The Utility of “Even if” Semifactual Explanation to Optimise Positive Outcomes
Authors: Eoin M. Kenny
Weipeng Fuzzy Huang
Venue: NeurIPS 2023 poster (2023)
Keywords: Semifactual Explanation
Counterfactual Explanation
Explainable AI
Recourse
User Study
TL;DR: We show how to use semifactual explanation to optimise positive outcomes.

Abstract: When users receive either a positive or negative outcome from an automated system, Explainable AI (XAI) has almost exclusively focused on how to mutate negative outcomes into positive ones by crossing a decision boundary using counterfactuals (e.g., *"If you earn 2k more, we will accept your loan application"*). Here, we instead focus on positive outcomes, and take the novel step of using XAI to optimise them (e.g., *"Even if you wish to half your down-payment, we will still accept your loan application"*). Explanations such as these that employ "even if..." reasoning, and do not cross a decision boundary, are known as semifactuals. To instantiate semifactuals in this context, we introduce the concept of *Gain* (i.e., how much a user stands to benefit from the explanation), and consider the first causal formalisation of semifactuals. Tests on benchmark datasets show our algorithms are better at maximising gain compared to prior work, and that causality is important in the process. Most importantly however, a user study supports our main hypothesis by showing people find semifactual explanations more useful than counterfactuals when they receive the positive outcome of a loan acceptance.

The Utility of "Even if..." Semifactual Explanation to Optimise Positive Outcomes
Eoin M. Kenny* Massachusetts Institute of Technology Cambridge, MA, U.S.A. akenny@mit.edd
Weipeng Huang" Tencent Security Big Data Lab Shenzhen, Guangdong Province, China fuzzzyuang@tencent.co com
Abstract
When users receive either positive or negative outcome from an automated system, Explainable AI (XAI) has almost exclusively focused on how to mutate negative outcomes into positive ones by crossing a decision boundary using counterfactuals (e.g., "If you earn 2k more, we will accept your loan application "). Here, we instead focus positive outcomes, and take the novel step of using XAI to optimise them (e.g.. "Even if you wish to half your poym-n,yyent, we will still accept your loan application" "). Explanations such these that employ "even if..." reasoning. and do not cross a decision boundary, are known as semifactuals. To instantiate semifactuals in this context, introduce the concept of Gain (i.e., how much user stands to benefit from the explanation), and consider the first causal formalisation better
of semifactuals. Tests benchmark datasets show our algorithms maximising gain compared to prior work, and that causality important the process. Most importantly however, user study supports our main hypothesis by showing people find semifactual explanations more useful than counterfactuals when they receive the positive outcome of loan acceptance.
Introduction
Explainable AI (XAI) is broadly categorised into factual [4, 27, 30] and contrastive explanations [29, 38]. Within contrastive XAI, despite being neglected in comparison counterfactuals, semifactuals are a major, fundamental part of human explanation, and have long been studied in psychology [9, 35, 49], philosophy [6, 20], and lately computer science [1-3, 29, 34, 52, 62]. They take the form of "Even if x happened, would still be the outcome". ". Such reasoning has many potential uses as demonstrated by these prior works, but here we are focused on how semifactuals can help optimise positive outcomes for users, which (to the best of our knowledge) remains completely unexplored.
Our definition of counterfactuals is in line with Wachter al. [55], where test instance classified as e must be mutated to cross a decision boundary into class c'. Likewise, as established in the literature [1, 29], define semifactual an instance classified such a way as to not cross a decision boundary (and hence remain class e) [29]. In recourse [25, 50], "negative outcomes" (e.g., loan rejection) are generally mutated to produce "positive outcomes" (c.g., loan acceptance) for users using counterfactuals. In our setting, we are assuming there was initially& a positive outcome, and are trying mutate features produce an even better situation for users, and in doing so not cross the boundary into the negative outcome (i.c., using semifactuals).
which must be modified in
Historically, counterfactuals have had obvious applications in computer science, such explaining how to have a bank loan accepted rather than rejected, but applications for semifactuals as less
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37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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clear. As such, the usage of semifactuals has often inadvertently defaulted to copying counterfactual research by also explaining negative outcomes (e.g., "Even you double your savings, your loan will still be rejected" [3. 29, 48)). However, such an application for semifactual explanation perhaps has two main issues. Firstly, is debatable if these explanations convey useful information [1], whilst a counterfactual explaining how to cross a decision boundary and have loan accepted has obvious utility [24]. Secondly, such explanations make the user's situation seem helpless [35], in that they cannot possibly have their loan accepted, which raises ethical concerns [l]. However, our proposed framework can be used to not only overcome both of these issues, but actively contribute to fairness.
Firstly, to try offer useful information for users, we flip the usual recourse problem and consider the user starting from positive (rather than a negative) outcome. In this setting, consider user that has had their loan accepted, but might prefer to make a smaller down -wwymeetentt on a loan application. In this situation, our framework could present an explanation such as "Even if you half your down- payment, your loan will still be accepted", which seems to be more useful than explaining negative outcomes (see Section 6). Secondly, because we are starting from a positive outcome, there is no danger of manipulating people into accepting negative outcome, which guarantees fairness is this regard. Now, with regards to optimising faimess even further, note that banks are not motivated to share such explanations even though they may help people, because (for example) larger down- payments are associated with lower risk on their behalf [7]. So, the usage of semifactuals in this application has clear potential to actively encourage fairness and transparency. As aside, it is worth noting that although the focus of this paper is on financial applications, this research has broad impact on any domain for which the optimisation of positive outcome is beneficial. For instance, in medical applications, our framework could present explanations of the form "Even if you half your dose of drug æ, you will still be at a low risk for disease y". This is once again important for optimising fairness because people are frequently over-prescribed medicine with adverse side-effects [47], but due to profit Big Pharma has no incentive to actively encourage this type of transparency. Similar usage of semifactuals have also been proposed in smart agriculture to combat climate change [29].
Our main contributions are: (1) the first explicit exploration of how to optimise positive outcomes with XAI, (2) the problem formulation for this which involved augmenting current semifactual research with the concept of Gain (see Section 3.3), and (3) the premiere user test in the XAI literature for semifactuals, showing a clear application in which users find them more useful than counterfactuals.
2 Literature Review
When using contrastive explanation to explain loan acceptance decisions, to the best of our knowledge, this has only been explored by McGrath al. [36]. Specifically, they suggest positive counterfactuals, which show "by how much" a user had their loan accepted to help inform them when making future financial decisions. While this is interesting information, we show that users find semifactual explanations more useful in loan acceptance situations than positive counterfactuals (see Section 6).
Semifactual explanation is growing in popularity [3], Kenny & Keane [29] first explored the idea, but focused only on images.' 2 Artelt & Hammer [1] used diverse semifactuals to explain why an AI system refuses to make predictions due to having an unacceptably low certainty, but ignore how to explain either positive or negative outcomes. Lu et al. [34] explain spurious patterns with semifactuals using a human-in-the-logg framework in NLP. Zhao et al. [62] proposed a class-to-class variational encoder (C2C-VAR with low computational cost that can generate semifactual images. Vats et al. [S2] used generative models to produce semifactual image explanations for classifications of ulcers. Lastly, for model exploration, Xie et al. [59] sampled semifactual images with a joint Gaussian mixture model, and Dandl et al. [15] proposed deriving semifactual explanations from interpretable region descriptors. In contrast to all these approaches, we are showcasing how semifactuals can be used to optimise positive outcomes for users (notably in causal settings).
From a user perspective, many have discussed the urgent need for comparative tests with semifactu- als [29, 37, 40, 48, 56], with Aryal & Keane [3] pointing to the 'paucity of user studies' in the area. However, the only such tests we are aware of are in the psychological literature over two decades ago [35]. Taking to this challenge, we conduct the first such test directly comparing semifactuals to counterfactuals in the XAI literature (see Section 6).
Note there is other work on a-fortiori explanations which have similar computational techniques to semifac- tuals [14, 19, 45J, they are justifications of the form "Because it true, y must also be true".
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Our research is related to algorithmic recourse [50] in that we are trying to ensure users are treated fairly by automated systems [24]. In this area, Mothilal al. [39] explored counterfactual diversity, in that we should be offering users several explanations. In addition, counterfactual robustness has been examined [18], which proposes that generated explanations should robust to distributional shifts. Lastly, causality has been argued as essential to providing plausible recourse [26]. We see these three facets as being important to our problem setting, and instantiate them in our framework. There are other areas in recourse such sequential decision making [16, 42], fairness [54], and privacy [43], but we leave their exploration within semifactual explanation for future work.
As an aside, the literature on sufficiency could be conflated with semifactual explanation, as it describes set of "sufficient" features for a prediction which, in the presence the other features mutating. mostly doesn't 't affect the outcome [17, 46, 57]. However the techniques offer no insights for how to generate a meaningful semifactual. More importantly though, if the sufficient features are the only actionable ones, then by definition we can't 't modify them to create semifactual.
Semifactual Framework
In this section, we describe the basic definitions and assumptions for our semifactual framework to optimise positive outcomes for users, before formalising under the concept of Gain (i.e., how much stands to benefit from the explanation) in causal setting, neither of which has been considered before. As an aside, we also show how the established concepts of plausibility, robustness, and diversity can be made fit into the objective to offer better explanations. Finally, we reflect on the theoretical properties of the framework.
3.1 Definitions
Let us denote an individual x E X' with & mutable features = {. Xg). Given the individual set of actions can be applied to where each action axx) is also k-dim vector. As in prior work [25], we apply a(x) and a exchangeably, since the individual x will always be fixed. We explicitly exclude features that are either immutable non-actionable. Adopting Pearl's do() operator [44], an action can be defined as a(x) = do(X = 0), or simply do()), to force a hard
Loun Accepted
Loan Rejected
X. az; M)
If you work 12hrs less week the loan will be rejected.
Even if you double your repayment term, and work per week, the loon will still be accepted.
Even if you triple your loan amount, it will still be accepted.
X
Even you double your loan amount, and double your repayment term. the lown will still be accepted.
SF(x, 0g: M))
SP(xx, M)
Semifactual Explanation
SB (x. M) Semifactual Function Robust E-ncighborhood
Min-Edit Counterfactual
Figure 1: Semifactual Explanation Optimise Positive Outcomes: An individual x has their loan accepted, but there are several semifactual explanations which can help optimise their outcome. Our algorithm produces set of semifactual explanations which maximise the distance between and the final explanation (x, a; M). This allows the largest Gain to be achieved so that the user gets the maximum benefit. In contrast, counterfactual algorithms are not suitable because they are designed to target the shortest path across a decision boundary. In addition, the semifactuals are robust to distributional shifts by constraining an c-neighborhood between them and the decision boundary. Note M is the Structural Causal Model (SCM), see Section 3.
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intervention of replacing x by θ where 0 E X. It implies that, for each feature, X; == 0g for the individual x. If the action do(0) imposes no change, x 0 holds. We further denote a set of human-constrained actionable ranges A = {a(x) = do v0 E &)). Note that the actions have to be mutable and explicitly exclude any action which keeps the individual in the same position.
The non-causal semifactual interaction between x and a(x) is defined by SI : X x A X. That is, the individual x taking action a(x) will lead another representation 0 E X representing that person's 's recourse. Now, a structural semifactual is defined which considers the dependence between the related features [18, 24]. We denote the structural causal model (SCM) by M (S, Pt) where S are a set of structural equations and Fy is the distribution over the exogenous variables U E U. Consider that in causal graph, there is a set of causal parents for each feature 2; E x, denoted by Pa,. We denote the structural equations as S = {x; == 9: (Pa.. U,) i I.....k: }}} where ge(+) a deterministic function that describes the causal relationship for 2;, and depends on the exogenous variable U; E U alongside the corresponding parent set Pag. Hence, S induces a mapping 8: 5:44 1 + X and its inverse mapping S-1 X' + U. Let f o g(x) = (x())) which can be extended to more functions. Hence, we specify the SCM-processed semifactual by (x MM) to denote the transition between the states by taking a certain action through an SCM M, where
0' SF( x, do 0): M) = S s
1----oo-- o SF do( ())
If all features are independently manipulable, we have 0' = 0 = (x (0)0)) = SF do)). Therefore, SF(x,, do(0); MM) is more generalised formulation which covers the non-causal case. Lastly, we assume a binary model that generates the score for the users is h, where h X + {0, 1) by which we can simply consider that 1 means a positive outcome (c.g., loan acceptance) and 0 is a negative outcome (e.g., a loan rejection). We set a lower threshold v that separates the decision boundary. For the form defined above, vi 0.5 is reasonable threshold that fits all situations well.
3.2 Framework
We define our semifactual framework as one centering on gain (G) that is weighted by plausibility (P), regularization in the form of diversity (R), and hard constraints in the form of robustness (H), indexed by j. All of the components are parameterized with x and a subset of suggestions {aI..... amm J. Letting f() be a function composed by gain and some weighting (i.e., plausibility for us), the causal semifactual framework is defined as
1 1
Σ f( a.), P(x, P(), (R((()(()))))) 0m))
max 01 you Bom
==1
s.t.
0. = SF ag; M), (0,) 2 0, Vi.j
where the regularisation and hard constraints can be multiple and indexed with i and j. respectively. One may define similar formulation for the non-causal case (see Section 4.2). We defer all details of the components until Section 3.4.
3.3 Optimising Positive Outcomes with Gain
For the core of the objective we appeal to the notion of gain. Note that gain is similar to the idea of cost commonly used in recourse [50], but there are three crucial differences. First, we are trying to maximise gain, rather than minimise cost [24]. Second, gain ideally considers the causal dependencies between features in its function, whilst cost typically only considers the user's action(s) [24]. Third, gain is further subdivided into positive and negative polarities. To elaborate on this last point, take the example of a user who has their loan application for buying a new house accepted. In this situation, if they desired to spend more time away from work with family, they would experience positive gain if they could work less hours per week and still have their loan accepted (see Figure 1). Conversely, if this person increased the number of hours they worked, they would experience negative gain. Notably, positive/negative gain is not necessarily connected to the model's probabilities (see ai in Figure/ moving away from the decision boundary). Similar to actionability constraints which can offer individualised recourse [50], what is positive/negative gain must be manually defined for each individual. As prior work on semifactuals simply maximised the L2 distance between a test instance and explanatory one to define good explanations [1, 29], we introduced the concept of gain to make them more meaningful in application.

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More formally, we define the gain function by G A R. By denoting 0 = (x, a; M), we decompose the function as follows:
G a) == PEr 8(x, 0) Psr 8(x, SIF (IF
where -): is an oracle function that computes the payoff based on the vectorised difference between x and 0, i.e., ò : X' x X' - R which is symmetrical difference function between the two feature representations. The subscript of PSF denotes semifactual. In interpretation. the gain function compares two states, (1) the original feature vector x, and (2) the SCM SCM-processed end state which was led through taking action a.
Why is Gain not necessarily equivalent to Cost? Formally, to enable the comparison, we write the cost function (denoted by C(x,a)) a)) as
C(x,a) - 8(x (F,xSxFFFFx (F(
which builds on the fact that cost solely considers the feature change. Note that SF is equivalent to the notion CF [25], what makes our approach different the consideration of positive outcomes and gain. Our finding is that gain in semifactuals (SFs) is not necessarily equivalent to cost in counterfactuals (CFs) where the equivalence ignores the sign of both quantities, as formally stated as follows.
Theorem 3.1. Even ifPsF(;,) = PP gain and cost are not necessarily equivalent ignoring the sign.
Proof. Note that SCMs are also considered in counterfactual recourse [18, 24, 26]. However, in this prior research SCMs are typically applied for enforcing hard plausibility constraints, not the computation of a user's s cost. In contrast, our gain function takes the SCM- CCM-ppocessed semifactual l' an input. We employ proof by contradiction here. Assume that cost and gain are equivalent ignoring sign that, without loss of generality,
|G a)l [C(x, a)l
> P O a; M)) = |P SF
(5) holds. However, SCMs result in possibly more features being changed since features could be others' causal parents and those causal children will change their values accordingly. By denoting 0 SF a) and 0' (x a'), we consider the general case as follows:
8(x, SIF( M)) 8(x, SF((, a))
16(x, - 18(xx 0')1 = Σ 1(xx 0)al - Σ (0)x
E
18(x. 0)l - 16(x, 0'), -0
Σ +0E
16(x, 0).1 (000)-10(x_00) 0' 80, (6)
=
which contradicts with Equation (5). Thus, even if the oracle function for calculating the payoff the same, gain and cost are still not necessarily equivalent. Also, the equality in Equation (6) holds when all features are independently manipulable or the changed features are independently manipulable of
the remaining features, so that (F a) SF M). The proof completes here.
L
3.4 Semifactual Components
Here, we detail how to incorporate the concepts of plausibility, robustness, and diversity into our framework for maximising gain, because they are agreed upon important in the literature and useful for evaluation. While plausibility and diversity have been explored semifactual explanation [1, 29]. robustness and causality (and indeed an objective balancing all together) have not, yet we argue and show that the subtleties of "even if... thinking are perhaps better captured in a casual setting.
Plausible Gain We define plausibility here as explanations which are within distribution. For example, an explanation saying person could earn less and still have their loan accepted should change their "debt-to-income ratio" feature also, or it will lie outside the data manifold. Prior work on semifactuals has only considered euclidean distance to training data as heuristic for this [29], in contrast posit (similar to the counterfactual literature [23]) that this is better approached with
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SCMs. Hence, we define the plausibility for x taking the action a by P(x, a) = Pr(a = do where x fixed for an individual and Pr(-) is density function. In our non-causal tests, we use the T/ norm to training data to approximate plausibility (i.e., being in distribution, similar to [29, 32, 51 J. However, this issue of plausibility is naturally taken care of in our causal tests thanks to the SCM ensuring plausible feature mutations, we don't explicitly consider plausibility there going forward.
Robust Gain Continuing with the example of person who has a loan accepted to buy a house, the semifactual should sometimes be robust to distribution shifts. For example, if the person uses the semifactual explanation triple their loan amount (recall Figure 1). they will likely need upwards of six months to locate new house during which the semifactual should hold if the person e.g. gets an additional credit card. Hence, we define our semifactual robustness such that while taking action a, any close neighbor of the generated semifactual SF (x(,, a; M) can still receive a positive outcome. The e-neighborhood of x centering around an individual
((x) = {0 = SF a; M): : Va E Sc
which covers all neighbors that can be reached from x by taking an actionable feature change through the SCM M. By definition. x is also neighbor of itself since x E B(x) holds given 5(x.x] x) = O. Let us represent (XF a; M)) by (x a) for simplicity. Given the predictive model h-)) and an individual x, an action is robust for individual if h()) > v, V0 E B, (x, a). which is equivalent to minges, 5, (x,a) h(0) & > O. For instance, &p = 0.5 works for a binary model case. We hence denote the term related to the robustness by
H(x, a)
min h(0) v
(8)
(x,,(
which will be useful for constructing the final objective.
Diverse Gain It is generally preferred offer a number of suggested actions {e1,-... Um J. rather than a single one [60]. Like prior work in counterfactuals, we define diversity as the average pair-wise
distance among set of entities [39, 61]. We reuse the distance function ē and define the diversity objective within a set of SFs {0, F==1 X as
2 m
L2 o 0(0,,) Lizi jii
m
> 1 m =
R 0 mm
which represents pairwise mean distance among the set of data points, based on the 1.2 norm. One may accommodate m 1 for the case when only single semifactual is desired.
3.4.1 Semifactual Objective
The final objective may be constructed as follows.
Definition 3.2 (Semifactual Objective). We consider a simple composition multiplication function for f(.). Considering gain, plausibility, robustness, and diversity, the semifactual objective function is:
max
P(x, 0; G,)) az) (ISF(xx ag M)
m
(10)
S.L. W= m m:a: 0:; E A, H 0,) > vb
In optimisation [10], an adversarial interpretation from the perspective of two-player zero sum game can further simplify Equation (10) to
T
I
min Am. 20
max
P(xx,))(xx,a) (XHGxxx,a++xx(xxxx,,xxxx (1
Ae E.A MIL
(11)
where Hxx, a) is the Lagrangian. The primal player tries to maximise the plausibility-weig weighted gain and diversity, with regard to whilst the dual player tries to minimise regarding set of X.
Since there are m suggestions, the constraints for robustness will be m times. Observing the objective, the robustness is hard constraint, whilst the diversity can be regarded regularisation. P can be seen as a scaling factor for G which helps to guarantee that high expected gain is only possible alongside high plausibility, simply adding them misses this special property.

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3.5 Properties of the Framework
Effective Solution Space. We discuss the set of meaningful solutions here and the result validates the re-formulation [i.e., Equation (11)] of the semifactual framework [i.e., Equation (10)]. First, we depict the lemma.
Lemma 3.3. Assume that the limit of the gain function and diversity temn are finite. Also, assume that A = {a E A G(x. a) 2 0} is non-empty for an individual x. The semifactual objective J 2 when Vi 1, .....,,......................... m. a4 A H(x o.) 2 0, otherwise =00.
See Section A.I for the proof. We can summarise that the action set which is able to provide the positive payoff can be defined by the named effective solution space for x: A {a E A [x, a) 2 0, G(x, a) > 0}. Hence, repeated suggestions will be produced when the number of actions in this space is smaller than the required m. Otherwise, the solution will provide more versatile options. There suggestions achieve effective semifactual(s) (i.e., with positive gain) if this solution space is empty. However, similar situations exist for counterfactuals when they are also impossible to generate, assuming similar set of actionable constraints defined.
4 Implementation Details
We now introduce our methods to solve Equation (11), henceforth called Semifactual-recourse GENeration (S-GEN), for both causal and non-causal domains. In the following paragraphs, we use G to denote an empirical approximation of G, and likewise for P, H, R, and J.
4.1 Causal Case
Assuming the presence of differentiable classifier h(..) and SCM M, (recall the latter guarantees plausibility), let (x) = (Pr(0) E (x,x, a:)) be the probability distribution over the 6- neighborhood of SFF [x a;; .). Also, let B; represent finite subset of (x, a:) sampled according to f2 (x). Our objective is:
1 m
max min
ArC (h(SFxx az M;;
Σ (x())))
aon Ae Am
m 1=1
B
O.ER;
+ P(x G(x, az) 7 (((SS( o;; M)
s.t.
Vi - 1,.,,,.,, ag A, >
(12)
where C is the binary cross entropy loss. For robustness, we used Monte Carlo (MC) sampling with an epsilon robust hypersphere, and if either the instance or sampling return negative outcome with h(.), we use the prior optimisation step as the solution. For diversity, is set to the number of actionable feature sets, and a solution obtained for each. We utilise the causal recourse approach of Karimi et al. [26] for sol ving the maximin. The actionable bounds are clipped each iteration, and À is iteratively decreased to put more emphasis on gain over time (see Algorithm 2).
4.2 Non-Causal Case
For the non-causal case, we use genetic algorithm [53, 58] which only assumes binary predictive model ht.). This approach follows the standard design for genetic algorithms, with some minor alterations specifically for semifactual generation, Appendix D for the pseudocode. Next present the fitness function which optimises our objective.
4.2.1 Fitness Function
For gain, the average distance between an individual x and each semifactual SF(x, a) is measured as C(x, a) = [SS((x a) - x||2. For robustness, we relax it to two constraints: Hp is the probabilistic robustness for the neighbor points where the generated semifactuals for a query are randomly perturbed using MC simulation to make sure the surrounding neighborhood robust, and H, the absolute robustness for the individual x (more detail in Section A.2). For the first constraint, score of (x) = 1 Σi [(h(x) h(0,)) where 0 Pr(B, a)), is returned. For
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the second constraint, score of Hx(x.a) = 1fh(x) = h o SF a}} is returned. Hence, the solution is rewarded for (1) the neighborhood samples, and (2) the semifactuals themselves being classified as h(x). For plausibility, we take from prior work and directly use the training data [29]. Specifically, considering the training data set DD we define the notion of plausibility using the distance of each semifactual generated to the nearest training data point. As the term must be maximised, we use a function which is monotonically decreasing with respect to the distance with P(x, a) = Pp(x, a) = exp(1 exp(1/(mingg (mmingg [[ a) 013 + 7p)} where "p is to account for when perfect match to the semifactual exists in the training data (thus the division is undefined), and Po(x, a) is empirical approximation (based on D) of the plausibility. Lastly, for diversity [39, 61], we take the mean distance between all m generated semifactuals with R ((SFF(x a))); which precisely follows Equation (9). This objective collapses to 0 when m = 1.
Certain objectives need to be weighted individually based on the problem. For example, explanations which can be acted upon immediately perhaps don't tt need robustness. Notably, the multiplier in Equation (11) is split to Xp and Aa. for Hp and Ha respectively. In this work, we treat them as hyperparameters. Also, they are used alongside y to balance the objectives. Since À and are selected as hyperparameters, they are removed under the min operator. Finally, the objective (fitness) function is defined as:
max 01 01 EA+
1 Σ
Polx, Po(x,a))(xxa)) + Xp(()(x,a)) + X.lla(x,
( M))
We selected the hyperparameters via a grid search, see Appendix C. Crucially, we also weight the fitness function output by (x,(x, a) to encourage solutions with more semifactuals (see Algorithm 1).
5 Experiments & Results
Here we test S-GEN in both causal and non-causal settings. We show the effectiveness of our method in optimising user's positive outcome compared to baselines and open source our code (see Appendix E). The actionability constraints are detailed in Appendix B. Basclines were modified to be appropriately compared, most importantly, we stopped counterfactual techniques before they crossed a decision boundary (thus generating semifactuals), and modified semifactual techniques to work tabular data, Appendix G details the peripheral modifications.
In the non-causal setting, we consider three datasets, Loan Application [33], German Credit [21], and BCSC [11]. All categorical variables are one hot encoded. Three models were used, a decision tree, logistic regression, and naive bayes, each with 30 random test data point explanation samples gotten by varying the random seed. Note that because the range varied on each dataset, the results were normalised and averaged for each, but Appendix F details each individual dataset for completeness. For baselines, we modify three techniques, DiCE by Mothilal et al. [39] (henceforth DiCE*), PIECE by Kenny& & Keane [29] (henceforth PIECE*), and Diverse semifactual Explanations of Reject by Artelt & Hammer [1] (henceforth > Plausibility is measured
the distance between generated semifactual(s) and the nearest training example; thus, the smaller the better. Robustness is 100 single feature perturbations of each semifactual 0,, predicting their class, and returning a float between 0-1 of the success rate as described in Section 4.2.
measured by MC sampling
In the causal setting, the Adult [31] and COMPAS [5] datasets are considered. The SCMs from Nabi Shpitser [41] were used, and the structural equations from Dominguez et al. [18]. All categorical features are treated as real-valued. We use the pre-trained MI.P classifiers from Dominguez et al. [18] and take 30 averaged samples from random seeds. As basclines we modify the technique of Karimi et al. [26] [henceforth Karimi et al. (202 and Dominguez et al. [18] [henceforth Dominguez et al. (2022*), the latter optimises with robustness in mind. We optimise the relevant techniques to be robust in an 0.1 hypersphere, and the C&W adversarial attack by Carlini & Wagner [12]
measures robustness by checking if the nearest adversarial attack is outside this radius.
For all tests, the main metric of concern is gain, that is, the mean distance between a query and its generated semifactual(s), the larger this number, the better. Diversity is also measured for all tests as the mean distance between all generated semifactuals for an individual the higher the number, the better. To be line with prior art, the l.2 norm used in non-causal tests [1], and the I,, in causal [18]. Note for causal tests the SCM guarantees plausibility so this metric is not reported.
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5.1 Non-Causal Results
Our purpose here to show that current methods are insufficient to meet the basic requirements for semifactual explanation discussed in Section 3.4. Specifically, technique needs to optimise gain, while remaining plausible, robust, and offering diverse explanations.
Observing the average normalised results across all datasets (note robustness was not normalised since it is already 0-1 range), Figure shows that S-GEN performed the best on all metrics for all values of m (1-10). The results demonstrate that traditional counterfactual approaches (DiCE*) are not suitable to achieve optimal gain, due to them focusing on minimising cost. Moreover, methods built for semifactual generation specifically (i.e.. DSER & PIECE*) that do actually maximise gain somewhat, still fail to match the results of S-GEN. This shows that S-GEN is superior to existing semifactual methods (and popular counterfactual approaches appropriately modified) for maximising user's 's gain positive outcomes. Moreover, it does so while maintaining superior plausibility, robustness, and diversity in all tests.
5.2 Causal Results
We evaluate our algorithm in a causal setting where the SCMs and structural equations are known. The primary purpose of this test is to demonstrate that the semantics of semifactual "even if" thinking is better captured in a causal setting due to dependencies being taken into account when calculating a person's 's gain. With regard to diversity, we fix to the maximum number of feature sets available from the actionable features (so only one m value is tested).
Figures 3a and 3b show the initial gain achieved by person after taking certain action (i.e., the Action Gain), and how this gain transforms after considering the causal relationship between features (i.e., the Causal Gain). Firstly, the total gain achieved by S-GEN is much larger than the baselines in both datasets and hence consistent with our non-causal tests. More importantly however, the change in gain a person achieves after considering the causal relations in the adult dataset is significantly higher both significance testing and effect size (0.055 + 0.001 v. 0.063 = 0.001; t-test p < 0.02: Cohen's d 2.24), showing it is beneficial to consider causality when calculating person's 's gain. The results of diversity put S-GEN first also (S-GEN = 0.84 = 0.09 v. Karimi = 0.43 = 0.03 Domineguez 0.34 = 0.03). In robustness, both S-GEN and Domineguz et al. (2022)* did reasonably well (S-GEN = 87% success V. Domineguz = 54% success), but Karimi et al. (2021)* did not (7.2% success), likely due to the latter not being designed for this.
6 User Evaluation
The primary motivation behind this work is the hypothesis that semifactual explanation would be preferred by users over counterfactuals in positive outcome settings. To test this assumption, we design the first user test in XAI directly comparing the two. Specifically, we show users three materials in which person has a bank loan accepted, and three in which they don't. 't. Users were then shown both explanation types for each material, and asked to rate a scale from 1-5 how useful each were. So, the study was a within subjects design, and the condition was the explanation type. Note that although we are studying the effect of the explanation type loan acceptance, the
Dain
Plausibilty
Robaatness
Diversity
0.75
8-GEN
0.50
DSER- PIECE
0.25
0.00
Figure 2: Results: The ability of S-GEN to create semifactuals is compared to DiCE*, DSER*, and PIECE*, Overall, S-GEN does the best, achieving significantly better results to all baselines in all tests. Note we normalised all results before averaging because each dataset has different scaling. Standard error bars are shown.

---

0.07
Melhod
Exclanatio Type Sandiactual
Percdivod Usohh lnses
0,06
Method
Karini (2021)"
Domingusz st (2022)"
Countar
0.04
SGEN
0.20
Kurim wi (2021)
Duninguuz (0022)'
0.15
0:10
0.01
Action Ciem
Cauasal Cwin
Action Carin
Calarll Cein
Lown Accepieed
Loser Rejecled
(a) Causal Experiment: Adult
(b) Causal Experiment: COMPAS
(c) User Study
Figure 3: Causal Experiment & User Study Results: (a/b) show the gain achieved by all methods both before and after considering the causal dependencies. Firstly, note that S-GEN achieves significantly more gain than the alternatively proposed approaches. Most importantly however, (a) shows there is significantly more gain achieved on the Adult data by S-GEN after taking causal dependencies into account, showing the importance of causal formalisation. (c) Shows the user study results, where people perceive semifactual explanation as being significantly more useful than counterfactuals in the positive outcome of having a loan accepted. Standard error bars are shown.
loan rejection scenarios were also included to balance people's view of the problem setting, and as attention checks to verify that users were engaging with the materials and varying their scores accordingly. For analysis, each user's scores for counterfactuals and semifactuals were averaged in both loan acceptance and rejection materials into four decimal scores per user, thus allowing us to analyse the discrete Likert scores with t-tests [28]. As is popular approach [22], don't 't explicitly define what "useful" means to users, but rather let them use their own natural interpretation, as the results returned were reasonably consistent across individuals, they appear have converged on an common interpretation of this word. The null hypothesis is that people will find both explanation types not significantly different in loan acceptance. The alternative is that people will find semifactuals significantly more useful in loan acceptance.
A power analysis [13] of two dependent means with an effect size dz 0.8, alpha 0.05, and power (1 - 3 err prob)=0.9 informed a sample of 15 was appropriate for t-tests. Users were gathered from Prolific.com, 8 males, females, aged 18+, native English speakers, and from the U.S. People were paid $ 1/h/r, which totalled $35. The semifactuals were generated with S-GEN, and the counterfactuals with DiCE [39], notably these are equivalent positive counterfactuals by McGrath et al. [36] for explaining loan acceptance situations. The study obtained IRB approval from MIT.
All users engaged and changed their ratings significantly depending whether loan was accepted or rejected, so none were excluded. Figure 3e shows users find semifactuals significantly more useful .005) compared to rejections when
in loan acceptance (S-GEN=3.6) 3.60-0..77 DiCE-2.3310.33 counterfactuals are preferred (S-GEN=2.6100) v. DiCE=4.53+0..1 17; p < ,0001). Hence we reject the null and lend credible evidence that semifactuals are more useful to explain positive outcomes.
7 Discussion
Although much XAI work has explored how to explain positive outcomes, the best of our knowl- edge, no consideration has been given towards explaining how to optimise them. Here, we have taken the novel step of exploring this, and showed how semifactuals are especially suited for the purpose. This required building on prior work in semifactuals by (1) introducing the concept of Gain, (2) re-framing them in causal setting, and (3) conducting their first user test XAI. Perhaps the notable limitation of our work is that although we have shown people do perceive semifactuals as being more useful in positive outcomes, we have not demonstrated this quantitatively, notably because of the difficulties acquiring an appropriate user base alongside the ethical considerations of such study. Moreover, considering casual formulation of semifactuals requires an SCM, which not always realistic, but we have provided non-causal algorithm for these situations. In future work, it would be interesting to formalise the utility of semifactuals for optimising positive outcomes in other domains such robotics, which likely requires other considerations.
10

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Acknowledgements
The authors would like to thank Neil Hurley, alongside Mark T. Keane and Ruth M.J. Byrne who both inspired early consideration of the ideas in this paper. The authors would also like to thank MIT for their support in this project. This research wasn't directly supported by any grants funding. We hope readers find the ideas interesting and useful for application.
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Towards
14

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A Property Analysis
A.1 Proof of Lemma 3.3
Proof. We can rewrite J to
m
max J
P(x, (xx
+ m
( M)}
min
al EA
m
2
(13)
We derive the fact that, for any i,
-0
H(x,
min A R((, = 2
otherwise
where -00 comes from setting A = oc and is obtained by setting Ap. = By the linearity of summation, can further derive
m
min A, 0;) 1-00 Bi, Hi, <0 A.20
77
otherwise
That is, if any constraint for the robustness is unsatisfied, the dual player will minimise the objective towards -oo; however, the primal player cannot optimise towards oo given that the limit of the gain function and the diversity are finite. other words, the constraints are satisfied, the primal player can freely optimise the objective. Once Hxx, a:) 2 O, Va, are satisfied, the objective becomes
:= max
a(x,a,)G(x, +
M)
EAt m
m
(14)
min
P(x,a,)((xaa) + yR ((SF(x. az M)
EAt m
as P(x, a:) 0 and G(x, a.) 2 for any a, E A+; also, R 2 0 holds. We conclude the proof here.
A.2 Probabilistic Relaxation of Robustness
Absolute robustness is difficult to guarantee, and common practice is to relax this via probabilistic approach [26].
Assume there is a distribution over the sample space (, a) denoted by Pr(B, a)). We write Pr(B. (x, a)) to indicate that is sampled from the set B, (x, a) under the density Pr(-). Let ([)(O))x, a] denote the expectation of 0 in this configuration. Hence, we modify Equation (8) to
E E[h(0)|x,a) a] 3
(15)
where v is function that adjusts the base score threshold th. It is crucial to have this threshold function in order consider the variance of scores the neighbor set. Particularly, we would like most neighbors to remain in a similarly "good" state, with low variance between them.
Moreover, we explicitly impose ((FF(x, M)) & O. places hard constraint avoid the case in which the neighbors of the semifactual are robust, but the "semifactual" itself has crossed the decision boundary become a counterfactual. Whilst somewhat unlikely, this situation is theoretically possible. and requires consideration. In this case, H is re-written as Hp, which represents combination of (i) the probabilistic robustness, and (ii) the absolute robustness for the semifactual H,, such that:
H a) = E[)())xx, a] - &
Ha(x, a) (SF a; M)) - -
(16)
In practice, Hp still non non-triviss triviilaal solve. Monte Carlo (MC) sampling common strategy to apply here such that, by sampling fixed sized batch B (0 0 Pr(B, (x, a))],
H,(x; a) [[](0)|x, a) (1/ BB) EpER h()) -
(17)
This implies that substitute an unbiased estimator for the population mean.
15

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B Actionability Constraints
B.1 Non-Causal
Here we define the actionability constraints used in the various domains. It may be assumed that the direction features are allowed to change corresponds with positive gain. We use various sized "action sets" to fully test all algorithms various setups. The German Credit data used 15 actionable features be closely line with Mothilal et al. [39] whom allowed all features be mutable. However, also used 7 on Lending Club, and 4 on Adult Census/Breast Cancer to test the algorithms in situations with smaller action spaces also for completeness.
We ordered categorical features in a sensible fashion to "direct" semifactual "even if" thinking, and when say categorical feature could decrease/increase, you are interested in the exact ordering, please refer to our code which contains all the lists, but here summarise. In reality however, user must specify their exact actionability constraints, what we have specified here is designed to be representative what is possible for the "average" individual.
referring to this pre-defined order.
B.1.1 German Credit Dataset
The continuous features used were 'duration', 'amount', 'age', the categorical ones were 'status', "credit historr', 'purpose', 'savings', "employment duration', " 'installment rate', 'per-
_acul__ttttu_seeee _ex',ex,, "other debtors', 'present_ residence', 'property', 'other installment lactllmmet__ppasss 'hous- ing', 'number
credits', ceumbeercreeits', for semifactual recourse, we considered the following:
'job', 'people_liable', liable', 'telephone', 'foreign_worfe" worker'. As actionable features
duration: We allowed people increase the duration of their loan.
amount: We allowed people increase the amount of their loan.
status: We allowed people to move towards having lower status.
_rsto_h allowed people move towards egg. having late payment if their credit history was otherwise good.
savings: This feature was allowed to decrease.
employment duration: This feature was allowed decrease in case people wanted to e.g. start a new job.
installment_rate This feature was allowed to move towards lower payments.
other debetrss this feature was allowed to add another co-applicant.
cressentresxdeene This feature was allowed to move towards e.g. renting in case the user desired do so whilst searching for new house with their loan.
property: this feature was allowed to move towards having property in case the user desired to sell their house/car etc to help pay for e.g. a downpayment.
other other_instaline installment plans: This feature was allowed add other installment plans.
housing: this feature was allowed to move towards renting away from e.g. owning.
number_ ccedits: This feature was allowed to increase the user desired to acquire more credit cards.
job: this feature was allowed to decrease in case the individual desired to get a different, less demanding job within their institution, or indeed quite their job to e.g. start business.
people_liable liable: This feature was allowed to move towards more people being liable.
B.1.2 Lending Club
The continuous features used were 'loan aaan_,,nt', 'pub_rec_hne rec__re________ gec_bankruptcies, 'annual inc", 'dti', the categorical ones were 'emp_ length'' 'term', 'grade', hoom_owwnrshhip 'purpose'. As actionable features for semifactual recourse, we considered the following:
home_ownership This feature was allowed to decrease towards e.g. renting.
annual_inc this feature was allowed decrease if the person desired to e.g. work less hours.
16

---

emp lep_lenghh: This feature was allowed to decrease in case the individual desired to change careers.
dii: dept to income ratio, this feature allowed to increase.
pub_red bec__re___ue bankruptcies: This feature was allowed increase in case the user decided they wanted to declare bankruptcy to e.g. try and keep some assets.
loan aonnt this feature was allowed to increase.
term: This feature was allowed to decrease.
B.1.3 Breast Cancer
The continuous features used were none, the categorical ones were 'agegrp', 'density', 'race', 'Hispanic', 'bmi', 'agefirst', 'nrelbe', 'brstproc', 'lastmamm', 'surgmeno', 'hrt'. As actionable features for semifactual recourse, we considered the following:
bmi: This feature was allowed to move towards less healthy BMI levels in case the patient
e.g. has hypothyroidism.
brstproc: this feature was allowed to
towards having had previous breast procecdure in case the patient would like to do so or was advised.
hrt: This feature was allowed to move towards starting HRT, case person may wish to alleviate synthoms of the menopause.
agegrp: this feature was allowed to get older in case the individual would like to take no action confident that it would not lead to cancer in the next few years/decades.
B.2 Causal
In the causal setting, we allowed a user's age to increase a maximum of 5 years to mimic the motivating examples the paper about a user having a bank loan accepted. In such a situation, the user may want to e.g. work less hours over the next 5 years whilst they repay the loan, and still have accepted.
Next, we detail the direction features are allowed to change, and what direction corresponds to positive gain.
B.2.1 Adult Income Census
We use the features "sex", "age", "native-country" "marital-status", "education-num" "hours-per- week". which are the variables in the causal graph of Nabi & Shpitser [41]. We consider "age" and "hours "houus-perrweek" as actionable. We allow "age" to increase a maximum of five years, and "hours shours-per-week. to decrease.
For positive gain, we considered: Age, marital status, and eduation-num increasing corresponding to positive gain, and hours-per-week decreasing corresponding positive gain. A persons as neutral gain.
B.2.2 COMPAS
We use the features "age", "race", "sex" and "priors count", which the variables in the causal graph of Nabi & Shpitser [41]. We consider "age" and "priors count" as actionable. As actionability constraints, we assume that both features are non-negative and can only be increase, Age specifically is only allowed to increase by 5 years for each individual.
For positive gain, considered: Age and priors count increasing corresponding to positive gain. A persons sex and race was seen as neutral gain.
C Hyperparameter Choices
In this section, we discuss the hyperparameter specifications for the causal and non-causal cases respectively.
17

---

C.1 Non-Causal
Here we the values for the hyperparameters used in our demonstrations, All were obtained though pilot grid-searches across each dataset. The hyperparameter choices are summarised in Table
Table 1: Hyperparameter Specifications
Data
Ap
As
German credit
30
10
le
Lending Club
30
10
le-1
Breast Cancer
10
10
le-1
For S-GEN itself, we used the same hyperparameters everywhere outside of the above table. The number of generations spent searching for a solution was 20. The population size was fixed at {12, 24, 48, 72, 96, 120). for diversity sizes of {1, 2, 4, 6, 8, 10), respectively. The mutation rate was for each generation was 4. The probability of
0.05. The number of "elite" solutions passed crossover happening was 0.5. The number of Monte Carlo trials for each instance was 100. The continuous features sampling a standard normal distribution with standard deviation equal to the max actionable feature value, minus the min actionable feature value, multiplied by 0.05.
perturbed (in mutation population initialization) by the output from
C.2 Causal
In our causal tests we chose À as 1.0, and this was gradually decreased by momentum 7=0.9 each iteration to put more emphases on the maximization of gain.
D Algorithm Pseudocode
Algorithm S-GEN: Genetic Algorithm to Generate semifactual Recourse with Robustness and Diversity in a Non-Causal Model Agnostic Setting
Require: x the user feature
Require: 1c(-) the predictive model
Require: the expected number of suggestions
Require: n the number of candidates, n > m
Ensure: ReF the set of semifactual(s)
1: Sample n candidates D + {0, ~ X)
2: while the stopping criterion is not satisfied do
3: Obtain the fitness scores f with respect to D
4: Save the fittest 0' D according f
5: Let evolve by natural selection according to f, crossover, mutation, and elitism with x"
6: end while
7: Collect the best m unique candidates from {0 D (()) = ((x)((xxx))))( = 1} to RsF, according to
the corresponding fitness scores in f
8: if |R < m then
9: Complement RsF m elements with 0 randomly drawn from RsF
10: end if
E
Code
For our full code used please see: https://gite
The ability to reproduce the results is given.
18

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Algorithm 2 S-GEN;
Algorithm to Generate Robust & Diverse Causal semifactual Explanations for Differentiable Classifiers
Require: x the user feature vector
Require: h(()) the predictive model
Require: M the differentiable SCM
Require: e the epsilon robustness
Require: 7) the momentum parameter
Require: T the learning rate
Require: Proj(-) projection function that ensures the action is actionable
Ensure: RşF the set of semifactual(s)
1: RşF <-
2: i 5-0
3: for A do
4: Move to next loop the SF generated with the initial a does not satisfy the constraints.
5: a; +
6: while not converged do
7: Sample batch of neighbors from B, [x, a.), denoted by B;
8: if (SFF(x, as; M)) = 0 or h(0) = O, 30 E B, then
9: break
10:
end if
J: + -x. (M(S(x az; M)), Eo,B, A (M))), P(x,as)G(x,0 a;)
12:
+ Proj (a; + aV,
X A
14:
end while
RSF + Rsr U
16:
i i-i+1
i m then
18:
break
19:
end if
20:
end for
F Individual Dataset Results
The results are presented in Figure 4.
Qan
Passilly
Rebestnew
DAER
MOCEE"
0.
The ability of S-GEN create sernifactulls compared to DiCE* and PIECE* Overall, S-GEN does the best, achieving significantly better results to both baselines on 11/16 tests. Moreover, S-GEN was only significantly worse than either baseline on single test (i.e., plausibility on German Credit), with the remaining four tests being competitive between methods. Standard error bars are shown.
Figure 4: Results:
19

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G Baselines
G.1
Non-Casual
Our modification to DiCE, starts by generating a counterfactual(s) for query. Next, we use the algorithm again, but on the generated counterfactualss), make them generate second counterfactual, which goes back over the decision boundary. In effect, this generates a semifactual(s) for query.
DiCE
PIECE Second, we use the PIECE framework by Kenny and Keane [29], but apply it to tabular data. Following the authors, we divide the training data into sets, the first corresponding those predicted as the original class and the second to those predicted as the counterfactual class c', these are again split into the respective features. Hence, if there are 2 classes, with 4 features, there are 2 x 4 = 8 sets of data. These sets were then modeled using the best fit found for Beta distribution on continuous features, and simple Categorical distribution for categorical features. To generate semifactual predicted
take the probability of each feature value in the query using the models of the counterfactual class Ç', and modify each to be its expected statistical value in é one-by-one (from the lowest probability to the highest), until the next would take it over the decision boundary. In the case of continuous features, as done by Kenny and Keane [29], we take the probability as being the minimum of the two integrals either side of the feature value in the distribution. In the case the
expected feature values lic outside the actionability range, we clip them to the closest value allowed.
For Diverse Explanation of Reject [1] (DSER) had to modify the the technique in two main ways. Most notably, the techniques doesn't deal with categorical features, so to overcome this, optimised treating all one hot encoded features as real-valued, and then projected each categorical feature onto its nearest value. Next, the method addresses diversity by iterating all different sets of possible features, in our domains this is computationally intractable. Hence, optimise one semifactual at a time, each time pushing each solution as far as possible from those already found.
DSER
G.2 Causal
Karimi et al. (2021) The method by Karimi et al. [25] is recourse method designed to minimise cost whilst traversing the decision boundary. To modify the technique, we simply stop the optimization when the next step would take it over the decision boundary.
Dominguez et al. (2022) The method by Dominguez et al. [18] is identical to Karimi et al. [25], but they add in robustness component. Namely, they take an individual x, and solve inner loss which means that an individual of distance e = 0.1 (in our tests) close to x, with the same recourse given, will also achieve recourse. We simply keep the same optimization process, but aim solve a different objective. The objective we solve is to move towards the decision boundary, but when the recourse option causes either x the individual close it cross the decision boundary, terminate the optimization one step prior to this.
H Computational Costs
All tests were run on a MacBook Pro, Apple MI Pro, 16 GB. Re-running tests will take less than 1 day.
I User Study
Here we show our entire user study for complete transparency. We used the German Credit dataset, but converted the currency into U.S. dollars since it was given to U.S. citizens to complete.
20

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Intro Brief
Thank you for clicking this study!
Do no take this study on a mobile phone, the tables and images wont display. correctly.
Please don't take this study if you did similar recently.
You are free to leave at any time.
The study will take around 8mins.
You will be paid S12 per hour for your efforts.
Thank you for your participation!
Enter ID
Please Enter Your Prolific ID Here
Introduction
Introduction
21

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LOAN APPLICATION
APPROVED ROGLENT
NEVEN
UNDER SUEVIRW
LOAN APPLICATION
Personal Inlomnation
MaaeeLLT PUBLIC
19090 MAIN
JOPUq
Services neaded
UNDER SEEVEEW
You are going to be shown six situations in which a person either has a loan application approved, or rejected.
You will then be shown two different pieces of information for each situation that a bank clerk could tell the person.
You are then asked to rate how useful each of these are. That is, could the information possibly be useful in any way? Or is it not that useful?
Each situation has 4 "features".
22

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Feature Explanation
Features Used in Decision
The four "features" used to decide if each person has their loan approved rejected
are:
Duration:
Over how long the applicant wishes pay back the loan.
Amount: How much are they asking loan from the bank.
3: Savings: How much money does the applicant have saved.
4: Credit Cards: How many credit cards does the applicant have.
These are the only features the bank clerk uses to make decisions.
Click Next
Now, Please Practice On The Next Question
Sample Question
Example Question
Lucas had his bank loan accepted, his features are:
Duration
Months
Amount
$2,000
Savings
$500
Credit Cards
23

---

The two possible things the bank clerk could tell him are:
Option Even if you want increase your Duration 14 months, and Amount $3,000, we will still accept your loan application.
Option your Savings were $100, and your Credit Cards 5, we would have rejected your loan application.
How useful each option?
Not Useful
Very Useful
Option
O
O
0
0
O
0
Option 2
O
O
Click Next 2
Please only participate in this study if you understand the instructions well
the
Block 15
Remember, the key question is how USEFUL is each option
option
Click Next 3
Click Next To Begin The Study
24

---

Question
Kate had her bank loan accepted, her features are:
Duration
6 Months
Amount
$932
Savings
Over $1000
Credit Cards
2-3
The two possible things the bank clerk could tell her are:
Option 1: Even if you want to increase your Amount to $2,841, and increase your number of Credit Cards to 4-5, will still accept your loan application.
your Duration was 44 months, and you had had Savings less than $100, we would have rejected your loan application.
Option 2:
How useful is each option?
Useful
Very Useful
Option
O
O
0
00
O
Option 2
O
O
O
Question
Paul had his bank loan accepted, his features are:
Duration
18 Months
Amount
$1,239
Savings
Over $1,000
Credit Cards
25

---

The two possible things the bank clerk could tell him are:
Option Even if you want increase your Duration 21 Months, and lower your Savings to $500-$1,000, we will still accept your loan application.
Option you asked for an Amount of $15,499 (or more), and had had Credit Cards, would have rejected your loan application.
How useful is each option?
Not Useful
Very Useful
Option
O
O
0
O
Option 2
O
O
O
O
O
Question
Xue had her bank loan accepted, her features are:
Duration
Months
Amount
$1549
Savings
Over $1,000
Credit Cards
The two possible things the bank clerk could tell her are:
Option 1: Even if you want to increase your Duration to 25 Months, and increase your Amount to $4,620, we will still accept your loan application.
Option 2: you had had 3 Credit Cards, and no Savings, we would have rejected your loan application.
26

---

How useful each option?
Not Useful
Very Useful
Option
O
O
0
0
O
0
Option
O
O
O
Question
Siddarth had his bank loan rejected, his features are:
Duration
48 Months
Amount
$6,143
Savings
None
Credit Cards
2-3
The two possible things the bank clerk could tell him are:
Option 1: Even if you increase your Savings to $100, and lower your number of Credit
Cards to 1, we will still reject your loan application.
Option 2: you lower your Duration to 15 Months, and lower your Amount to $4,627, we will accept your loan application.
How useful each option?
27

---

Not Useful
Very Useful
Option
O
O
0
O
O
0
Option
O
O
0
O
Question
Camila had her bank loan rejected, her features are:
Duration
60 Months
Amount
$15,653
Savings
None
Credit Cards
2-3
The two possible things the bank clerk could tell her are:
Option 1: Even if you increase your Duration to 70 Months, and reduce your number
Credit Cards to 1, we will still reject your loan application.
Option you reduce your Amount $7,296 (or less), and you get $1, $1 Savings, we will accept your loan application.
How useful each option?
Not Useful
Very Useful
Option Option 2
O
00
0
0
0
O
O
O
O
Question
28

---

Angelo had his bank loan rejected, his features are:
Duration
60 Months
Amount
$7,408
Savings
Less than $100
Credit Cards
2
The two possible things the bank clerk could tell him are:
Option 1: Even if you decrease your Amount to $6,505, and increase your Savings to over $1000, we will still reject your loan application.
Option you lower your Duration Months, and you reduce your number of Credit Cards to 1, we will accept your loan application.
How useful is each option?
Useful
Very Useful
Option
O
0
0
:
O
Option 2
O
O
Debrief
Debrief: You Have Reached The End
Thank you for your participation, this study was designed to evaluate what kind of explanation people prefer from an artificial intelligence system.
29


Reviews:
[{"review_id": "mdGXmqzXl7", "reviewer": "Reviewer_LWa1", "summary": "This paper explores techniques for optimizing positive outcomes using semi-factuals. To do so, they propose the concept of \"gain\" and set up a small user test to demonstrate the usefulness of their technique.", "questions": "-- I think most importantly, the authors need to better motivate their work, especially in the Introduction. I think it might help to give more context of previous work in the Introduction: in what contexts is this kind of work used? what's come before? how does this work add on to it? The answers to these questions are scattered throughout the rest of the paper, and in Related Work, but it should be very crisply and tightly explained in the Introduction. In general, the language in the Introduction is a little too loose. I'd love a very clear distinction between counterfactuals and semi-factuals, and between positive outcomes and negative outcomes. I see that loans are one context in which this work could be useful, but what are other contexts in which this work could be useful? This is just a smattering of ideas for cleaning up the Introduction, but I think in general, the Introduction feels a little too philosophical. At some point in the Introduction, I'd like it to be made clear how this is used in the context of ML.\n-- There are also some strong claims made that I would want justified (e.g. \"completely unexplored\" in line 22).\n-- I wasn't convinced by the two main issues outlined in the second paragraph of the Introduction. These are the two main issues according to who? Fairness is mentioned in line 37, but it wasn't clear how fairness and ethics is really an issue in this paragraph. (I understand what the authors are trying to say, but the writing doesn't make it as clear as it could be.)\n-- There are typos scattered throughout the manuscript, which should be proof-read carefully.\n-- It might help to move part of lines 61-65 into the Introduction. Also, I think describing the \"urgent need\" in line 57 would help motivate this work, too.", "limitations": "The authors do mention the limitations of their user evaluation in the Discussion, but I feel they could expand on the limitations of their approach a bit further.", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 1, "contribution": 3, "strengths": "The authors offer an interesting and compelling framework for optimizing positive outcomes, with strong formal notation and proofs.", "weaknesses": "My biggest concern with the paper is its presentation. I think the authors still have much work to do in cleaning up the writing and presentation of this work. See below.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for noting the compelling nature of our framework, and on account of voting for the acceptance of the work. Here, we address your specific questions and concerns.\n\n***\n\n*I think... the authors need to better motivate their work, especially in the Introduction... the language... is a little too loose. I'd love a very clear distinction between cfs and sfs, and between positive v. negative outcomes.*\n\nThank you for pointing this out. We have added clearer definitions for CFs v. SFs and positive v. negative outcomes to the intro. the new sentences added state: *“Our definition of counterfactuals is in line with the literature and Wachter et al. [4], where a given test instance $x$ classified as class $c$ must be modified in such a way as to cross a decision boundary into class $c'$, and a semi-factual $x$ classified as $c$ must be modified in such a way as to not cross a decision boundary (and hence remain class $c$) [5]”*, and *“In recourse, bad outcomes (e.g., a loan rejection) are generally mutated to produce good outcomes (e.g., a loan acceptance) for users using counterfactuals. In our setting, we are assuming there was initially a good outcome, and we are trying to mutate features to produce an even better outcome for users, but in doing so not cross the boundary into the bad outcome (i.e., using semi-factuals).”* \n\n**Revision:** We added the above sentences to the intro and “tightened up” the language to be less loose in general and more formal.\n\n***\n\n*Where else is this work useful?*\n\nGood question, we envision that it will also be quite useful in medical applications (partly why we used the BCSC dataset). For example, patients are often over prescribed drugs with adverse side effects [1]. So we envision a semi-factual could say *“Even if you half your intake of drug x, you will still be at a low risk for disease y”*. This way the patient can hopefully avoid the side effects of taking so much of the drug, whilst still maintaining low disease risk (i.e., the good outcome). But the explanations will be likely useful in any application that can be parsed as positive/negative outcomes. \n\n**Revision:** We note the possible medical applications for our work in the second paragraph of Sec 1.\n***\n*In the Introduction, I'd like it to be made clear how this is used in the context of ML.*\n\nWe understand, we have mentioned how we envision the explanations being useful for financial and medical applications as discussed above, as well as a clear definition of CF v. SF and positive v. negative outcomes. Apologies for the lack of clarity.\n\n**Revision:** See above two revisions.\n***\n*Justify \"completely unexplored\" in line 22.*\n\nNoted, we conducted a literature review to the best of our ability and couldn’t find comparable work. \n\n**Revision:** We have rephrased this to *“... which (to the best of our knowledge) remains completely unexplored”* to be more conservative.\n***\n*I wasn't convinced by the two main issues outlined in the second paragraph of the Introduction. These are the two main issues according to who?*\n\nArtelt & Hammer [2]  discuss the second issue as a concern (i.e., manipulating people to accepting bad outcomes when they don’t have to) which we cite (line 22). Now, regarding the first issue of “usefulness”, we found this through our user study both through (1) the results, and (2) initial pilot testing to gauge people’s feelings about the explanations. However there are other possible issues such as how they can negatively influence people’s causal understanding of AI systems [3], but this was beyond our scope to investigate here. \n\n**Revision:** We have fixed this by changing line 26 to:  *“However, such a use case for semi-factual explanation **perhaps** has two main issues”*. Also, we foreshadow the user study results to make this statement about “usefulness” more convincing.\n \n***\n\n*...it wasn't clear how fairness and ethics is really an issue… It might help to move part of lines 61-65 into the Introduction.*\n\nThank you for this. We have done as you suggested and we agree it does indeed make the fairness issues much clearer in the introduction, thanks for the suggestion.\n\n**Revision:** We moved lines 61-65 to Paragraph 2 in the Intro.\n\n***\n\n*There are typos scattered throughout the manuscript, which should be proof-read carefully.*\n\nApologies for those mistakes, fixed.\n\n***\n\n*Also, I think describing the \"urgent need\" in line 57 would help motivate this work, too.*\n\nGood point, we have elaborated with the recently published work [3] as motivation here as they discuss *“the paucity of user studies”* as a severe limitation for semi-factuals; in that until some carefully-controlled studies are carried out, we do not really know how users will respond to these explanations in the ML context.\n\n***\n\n*The authors do mention the limitations of their user evaluation in the Discussion, but I feel they could expand on the limitations of their approach a bit further.*\n\nAgreed, we added the limitation regarding the need for a causal model of the domain in order to use causality when calculating gain (although you may still do this without the SCM in our non-causal algorithm).\n\n***\n\n[1] Safer, Daniel J. \"Overprescribed medications for US adults: four major examples.\" Journal of clinical medicine research 11.9 (2019): 617.\n\n[2] Artelt, André, and Barbara Hammer. \"“Even if…”–Diverse Semifactual Explanations of Reject.\" 2022 IEEE Symposium Series on Computational Intelligence (SSCI). IEEE, 2022.\n\n[3] Aryal, Saugat, and Mark T. Keane. \"Even if explanations: Prior work, desiderata & benchmarks for semi-factual XAI.\" IJCAI Survey Track – 2023\n\n[4] Wachter, Sandra, Brent Mittelstadt, and Chris Russell. \"Counterfactual explanations without opening the black box: Automated decisions and the GDPR.\" Harv. JL & Tech. 31 (2017): 841.\n\n[5] Kenny, Eoin M., and Mark T. Keane. \"On generating plausible counterfactual and semi-factual explanations for deep learning.\" AAAI\n\n"}, {"review_id": "wyCAseeEcS", "reviewer": "Reviewer_6hWY", "summary": "The paper formalizes the notion of semi-factual for positive outcomes. Given a black-box classifier, a semi-factual is a statement which informs a user about “even if” scenarios in which they would maintain the assigned positive prediction, increasing their “gain”. The authors formalize the problem for both a causal and non-causal setting. They show the validity of their method, S-GEN, via experiments on real-world datasets. Lastly, they also performed a first user study which shows the usefulness of semi-factual for users.", "questions": "- If we consider the notion of Gain, how does it relate also to the concept of Shapley value [1]? Which are the differences/similarities between them?\n- In Section 3,.4, on line 181, how do you define such distribution density? Namely, given the SCM, you can define the interventional joint distribution after a do action $Pr(\\mathbf{X} | do(\\mathbf{a}))$. Is it the same thing?\n- In Section 4, how do you define the “empirical approximation” of $G$, $P$, $H$, $R$ and $J$? How can you learn such approximations?\n- In Section 6, on line 333, the paper says “...rate on a scale from 1-5 how useful each where.”. Could you elaborate more on this point? Useful to what?\n\n[1] Lundberg, Scott M., and Su-In Lee. \"A unified approach to interpreting model predictions.\" Advances in neural information processing systems 30 (2017).\n", "limitations": "The authors adequately address all the limitations.\n", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "To the best of my knowledge, the topic is novel and the proposed ideas are interesting for the algorithmic recourse field. The paper is clear and it can be understood quite well. The related works are discussed extensively and the problem is nicely framed in the context of the state-of-the-art literature. I liked the causal formulation of the semi-factual problem and the user evaluation. More specifically, the user study provides strong hints that semi-factuals have merits, thus it grounds the algorithmic recourse research in the real world.", "weaknesses": "The notation and the various proofs can be hard to read at first glance. For example, in Section 3.1., between lines 95 and 112, having $S$ to be the counterfactual and $\\textbf{S}$ to be the structural equation instead can cause confusion.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We greatly thank the reviewer for the extremely positive comments, noting the novelty of our work, and voting for acceptance. We now address all questions and concerns:\n\n***\n\n*The notation and the various proofs can be hard to read at first glance. For example, in Section 3.1., between lines 95 and 112, having $S$ to be the counterfactual and $\\mathbf{S}$ to be the structural equation instead can cause confusion.*\n\nThanks for making us aware of this, we have changed $S$ to $SF$ for representing the semi-factual function to avoid confusion here. If there is anything else specific which stands out we can modify that also, please just let us know. \n\n**Revision:** We modified the notation for $S$ => $SF$ in Section 3.1\n\n***\n\n*If we consider the notion of Gain, how does it relate also to the concept of Shapley value [1]? Which are the differences/similarities between them?*\n\nIn XAI, Shapley values are usually used to explain the contribution of each feature to a prediction (or score) either for a single sample or a set of samples. Gain is a function which outputs a real-valued number representing how much a user benefited (positively or negatively) from feature mutations. We feel the two have few if any similarities.\n\n***\n\n*In Section 3,.4, on line 181, how do you define such distribution density? Namely, given the SCM, you can define the interventional joint distribution after a do action $Pr(\\mathbf{X}| do(\\mathbf{a}))$. Is it the same thing?*\n\nYes you are correct it is the same thing. Assume that we read $Pr(\\mathbf{X}| do(\\mathbf{a}))$ as the probability of $\\mathbf{x}$ resulting from some $\\mathbf{x_0}$ taking action $\\mathbf{a}$. $Pr(S_{M}(\\mathbf{x}, \\mathbf{a}))$ is the probability of some $\\mathbf{x}’$ resulting from $\\mathbf{x}$ taking action $\\mathbf{a}$. Essentially, $\\mathbf{x_0}$ and $\\mathbf{x}$ are the same as $\\mathbf{x}$ and $\\mathbf{x’}$ in the notation, respectively. If the reviewer prefers we can easily change the notation to match what you wrote above, or simply note this in a footnote, please let us know your preference.\n\n**Revision:** We await your feedback.\n\n***\n\n*In Section 4, how do you define the “empirical approximation” of G, P, H, R and J? How can you learn such approximations?*\n\nWe define the empirical approximations as follows: $G$ is a simple euclidean distance between a test instance and the generated semi-factual(s) (see lines 294-297). $P$ is defined (in non-causal tests) in line 257, it is a simple function to return high values when the test instance is close to a training datapoint (high plausibility), and low values when it’s not (low plausibility), it is naturally taken care of in causal domains thanks to the SCM. $H$ is the Lagrangian interpretation of robustness $R$, it is approximated with $\\hat{H}_p$ which uses MC sampling to see if the neighborhood around the semi-factual explanation(s) is robust, and $\\hat{H}_a$ which checks the robustness of the semi-factual(s) itself by classifying it with the model (i.e., it doesn’t need to be approximated). $J$ is just the objective. \n\nThere is a vast literature on learning optimal weights in multi-objective optimization [4], we chose one of the most popular approaches of a grid search (see line 237), but other options involving reinforcement learning, bayesian optimization etc. are options. But in our case it wasn’t necessary to achieve the desired performance.\n\n***\n\n*In Section 6, on line 333, the paper says “...rate on a scale from 1-5 how useful each where.”. Could you elaborate more on this point? Useful to what?*\n\t\t\t\nNo problem, there are many things the semi-factuals could be useful for in the context of a loan application. Our take is that they are useful for things such as e.g. spending more time with family (see lines 131-133), or reducing your downpayment to have more flexible financial options. We didn’t want to explicitly define what was meant by “useful” to users (indeed we avoided it in the user study on purpose), but rather let people use their own “natural” interpretation. This is standard practice in cognitive studies in our experience, and indeed the same approach is taken in the DARPA “trust” and “satisfaction” measures too which are popular in XAI [2]. A sign this has failed is if the results are not consistent (e.g. see [3] and their \"reasonableness\" measure), but here we see a clear result in that people seem to converge on what they interpret as “useful” for the setting.\n\n**Revision:** We explain our definition of “useful” in Section 6.\n\n***\n\n[1] Lundberg, Scott M., and Su-In Lee. \"A unified approach to interpreting model predictions.\" Advances in neural information processing systems 30 (2017).\n\n[2] Hoffman, Robert R., et al. \"Metrics for explainable AI: Challenges and prospects.\" arXiv preprint arXiv:1812.04608 (2018).\n\n[3] Kenny, Eoin M., et al. \"Explaining black-box classifiers using post-hoc explanations-by-example: The effect of explanations and error-rates in XAI user studies.\" Artificial Intelligence 294 (2021): 103459.\n\n[4] Gennert, Michael A., and Alan L. Yuille. \"Determining the optimal weights in multiple objective function optimization.\" ICCV. 1988\n"}, {"review_id": "WOeKPKwcbw", "reviewer": "Reviewer_vFfC", "summary": "This paper looks into the mirror problem of counterfactual recourse. In the latter the goal is to determine what is needed to change a negative outcome to a positive such as in loans. In the former the goal is to determine to what extend a positive decision would remain and to what extend the end user could have pushed their features. The paper introduces this novel view with motivations from psychology and also investigates this with a user study. In order to achieve this they use similar ideas from causal algorithm recourse with constraints on the proposed explanations.", "questions": "see above", "limitations": "NA", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The paper tackles a new problem that has not had much attention in the XAI community. We are often focused on the algorithm recourse, however, this paper proposes a new twist to explanations when users in fact got a positive outcome. The user study albeit only 15 samples (might be wrong on this) shows that people do in fact prefer this type of explanation when in fact getting a positive outcome. Which tbh makes sense as in why would they care about what would have happened if they didn't get it?\n- Another plus is the simplicity of the idea which i like very much given that it is well-motivated.\n- The paper also clearly introduces their desiderata for the explanations as well as the objective they aim to solve.\n- Lastly, just like in the previous causal algorithm recourse, they also provide a causal view of this problem.", "weaknesses": "- The paper seems to take too much inspiration from the causal algorithm recourse paper to a point where it becomes just a rewriting of that paper from a different view with the exact same/similar techniques.\n- I also don't understand how gain and cost are not equivalent as you can have causal costs as well. in your line 145 you say the main difference is the causal part which I don't get as this can also be done with cost no?\n- In Figure 2, what happened to plausibility? does that mean the explanations are all non-plausible?\n- ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for noting the novelty and simplicity (in a good way) of our idea and its relevance to the field. We now address all questions and concerns:\n\n***\n\n*The paper seems to take too much inspiration from the causal algorithm recourse paper to a point where it becomes just a rewriting of that paper from a different view with the exact same/similar techniques.*\n\nOur causal algorithm is different to prior work because it uses (1) a maximization objective, and (2) a \"Gain\" function. Together, these aim to produce the nuances of “even if” thinking we are intending for (which other approaches don’t).\n\nHowever, we would like the emphasize that this algorithm is not our main contribution, it was instead (see line 38) the conceptual idea to use semi-factuals (and indeed XAI in general) for explaining how to optimize positive outcomes (see lines 30-33). Moreover, this involved creating an approach for when the causal structure is not known (i.e., our non-causal algorithm), verifying people find the explanations useful (i.e., our user study), and laying a strong theoretical foundation for semi-factual research going forward (i.e., our proofs and derivations).\n\nThe causal experiment primarily exists to show how the nuances of “even if” thinking are better captured with causality (see lines 313-316), and indeed there was a very strong effect size (lines 322-323), which gives practical validity to our proof in Theorem 3.1.\n\n***\n\n*I also don't understand how gain and cost are not equivalent as you can have causal costs as well. in your line 145 you say the main difference is the causal part which I don't get as this can also be done with cost no?*\n\nApologies for the lack of clarity, our gain function requires a set of human defined directions for features to mutate in order to be considered positive/negative gain (e.g., if a user desires to work less hours per week then lowering this feature = positive gain, and raising it = negative gain; see lines 131-133), which cost does not consider, this alone means it can be different (see line 136).\n\nTo answer your question however, our experience is that cost is usually formulated as the actions an end user takes to achieve recourse [1,2]. This makes sense since even if other features change causally, it didn’t “cost” the user anything to modify them in terms of effort. However, gain doesn’t only care about the “effort” taken by the user, but also how much they stand to gain from the resultant causal mutation, so ideally, gain will *always* consider causality, whilst cost doesn't necessarily (although we understand that isn't always practical, so we included the non-causal algorithm too).\n\n**Revision:** We emphasise how a Gain function (in contrast to cost) requires the specification of positive/negative directions for each feature in our “Why is gain not necessarily equivalent to cost?” paragraph.\n\n***\n\n*In Figure 2, what happened to plausibility? does that mean the explanations are all non-plausible?*\n\nApologies but we assume the reviewer is referring to Figure 3? Plausibility is shown in Figure 2. In this work we define plausibility as the generation of explanations which are within distribution. Thanks to the SCM, this is guaranteed in the causal experiments, so we don’t measure plausibility there. \n\n**Revision:** We made this clear at the end of Section 5 by adding *“Note for causal tests the SCM guarantees plausibility so this metric is not reported”*.\n\n***\n\n[1] Karimi, Amir-Hossein, et al. \"A survey of algorithmic recourse: definitions, formulations, solutions, and prospects.\"\n\n[2] Dominguez-Olmedo, Ricardo, Amir H. Karimi, and Bernhard Schölkopf. \"On the adversarial robustness of causal algorithmic recourse.\" International Conference on Machine Learning. PMLR, 2022.."}, {"review_id": "EzVvGuUP2V", "reviewer": "Reviewer_4usU", "summary": "This paper proposes a framework for \"semi-factual\" explanations - explanations given to an applicant who receives an accept decision, about what they could do which would allow them to maintain that decision. The framework and objective propose utilize a causal model of the action space, and consider the applicant's gain in utility, the plausibility of the output, the diversity of suggestions and the robustness of the suggested actions. Experimentally, they demonstrate an improvement of their approach over baselines and also conduct a human study showing their approach is perceived to produce more useful explanations.", "questions": "- Is there work from the counterfactual (rather than semi factual) literature that uses causal modelling? I'm not sure how deep the novelty is supposed to run here", "limitations": "I recommend against using COMPAS as a dataset in papers which are not in a criminal justice context - see \"It's COMPASlicated: The Messy Relationship between RAI Datasets and Algorithmic Fairness Benchmarks\" for more, and it's probably worth writing a piece about this.", "rating": 3, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "- incorporation of causal models in counterfactual-type explanations is a good direction and a sensible idea\n- the presented framework is quite thorough in terms of the desiderata you'd hope for in semi-factual explanations\n- the human study at the end is very cool - I haven't seen so many of these in this literature and I think that really sets this paper apart", "weaknesses": "- Overall, I find myself lacking quite a bit of clarity around many of the formalisms in pages 3-6 as the framework is described. Some of this points of confusion are: \n1. I find 3.1 hard to follow - for instance, it's not clear to me what kind of object A is: it seems to be from a subset of the domain of X, but then it's stated later that they're not simply added, so I'm not sure how to interpret its value, or what it means that \"actions have to be mutable\" (line 94); I'm not so sure why we need S if we have S_M, what is S supposed to be?\n2. It's not obvious to me what P is supposed to be - should it be conditional on x rather than a marginal? is it defined with reference to some action history? How do we define it in a way which serves us? Also, I'm not sure why P and G should be multiplied - may need more explanation for that\n- I found Section 4 challenging to follow as well:\n1. it's not clear to me what a \"causal\" vs a \"non-causal\" domain is supposed to be\n2. I didn't follow the methods for empirical approximations of H or M, and I'm not sure why P needs an approximation if it's just calculating a known density that we define?\n3. I don't really follow the fitness function section either but I'm not so familiar with genetic algorithms so that could be why\n\n- I'm not sure that Lemma 3.3 isn't trivial - if H >= 0 for all a, and we assume G > 0 and P, R, \\lambda are > 0 by definition, doesn't J have to be >= 0 by definition as well?\n\n\n\nSmaller notes:\n- how dow we know S inverse exists? (line 105)\n- line 120: a little confusing referring to post robustness before it is defined\n- line 216: depict is an odd word here, present may be better\n- how are the results in Fig 2 normalized? It's not clear to me that they should be\n- why is P lowers for S-GEN? Is this good or bad?\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for noting the relevance of our user study and how it sets the paper apart. We are truly grateful for your suggestions which have improved the paper, we now respond to your questions & concerns. Apologies for the brevity of some points, but space was limited.\n\n***\n\n### Reviewer's Main Question:\n\nYes there is some work recently, perhaps most relevant is the two works we cited and compared to [1,2]. \n\nOur causal algorithm is different to prior work because it uses (1) a maximization objective, and (2) a \"Gain\" function. Together, these aim to produce the nuances of “even if” thinking we are intending for (which other approaches don’t).\n\nHowever, we would like the emphasize that this algorithm is not our main contribution, it was instead the conceptual idea to use semi-factuals (and indeed XAI in general) for explaining how to optimize positive outcomes (lines 30-33). Moreover, this involved creating an approach for when the causal structure is not known (i.e., our non-causal evolutionary algorithm), verifying people find the explanations useful (i.e., our user study), and laying a strong theoretical foundation for semi-factual research going forward (i.e., our proofs & derivations).\n\nThe causal experiment primarily exists to show how the nuances of “even if” thinking are better captured with causality (lines 313-316), and indeed there was a very strong effect size (lines 322-323), which gives practical validity to Theorem 3.1.\n\n***\n### Other Q's\n\n**Object A & Mutable:** A is simply a set of actions the user “does” (e.g., increase their downpayment on a loan). Regarding the addition, we feel that a “+” operator is not the best notation to use here because categorical features are involved. We’d be happy if the reviewer could recommend a better way. By “actions have to be mutable” we mean that the feature should be possible for the user to change, e.g. not “Race”. We have made these points clearer.\n\n**Clarify S and S_M:** S is for the non-causal setting while S_M is for the causal setting. We explicitly mention this now.\n\n**What is P?**  P is a plausibility function, here we realise this with the NN-Dist metric from [3] in non-causal settings, and the SCM in causal ones (line 233). We now clearly note that the SCM guarantees plausibility in casual settings in Sec 4.1 and the “Plausible Gain” paragraph in Sec 3.4\n\n**Should P be conditional on x?**  Correct, that is actually how we implemented P (line 257).\n\n**Is x defined with action history:** No, we left sequential decision making for future work (see line 71).\n\n**How to define P to serve us?** This is an open question, prior work has shown the NN-Dist metric works well [3], but the advantage of SCMs is that we don’t have to worry about it.\n\n**Why P & G multiplied?** We regard P as a scaling factor for G, together it can be seen as an expected gain where a high plausibility ensures high expected gain. Likewise, 0 plausibility ensures that we will have a 0 expected gain. You can add them, but doing so misses this special property. We mention this now.\n\n**Explain causal\" v. non-causal:** The causal setting here is when the SCM and structural equations are known, whilst the latter is when you ignore causal dependencies.\n\n**How approximate H?** It has two parts: Firstly, $\\hat{H}_p$ refers to the robust $e$-neighborhood around a semi-factual that is approximated with MC sampling. Secondly, $\\hat{H}_a$ is the robustness of the semi-factual itself being classified correctly. We made this clearer in Sec 4.\n \n**How approximate M?** We assume the reviewer is using M to refer to the causal model? We were not approximating it as it is pre-defined.\n\n**Why P need approximation?** Because the distribution of most real world data is unknown. Line 181 is just a theoretical formulation, in practice we need to estimate it, for us this is the NN-Dist metric from [3], whilst the SCMs guarantee it in casual situations. We now note this about P in Sec 3.4. \n\n**Fitness Function Clarity:**  We included an additional sentence in Sec 4.2. explaining fitness function background.\n\n**Lemma 3.3 Trivial?** The main focus of Lemma 3.3 is to verify that our simplification from Eq. (8) to formulation Eq. (9) is correct. As an aside, we also want to show that if the constraints are satisfied, the adversarial objective will just go to infinity. Another way to think about it is the score becomes reasonable when the constraints are all satisfied, and in our task this means that robustness must be satisfied to get reasonable Gain. \n\n**S inverse exists?** S inverse does not always exist and this is usually decided by the specific dataset where a causal graph exists. It is not a rigorous function but can simply be some rule-based mapping. It follows the same flow of thought in related work like [1]. \n\n**Post robustness & line 216:** fixed both.\n\n**Fig 2 normalization:** We used min-max normalisation. We did it because all datasets have different ranges in their results, so we can’t just average them normally. We now note this in Fig 2 and added the non-normalized plots in the Appendix. We also uploaded this as a PDF in the global response above for the reviewer's convenience.\n\n**Why P lower for S-GEN?** In our non-causal tests lower plausibility is better, so yes S-GEN performs best in Fig 2. Plausibility here is the distance of the generated semi-factual to the nearest training datapoint, we borrowed the metric from [3]. We now note this in the Fig caption.\n\n**COMPAS:** Thanks. We used it because of the availability of high quality SCMs and structural equations, as is common practice [1]. In the future we will avoid using this dataset.\n\n***\n\n[1] Dominguez-Olmedo, et al. \"On the adversarial robustness of causal algorithmic recourse.\" ICML\n\n[2] Karimi et al. \"Algorithmic recourse: from counterfactual explanations to interventions.\" FAccT.\n\n[3] Kenny & Keane. \"On generating plausible counterfactual and semi-factual explanations for deep learning.\" AAAI"}]

Author rebuttal:
We would like to thank the reviewers for taking the time to read our paper and give constructive feedback. We have taken seriously the suggestions of all reviewers and either amended the manuscript accordingly to address the concerns, or discussed it here in our response.

Firstly however, we thank the reviewers for their positive comments, and overall vouching for acceptance, **Reviewer-6hWY** said *“To the best of my knowledge, the topic is novel and the proposed ideas are interesting for the algorithmic recourse field. The paper is clear and it can be understood quite well. The related works are discussed extensively and the problem is nicely framed in the context of the state-of-the-art literature. I liked the causal formulation of the semi-factual problem and the user evaluation. More specifically, the user study provides strong hints that semi-factuals have merits, thus it grounds the algorithmic recourse research in the real world.”* **Reviewer-vFfC** pointed out that *“the paper tackles a new problem that has not had much attention in the XAI community – Another plus is the simplicity of the idea which i like very much given that it is well-motivated”*, **Reviewer-LWa1** stated *“The authors offer an interesting and compelling framework for optimizing positive outcomes, with strong formal notation and proofs.”* lastly **Reviewer-4usU** said *“the human study at the end is very cool - I haven't seen so many of these in this literature and I think that really sets this paper apart”*.

We are truly grateful for these (and other) encouraging comments made.

Beyond the positive reviews, we are grateful for many of the suggestions provided (mainly presentation related) and have incorporated them into the latest revision to improve the manuscript. Our changes include:

* We took the suggestions of **Reviewer LWa1** and improved the clarity of our writing in the introduction, literature review, and limitations (details in our response to the reviewer).
* Regarding the feedback of **Reviewer 6hWY**, we modified the notation of our semi-factual function and also made our notion of what “useful” means clear in the user study.
* Thanks to **Reviewer vFfC** we make it clearer why gain and cost are not the same and how SCMs guarantee our notion of plausibility.
* **Reviewer 4usU** overall felt the paper lacked clarity in pages 3-6, so we added additional detail around areas of confusion alongside other suggestions from the reviewer to improve the presentation (details in response to the reviewer).
* For **Reviewer 4usU** we also uploaded the non-normalized results of Experiment 1 for clarity here as a PDF file (our non-causal tests), we will add this to the final appendices in case readers are interested.

**Importantly, all of these were quite minor edits that we completed the past week**. Although we are not allowed to upload the updated manuscript, we detail as much as appropriate the changes made below.

Best wishes,

Anonymous Author(s)


Meta-review: The authors formalizes the notion of semi-factual for positive outcomes and propose a methodology to optimize them. A majority of reviewers were positive about the paper. There was one reviewer who was particularly negative about the paper, however, the reviewer's concerns were mainly due to a lack of clarity in terms of presentation, which was also shared by other reviewers. I request the authors to revise their manuscript very carefully following the reviews/rebuttal when preparing the final version of the paper.
Decision: Accept (poster)