Title: The Utility of “Even if” Semifactual Explanation to Optimise Positive Outcomes
Venue: NeurIPS 2023
Accepted: yes
Keywords: Semifactual Explanation

Counterfactual Explanation

Explainable AI

Recourse

User Study
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

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.

1 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 to counterfactuals, semifactuals are a major, fundamental part of human explanation, and have long been studied in psychology [9, 35, 49], philosophy [6, 8, 20], and lately computer science [1-3, 29, 34, 52, 62]. They take the form of 'Even if x happened, y 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 et al. [55], where a test instance classified as c must be mutated to cross a decision boundary into class c ′ . Likewise, as established in the literature [1, 29], we define a semifactual as an instance classified as c , which must be modified in such a way as to not cross a decision boundary (and hence remain class c ) [29]. In recourse [25, 50], 'negative outcomes' (e.g., a loan rejection) are generally mutated to produce 'positive outcomes' (e.g., a loan acceptance) for users using counterfactuals. In our setting, we are assuming there was initially a positive outcome, and we are trying to mutate features to produce an even better situation for users, and in doing so not cross the boundary into the negative outcome (i.e., using semifactuals).

Historically, counterfactuals have had obvious applications in computer science, such as explaining how to have a bank loan accepted rather than rejected, but applications for semifactuals as less

Weipeng Huang ∗

clear. As such, the usage of semifactuals has often inadvertently defaulted to copying counterfactual research by also explaining negative outcomes (e.g., 'Even if 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, it is debatable if these explanations convey useful information [1], whilst a counterfactual explaining how to cross a decision boundary and have a 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 [1]. 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 a positive (rather than a negative) outcome. In this setting, consider a user that has had their loan accepted, but might prefer to make a smaller down-payment on a loan application. In this situation, our framework could present an explanation such as 'Even if you half your downpayment, 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 a negative outcome, which guarantees fairness is this regard. Now, with regards to optimising fairness even further, note that banks are not motivated to share such explanations even though they may help people, because (for example) larger downpayments 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 an 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 a 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 x , 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 et 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-loop 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. [52] 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 semifactuals [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).

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 et 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 be 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 as 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 a set of 'sufficient' features for a prediction which, in the presence of the other features mutating, mostly doesn'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 modify them to create a semifactual. explicitly exclude features that are either immutable or non-actionable . Adopting Pearl's do () operator [36], an action can be defined as a ( x ) = do ( X = θ ) , or simply do ( θ ) , to force a hard intervention of replacing x by θ where θ ∈ X . It implies that, for each feature, X i := θ i for the individual x . If the action do ( θ ) imposes no change, we have x = θ . We further denote a set of human-constrained actionable ranges A = { a ( x ) = do ( θ ) : ∀ θ ∈ X} . Moreover, the actions have to be mutable and explicitly exclude any action which keeps the individual in the same position.

3 Semifactual Framework The non-causal semi-factual interaction between is, the individual x taking action a ( x )

will lead to another representation

θ

SF

(

x

)

is defined by

:

∈ X

x

and

a

explicitly exclude features that are either

X × A ↦→ X

. That immutable

representing that

In this section, we describe the basic definitions and assumptions for our semifactual framework to optimise positive outcomes for users, before formalising it under the concept of Gain (i.e., how much a user stands to benefit from the explanation) in a 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. explicitly exclude features that are either immutable or non-actionable . Adopting Pearl's do () operator [36], an action can be defined as a ( x ) = do ( X = θ ) , or simply do ( θ ) , to force a hard intervention of replacing x by θ where θ ∈ X . It implies that, for each feature, X i := θ i for the individual x . If the action do ( θ ) imposes no change, we have x = θ . We further denote a set of human-constrained actionable ranges A = { a ( x ) = do ( θ ) : ∀ θ ∈ X} . Moreover, the actions have to be mutable and explicitly exclude any action which keeps the individual in the same position. The non-causal semi-factual interaction between x and a ( x ) is defined by SF : X × A ↦→ X . That is, the individual x taking action a ( x ) will lead to another representation θ ∈ X representing that person's recourse. Moreover, a structural semi-factual is defined which considers the dependence between the related features [15, 19]. We denote the structural causal model (SCM) by M = ( S , P U ) where S are a set of structural equations and P U is the distribution over the exogenous variables U ∈ U . Consider that in a causal graph, there is a set of causal parents for each feature x i ∈ x , denoted by Pa i . We denote the structural equations as S = { x i := g i (Pa i , U i ) : i = 1 , . . . , k } where g i ( · ) is a deterministic function that describes the causal relationship for x i , and depends on the exogenous variable U i ∈ U alongside the corresponding parent set Pa i . Hence, S induces a mapping S : U ↦→ X ∗ and its inverse mapping S -1 : X ↦→ U . Let f ◦ g ( x ) = f ( g ( x )) which can be extended to more functions. Hence, we specify the SCM-equipped semi-factual by SF ( x , do ( θ ); M ) to denote the transition between the states by taking a certain action through an SCM , where person's recourse. Moreover, a structural semi-factual is defined which considers the dependence

operator [36], an action can be defined as

x

It implies that, for each feature,

(

by

x

θ

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do

(

X

=

θ

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(

θ

a

(

x

)

, or simply or

non-actionable

.

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);

where

θ

) :=

.

-

where do

.

1

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θ

)

do

(

M

M

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(

are a set of structural equations and

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(

θ

);

)

.

SF

S

U

, to force a hard

.

X

i

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θ

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x

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S

, we have

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=

θ

=

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(

x

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(

θ

);

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(

3.1 Definitions intervention of replacing individual x . If the action θ ′ =

∈ U

denoted by

g

i

(

(

θ

)

∈ U

′

M

◦

·

M

Consider that in a causal graph, there is a set of causal parents for each feature

= (

S

do

,

()

P

U

is the distribution over the exogenous variables

P

(1)

θ

) =

x

, do

(

U

))

.

human-constrained actionable ranges

Let us denote an individual x ∈ X with k mutable features X = { X 1 , . . . , X k } . Given the individual x , a set of actions can be applied to x where each action a ( x ) is also a 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 or non-actionable . Adopting Pearl's do () operator [44], an action can be defined as a ( x ) = do ( X := θ ) , or simply do ( θ ) , to force a hard !Χ#$% &'(ℳ explicitly exclude features that are either immutable or non-actionable . operator [36], an action can be defined as a ( x ) = do ( X = θ ) , or simply do ( θ intervention of replacing x by θ where θ ∈ X . It implies that, for each feature, individual x . If the action do ( θ ) imposes no change, we have x = θ . human-constrained actionable ranges A = { a ( x ) = do ( θ ) : ∀ θ ∈ X} . Moreover, the actions have to be mutable and explicitly exclude any action which keeps the individual in the same position. The non-causal semi-factual interaction between x and a ( x ) is defined by SF : X × A ↦→ X . That is, the individual x taking action a ( x ) will lead to another representation θ ∈ X representing that person's recourse. Moreover, a structural semi-factual is defined which considers the dependence between the related features [15, 19]. We denote the structural causal model (SCM) by M = ( S , P U ) where S are a set of structural equations and P U is the distribution over the exogenous variables U . Consider that in a causal graph, there is a set of causal parents for each feature denoted by Pa i . We denote the structural equations as S = { x i := g i (Pa i , U i ) : i = 1 g i ( · ) is a deterministic function that describes the causal relationship for x i exogenous variable U i ∈ U alongside the corresponding parent set Pa i . Hence, S S : U ↦→ X ∗ and its inverse mapping S -1 : X ↦→ U . Let f ◦ g ( x ) = f ( g ( x )) to more functions. Hence, we specify the SCM-equipped semi-factual by SF ( x , do ( θ ); the transition between the states by taking a certain action through an SCM M , where θ ′ = SF ( x , do ( θ ); ) := S S -1 SF ( x , do ( θ ); ) . If all features are independently manipulable M Therefore, SF ( x , do ( θ ); M ) is a more generalized 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 (e.g., a loan acceptance) and 0 is a negative outcome (e.g., a loan rejection). We set a lower threshold ψ that separates the decision boundary. For the form defined above, ψ = 0 . 5 is a reasonable threshold that fits all situations well. θ ′ = SF ( x , a ; ) := S S -1 SF ( x , do ( θ ); ) . (2)

Pa

x

A

M

◦

x

)

i

x

,

, . . . , k

}

where

∈

, and depends on the which can be extended

induces a mapping

Adopting Pearl's do

()

)

to denote

, to force a hard

)

X

for the

i

i

:=

θ

M

We further denote a set of

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) :

do

(

θ

θ

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′

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=

=

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∈ X}

M

. Moreover, the actions have to

(

x

, do

(

θ

);

(1)

be mutable and explicitly exclude any action which keeps the individual in the same position.

M

i

:=

=

x

i

If all features are

. We denote the structural equations as

S

1

i

x

x

=

,

, U

g

i

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i

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i

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i

= 1

, . . . , k

∈

a

independently manipulable

.

(

x

, do

(

θ

))

{

}

where

{

The non-causal semi-factual interaction between is a more generalized formulation which covers the non-causal case.

:

is defined by

SF

M

) =

θ

X × A ↦→ X

x

a

(

x

)

and

. That

SF

, where

◦

is a deterministic function that describes the causal relationship for

, and depends on the person's recourse. Moreover, a structural semi-factual is defined which considers the dependence

∈ X

representing that means a positive outcome (e.g., a loan acceptance) and

h

:

0

,

1

0

X ↦→ {

}

that separates the decision

P

M

= (

S

,

is is the distribution over the exogenous variables

x

is a reasonable threshold that fits all situations well.

Consider that in a causal graph, there is a set of causal parents for each feature

(Pa

, U

i

U

)

x

,

) :

x

i

= 1

, . . . , k where

∈

i

i

, and depends on the

}

S

i

. Hence, induces a mapping

(2)

(

g

(

x

))

which can be extended

(

SF

);

x

M

, do

(

θ

);

)

M

, where

M

(3)

)

.

to denote

(1)

, do is a more generalized formulation which covers the non-causal case.

(

θ

);

h

M

) =

SF

(

x

, do

(

θ

.

))

(4)

, where

,

1

0

means a positive outcome (e.g., a loan acceptance) and

h

:

X ↦→ {

0

ψ

is

}

that separates the decision

(5)

is a reasonable threshold that fits all situations well.

M

)

.

. All of the components are parameterized

, we have

(

θ

i

Semifactual Explanation

Min-Edit Counterfactual

%

Robust    -neighborhood

Semifactual Function

Even if you

triple your

loan

amount

, it

will still be

accepted.

Even if you double

your

loan

amount

, and

double your

repayment term

,

the loan will still

be

accepted.

Even if you double

your

repayment

term

, and work

8hrs

less

per week,

the loan will still be

accepted.

If you work

12hrs

less per

week, the loan

will be

rejected

.

Loan Rejected

Loan Accepted

"

is, the individual

x

taking action

a

(

x

)

will lead to another representation

between the related features [15, 19]. We denote the structural causal model (SCM) by

where

S

are a set of structural equations and

P

U

U

∈ U

.

denoted by

Pa

i

. We denote the structural equations as

S

=

{

x

i

:=

g

i

g

i

(

·

)

is a deterministic function that describes the causal relationship for

exogenous variable

U

i

∈ U

alongside the corresponding parent set

Pa

i

S

:

U ↦→ X

∗

and its inverse mapping

S

-

1

:

X ↦→ U

. Let

f

◦

g

(

x

) =

f

to more functions. Hence, we specify the SCM-equipped semi-factual by

the transition between the states by taking a certain action through an SCM

θ

′

=

SF

(

x

, do

(

θ

);

M

) :=

S

◦

S

-

1

◦

SF

(

x

, do

(

θ

If all features are

independently manipulable

, we have

θ

′

=

θ

=

SF

(

x

Therefore,

SF

(

x

, do

(

θ

);

M

)

Lastly, we assume a binary model that generates the score for the users is

by which we can simply consider that

1

a negative outcome (e.g., a loan rejection). We set a lower threshold

boundary. For the form defined above,

ψ

= 0

.

5

θ

′

=

SF

(

x

, a

;

M

) :=

S

◦

S

-

1

◦

SF

(

x

, do

(

θ

);

exogenous variable

U

i

∈ U

alongside the corresponding parent set

Pa

i

. Hence,

S

induces a mapping

S

:

U ↦→ X

∗

and its inverse mapping

S

-

1

:

X ↦→ U

. Let

f

◦

g

(

x

) =

f

(

g

(

x

))

which can be extended

to more functions. Hence, we specify the SCM-equipped semi-factual by

SF

(

x

, do

(

θ

);

M

)

to denote

the transition between the states by taking a certain action through an SCM

M

, where

θ

′

=

SF

(

x

, do

(

θ

);

M

) :=

S

◦

S

-

1

◦

SF

(

x

, do

(

θ

);

M

)

.

(1)

If all features are

independently manipulable

, we have

θ

′

=

θ

=

SF

(

x

, do

(

θ

);

M

) =

SF

(

x

, do

(

θ

))

.

Therefore,

SF

(

x

, do

(

θ

);

M

)

is a more generalized 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 (e.g., a loan acceptance) and

0

is

a negative outcome (e.g., a loan rejection). We set a lower threshold

ψ

that separates the decision

boundary. For the form defined above,

ψ

= 0

.

5

is a reasonable threshold that fits all situations well.

θ

′

=

SF

(

x

, a

;

M

) :=

S

◦

S

-

1

◦

SF

(

x

, do

(

θ

);

M

)

.

(2)

θ

′

=

SF

(

x

, a

1

;

M

) :=

S

◦

S

-

1

◦

SF

(

x

, do

(

θ

);

M

)

.

(3)

θ

′

=

SF

(

x

, a

2

;

M

) :=

S

◦

S

-

1

◦

SF

(

x

, do

(

θ

);

M

)

.

(4)

(

x

, do

(

θ

);

M

)

Lastly, we assume a binary model that generates the score for the users is

h

by which we can simply consider that

1

a negative outcome (e.g., a loan rejection). We set a lower threshold

ψ

boundary. For the form defined above,

ψ

= 0

.

5

θ

′

=

SF

(

x

, a

;

M

) :=

S

◦

S

-

1

◦

SF

(

x

, do

(

θ

);

M

)

.

θ

′

=

SF

(

x

, a

1

;

M

) :=

S

◦

S

-

1

◦

SF

(

x

, do

(

θ

);

M

)

.

θ

′

=

SF

(

x

, a

2

;

M

) :=

S

◦

S

-

1

◦

SF

(

x

, do

(

θ

);

M

)

.

θ

′

=

SF

(

x

, a

3

;

M

) :=

S

◦

S

-

1

◦

SF

(

x

, do

(

θ

);

M

)

.

3.2

Framework

Let us define the semi-factual framework centering on Gain (

G

regularization (

R

), and hard constraints (

H

), indexed by

j

θ

′

=

SF

(

x

, a

1

;

M

) :=

S

◦

S

-

◦

SF

(

x

, do

(

θ

);

M

)

.

(3)

θ

′

=

SF

(

x

, a

2

;

M

) :=

S

◦

S

-

1

◦

SF

(

x

, do

(

θ

);

M

)

.

(4)

θ

′

=

SF

(

x

, a

3

;

M

) :=

S

◦

S

-

1

◦

SF

(

x

, do

(

θ

);

M

)

.

(5)

3.2

Framework

Let us define the semi-factual framework centering on Gain (

G

), with additional plausibility (

P

),

regularization (

R

), and hard constraints (

H

), indexed by

j

. All of the components are parameterized

with

x

and a subset of suggestions

{

a

1

, . . . , a

m

}

. Letting

f

(

·

)

be a function composed by Gain and

weight, the causal semi-factual framework is defined as

a

1

,...,a

m

1

m

m

∑

i

=1

f

(

P

(

x

, a

i

)

, G

(

x

, a

i

)) +

γR

(

{

θ

1

, . . . ,

θ

m

}

)

θ

i

=

SF

(

x

, a

i

;

M

)

, H

j

(

θ

i

)

0

,

∀

i, j

(6)

where the regularization and hard constraints can be multiple and be indexed with

i

and

j

respectively.

One may define a similar formulation for the non-causal case. In the paper, we implement a

Gain

function, with plausibility as weight, diversity as regularization, and robustness as hard constraints.

Therefore, with

)

, H

3.3

3.2

Framework and a subset of suggestions

), with additional plausibility (

P

), be a function composed by Gain and

a

, . . . , a

. Letting

f

(

(6)

(2)

i

i

j

i

1

x

(

)

We defer all details of the components until Section 3.4.

θ

=

(

x

, a

;

θ

)

0

,

i, j

Optimizing Positive Outcomes With 'Gain'

M

∀

3.3

Optimizing Positive Outcomes With 'Gain'

j

i

SF

where the regularization and hard constraints can be multiple and be indexed with and

respectively.

For the core of the objective we appeal to the notion of

One may define a similar formulation for the non-causal case. In the paper, we implement a of

We defer all details of the components until Section 3.4.

3.3

Cost

Optimizing Positive Outcomes With 'Gain'

Gain

For the core of the objective we appeal to the notion of function, with plausibility as weight, diversity as regularization, and robustness as hard constraints.

of

Gain

. Note that gain is similar to the idea commonly used in recourse [41], but the crucial difference here is that we are trying to

Cost

. Note that gain is similar to the idea commonly used in recourse [41], but the crucial difference here is that we are trying to

maximize gain, rather than

Gain cost (which is typically done). Moreover, gain ideally considers

minimize

4

the causal dependencies between features in its function, whilst cost typically only considers the

)

M

∈ X

intervention of replacing x by θ where θ ∈ X . It implies that, for each feature, X i := θ i for the individual x . If the action do ( θ ) imposes no change, x = θ holds. We further denote a set of human-constrained actionable ranges A = { a ( x ) = do ( θ ) : ∀ θ ∈ X} . 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 SF : X × A ↦→ X . That is, the individual x taking action a ( x ) will lead to another representation θ ∈ X representing that person'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 , P U ) where S are a set of structural equations and P U is the distribution over the exogenous variables U ∈ U . Consider that in a causal graph, there is a set of causal parents for each feature x i ∈ x , denoted by Pa i . We denote the structural equations as S = { x i := g i (Pa i , U i ) : i = 1 , . . . , k } where g i ( · ) is a deterministic function that describes the causal relationship for x i , and depends on the exogenous variable U i ∈ U alongside the corresponding parent set Pa i . Hence, S induces a mapping S : U ↦→ X ∗ and its inverse mapping S -1 : X ↦→ U . Let f ◦ g ( x ) = f ( g ( x )) which can be extended to more functions. Hence, we specify the SCM-processed semifactual by SF ( x , do ( θ ); M ) to denote the transition between the states by taking a certain action through an SCM M , where

If all features are independently manipulable , we have θ ′ = θ = SF ( x , do ( θ ); M ) = SF ( x , do ( θ )) . Therefore, SF ( x , do ( θ ); M ) is a 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 (e.g., a loan acceptance) and 0 is a negative outcome (e.g., a loan rejection). We set a lower threshold ψ that separates the decision boundary. For the form defined above, ψ = 0 . 5 is a 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 { a 1 , . . . , a m } . Letting f ( · ) be a function composed by gain and some weighting (i.e., plausibility for us), the causal semifactual framework is defined as

where the regularisation and hard constraints can be multiple and indexed with i and j , respectively. One may define a 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 a 1 in Figure 1 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 L 2 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.

More formally, we define the gain function by G : X × A → R . By denoting θ = SF ( x , a ; M ) , we decompose the function as follows:

where P ( · , · ) is an oracle function that computes the payoff based on the vectorised difference between x and θ , i.e., δ : X × X ↦→ R k which is a symmetrical difference function between the two feature representations. The subscript of P SF denotes a semifactual. In interpretation, the gain function compares two states, (1) the original feature vector x , and (2) the SCM-processed end state θ which was led to through x 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 ) ) as

which builds on the fact that cost solely considers the feature change. Note that SF is equivalent to the notion CF in [25], what makes our approach different is 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 if P SF ( · , · ) ≡ P CF ( · , · ) , 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 in the computation of a user's cost. In contrast, our gain function takes the SCM-processed semifactual θ ′ as an input. We employ proof by contradiction here. Assume that cost and gain are equivalent ignoring sign so that, without loss of generality,

holds. However, SCMs can result in possibly more features being changed since some features could be others' causal parents and those causal children will change their values accordingly. By denoting θ = SF ( x , a ) and θ ′ = SF ( x , a ′ ) , we consider the general case as follows:

̸

̸

which contradicts with Equation (5). Thus, even if the oracle function for calculating the payoff is 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 SF ( x , a ) = SF ( x , a ; M ) . The proof completes here.

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 as important in the literature and useful for evaluation. While plausibility and diversity have been explored in 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 a 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 a heuristic for this [29], in contrast we posit (similar to the counterfactual literature [23]) that this is better approached with

SCMs. Hence, we define the plausibility for x taking the action a by P ( x , a ) = Pr( a = do ( θ ) | x ) where x is fixed for an individual and Pr( · ) is a density function. In our non-causal tests, we use the L 2 norm to training data to approximate plausibility (i.e., being in distribution, similar to [29, 32, 51]). However, this issue of plausibility is naturally taken care of in our causal tests thanks to the SCM ensuring plausible feature mutations, so we don't explicitly consider plausibility there going forward.

Robust Gain Continuing with the example of a 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 to triple their loan amount (recall Figure 1), they will likely need upwards of six months to locate a 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 ϵ -neighborhood of x centering around an individual x is

which covers all neighbors that can be reached from x by taking an actionable feature change a through the SCM M . By definition, x is also a neighbor of itself since x ∈ B ( x ) holds given δ ( x , x ) = 0 . Let us represent B ( SF ( x , a ; M )) by B s ( x , a ) for simplicity. Given the predictive model h ( · ) and an individual x , an action a is robust for individual x if h ( θ ) > ψ, ∀ θ ∈ B s ( x , a ) , which is equivalent to min θ ∈B s ( x ,a ) h ( θ ) -ψ > 0 . For instance, ψ = 0 . 5 works for a binary model case. We hence denote the term related to the robustness by

which will be useful for constructing the final objective.

Diverse Gain It is generally preferred to offer a number of suggested actions { a 1 , . . . , a m } , rather than a single one [60]. Like prior work in counterfactuals, we define diversity as the average pair-wise distance among a set of entities [39, 61]. We reuse the distance function δ and define the diversity objective within a set of SFs { θ i } i m =1 ⊆ X m as

which represents a pairwise mean distance among the set of data points, based on the L 2 norm. One may accommodate m = 1 for the case when only a 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:

In optimisation [10], an adversarial interpretation from the perspective of a two-player zero sum game can further simplify Equation (10) to

where H ( x , a ) is the Lagrangian. The primal player tries to maximise the plausibility-weighted gain and diversity, with regard to a , whilst the dual player tries to minimise regarding a set of λ .

Since there are m suggestions, the constraints for robustness will be m times. Observing the objective, the robustness is a hard constraint, whilst the diversity can be regarded as 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.

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 term are finite. Also, assume that A + := { a ∈ A : G ( x , a ) ≥ 0 } is non-empty for an individual x . The semifactual objective J ≥ 0 when ∀ i = 1 , . . . , m, a i ∈ A + : H ( x , a i ) ≥ 0 , otherwise J = -∞ .

See Section A.1 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 ∈ A : H ( x , a ) ≥ 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 are no suggestions to achieve an 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 a similar set of actionable constraints are 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 ˆ G to denote an empirical approximation of G , and likewise for P , H , R , and J .

4.1 Causal Case

Assuming the presence of a differentiable classifier h ( · ) and SCM M , (recall the latter guarantees plausibility), let Ω i ( x ) = { Pr( θ ) : θ ∈ B s ( x , a i ) } be the probability distribution over the ϵ -neighborhood of SF ( x , a i ; M ) . Also, let B i represent a finite subset of B s ( x , a i ) sampled according to Ω i ( x ) . Our objective is:

where L is the binary cross entropy loss. For robustness, we used Monte Carlo (MC) sampling with an epsilon ϵ robust hypersphere, and if either the instance or sampling return a negative outcome with h ( · ) , we use the prior optimisation step as the solution. For diversity, m is set to the number of actionable feature sets, and a solution is obtained for each. We utilise the causal recourse approach of Karimi et al. [26] for solving the maximin. The actionable bounds are clipped each iteration, and λ 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 a genetic algorithm [53, 58] which only assumes a binary predictive model h ( · ) . This approach follows the standard design for genetic algorithms, with some minor alterations specifically for semifactual generation, see Appendix D for the pseudocode. Next we 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 ˆ G ( x , a ) = ∥ SF ( x , a ) -x ∥ 2 . For robustness, we relax it to two constraints: H p 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 is robust, and H a the absolute robustness for the individual x (more detail in Section A.2). For the first constraint, a score of ˆ H p ( x , a ) = 1 n ∑ n i =1 ✶ { h ( x ) = h ( θ i ) } where θ ∼ Pr( B s ( x , a )) , is returned. For

the second constraint, a score of ˆ H a ( x , a ) = ✶ { h ( x ) = h ◦ SF ( x , 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 D , 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 ) ≈ ˆ P D ( x , a ) = exp { 1 / (min θ ∈D ∥ SF ( x , a ) -θ ∥ 2 2 + γ p ) } where γ p is to account for when a perfect match to the semifactual exists in the training data (thus the division is undefined), and ˆ P D ( x , a ) is an empirical approximation (based on D ) of the plausibility. Lastly, for diversity [39, 61], we take the mean distance between all m generated semifactuals with ˆ R ( { SF ( x , a i ) } i m =1 ) 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 need robustness. Notably, the multiplier λ in Equation (11) is split to λ p and λ a , for H p and H a respectively. In this work, we treat them as hyperparameters. Also, they are used alongside γ to balance the objectives. Since λ and γ are selected as hyperparameters, they are removed under the min operator. Finally, the objective (fitness) function is defined as:

We selected the hyperparameters via a grid search, see Appendix C. Crucially, we also weight the fitness function output by ˆ H p ( 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 a user's positive outcome compared to baselines and open source our code (see Appendix E). The actionability constraints are detailed in Appendix B. Baselines 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 on 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 naïve 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 DSER*). Plausibility is measured as the distance between a generated semifactual(s) and the nearest training example; thus, the smaller the better. Robustness is measured by MC sampling n = 100 single feature perturbations of each semifactual θ i , predicting their class, and returning a float between 0-1 of the success rate as described in Section 4.2.1.

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 MLP classifiers from Dominguez et al. [18] and take 30 averaged samples from 5 random seeds. As baselines we modify the technique of Karimi et al. [26] [henceforth Karimi et al.(2021)*], 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 ϵ = 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 m generated semifactuals for an individual x , the higher the number, the better. To be in line with prior art, the L 2 norm is used in non-causal tests [1], and the L 1 in causal [18]. Note for causal tests the SCM guarantees plausibility so this metric is not reported.

5.1 Non-Causal Results

Our purpose here is to show that current methods are insufficient to meet the basic requirements for semifactual explanation discussed in Section 3.4. Specifically, a 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 2 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 a user's gain in 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 gain. With regard to diversity, we fix m 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 a person after taking a 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 in 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 a person's gain. The results of diversity put S-GEN first also (S-GEN = 0 . 84 ± 0 . 09 v. Karimi = 0 . 43 ± 0 . 03 v. 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 a person has a bank loan accepted, and three in which they don't. Users were then shown both explanation types for each material, and asked to rate on 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 on loan acceptance, the



















































































Action Gain

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Gain

Method

S-GEN

Dominguez et al. (2022)*

(a) Causal Experiment: Adult

Action Gain

0.05

0.10

0.15

0.20

0.25

0.30

Gain

Method

S-GEN

Dominguez et al. (2022)*

(b) Causal Experiment: COMPAS

Loan Accepted

Loan Rejected

1

2

3

4

5

Perceived Usefulness

Explanation Type

Semifactual

Counterfactual

(c) User Study

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 a popular approach [22], we don'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 to 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 α = 0 . 05 , and power ( 1 -β err prob)=0.9 informed a sample of 15 was appropriate for t-tests. Users were gathered from Prolific.com, 8 males, 7 females, aged 18+, native English speakers, and from the U.S. People were paid $12/hr, which totalled $35. The semifactuals were generated with S-GEN, and the counterfactuals with DiCE [39], notably these are equivalent to 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 on whether a loan was accepted or rejected, so none were excluded. Figure 3c shows users find semifactuals significantly more useful in loan acceptance (S-GEN=3.60 ± 0.27 v. DiCE=2.33 ± 0.34; p < .005) compared to rejections when counterfactuals are preferred (S-GEN=2.6 ± 0.32 v. DiCE=4.53 ± 0.17; 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, to the best of our knowledge, 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 a causal setting, and (3) conducting their first user test in 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 a study. Moreover, considering a casual formulation of semifactuals requires an SCM, which is not always realistic, but we have provided a 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 as robotics, which likely requires other considerations.

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A Property Analysis

A.1 Proof of Lemma 3.3

Proof. We can rewrite J to

We derive the fact that, for any i , where -∞ comes from setting λ i = ∞ and 0 is obtained by setting λ p i = 0 . By the linearity of summation, we can further derive

.

That is, if any constraint for the robustness is unsatisfied, the dual player will minimise the objective towards -∞ ; however, the primal player cannot optimise towards ∞ given that the limit of the gain function and the diversity are finite. In other words, if the constraints are satisfied, the primal player can freely optimise the objective. Once H ( x , a i ) ≥ 0 , ∀ a i are satisfied, the objective becomes

as P ( x , a i ) > 0 and G ( x , a i ) ≥ 0 for any a i ∈ A + ; also, R ≥ 0 holds. We conclude the proof here.

A.2 A Probabilistic Relaxation of Robustness

Absolute robustness is difficult to guarantee, and common practice is to relax this via a probabilistic approach [26].

Assume there is a distribution over the sample space B s ( x , a ) denoted by Pr( B s ( x , a )) . We write θ ∼ Pr( B s ( x , a )) to indicate that θ is sampled from the set B s ( x , a ) under the density Pr( · ) . Let E [ h ( θ ) | x , a ] denote the expectation of θ in this configuration. Hence, we modify Equation (8) to where ˜ ψ is a function that adjusts the base score threshold ψ . It is crucial to have this threshold function in order to consider the variance of scores in 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 h ( SF ( x , a ; M )) -ψ > 0 . It places a hard constraint to avoid the case in which the neighbors of the semifactual are robust, but the 'semifactual" itself has crossed the decision boundary to become a counterfactual. Whilst somewhat unlikely, this situation is theoretically possible, and requires consideration. In this case, H is re-written as H p , which represents a combination of (i) the probabilistic robustness, and (ii) the absolute robustness for the semifactual H a such that:

In practice, H p is still non-trivial to solve. Monte Carlo (MC) sampling is a common strategy to apply here such that, by sampling a fixed sized batch B = { θ : θ ∼ Pr( B s ( x , a )) } ,

This implies that we substitute an unbiased estimator for the population mean.

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 in various setups. The German Credit data used 15 actionable features to be closely in line with Mothilal et al. [39] whom allowed all features to be mutable. However, we 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 we say a categorical feature could decrease/increase, we are referring to this pre-defined order. If you are interested in the exact ordering, please refer to our code which contains all the lists, but here we summarise. In reality however, a user must specify their exact actionability constraints, what we have specified here is designed to be representative what is possible for the 'average" individual.

B.1.1 German Credit Dataset

The continuous features used were 'duration', 'amount', 'age', the categorical ones were 'status', 'credit_history', 'purpose', 'savings', 'employment_duration', 'installment_rate', 'per-sonal_status_sex', 'other_debtors', 'present_residence', 'property', 'other_installment_plans', 'housing', 'number_credits', 'job', 'people_liable', 'telephone', 'foreign_worker'. As actionable features for semifactual recourse, we considered the following:

duration : We allowed people to increase the duration of their loan.

amount : We allowed people to increase the amount of their loan.

status : We allowed people to move towards having lower status.

credit_history : We allowed people to move towards e.g. having a late payment if their credit history was otherwise good.

savings : This feature was allowed to decrease.

employment_duration : This feature was allowed to decrease in case people wanted to e.g. start a new job.

installment_rate : This feature was allowed to move towards lower payments.

other_debtors : this feature was allowed to add another co-applicant.

present_residence : This feature was allowed to move towards e.g. renting in case the user desired to do so whilst searching for a new house with their loan.

property : this feature was allowed to move towards having no property in case the user desired to sell their house/car etc to help pay for e.g. a downpayment.

other_installment_plans : This feature was allowed to add other installment plans.

housing : this feature was allowed to move towards renting away from e.g. owning.

number_credits : This feature was allowed to increase if 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 a business.

people_liable : This feature was allowed to move towards more people being liable.

B.1.2 Lending Club

The continuous features used were 'loan_amnt', 'pub_rec_bankruptcies', 'annual_inc', 'dti', the categorical ones were 'emp_length', 'term', 'grade', 'home_ownership', '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 to decrease if the person desired to e.g. work less hours.

emp_length : This feature was allowed to decrease in case the individual desired to change careers.

dti : dept to income ratio, this feature was allowed to increase.

pub_rec_bankruptcies : This feature was allowed to increase in case the user decided they wanted to declare bankruptcy to e.g. try and keep some assets.

loan_amnt : 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', 'nrelbc', '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 move towards having had a previous breast proceedure in case the patient would like to do so or was advised.

hrt : This feature was allowed to move towards starting HRT, in case a 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 in 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 it 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-perweek', which are the variables in the causal graph of Nabi & Shpitser [41]. We consider 'age' and 'hours-per-week' as actionable. We allow 'age' to increase a maximum of five years, and 'hours-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 to positive gain. A persons sex was seen as neutral gain.

B.2.2 COMPAS

We use the features 'age', 'race', 'sex' and 'priors count', which are 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, we 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.

C.1 Non-Causal

Here we note 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 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 0.05. The number of 'elite" solutions passed on for each generation was 4. The probability of a crossover happening was 0.5. The number of Monte Carlo trials for each instance was 100. The continuous features were perturbed (in mutation or population initialization) by the output from 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.

C.2 Causal

In our causal tests we chose λ as 1.0, and this was gradually decreased by a momentum of η =0.9 each iteration to put more emphases on the maximization of gain.

D Algorithm Pseudocode

Algorithm 1 S-GEN: Genetic Algorithm to Generate semifactual Recourse with Robustness and Diversity in a Non-Causal Model Agnostic Setting

Require: x the user feature

1: Sample n candidates D ←{ θ i ∼ X} n i =1

Require: ·

h ( ) the predictive model

Require: m the expected number of suggestions

Require: n the number of candidates, n > m

Ensure:

R SF the set of semifactual(s)

2: while the stopping criterion is not satisfied do

4: Save the fittest θ ∗ ∈ D according to f

3: Obtain the fitness scores f with respect to D

5: Let D evolve by natural selection according to f , crossover , mutation , and elitism with x ∗ 6: end while

8: if | R SF | < m then

7: Collect the best m unique candidates from { θ ∈ D : h ( θ ) = h ( x ) = 1 } to R SF , according to the corresponding fitness scores in f

9: Complement R SF to m elements with θ randomly drawn from R SF

10: end if

E Code

For our full code used please see:

https://github.com/EoinKenny/Semifactual_Recourse_Generation

The ability to reproduce the results is given.

Algorithm 2 S-GEN: Algorithm to Generate Robust & Diverse Causal semifactual Explanations for

F Individual Dataset Results

The results are presented in Figure 4.

























































































































































































G Baselines

G.1 Non-Casual

DiCE Our modification to DiCE, starts by generating a counterfactual(s) for a query. Next, we use the algorithm again, but on the generated counterfactuals(s), to make them generate a second counterfactual, which goes back over the decision boundary. In effect, this generates a semifactual(s) for a query.

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 two sets, the first corresponding to those predicted as the original class c , 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 × 4 = 8 sets of data. These sets were then modeled using the best fit found for a Beta distribution on continuous features, and a simple Categorical distribution for categorical features. To generate a semifactual predicted as c , we take the probability of each feature value in the query using the models of the counterfactual class c ′ , and modify each to be its expected statistical value in c ′ 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 lie outside the actionability range, we clip them to the closest value allowed.

DSER For Diverse Explanation of Reject [1] (DSER) we had to modify the the technique in two main ways. Most notably, the techniques doesn't deal with categorical features, so to overcome this, we 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, we optimise one semifactual at a time, each time pushing each solution as far as possible from those already found.

G.2 Causal

Karimi et al. (2021) The method by Karimi et al. [25] is a 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 a robustness component. Namely, they take an individual x , and solve an inner loss which means that an individual of distance ϵ = 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 to solve a different objective. The objective we solve is to move towards the decision boundary, but when the recourse option causes either x or the individual close to it to cross the decision boundary, we terminate the optimization one step prior to this.

H Computational Costs

All tests were run on a MacBook Pro, Apple M1 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.

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LOAN

APPLICATION

Personal lnformation

APPROVED

ONDBRREVIEW

Addss(MngAddrs)

12345MAIN

REVEW

JQPJQI

SUBJBCT

Servicesneeded

UNDERREVIEW

Current Income

aateny or

LOAN

APPLICATION

Personal Information

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otherTelephoe

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Name(Last)

PUBLIC

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UNDERREVIEW

Address(Mailing Address)

12345MAIN

SUBJECT

REVIEW

Servicesneeded

UNDERREVIEW

Maior

IGED)Test Passed?Yes No

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS:\n- 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\n\nWEAKNESSES:\n- 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\nQUESTIONS:\n- 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\n\nLIMITATIONS:\nI 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.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2023-06-23"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS:\n- 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.\n\nWEAKNESSES:\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- 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-\n\nQUESTIONS:\nsee above\n\nLIMITATIONS:\nNA", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS:\nTo 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.\n\nWEAKNESSES:\nThe 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.\n\nQUESTIONS:\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- 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\nLIMITATIONS:\nThe authors adequately address all the limitations.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-10"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS:\nThe authors offer an interesting and compelling framework for optimizing positive outcomes, with strong formal notation and proofs.\n\nWEAKNESSES:\nMy 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.\n\nQUESTIONS:\n-- 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.\n\nLIMITATIONS:\nThe 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.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-15"}