# Reconciling Predictive and Statistical Parity:
A Causal Approach

Drago Plecko and Elias Bareinboim
Department of Computer Science
Columbia University
<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

Since the rise of fair machine learning as a critical field of inquiry, many
different notions on how to quantify and measure discrimination have been
proposed in the literature. Some of these notions, however, were shown to be
mutually incompatible. Such findings make it appear that numerous different
kinds of fairness exist, thereby making a consensus on the appropriate measure
of fairness harder to reach, hindering the applications of these tools in
practice. In this paper, we investigate one of these key impossibility results
that relates the notions of statistical and predictive parity. Specifically,
we derive a new causal decomposition formula for the fairness measures
associated with predictive parity, and obtain a novel insight into how this
criterion is related to statistical parity through the legal doctrines of
disparate treatment, disparate impact, and the notion of business necessity.
Our results show that through a more careful causal analysis, the notions of
statistical and predictive parity are not really mutually exclusive, but
complementary and spanning a spectrum of fairness notions through the concept
of business necessity. Finally, we demonstrate the importance of our findings
on a real-world example.

##  1 Introduction

As society increasingly relies on AI-based tools, an ever larger number of
decisions that were once made by humans are now delegated to automated
systems, and this trend is likely to only accelerate in the coming years. Such
automated systems may exhibit discrimination based on gender, race, religion,
or other sensitive attributes, as witnessed by various examples in criminal
justice (Angwin et al., 2016), facial recognition (Harwell, 2019; Buolamwini
and Gebru, 2018), targeted advertising (Detrixhe and Merrill, 2019), and
medical treatment allocation (Rajkomar et al., 2018), to name a few.

In light of these challenges, a large amount of effort has been invested in
attempts to detect and quantify undesired discrimination based on society’s
current ethical standards, and then design learning methods capable of
removing possible unfairness from future predictions and decisions. During
this process, many different notions on how to quantify discrimination have
been proposed. In fact, the current literature is abundant with different
fairness metrics, some of which are mutually incompatible (Corbett-Davies and
Goel, 2018). The incompatibility of these measures can create a serious
obstacle for practitioners since choosing among them, even for the system
designer, is usually a non-trivial task.

In the real world, issues of discrimination and unfairness are analyzed
through two major legal doctrines. The first one is disparate treatment, which
enforces the equality of treatment of different groups and prohibits the use
of the protected attribute (e.g., race) during the decision process. One of
the legal formulations for showing disparate treatment is that “a similarly
situated person who is not a member of the protected class would not have
suffered the same fate” (Barocas and Selbst, 2016). Disparate treatment is
commonly associated with the notion of direct effects in the causal
literature. The second doctrine is known as disparate impact and focuses on
outcome fairness, namely, the equality of outcomes among protected groups.
Discrimination through disparate impact occurs if a facially neutral practice
has an adverse impact on members of the protected group, including cases where
discrimination is unintended or implicit. In practice, the law may not
necessarily prohibit the usage of all characteristics correlated with the
protected attribute due to their relevance to the business itself, which is
legally known as “business necessity” (labelled BN from now on) or “job-
relatedness”. Therefore, some of the variables may be used to distinguish
between individuals, even if they are associated with the protected attribute
(Kilbertus et al., 2017). From a causal perspective, disparate impact is
realized through indirect forms of discrimination, and taking into account BN
considerations is the essence of this doctrine (Barocas and Selbst, 2016).

We now note how BN requirements span a range of fairness notions between
statistical (SP) and predictive parity (PP). Consider a set of causal pathways
between the attribute $X$ and the predictor $\widehat{Y}$, labeled
$\mathcal{C}_{1},\dots,\mathcal{C}_{k}$. For example, these pathways could
represent the direct, indirect, and spurious effects of $X$ on $\widehat{Y}$,
or more generally any set of path-specific effects. For each
$\mathcal{C}_{i}$, the system designer needs to decide whether the causal
pathway in question is considered as discriminatory (i.e., not in the BN set),
or if it considered non-discriminatory (i.e., in the BN set), as shown in Fig.
1 left. If $\mathcal{C}_{i}$ is not in the BN set, then the causal effect
transmitted along this pathway should equal zero, written
$\mathcal{C}_{i}(X,\widehat{Y})=0$ (which is the main focus of previous works
on path-specific counterfactual fairness (Wu et al., 2019; Chiappa, 2019)).
However, if $\mathcal{C}_{i}$ is in the BN set, then the transmitted causal
effect does not need to equal $0$, i.e., $\mathcal{C}(X,\widehat{Y})\neq 0$
may be allowed. Interestingly, the transmitted causal effect should not take
an arbitrary value in this case. Rather, it should equal the transmitted
effect from $X$ to the original outcome $Y$ (observed in the real world) along
the same pathway, written as
$\mathcal{C}_{i}(X,\widehat{Y})=\mathcal{C}_{i}(X,Y)$. Therefore,
considerations of BN can be summarized as a $0/1$ vector of length $k$, where
each pathway $\mathcal{C}_{i}$ is represented by an entry.

ptCausal PathwaysptNot BN ✘ptBN
✔pt$\mathcal{C}_{1}$pt$\mathcal{C}_{2}$pt$\mathcal{C}_{3}$pt$\mathcal{C}_{4}$pt$\mathcal{C}_{5}$pt$\mathcal{C}_{6}$Statistical
Parity$(0,\dots,0)$Predictive Parity$(1,\dots,1)$BN Specification

Figure 1: BN specification (left) and the spectrum between statistical and
predictive parity (right).

As we demonstrate both intuitively and formally throughout the paper, the
choice of the BN set being empty, written $(0,\dots,0)$, will ensure the
notion of statistical parity. The choice of the BN set that includes all
pathways, written $(1,\dots,1)$, will lead to predictive parity. Crucially,
various intermediate fairness notions between the two ends of the spectrum are
possible, depending on what is allowed or not; for an illustration, see Fig. 1
(right side).

The unification of the principles behind statistical and predictive parity
through the concept of BN has the potential to bridge the gap between the two
notions and improve the current state-of-the-art, by providing an argument
against the seemingly discouraging impossibility result between the two
notions. The practitioner is no longer faced with a false dichotomy of
choosing between statistical or predictive parity but rather faces a spectrum
of different fairness notions determined by the choice of the business
necessity set, which is usually fixed through societal consensus and legal
requirements.

###  1.1 Organization & Contributions

In Sec. 2, we introduce important preliminary notions in causal inference, and
the formal tools for specifying causal pathways
$\mathcal{C}_{1},\dots,\mathcal{C}_{k}$ described above. Further, we discuss
the notions of statistical and predictive parity, together with the
impossibility result that separates them. In Sec. 2.2 we discuss how different
causal variations with the statistical parity measure can be disentangled
using an additive decomposition (Zhang and Bareinboim, 2018). In Sec. 3, we
develop a novel decomposition of the predictive parity measure in terms of the
underlying causal mechanisms and discuss how this notion is in fact
complementary to statistical parity through the concept of business necessity.
In Sec. 3.1, we unify the theoretical findings by introducing a formal
procedure that shows how to assess the legal doctrines of discrimination by
leveraging both the concepts of causal predictive parity and causal
statistical parity. In Sec. 4, we apply our approach in the context of
criminal justice using the COMPAS dataset (Angwin et al., 2016), and
demonstrate empirically the trade-off between SP and PP. Our key formal
contributions are the following:

  * •

We develop the first non-parametric decomposition of the predictive parity
measure in terms of the underlying causal mechanisms (Thm. 1).

  * •

Building on the previous result, we define a natural notion of causal
predictive parity (Def. 5). We then develop a procedure (Alg. 1) for
evaluating if a classifier satisfies the desired notions of causal statistical
parity and causal predictive parity, which provides a unified framework for
incorporating desiderata from both predictive and statistical parity and sheds
light on the impossibility theorem that relates the two notions.

##  2 Background

We use the language of structural causal models (SCMs) as our basic semantical
framework (Pearl, 2000). A structural causal model (SCM) is a tuple
$\mathcal{M}:=\langle V,U,\mathcal{F},P(u)\rangle$ , where $V$, $U$ are sets
of endogenous (observables) and exogenous (latent) variables, respectively,
$\mathcal{F}$ is a set of functions $f_{V_{i}}$, one for each $V_{i}\in V$,
where $V_{i}\leftarrow f_{V_{i}}(\mathrm{pa}(V_{i}),U_{V_{i}})$ for some
$\mathrm{pa}(V_{i})\subseteq V$ and $U_{V_{i}}\subseteq U$. $P(u)$ is a
strictly positive probability measure over $U$. Each SCM $\mathcal{M}$ is
associated to a causal diagram $\mathcal{G}$ (Pearl, 2000) over the node set
$V$ where $V_{i}\rightarrow V_{j}$ if $V_{i}$ is an argument of $f_{V_{j}}$,
and $V_{i}\dashleftarrow\\!\\!\\!\\!\\!\\!\dashrightarrow V_{j}$ if the
corresponding $U_{V_{i}},U_{V_{j}}$ are not independent. An instantiation of
the exogenous variables $U=u$ is called a unit. By $Y_{x}(u)$ we denote the
potential response of $Y$ when setting $X=x$ for the unit $u$, which is the
solution for $Y(u)$ to the set of equations obtained by evaluating the unit
$u$ in the submodel $\mathcal{M}_{x}$, in which all equations in $\mathcal{F}$
associated with $X$ are replaced by $X=x$. Building on the notion of a
potential response, one can further define the notions of counterfactual and
factual contrasts, given by:

######  Definition 1 (Contrasts (Plečko and Bareinboim, 2022)).

Given an SCM $\mathcal{M}$, a contrast $\mathcal{C}$ is any quantity of the
form

$\mathcal{C}(C_{0},C_{1},E_{0},E_{1})=\mathbbm{E}[y_{C_{1}}\mid
E_{1}]-\mathbbm{E}[y_{C_{0}}\mid E_{0}],$ (1)

where $E_{0},E_{1}$ are observed (factual) clauses and $C_{0},C_{1}$ are
counterfactual clauses to which the outcome $Y$ responds. Furthermore,
whenever

  1. (a)

$E_{0}=E_{1}$, the contrast $\mathcal{C}$ is said to be counterfactual;

  2. (b)

$C_{0}=C_{1}$, the contrast $\mathcal{C}$ is said to be factual.

For instance, the contrast
$(C_{0}=\\{x_{0}\\},C_{1}=\\{x_{1}\\},E_{0}=\emptyset,E_{1}=\emptyset)$
corresponds to the average treatment effect (ATE)
$\mathbbm{E}[y_{x_{1}}-y_{x_{0}}]$. Similarly, the contrast
$(C_{0}=\\{x_{0}\\},C_{1}=\\{x_{1}\\},E_{0}=\\{x_{0}\\},E_{1}=\\{x_{0}\\})$
corresponds to the effect of treatment on the treated (ETT)
$\mathbbm{E}[y_{x_{1}}-y_{x_{0}}\mid x_{0}]$. Many other important causal
quantities can be represented as contrasts, as exemplified later on.

Throughout this manuscript, we assume a specific cluster causal diagram
$\mathcal{G}_{\text{SFM}}$ known as the standard fairness model (SFM) (Plečko
and Bareinboim, 2022) over endogenous variables $\\{X,Z,W,Y,\widehat{Y}\\}$
shown in Fig. 3. The SFM consists of the following: protected attribute,
labeled $X$ (e.g., gender, race, religion), assumed to be binary; the set of
confounding variables $Z$, which are not causally influenced by the attribute
$X$ (e.g., demographic information, zip code); the set of mediator variables
$W$ that are possibly causally influenced by the attribute (e.g., educational
level or other job-related information); the outcome variable $Y$ (e.g., GPA,
salary); the predictor of the outcome $\widehat{Y}$ (e.g., predicted GPA,
predicted salary). The SFM also encodes the assumptions typically used in the
causal inference literature about the lack of hidden confounding111Partial
identification techniques for bounding effects can be used for relaxing these
assumptions (Zhang et al., 2022)..

$Z$$X$$W$$Y$$\widehat{Y}$

Figure 2: Standard Fairness Model.

$X$$W$$Y$$\widehat{Y}$ Figure 3: Standard Fairness Model with $Z=\emptyset$
from Thm. 1, extended with the predictor $\widehat{Y}$.

We next introduce the key notions and results from the fair ML literature
needed for our discussion.

###  2.1 Statistical & Predictive Parity Notions

The notions of statistical parity and predictive parity are defined as
follows222The paper (Chouldechova, 2017) introduces predictive parity for a
binary predictor $\widehat{Y}$, and calibration for a continuous
classification score $\widehat{Y}$. In this paper we are agnostic to this
distinction, and thus use predictive parity for both.:

######  Definition 2 (Statistical (Darlington, 1971) and Predictive
(Chouldechova, 2017) Parity).

Let $X$ be the protected attribute, $Y$ the true outcome, and $\widehat{Y}$
the outcome predictor. The predictor $\widehat{Y}$ satisfies statistical
parity (SP) with respect to $X$ if $X{\perp\\!\\!\\!\perp}\widehat{Y}$, or
equivalently if the measure

$\displaystyle\text{SPM}_{x_{0},x_{1}}(\widehat{y}):=P(\widehat{y}\mid
x_{1})-P(\widehat{y}\mid x_{0})=0.$ (2)

Further, $\widehat{Y}$ satisfies predictive parity (PP) with respect to $X,Y$
if $Y{\perp\\!\\!\\!\perp}X\mid\widehat{Y}$, or equivalently if

$\displaystyle\text{PPM}_{x_{0},x_{1}}(y\mid\widehat{y}):=P(y\mid
x_{1},\widehat{y})-P(y\mid x_{0},\widehat{y})=0\;\forall\widehat{y}.$ (3)

Statistical parity ($\widehat{Y}{\perp\\!\\!\\!\perp}X$) requires that the
predictor $\widehat{Y}$ contains no information about the protected attribute
$X$. In contrast to this, the notion of predictive parity
($Y{\perp\\!\\!\\!\perp}X\mid\widehat{Y}$), requires that $\widehat{Y}$ should
“exhaust”, or capture all the variations of $X$ coming into the outcome $Y$ in
the current real world. An important known result that further corroborates
this point is the following (see proof in Appendix C):

######  Proposition 1 (PP and Efficient Learning).

Let $\mathcal{M}$ be an SCM compatible with the Standard Fairness Model (SFM).
Suppose that the predictor $\widehat{Y}$ is based on the features $X,Z,W$.
Suppose also that $\widehat{Y}$ is an efficient learner, meaning that:

$\displaystyle\widehat{Y}(x,z,w)=P(y\mid x,z,w).$ (4)

Then, it follows that $\widehat{Y}$ satisfies predictive parity w.r.t. $X$ and
$Y$.

Prop. 1 shows that PP is expected to hold whenever we learn the true
conditional distribution $P(y\mid x,z,w)$. This is in stark contrast with the
SP notion. For the SP to hold, the predictor $\widehat{Y}$ should not be
associated at all with the attribute $X$, whereas for PP to hold,
$\widehat{Y}$ should contain all variations coming from $X$. Perhaps
unsurprisingly after this discussion, the following result holds:

######  Proposition 2 (SP and PP impossibility (Barocas et al., 2017)).

Statistical parity ($\widehat{Y}{\perp\\!\\!\\!\perp}X$) and predictive parity
($Y{\perp\\!\\!\\!\perp}X\mid\widehat{Y}$) are mutually exclusive except in
degenerate cases when $Y{\perp\\!\\!\\!\perp}X$.

The above theorem might lead the reader to believe that SP and PP criteria
come from two different realities, and bear no relation to each other. After
all, the theorem states that it is not possible for the predictor
$\widehat{Y}$ to include all variations of $X$ coming into $Y$ and
simultaneously include no variations of $X$ coming into $Y$. This realization
is the starting point of our discussion in the rest of the manuscript.

###  2.2 Statistical Parity Decomposition

We continue by analyzing the decomposition of the statistical parity measure
(also known as parity gap, or total variation), which is commonly used to
determine if statistical parity is satisfied. The causal decomposition of the
SPM requires the usage of causal measures known as counterfactual direct,
indirect, and spurious effects, which are defined as:

######  Definition 3 (Counterfactual Causal Measures).

The counterfactual-{direct, indirect, spurious} effects of $X$ on
$\widehat{Y}$ are defined as follows

$\displaystyle\text{Ctf-DE}_{x_{0},x_{1}}(\widehat{y}\mid x)$
$\displaystyle=P(\widehat{y}_{x_{1},W_{x_{0}}}\mid
x)-P(\widehat{y}_{x_{0}}\mid x)$ (5) $\displaystyle\text{Ctf-
IE}_{x_{1},x_{0}}(\widehat{y}\mid x)$
$\displaystyle=P(\widehat{y}_{x_{1},W_{x_{0}}}\mid
x)-P(\widehat{y}_{x_{1}}\mid x)$ (6) $\displaystyle\text{Ctf-
SE}_{x_{1},x_{0}}(\widehat{y})$ $\displaystyle=P(\widehat{y}_{x_{1}}\mid
x_{0})-P(\widehat{y}_{x_{1}}\mid x_{1}).$ (7)

The measures capture the variations from $X$ to $\widehat{Y}$ going through:
(i) the direct mechanism $X\rightarrow\widehat{Y}$; (ii) the indirect
mechanism $X\rightarrow W\rightarrow\widehat{Y}$; (iii) the spurious
mechanisms $X\dashleftarrow\\!\\!\\!\\!\\!\\!\dashrightarrow
Z\rightarrow\widehat{Y}$, and
$X\dashleftarrow\\!\\!\\!\\!\\!\\!\dashrightarrow Z\rightarrow
W\rightarrow\widehat{Y}$, respectively (see Fig. 3). Based on the defined
measures, SPM${}_{x_{0},x_{1}}(\widehat{y})$ admits an additive decomposition,
first obtained in (Zhang and Bareinboim, 2018), given in the following
theorem333We note that the same decomposition can also be applied for the true
outcome $Y$ instead of the predictor $\widehat{Y}$, as will be relevant in
later sections.:

######  Proposition 3 (Causal Decomposition of Statistical Parity (Zhang and
Bareinboim, 2018)).

The statistical parity measure admits a decomposition into its direct,
indirect, and spurious variations:

$\displaystyle\text{SPM}_{x_{0},x_{1}}(\widehat{y})=$
$\displaystyle\;{\text{Ctf-DE}_{x_{0},x_{1}}(\widehat{y}\mid
x_{0})}-{\text{Ctf-IE}_{x_{1},x_{0}}(\widehat{y}\mid x_{0})}-{\text{Ctf-
SE}_{x_{1},x_{0}}(\widehat{y})}.$ (8)

This result shows how we can disentangle direct, indirect, and spurious
variations within the SPM. We emphasize the importance of this result in the
context of assessing the legal doctrines of discrimination. If a causal
pathway (direct, indirect, or spurious) does not lie in the business necessity
set, then the corresponding counterfactual measure (Ctf-DE, IE, or SE) needs
to equal $0$. To formalize this notion, we can now introduce the criterion of
causal statistical parity:

######  Definition 4 (Causal Statistical Parity).

We say that $\widehat{Y}$ satisfies causal statistical parity with respect to
the protected attribute $X$ if

$\displaystyle{\text{Ctf-DE}_{x_{0},x_{1}}(\widehat{y}\mid x_{0})}$
$\displaystyle={\text{Ctf-IE}_{x_{1},x_{0}}(\widehat{y}\mid
x_{0})}={\text{Ctf-SE}_{x_{1},x_{0}}(\widehat{y})}=0.$ (9)

In practice, causal statistical parity can be a strong requirement, but the
notion can be easily relaxed to include only a subset of the Ctf-{DE, IE, or
SE} measures, under BN requirements.

##  3 Predictive Parity Decomposition

After discussing the decomposition of the SPM, our aim is to obtain a causal
understanding of the predictive parity criterion,
$Y{\perp\\!\\!\\!\perp}X\mid\widehat{Y}$. To do so, we derive a formal
decomposition result of the PP measure that involves both $Y$ and
$\widehat{Y}$, shown in the following theorem:

######  Theorem 1 (Causal Decomposition of Predictive Parity).

Let $\mathcal{M}$ be an SCM compatible with the causal graph in Fig. 3 (i.e.,
SFM with $Z=\emptyset$). Then, it follows that the
PPM${}_{x_{0},x_{1}}(y\mid\widehat{y})=P(y\mid x_{1},\widehat{y})-P(y\mid
x_{0},\widehat{y})$ can be decomposed into its causal and spurious anti-causal
variations as:

$\displaystyle\text{PPM}_{x_{0},x_{1}}(y\mid\widehat{y})=$ $\displaystyle
P(y_{x_{1}}\mid x_{1},\widehat{y})-P(y_{x_{0}}\mid
x_{1},\widehat{y})+P(y_{x_{0}}\mid\widehat{y}_{x_{1}})-P(y_{x_{0}}\mid\widehat{y}_{x_{0}}).$
(10)

Thm. 1 offers a non-parametric decomposition result of the predictive parity
measure that can be applied to any SCM compatible with the graph in Fig. 3. In
Appendix A.1 we provide a proof of the theorem (together with the proof of
Cor. 1 stated below), and in Appendix A.2 we perform an empirical study to
verify the decomposition result of the theorem. For the additional special
case of linear SCMs, the terms appearing in the decomposition in Eq. 10 can be
computed explicitly:

######  Corollary 1 (Causal Decomposition of Predictive Parity in the Linear
Case).

Under the additional assumptions that (i) the SCM $\mathcal{M}$ is linear and
$Y$ is continuous; (ii) the learner $\widehat{Y}$ is efficient, then

$\displaystyle\mathbbm{E}(y_{x_{1}}\mid
x_{1},\widehat{y})-\mathbbm{E}(y_{x_{0}}\mid x_{1},\widehat{y})$
$\displaystyle=\alpha_{XW}\alpha_{WY}+\alpha_{XY}$ (11)
$\displaystyle\mathbbm{E}(y_{x_{0}}\mid
x_{1},\widehat{y}_{x_{1}})-\mathbbm{E}(y_{x_{0}}\mid
x_{1},\widehat{y}_{x_{0}})$
$\displaystyle=-(\alpha_{XW}\alpha_{WY}+\alpha_{XY}),$ (12)

where $\alpha_{V_{i}V_{j}}$ is the linear coefficient between variables
$V_{i},V_{j}$.

We now carefully unpack the key insight from Thm. 1. In particular, we showed
that in the case of an SFM with $Z=\emptyset$444We remark that the essence of
the argument is unchanged in the case with $Z\neq\emptyset$, but handling this
case limits the clarity of presentation. the predictive parity measure can be
written as:

$\displaystyle\text{PPM}=$ $\displaystyle\underbrace{P(y_{x_{1}}\mid
x_{1},\widehat{y})-P(y_{x_{0}}\mid x_{1},\widehat{y})}_{\text{Term (I)
causal}}+\underbrace{P(y_{x_{0}}\mid\widehat{y}_{x_{1}})-P(y_{x_{0}}\mid\widehat{y}_{x_{0}})}_{\text{Term
(II) reverse-causal spurious}}.$ (13)

The first, causal term can be expanded as

$\displaystyle P(y_{x_{1}}\mid x_{1},\widehat{y})-P(y_{x_{0}}\mid
x_{1},\widehat{y})=\sum_{u}\big{[}\underbrace{y_{x_{1}}(u)-y_{x_{0}}(u)}_{\text{unit-
level difference}}\big{]}\underbrace{P(u\mid
x_{1},\widehat{y})}_{\text{posterior}}.$ (14)

The expansion shows us that the term is a weighted average of unit-level
differences $y_{x_{1}}(u)-y_{x_{0}}(u)$. Each unit-level difference measures
the causal effect of a transition $x_{0}\to x_{1}$ on $Y$. The associated
weights are given by the posterior $P(u\mid x_{1},\widehat{y})$, which
determines the probability mass corresponding to a unit $u$ within the set of
all units compatible with $x_{1},\widehat{y}$. Therefore, the term represents
an average causal effect of $X$ on $Y$ for a specific group of units.
Interestingly, for any set of units selected by $x_{1},\widehat{y}$, the
effect in Term (I) does not depend on the constructed predictor $\widehat{Y}$,
but only on the underlying system, i.e., it is not under the control of the
predictor designer. The additional linear result in Cor. 1 may also help the
reader ground this idea, since it shows that Term (I) indeed captures the
causal effect, which, in the linear case, can be obtained using the path-
analysis of (Wright, 1934).

To achieve the criterion $\text{PPM}=0$, the second term needs to be exactly
the reverse of the causal effect, captured by the spurious variations induced
by changing $\widehat{y}_{x_{0}}\to\widehat{y}_{x_{1}}$ in the selection of
units. The second term, which is in the control of the predictor $\widehat{Y}$
designer, needs to cancel out the causal effect measured by the first term for
PPM to vanish. Therefore, we see that achieving predictive parity is about
constructing $\widehat{Y}$ in a way that reflects the causal effect of $X$ on
$Y$, across various groups of units. This key observation motivates a novel
definition that we call causal predictive parity:

######  Definition 5 (Causal Predictive Parity).

Let $\widehat{Y}$ be a predictor of the outcome $Y$, and let $X$ be the
protected attribute. Then, $\widehat{Y}$ is said to satisfy causal predictive
parity (CPP) with respect to a counterfactual contrast $(C_{0},C_{1},E,E)$ if

$\displaystyle\mathbbm{E}[y_{C_{1}}\mid E]-\mathbbm{E}[y_{C_{0}}\mid
E]=\mathbbm{E}[\widehat{y}_{C_{1}}\mid E]-\mathbbm{E}[\widehat{y}_{C_{0}}\mid
E].$ (15)

Furthermore, $\widehat{Y}$ is said to satisfy CPP with respect to a factual
contrast $(C,C,E_{0},E_{1})$ if

$\displaystyle\mathbbm{E}[y_{C}\mid E_{1}]-\mathbbm{E}[y_{C}\mid
E_{0}]=\mathbbm{E}[\widehat{y}_{C}\mid E_{1}]-\mathbbm{E}[\widehat{y}_{C}\mid
E_{0}].$ (16)

The intuition behind the notion of causal predictive parity captures the
intuition behind predictive parity. If a contrast $\mathcal{C}$ describes some
amount of variation in the outcome $Y$, then it should describe the same
amount of variation in the predicted outcome $\widehat{Y}$. For any of the
contrasts Ctf-{DE, IE, SE} corresponding to a causal pathway, causal
predictive parity would require that
$\mathcal{C}(X,\widehat{Y})=\mathcal{C}(X,Y)$.

###  3.1 Reconciling Statistical and Predictive Parity

Algorithm 1 Business Necessity Cookbook

1: Input: data $\mathcal{D}$, BN-Set
$\text{BN}\subseteq\\{\text{DE},\text{IE},\text{SE}\\}$

2: for CE $\in\\{\text{DE},\text{IE},\text{SE}\\}$ do

3: if CE $\in$ BN then

4: Compute the effects $\text{Ctf-CE}(y)$, $\text{Ctf-CE}(\widehat{y})$

5: Assert that $\text{Ctf-CE}(y)=\text{Ctf-CE}(\widehat{y})$, otherwise FAIL

6: else

7: Compute the effect $\text{Ctf-CE}(\widehat{y})$

8: Assert that $\text{Ctf-CE}(\widehat{y})=0$, otherwise FAIL

9: end if

10: end for

11: if not FAIL then

12: return SUCCESS

13: end if

14: Output: SUCCESS or FAIL of ensuring that disparate impact and treatment
hold under BN.

We now tie the notions of statistical and predictive parity through the
concept of business necessity. In particular, if a contrast $\mathcal{C}$ is
associated with variations that are not in the business necessity set, then
the value of this contrast should be $\mathcal{C}(X,\widehat{Y})=0$, following
the intuition of causal statistical parity from Def. 4. However, if the
variations associated with the contrast are in the business necessity set,
then the value of that contrast should be equal for the predictor to the value
for the true outcome

$\displaystyle\mathcal{C}(X,\widehat{Y})=\mathcal{C}(X,Y),$ (17)

following the intuition of causal predictive parity. Combining these two
notions through business necessity results in Alg. 1. The algorithm requires
the user to compute the measures

$\displaystyle\text{Ctf-\\{DE, IE, SE\\}}(y),\text{Ctf-\\{DE, IE,
SE\\}}(\widehat{y}).$ (18)

Importantly, under the SFM, these measures are identifiable from observational
data:

######  Proposition 4.

Under the assumptions of the standard fairness model in Fig. 3, the causal
measures of fairness $\text{Ctf-\\{DE, IE, SE\\}}(y),\text{Ctf-\\{DE, IE,
SE\\}}(\widehat{y})$ are identifiable from observational data, that is, they
can be computed uniquely from the observational distribution $P(V)$.

The explicit identification expression for each of the measures is given in
Appendix B. The above result guarantees that the procedure of Alg. 1 is
applicable to practical data analysis, in a fully non-parametric nature.

Further, for each of the DE, IE, and SE effects, the user needs to determine
whether the causal effect (CE) in question falls into the business necessity
set. If yes, then the algorithm asserts that

$\displaystyle\text{Ctf-CE}(y)=\text{Ctf-CE}(\widehat{y}).$ (19)

In the other case, when the causal effect is not in the business necessity
set, the algorithm asserts that

$\displaystyle\text{Ctf-CE}(\widehat{y})=0.$ (20)

We remark that Alg. 1 is written in its population level version, in which the
causal effects are estimated perfectly with no uncertainty. In the finite
sample case, one needs to perform hypothesis testing to see whether the
effects differ. If one is also interested in constructing a new fair predictor
before using Alg. 1 (instead of testing an existing one), one may use tools
for causally removing discrimination, such as (Chiappa, 2019) or (Plečko and
Meinshausen, 2020; Plečko et al., 2021).

##  4 Experiments

We now apply Alg. 1 to the COMPAS dataset (Angwin et al., 2016), as described
in the following example.

Courts in Broward County, Florida use machine learning algorithms, developed
by Northpointe, to predict whether individuals released on parole are at high
risk of re-offending within 2 years ($Y$). The algorithm is based on the
demographic information $Z$ ($Z_{1}$ for gender, $Z_{2}$ for age), race $X$
($x_{0}$ denoting White, $x_{1}$ Non-White), juvenile offense counts $J$,
prior offense count $P$, and degree of charge $D$.

We construct the standard fairness model (SFM) for this example, which is
shown in Fig. 5. The bidirected arrow between $X$ and $\\{Z_{1},Z_{2}\\}$
indicates possible co-variations of race with age and sex, which may not be
causal in nature555The causal model is non-committal regarding the complex
historical/social processes that lead to such co-variations.. Furthermore,
$\\{Z_{1},Z_{2}\\}$ are the confounders, not causally affected by race $X$.
The set of mediators $\\{J,P,D\\}$, however, may be affected by race, due to
an existing societal bias in policing and criminal justice. Finally, all of
the above mentioned variables may influence the outcome $Y$.

${Z_{1}}$${Z_{2}}$$X$$J$$P$${D}$$Y$pt$Z$-setpt$W$-set

Figure 4: SFM for the COMPAS dataset.

Figure 5: Fairness measures on COMPAS.

Having access to data from Broward County, and equipped with Alg. 1, we want
to prove that the recidivism predictions produced by Northpointe (labeled
$\widehat{Y}^{NP}$) violate legal doctrines of anti-discrimination. Suppose
that in an initial hearing, the Broward County district court determines that
the direct and indirect effects are not in the business necessity set, while
the spurious effect is. In words, gender ($Z_{1}$) and age ($Z_{2}$) are
allowed to be used to distinguish between the minority and majority groups
when predicting recidivism, while other variables are not. If Northpointe’s
predictions are found to be discriminatory, we are required by the court to
produce better, non-discriminatory predictions.

In light of this information, we proceed as follows (see source code). We
first obtain a causally fair predictor $\widehat{Y}^{FP}$ using the fairadapt
package, which sequentially performs optimal transport to match conditional
distributions between groups for each variable in the causal diagram. Then, we
compute the counterfactual causal measures of fairness for the true outcome
$Y$, Northpointe’s predictions $\widehat{Y}^{NP}$, and the fair predictions
$\widehat{Y}^{FP}$ (see Fig. 5). For the direct effect, we have:

$\displaystyle\text{Ctf-DE}_{x_{0},x_{1}}(y\mid x_{0})$
$\displaystyle=-0.08\%\pm 2.59\%,$ (21) $\displaystyle\text{Ctf-
DE}_{x_{0},x_{1}}(\widehat{y}^{NP}\mid x_{0})$ $\displaystyle=6\%\pm 2.96\%,$
(22) $\displaystyle\text{Ctf-DE}_{x_{0},x_{1}}(\widehat{y}^{FP}\mid x_{0})$
$\displaystyle=-0.72\%\pm 1.11\%.$ (23)

The indicated 95% confidence intervals are computed using repeated bootstrap
repetitions of the dataset. Since the direct effect is not in the business
necessity set, Northpointe’s predictions clearly violate the disparate
treatment doctrine (green bar for the Ctf-DE measure in Fig. 5). Our
predictions, however, do not exhibit a statistically significant direct effect
of race on the outcome, so they do not violate the criterion (blue bar). Next,
for the indirect effects, we obtain:

$\displaystyle\text{Ctf-IE}_{x_{1},x_{0}}(y\mid x_{0})$
$\displaystyle=-5.06\%\pm 1.24\%,$ (24) $\displaystyle\text{Ctf-
IE}_{x_{1},x_{0}}(\widehat{y}^{NP}\mid x_{0})$ $\displaystyle=-7.73\%\pm
1.53\%,$ (25) $\displaystyle\text{Ctf-IE}_{x_{1},x_{0}}(\widehat{y}^{FP}\mid
x_{0})$ $\displaystyle=-0.25\%\pm 1.98\%.$ (26)

Once again, the indirect effect, which is not in the business necessity set,
is different from $0$ for the Northpointe’s predictions (violating disparate
impact, see green bar for Ctf-IE in Fig. 5), but not statistically different
from $0$ for our predictions (blue bar). Interestingly, the indirect effect is
different from $0$ for the true outcome (red bar), indicating a bias in the
current real world. Finally, for the spurious effects, we obtain

$\displaystyle\text{Ctf-SE}_{x_{1},x_{0}}(y)$ $\displaystyle=-3.17\%\pm
1.53\%,$ (27) $\displaystyle\text{Ctf-SE}_{x_{1},x_{0}}(\widehat{y}^{NP})$
$\displaystyle=-3.75\%\pm 1.58\%,$ (28) $\displaystyle\text{Ctf-
SE}_{x_{1},x_{0}}(\widehat{y}^{FP})$ $\displaystyle=-2.75\%\pm 1.22\%.$ (29)

Since the spurious effect is in the business necessity set and each confidence
interval contains all three point estimates, no violations with respect to
spurious effects are found. The estimated effects for the outcome
$Y,\widehat{Y}^{NP},\text{ and }\widehat{Y}^{FP}$ are also shown graphically
in Fig. 5. We conclude that Northpointe’s predictions $\widehat{Y}^{NP}$
violate the legal doctrines of fairness, while our predictions
$\widehat{Y}^{FP}$ do not. Importantly, tying back to the original discussion
that motivated our approach, Northpointe’s predictions $\widehat{Y}^{NP}$ are
further away from statistical parity than our predictions $\widehat{Y}^{FP}$
according to the SPM (see SPM column in Fig. 5), while at the same time better
calibrated according to the integrated PP measure (iPPM) that averages the PP
measures from Eq. 3 across different values of $\widehat{y}$ (see iPPM column
in the figure). This observation demonstrates the trade-off between
statistical and predictive parity through business necessity.

The choice of the business necessity set in the example above, BN $=\\{W\\}$,
was arbitrary. In general, the BN set can be any of $\emptyset,Z,W,\\{Z,W\\}$,
which is the setting we explore next.

#### Statistical vs. Predictive Parity Pareto Frontier.

We now investigate other possible choices of the BN set, namely BN sets
$\emptyset,Z,W,\\{Z,W\\}$. Based on the theoretical analysis of Sec. 3, we
expect that different choices of the BN set will lead to different trade-offs
between SP and PP.

In particular, with the BN set being empty, any variation from $X$ to $Y$
would be considered discriminatory, so one would expect to be close to
statistical parity. In contrast, if the BN set is $\\{Z,W\\}$, then all
variations (apart from the direct effect) are considered as non-
discriminatory, so one would expect to be closer to satisfying predictive
parity. The remaining options, BN$=W$, and BN$=Z$ should interpolate between
these two extremes.

Based on the above intuition, we proceed as follows. For each choice of the
business necessity set,

$\displaystyle\text{BN}\in\\{\emptyset,Z,W,\\{Z,W\\}\\},$ (30)

we compute the adjusted version of the data again using the fairadapt package,
with the causal effects in the complement of the BN set being removed. That
is, if BN = $W$, then our procedure removes the spurious effect, but keeps the
indirect effect intact (other choices of the BN set are similar). Therefore,
for each BN set, we obtain an appropriately adjusted predictor
$\widehat{Y}_{\text{BN}}^{FP}$, and in particular compute the predictors
$\widehat{Y}_{\emptyset}^{FP},\widehat{Y}_{Z}^{FP},\widehat{Y}_{W}^{FP},\text{
and }\widehat{Y}_{\\{Z,W\\}}^{FP}.$We note that the fair predictor
$\widehat{Y}^{FP}$ from the first part of the example is the predictor with
the BN set equal to $Z$, i.e., it corresponds to $\widehat{Y}_{Z}^{FP}$. For
each $\widehat{Y}_{\text{BN}}^{FP}$, we in particular compute the SPM and iPPM
measures, namely:

$\displaystyle\text{SPM}_{x_{0},x_{1}}(\widehat{y}_{\text{BN}}^{FP}),\text{iPPM}_{x_{0},x_{1}}(\widehat{y}_{\text{BN}}^{FP}),$
(31)

across 10 different repetitions of the adjustment procedure that yields
$\widehat{Y}_{\text{BN}}^{FP}$. For each business necessity set, this allows
us to compute the SPM (measuring statistical parity), and iPPM (measuring
predictive parity).

Figure 6: SP vs. PP Pareto frontier on COMPAS.

The results of the experiment are shown in Fig. 6, where the error bars
indicate the standard deviation over different repetitions. As predicted by
our theoretical analysis, the choice of BN$=\emptyset$ yields the lowest SPM,
but the largest iPPM. Conversely, BN$=\\{Z,W\\}$ yields the lowest iPPM, but
the largest SPM. The BN sets $Z,W$ interpolate between the two notions, but
the data indicates the spurious effect explained by $Z$ does not have a major
contribution. Fig. 6, therefore, shows a trade-off between statistical and
predictive parity described through different business necessity options, and
gives an empirical validation of the hypothesized spectrum of fairness notions
in Fig. 1.

##  5 Conclusions

The literature in fair ML is abundant with fairness measures (Corbett-Davies
and Goel, 2018), many of which are mutually incompatible. Nonetheless, it is
doubtful that each of these measures corresponds to a fundamentally different
ethical conception of fairness. The multitude of possible approaches to
quantifying discrimination makes the consensus on an appropriate notion of
fairness unattainable. Further, the impossibility results between different
measures may be discouraging to data scientists who wish to quantify and
remove discrimination, but are immediately faced with a choice of which
measure they wish to subscribe to.

In this work, we attempt to remedy a part of this issue by focusing on the
impossibility of simultaneously achieving SP and PP. As our discussion shows,
the guiding idea behind SP is that variations transmitted along causal
pathways from the protected attribute to the predictor should equal $0$, i.e.,
the decision should not depend on the protected attribute through the causal
pathway in question (Def. 4). Complementary to this notion, and based on Thm.
1, the guiding principle behind PP is that variations transmitted along a
causal pathway should be equal for the predictor as they are for the outcome
in the real world (Def. 5). SP will therefore be satisfied when the BN set
includes all variations coming from $X$ to $Y$, while PP will be satisfied
when the BN set is empty. The choice of the BN set interpolates between SP and
PP forming a spectrum of fairness notions (see Fig. 1), in a way that can be
formally assessed based on Alg. 1.

Therefore, our work complements the previous literature by reconciling the
impossibility result between SP and PP (Barocas et al., 2017). Furthermore, it
complements the existing literature on path-specific notions of fairness
(Kilbertus et al., 2017; Wu et al., 2019; Chiappa, 2019), which does not
consider the true outcome $Y$ and the predictor $\widehat{Y}$ simultaneously,
and does not explicitly specify which levels of discrimination are deemed
acceptable along causal pathways in the BN set. Finally, we also mention the
work on counterfactual predictive parity (Ctf-PP) (Coston et al., 2020) that
is similar in name to our notion of causal predictive parity, but is in fact a
very different notion. Ctf-PP deals with the setting of decision-making and
considers counterfactuals of the outcome $Y$ with respect to a treatment
decision $D$ that precedes it, while our work considers counterfactuals of the
outcome $Y$ and the predictor $\widehat{Y}$ with respect to the protected
attribute $X$, in the context of fair predictions, and thus offers a different
line of reasoning.

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## Supplementary Material for Reconciling Predictive and Statistical Parity: A
Causal Approach

The source code for reproducing all the experiments can be found in the
anonymized repository. The code is also included with the supplementary
materials, in the folder source-code.

##  Appendix A Thm. 1 Proof and Evaluation

###  A.1 Proof of Thm. 1

Proof  We prove the theorem and the corollary jointly. Note that
$\mathbbm{E}(y\mid x_{1},\widehat{y})-\mathbbm{E}(y\mid x_{0},\widehat{y})$
equals

$\displaystyle\mathbbm{E}(y_{x_{1}}\mid
x_{1},\widehat{y}_{x_{1}})-\mathbbm{E}(y_{x_{0}}\mid
x_{0},\widehat{y}_{x_{0}})$
$\displaystyle=\underbrace{\mathbbm{E}(y_{x_{1}}\mid
x_{1},\widehat{y}_{x_{1}})-\mathbbm{E}(y_{x_{0}}\mid
x_{1},\widehat{y}_{x_{1}})}_{\text{Term (I)}}$ (32)
$\displaystyle+\underbrace{\mathbbm{E}(y_{x_{0}}\mid
x_{1},\widehat{y}_{x_{1}})-\mathbbm{E}(y_{x_{0}}\mid
x_{1},\widehat{y}_{x_{0}})}_{\text{Term (II)}}$ (33)
$\displaystyle+\underbrace{\mathbbm{E}(y_{x_{0}}\mid
x_{1},\widehat{y}_{x_{0}})-\mathbbm{E}(y_{x_{0}}\mid
x_{0},\widehat{y}_{x_{0}})}_{\text{Term (III)}}.$ (34)

Since there are no backdoor paths between $X$ and $Y,\widehat{Y}$, Term (III)
vanishes. By noting that $\mathbbm{E}(y_{x}\mid
x_{1},\widehat{y}_{x_{1}})=\mathbbm{E}(y_{x}\mid x_{1},\widehat{y})\;\;\forall
x$ by consistency axiom [Pearl, 2000, Ch.7] (and applying it to Term (I)), and
also that $Y_{x}{\perp\\!\\!\\!\perp}X$ (and applying it to Term (II)) gives
us the theorem. For the corollary, we further assume that the SCM is linear,
and that the predictor $\widehat{Y}$ is efficient, i.e.,
$\widehat{Y}(x,w)=\mathbbm{E}[Y\mid x,w]$. In this case, the efficiency simply
translates to the fact that

$\displaystyle\alpha_{W\widehat{Y}}=\alpha_{WY},\alpha_{X\widehat{Y}}=\alpha_{XY}.$
(35)

Due to linearity, for every unit $u$, we have that

$\displaystyle y_{x_{1}}(u)-y_{x_{0}}(u)=\alpha_{XW}\alpha_{WY}+\alpha_{XY},$
(36)

and since Term (I) can be written as
$\sum_{u}[y_{x_{1}}(u)-y_{x_{0}}(u)]P(u\mid x_{1},\widehat{y})$ using the
unit-level expansion of counterfactual distributions [Tian and Pearl, 2000,
Bareinboim et al., 2022], Eq. 11 follows. Similarly, Term (II) can be expanded
as

$\displaystyle\sum_{u}\widehat{y}_{x_{0}}(u)[P(u\mid\widehat{y}_{x_{1}})-P(u\mid\widehat{y}_{x_{0}})].$
(37)

We now look at units $u$ which are compatible with
$\widehat{Y}_{x_{1}}(u)=\widehat{y}$ and $\widehat{Y}_{x_{0}}(u)=\widehat{y}$.
We can expand $\widehat{Y}_{x_{1}}(u)$ as

$\displaystyle\widehat{Y}_{x_{1}}(u)=\alpha_{X\widehat{Y}}+\alpha_{XW}\alpha_{W\widehat{Y}}+\alpha_{W\widehat{Y}}u_{W}.$
(38)

Thus, we have that

$\displaystyle\widehat{Y}_{x_{1}}(u)=\widehat{y}\implies\alpha_{W\widehat{Y}}u_{W}=\widehat{y}-\alpha_{X\widehat{Y}}+\alpha_{XW}\alpha_{W\widehat{Y}}.$
(39)

Similarly, we also obtain that

$\displaystyle\widehat{Y}_{x_{0}}(u)=\widehat{y}\implies\alpha_{W\widehat{Y}}u_{W}=\widehat{y}.$
(40)

Due to the efficiency of learning which implies that
$\alpha_{W\widehat{Y}}=\alpha_{WY}$ and $\alpha_{X\widehat{Y}}=\alpha_{XY}$,
Eq. 39 and 40 imply

$\displaystyle y_{x_{0}}(u)$
$\displaystyle=\widehat{y}-(\alpha_{XY}+\alpha_{XW}\alpha_{WY})\;\;\forall
u\text{ s.t. }\widehat{Y}_{x_{1}}(u)=\widehat{y},$ (41) $\displaystyle
y_{x_{0}}(u)$ $\displaystyle=\widehat{y}\;\;\forall u\text{ s.t.
}\widehat{Y}_{x_{0}}(u)=\widehat{y},$ (42)

which in turn shows that

$\displaystyle\mathbbm{E}(y_{x_{0}}\mid\widehat{y}_{x_{1}})-\mathbbm{E}(y_{x_{0}}\mid\widehat{y}_{x_{0}})=-\alpha_{XY}-\alpha_{XW}\alpha_{WY}.$
(43)

###  A.2 Empirical Evaluation of Thm. 1

In this section, we empirically demonstrate the validity of Thm. 1 through an
example (see source code for reproducing the results). In particular, we
consider the following structural causal model:

$\displaystyle X$ $\displaystyle\leftarrow U_{X}$ (44) $\displaystyle W_{1}$
$\displaystyle\leftarrow X+U_{1}$ (45) $\displaystyle W_{2}$
$\displaystyle\leftarrow\frac{1}{4}W_{1}^{2}-\frac{1}{3}X+U_{2}$ (46)
$\displaystyle Y$
$\displaystyle\leftarrow\frac{1}{6}W_{1}W_{2}+W_{1}+\frac{1}{2}X+U_{Y}.$ (47)
$\displaystyle U_{X}$ $\displaystyle\in\\{0,1\\},P(U_{X}=1)=0.5,$ (48)
$\displaystyle U_{1}$ $\displaystyle,U_{2},U_{Y}\sim N(0,1),$ (49)

with the following causal diagram:

$X$$W_{1}$$W_{2}$$Y$

.

We proceed as follows. We investigate the decomposition in Thm. 1 with respect
to a varying number of samples available, with
$n\in\\{1000,2000,3500,5000,7500,10000\\}$. For each value, we generate $n$
samples from the SCM in Eqs. 44-49. Then, based on the samples, we construct a
predictor $\widehat{Y}$ using xgboost [Chen and Guestrin, 2016] (with a fixed
$\eta=0.1$ and number of rounds chosen using cross-validation with 10 folds).
Then, we bin the samples into 20 intervals corresponding to the quantiles of
the predictor $\widehat{Y}$:

$b_{1}=[0\%\text{-quant}(\widehat{Y}),5\%\text{-quant}(\widehat{Y})),\dots,b_{20}=[95\%\text{-quant}(\widehat{Y}),100\%\text{-quant}(\widehat{Y})].$

Further, we estimate the potential outcome $Y_{x_{0}}$ for each sample using
the fairadapt package [Plečko et al., 2021], and the estimator is labeled
$Y^{*}_{x_{0}}$ (also note that $Y_{x_{0}}(u)=Y(u)$ for all $u$ s.t.
$X(u)=x_{0}$). Then, within each bin $b_{i}$, we compute

Figure 7: iTerm I, iTerm II, iPPM estimated values.

Figure 8: iTerm I and $\widehat{\text{Term I}}_{i}$ values.

$\displaystyle\widehat{\text{Term I}}_{i}$
$\displaystyle=\widehat{\mathbbm{E}}[Y_{x_{1}}\mid x_{1},\widehat{y}\in
b_{i}]-\widehat{\mathbbm{E}}[Y^{*}_{x_{0}}\mid x_{1},\widehat{y}\in b_{i}],$
(50) $\displaystyle\widehat{\text{Term II}}_{i}$
$\displaystyle=\widehat{\mathbbm{E}}[Y^{*}_{x_{0}}\mid x_{1},\widehat{y}\in
b_{i}]-\widehat{\mathbbm{E}}[Y_{x_{0}}\mid x_{0},\widehat{y}\in b_{i}],$ (51)
$\displaystyle\widehat{\text{iPPM}}_{i}$
$\displaystyle=\widehat{\mathbbm{E}}[Y_{x_{1}}\mid x_{1},\widehat{y}\in
b_{i}]-\widehat{\mathbbm{E}}[Y_{x_{0}}\mid x_{0},\widehat{y}\in b_{i}],$ (52)

where $\widehat{\mathbbm{E}}$ denotes the empirical expectation. We then
compute $\widehat{\text{iTerm
I}}=\frac{1}{20}\sum_{i=1}^{20}\widehat{\text{Term I}}_{i}$, where iTerm I is
the “integrated” value of Term I across all 20 bins. We similarly estimate
$\widehat{\text{iTerm II}}$ and $\widehat{\text{iPPM}}$. For each dataset size
$n$, the experiment is repeated 20 times. In Fig. 8, we plot the obtained
estimates of iTerm I, iTerm II, and the iPPM, where the shaded region
represents the standard deviation of the estimates across 20 repetitions. The
figure illustrates that as the number of samples increases, the predictor
$\widehat{Y}$ is closer to calibration (iPPM is closer to $0$), as predicted
by Prop. 1. Furthermore, we can see that iTerm I and iTerm II are different
from $0$, and cancel each other out, as predicted by Thm. 1.

Furthermore, we verify if the estimates $\widehat{\text{Term I}}_{i}$ obtained
in Eq. 50 correspond to the ground truth. To this end, instead of estimating
the potential outcome $Y_{x_{0}}$, we obtain the true potential outcome values
$Y_{x_{0}}$ from the SCM. Then, we compute

$\displaystyle\text{Term I}_{i}$ $\displaystyle=\mathbbm{E}[Y_{x_{1}}\mid
x_{1},\widehat{y}\in b_{i}]-\mathbbm{E}[Y_{x_{0}}\mid x_{1},\widehat{y}\in
b_{i}].$ (53)

We take 50 repetitions of $n=5000$ samples and obtain the values of
$\widehat{\text{Term I}}_{i}$ and $\text{Term I}_{i}$. The obtained results
are shown in Fig. 8. In red, we can see the ground truth values for each
$\text{Term I}_{i}$, and in black we see the estimated values of
$\widehat{\text{Term I}}_{i}$, where the shaded area indicates the standard
deviation of the estimate across the 50 repetitions. Once again, we see that
the terms $\text{Term I}_{i}$ are estimated correctly, which in combination
with Fig. 8 demonstrates empirically the theoretical result predicted by Thm.
1.

##  Appendix B Identification Expressions of Prop. 4

We provide the identification expressions for the counterfactual measures
$\text{Ctf-DE}_{x_{0},x_{1}}(y\mid x)$, $\text{Ctf-IE}_{x_{1},x_{0}}(y\mid
x)$, and $\text{Ctf-SE}_{x_{1},x_{0}}(y)$. For the measures related measures
of $\widehat{y}$, the expressions are analogous, with $y$ replaced by
$\widehat{y}$ throughout:

$\displaystyle\text{Ctf-DE}_{x_{0},x_{1}}(y\mid x)$
$\displaystyle=\sum_{z,w}[P(y\mid x_{1},z,w)-P(y\mid x_{0},z,w)]P(w\mid
x_{0},z)P(z\mid x),$ (54) $\displaystyle\text{Ctf-IE}_{x_{1},x_{0}}(y\mid x)$
$\displaystyle=\sum_{z,w}P(y\mid x_{1},z,w)[P(w\mid x_{0},z)-P(w\mid
x_{1},z)]P(z\mid x),$ (55) $\displaystyle\text{Ctf-SE}_{x_{1},x_{0}}(y)$
$\displaystyle=\sum_{z}P(y\mid x_{1},z)[P(z\mid x_{1})-P(z\mid x_{0})].$ (56)

##  Appendix C Proof of Prop. 1

Proof  Notice that for any $X=x$

$\displaystyle P(y\mid x,\widehat{y})$
$\displaystyle=\sum_{\begin{subarray}{c}z,w:\\\
\widehat{Y}(x,z,w)=\widehat{y}\end{subarray}}P(y\mid
x,z,w,\widehat{y})P(z,w\mid x,\widehat{y})$ (57)
$\displaystyle=\sum_{\begin{subarray}{c}z,w:\\\
\widehat{Y}(x,z,w)=\widehat{y}\end{subarray}}P(y\mid x,z,w)P(z,w\mid
x,\widehat{y})$ (58)
$\displaystyle=\widehat{y}*\sum_{\begin{subarray}{c}z,w:\\\
\widehat{Y}(x,z,w)=\widehat{y}\end{subarray}}P(z,w\mid
x,\widehat{y})=\widehat{y}.$ (59)

The first step (Eq. 57) follows from the law of total probability, the second
(Eq. 58) from noting that $\widehat{Y}{\perp\\!\\!\\!\perp}Y\mid X,Z,W$, and
the third (Eq. 59) from the efficiency of the learner (Eq. 4). Therefore, it
follows that $P(y\mid x_{1},\widehat{y})=P(y\mid x_{0},\widehat{y})$, meaning
that $\widehat{Y}$ satisfies PP.