝑖 = 𝑐𝑖1 ); 𝑐𝑖 is the value for 𝐶𝑖 whose probability 𝑃 exceeds the threshold (𝜃). Assuming that
the threshold is suitably chosen so that 𝑐𝑖 is uniquely defined for each 𝐶𝑖 , the classifier can be
equated to the function ℳ such that ℳ(x) = (𝑐1 … , 𝑐𝑘 ).
this paper, we assume to be binary, in a given set C. Thus, the classification task may
be seen as a mapping ℳ(x) returning, for assignment x to input variables, either 1 or
0 (for the classification variables in C) depending on whether the probability exceeds a
given threshold 𝜃. We summarise the classification process in Figure 2. Note that the
choice of threshold is crucial to guarantee that a single value 𝑐𝑖 is determined by ℳ for
each classification variable 𝐶𝑖 : if 𝜃 is too high, then no value may be computed, whereas
if 𝜃 is too low, the probability of both values may exceed it. Note also that, in the case
of NNs, the probabilities may result from using, e.g., a softmax activation for the output
layer. Furthermore, note that for the purposes of this paper, the underpinning details of
these classifiers and how they can be obtained are irrelevant and will be ignored. In other
words, we treat the classifier as a black-box, as standard in much of the XAI literature,
and explain its outputs in terms of its inputs.
We represent the classification task by a (naive) BC or by a NN with the following
causal model:
Definition 5. A causal model for a naive BC or classifier built from a NN is a causal
model ⟨𝑈𝐶 , 𝑉𝐶 , 𝐸𝐶 ⟩, where:
• 𝑈𝐶 consists of the input variables I of the classifier, with their respective domains;
• 𝑉𝐶 = C such that, for each 𝐶𝑖 ∈ C, 𝒟 (𝐶𝑖 ) = {𝑐𝑖1 , 𝑐𝑖0 };
• 𝐸𝐶 corresponds to the computation of the probability values 𝑃(𝐶𝑖 = 𝑐𝑖1 )) by the
classifier (see Figure 2).
ℐ𝐶 = 𝑈𝐶 × 𝑉𝐶 represents the influences in the causal model for the classifier; these are
such that the exogenous variables 𝑈𝐶 are densely connected to the endogenous variables
𝑉𝐶 .
In line with our assumptions for RXs, we assume that the variables’ domains are
equipped with a pre-order.
On this basis, we envisage the following use of RXs as actionable explanations in
contexts where the user has a classification goal to reach and has control on (some of)
the input variables.
Given a situation with an undesired classification outcome (e.g. a rejected loan
application) and an explanation indicating the relevant attackers and supporters, if a
user would like to decrease the probability of the current classification, s/he would look
to increase (decrease) the value of the corresponding variable’s attackers (supporters,
respectively), in line with Property 1.
A broader investigation of the possible uses of RXs is left to future work.
6. Related Work
The role of causality within explanations for AI models has received increasing attention
of late. [2] define a framework for determining the causal effects between features
and predictions using a variational autoencoder. The detection of causal relations and
explanations between arguments within text has also proven effective within NLP [36].
[3] give causal explanations for NNs in that they train a separate NN by masking features
to determine causal relations (in the original NN) from the features to the classifications.
Generative causal explanations of black box classifiers [37] are built by learning the latent
factors involved in a classification, which are then included in a causal model. [38] take
a different approach, proposing a general framework for constructing structural causal
models with deep learning components, allowing tractable counterfactual inference. Other
approaches towards explaining NNs, e.g., [39, 40], take into account causal relations when
calculating features’ attribution values for explanation. Meanwhile, [4] introduce causal
explanations for reinforcement learning models based on [5].
Computational argumentation has been widely used in the literature as a mechanism for
explaining AI models, from data-driven explanations of classifiers’ outputs [41], powered
by AA-CBR [42], to the explanation of the PageRank algorithm [43] via bipolar AFs
[17]. The outputs of Bayesian networks have been explained by SAFs [16], while decision-
making [44] and scheduling [45] have also been targeted. Property-driven explanations
based on bipolar [20] and tripolar [35] AFs have been extracted for recommendations,
where the properties driving the extraction are defined in the orthodox manner (with
respect to the resulting frameworks), rather than inversions thereof, as we propose.
Other forms of argumentation have also proven effective in providing explanations for
recommender systems [14], decision making [46] and planning [47].
Various works have explored the links between causality and argumentation. [48] shows
that a propositional argumentation system in a full classical language is equivalent to a
causal reasoning system, while [49] develops a formal theory combining “causal stories”
and evidential arguments. Somewhat similarly to us, [50] present a method for extracting
argumentative explanations for the outputs of causal models. However, their method
requires more information than the causal model alone, namely, ontological links, and
the argumentation supplements the rule-based explanations, rather than being the main
constituent, as is the case in our approach.
7. Conclusions
We have introduced a novel approach for extracting AFs from causal models in order
to explain the latter’s outputs. We have shown how explanation moulds can be defined
for particular explanatory requirements in order to generate argumentative explanations.
We focused, in particular, on inverting the existing property of argumentation semantics
of bi-variate reinforcement to create an explanation mould, before demonstrating how
the resulting reinforcement explanations (RXs) can be used to explain causal models
representing different machine-learning-based classifiers.
One of the most promising aspects of this preliminary work is the vast array of directions
for future investigation it suggests. First, an experimental validation of the proposed
ideas and a comparison with other explanation approaches on a sufficiently large variety
of case studies is needed.
Clearly, the wide-ranging applicability of causal models broadens the scope of explana-
tion moulds and argumentative explanations well beyond machine learning models, and
we plan to undertake an investigation into other contexts in which they may be useful,
for example for decision support in healthcare.
We also plan to study inversions of different properties of argumentation semantics and
different forms of AFs to understand their potential, e.g. counting for AAFs [51]. Within
the context of explaining machine learning models, we plan to assess RXs’ suitability
for different data structures and different classifiers, considering in particular deeper
explanations, e.g. including influences amongst input variables and/or intermediate, in
addition to input and output, variables, in the spirit of [52, 25]. This may be aided by
the deployment of methods for the extraction of more sophisticated causal models from
classifiers, e.g., [53] for NNs.
Finally, while we posit that, when properly defined, the meaning and explanatory role
of the dialectical relations can be rather intuitive at a general level, providing effective
explanations to users through AFs will require the investigation of proper presentation
and visualization methods, possibly tailored to users’ competences and goals and to
different application domains.
Acknowledgments
Toni was partially funded by the European Research Council (ERC) under the Euro-
pean Union’s Horizon 2020 research and innovation programme (grant agreement No.
101020934). Further, Russo was supported by UK Research and Innovation [grant number
EP/S023356/1], in the UKRI Centre for Doctoral Training in Safe and Trusted Artificial
Intelligence (www.safeandtrustedai.org). Finally, Rago and Toni were partially funded by
J.P. Morgan and by the Royal Academy of Engineering under the Research Chairs and
Senior Research Fellowships scheme. Any views or opinions expressed herein are solely
those of the authors listed, and may differ, in particular, from the views and opinions
expressed by J.P. Morgan or its affiliates. This material is not a product of the Research
Department of J.P. Morgan Securities LLC. This material should not be construed as an
individual recommendation for any particular client and is not intended as a recommen-
dation of particular securities, financial instruments or strategies for a particular client.
This material does not constitute a solicitation or offer in any jurisdiction.
References
[1] R. Guidotti, A. Monreale, S. Ruggieri, F. Turini, F. Giannotti, D. Pedreschi, A
survey of methods for explaining black box models, ACM Computing Surveys 51
(2019) 93:1–93:42.
[2] D. Alvarez-Melis, T. S. Jaakkola, A causal framework for explaining the predictions
of black-box sequence-to-sequence models, in: Proc. EMNLP, 2017, pp. 412–421.
[3] P. Schwab, W. Karlen, CXPlain: Causal explanations for model interpretation under
uncertainty, in: Proc. NeurIPS, 2019, pp. 10220–10230.
[4] P. Madumal, T. Miller, L. Sonenberg, F. Vetere, Explainable reinforcement learning
through a causal lens, in: Proc. AAAI, 2020, pp. 2493–2500.
[5] J. Y. Halpern, J. Pearl, Causes and explanations: A structural-model approach:
Part 1: Causes, in: UAI, 2001, pp. 194–202.
[6] J. Woodward, Explanation, invariance, and intervention, Philosophy of Science 64
(1997) S26–S41.
[7] M. M. A. de Graaf, B. F. Malle, How people explain action (and autonomous
intelligent systems should too), in: Proc. AAAI Fall Symposia, 2017, pp. 19–26.
[8] T. Miller, Explanation in artificial intelligence: Insights from the social sciences,
Artificial Intelligence 267 (2019) 1–38.
[9] M. T. Ribeiro, S. Singh, C. Guestrin, ”why should I trust you?”: Explaining the
predictions of any classifier, in: Proc. ACM SIGKDD, 2016, pp. 1135–1144.
[10] S. M. Lundberg, S. Lee, A unified approach to interpreting model predictions, in:
Proc. NeurIPS, 2017, pp. 4765–4774.
[11] A. Ignatiev, Towards trustable explainable AI, in: Proc. IJCAI, 2020, pp. 5154–5158.
[12] K. Atkinson, P. Baroni, M. Giacomin, A. Hunter, H. Prakken, C. Reed, G. R. Simari,
M. Thimm, S. Villata, Towards artificial argumentation, AI Magazine 38 (2017)
25–36.
[13] P. Baroni, D. Gabbay, M. Giacomin, L. van der Torre (Eds.), Handbook of Formal
Argumentation, College Publications, 2018.
[14] J. C. Teze, L. Godo, G. R. Simari, An argumentative recommendation approach
based on contextual aspects, in: Proc. SUM, 2018, pp. 405–412.
[15] A. Dejl, P. He, P. Mangal, H. Mohsin, B. Surdu, E. Voinea, E. Albini, P. Lertvit-
tayakumjorn, A. Rago, F. Toni, Argflow: A toolkit for deep argumentative explana-
tions for neural networks, in: Proc. AAMAS, 2021.
[16] S. T. Timmer, J. C. Meyer, H. Prakken, S. Renooij, B. Verheij, Explaining Bayesian
networks using argumentation, in: Proc. ECSQARU, 2015, pp. 83–92.
[17] E. Albini, P. Baroni, A. Rago, F. Toni, PageRank as an argumentation semantics,
in: Proc. COMMA, 2020, pp. 55–66.
[18] C. Antaki, I. Leudar, Explaining in conversation: Towards an argument model,
Europ. J. of Social Psychology 22 (1992) 181–194.
[19] P. Madumal, T. Miller, L. Sonenberg, F. Vetere, A grounded interaction protocol
for explainable artificial intelligence, in: Proc. AAMAS, 2019, pp. 1033–1041.
[20] A. Rago, O. Cocarascu, C. Bechlivanidis, F. Toni, Argumentation as a framework
for interactive explanations for recommendations, in: Proc. KR, 2020, pp. 805–815.
[21] K. Cyras, A. Rago, E. Albini, P. Baroni, F. Toni, Argumentative XAI: A survey, in:
Proc. IJCAI, 2021, pp. 4392–4399.
[22] A. Vassiliades, N. Bassiliades, T. Patkos, Argumentation and Explainable Artificial
Intelligence: A Survey, Knowledge Eng. Rev. 36 (2021).
[23] D. M. Gabbay, Logical foundations for bipolar and tripolar argumentation networks:
preliminary results, J. Log. Comput. 26 (2016) 247–292.
[24] P. Baroni, G. Comini, A. Rago, F. Toni, Abstract games of argumentation strategy
and game-theoretical argument strength, in: Proc. PRIMA, 2017, pp. 403–419.
[25] E. Albini, A. Rago, P. Baroni, F. Toni, Relation-based counterfactual explanations
for bayesian network classifiers, in: Proc. IJCAI, 2020, pp. 451–457.
[26] L. Amgoud, J. Ben-Naim, Weighted bipolar argumentation graphs: Axioms and
semantics, in: Proc. IJCAI, 2018, pp. 5194–5198.
[27] C. Cayrol, M.-C. Lagasquie-Schiex, On the acceptability of arguments in bipolar
argumentation frameworks, in: Proc. ECSQARU, 2005, pp. 378–389.
[28] J. Pearl, Reasoning with cause and effect, in: Proc. IJCAI, 1999, pp. 1437–1449.
[29] J. Pearl, The do-calculus revisited, in: Proc. UAI, 2012, pp. 3–11.
[30] P. M. Dung, On the Acceptability of Arguments and its Fundamental Role in
Nonmonotonic Reasoning, Logic Programming and n-Person Games, Artificial
Intelligence 77 (1995) 321–358.
[31] L. Amgoud, J. Ben-Naim, Evaluation of arguments from support relations: Axioms
and semantics, in: Proc. IJCAI, 2016, pp. 900–906.
[32] P. Baroni, A. Rago, F. Toni, How many properties do we need for gradual argumen-
tation?, in: Proc. AAAI, 2018, pp. 1736–1743.
[33] J. Pearl, Causal diagrams for empirical research, Biometrika 82 (1995) 669–710.
[34] N. Potyka, Interpreting neural networks as quantitative argumentation frameworks,
in: Proc. AAAI, 2021, pp. 6463–6470.
[35] A. Rago, O. Cocarascu, F. Toni, Argumentation-based recommendations: Fantastic
explanations and how to find them, ????
[36] Y. Son, N. Bayas, H. A. Schwartz, Causal explanation analysis on social media, in:
Proc. EMNLP, 2018, pp. 3350–3359.
[37] M. R. O’Shaughnessy, G. Canal, M. Connor, C. Rozell, M. A. Davenport, Generative
causal explanations of black-box classifiers, in: Proc. NeurIPS, 2020.
[38] N. Pawlowski, D. C. de Castro, B. Glocker, Deep structural causal models for
tractable counterfactual inference, in: Proc. NeurIPS, 2020.
[39] A. Chattopadhyay, P. Manupriya, A. Sarkar, V. N. Balasubramanian, Neural
network attributions: A causal perspective, in: Proc. ICML, 2019, pp. 981–990.
[40] T. Heskes, E. Sijben, I. G. Bucur, T. Claassen, Causal Shapley values: Exploiting
causal knowledge to explain individual predictions of complex models, in: Proc.
NeurIPS, 2020.
[41] O. Cocarascu, A. Stylianou, K. Cyras, F. Toni, Data-empowered argumentation for
dialectically explainable predictions, in: Proc. ECAI, 2020, pp. 2449–2456.
[42] K. Cyras, K. Satoh, F. Toni, Abstract argumentation for case-based reasoning, in:
Proc. KR, 2016, pp. 549–552.
[43] L. Page, S. Brin, R. Motwani, T. Winograd, The PageRank Citation Ranking:
Bringing Order to the Web, WWW: Internet and Web Inf. Syst. 54 (1998) 1–17.
[44] L. Amgoud, H. Prade, Using arguments for making and explaining decisions,
Artificial Intelligence 173 (2009) 413–436.
[45] K. Cyras, D. Letsios, R. Misener, F. Toni, Argumentation for explainable scheduling,
in: Proc. AAAI, 2019, pp. 2752–2759.
[46] Q. Zhong, X. Fan, X. Luo, F. Toni, An explainable multi-attribute decision model
based on argumentation, Exp. Syst. Appl. 117 (2019) 42–61.
[47] N. Oren, K. van Deemter, W. W. Vasconcelos, Argument-based plan explanation,
in: Knowledge Engineering Tools and Techniques for AI Planning, Springer, 2020,
pp. 173–188.
[48] A. Bochman, Propositional argumentation and causal reasoning, in: Proc. IJCAI,
2005, pp. 388–393.
[49] F. Bex, An integrated theory of causal stories and evidential arguments, in: Proc.
ICAIL, 2015, pp. 13–22.
[50] P. Besnard, M. Cordier, Y. Moinard, Arguments using ontological and causal
knowledge, in: Proc. FoIKS, 2014, pp. 79–96.
[51] L. Amgoud, J. Ben-Naim, Axiomatic foundations of acceptability semantics, in:
Proc. KR, 2016, pp. 2–11.
[52] C. Olah, A. Satyanarayan, I. Johnson, S. Carter, L. Schubert, K. Ye, A. Mordvintsev,
The building blocks of interpretability, Distill 3 (2018) e10.
[53] T. Kyono, Y. Zhang, M. van der Schaar, CASTLE: regularization via auxiliary
causal graph discovery, in: Proc. NeurIPS, 2020.