11affiliationtext: RAND Corporation; E-mail:
<EMAIL_ADDRESS>Department of Criminology and
Department of Statistics and Data Science, University of Pennsylvania; E-mail:
<EMAIL_ADDRESS>Department of Biostatistics, Columbia
University; E-mail<EMAIL_ADDRESS>
# Mediated probabilities of causation

Max Rubinstein  Maria Cuellar  Daniel Malinsky

###### Abstract

We propose a set of causal estimands that we call “the mediated probabilities
of causation.” These estimands quantify the probabilities that an observed
negative outcome was induced via a mediating pathway versus a direct pathway
in a stylized setting involving a binary exposure or intervention, a single
binary mediator, and a binary outcome. We outline a set of conditions
sufficient to identify these effects given observed data, and propose a
doubly-robust projection based estimation strategy that allows for the use of
flexible non-parametric and machine learning methods for estimation. We argue
that these effects may be more relevant than the probability of causation,
particularly in settings where we observe both some negative outcome and
negative mediating event, and we wish to distinguish between settings where
the outcome was induced via the exposure inducing the mediator versus the
exposure inducing the outcome directly. We motivate our quantities of interest
by discussing applications to legal and medical questions of causal
attribution.

Keywords: Mediation analysis, probability of causation, machine learning, non-
parametrics, causal inference

##  1 Introduction

A common goal in research is to quantify the average effect of some
intervention on an outcome. Knowledge of these “causal effects” can help
inform decision makers responsible for implementing various policies or
procedures. However, average effects are not always the most relevant
quantities. In legal settings, for example, judges and lawyers often instead
wish to know the probability that some intervention or exposure caused an
observed negative outcome. As one example, to receive compensation under the
National Vaccine Injury Compensation Program (“NVIC”), claimants must show
that their alleged vaccine-related injuries were “more likely than not” caused
by the vaccine. Attributing rare adverse events to vaccinations may also
generally be of clinical interest. The relevant causal quantity for these
problems is known as the “probability of causation.” The probability of
causation, unlike the average causal effect, conditions on the outcome, and
thus concerns a sub-population of individuals who experience the negative
outcome under the exposure (Cuellar and Kennedy,, 2020; Lagakos and
Mosteller,, 1986; Tian and Pearl,, 2000; Dawid et al.,, 2017; Dawid and
Musio,, 2022).

However, the probability of causation may be too coarse of a measure for some
settings. Specifically, it fails to pinpoint any specific mechanism or
hypothesized causal pathway that induces the outcome. To illustrate, consider
the recent lawsuit against Harvard University alleging discrimination against
Asian American applicants.111https://www.nytimes.com/2022/12/02/us/asian-
american-college-applications.html The probability of causation ostensibly
helps formalize the plaintiffs’ discrimination claim. Specifically, we can
imagine a world where a rejected Asian American applicant were not (perceived
by the admissions officer to be) Asian American, and ask whether, in this
counterfactual world, their application would have been accepted. However, we
might also wonder by what mechanism might the applicant’s identity cause the
admissions decision? One hypothesized pathway posited in this case is that
(perceived) Asian American racial identity led admissions officers to give
worse subjective personality assessments, and in turn deny applications. From
the plaintiff’s perspective, admissions rejections via this “indirect” effect
would be impermissible. Additionally, there may exist other relevant pathways
that may be considered legally or morally permissible or not, depending on
one’s interpretation of the relevant legal statutes or background ethical
commitments.

Other legal and scientific questions also hinge on whether some intermediate
factor induces an outcome. For example, in jury selection attorneys sometime
question whether a juror’s exclusion might be due to factors causally related
to – or “downstream from” – race. In evaluating the harmful effect of some
chemical exposure on health outcomes, scientists or advocates may posit a
specific mediating element such as a negative physiological response – for
example, a change in hormone levels or gene expression – that ultimately leads
to disease or death. In clinical contexts, a mediating negative event may also
arise in the context of attributing “cause of death:” for example, someone may
hypothesize that a patient’s death was caused by an adverse reaction to
vaccination via a cardiovascular mechanism, having observed post-vaccination
myocarditis. However, the probability of causation does not distinguish
between possible pathways.

We therefore introduce and examine a new class of causal estimands that we
call the “mediated probabilities of causation.” These estimands separately
quantify the probability of causation via direct and indirect pathways, and
together sum to a “total” mediated probability of causation. Through
considering the simplified setting of a single binary exposure, mediator, and
outcome, our primary contribution is to define these causal estimands and
outline assumptions sufficient to identify these estimands in terms of
observed data. As a second contribution, we propose a doubly-robust estimation
strategy that targets projections of the mediated probabilities of causation,
extending the estimation method outlined in Cuellar and Kennedy, (2020), who
introduced this approach to estimating the probability of causation. This
strategy allows for the use of modern non-parametric and machine learning
methods that require minimal modeling assumptions, while still yielding root-n
consistent and asymptotically normal estimates under relatively mild
conditions.

Our paper proceeds as follows: in Section 2 we briefly review the probability
of causation, including the formal definition, the typical identifying
assumptions, and an expression for the identified estimand. In Section 3 we
propose the three causal estimands that we call the mediated probabilities of
causation: the probability of indirect causation, the probability of direct
causation, and the total mediated probability of causation. We outline a set
of assumptions sufficient to identify these estimands in observed data and
provide expressions for these identified estimands. In Section 4 we propose a
non-parametric projection-based approach to estimate these quantities,
following Cuellar and Kennedy, (2020). In Section 5 we illustrate our proposed
methods using a simulation study motivated by the Harvard discrimination case.
In Section 6 we conclude with general discussion of the problem, limitations
of our study, and possible areas for future research.

##  2 Background

We begin by reviewing the probability of causation, the identifying
assumptions sufficient to estimate this quantity in observed data, and the
identified data functional (Cuellar and Kennedy,, 2020). We use a stylized
description of the Harvard discrimination lawsuit as a motivating example.
Importantly, the model we posit simplifies what are certainly more complex
causal structures and we ignore actual legal considerations. We do not claim
that a formal analysis of this particular problem using our exact approach has
or should have ethical or legal merit, but rather use the example to
illustrate the concepts.

Let $Y$ denote a binary variable where $Y=1$ reflects a negative outcome. Let
$A$ denote a binary exposure, and $X$ denote a d-dimensional covariate vector,
which may include continuous variables. Let $Y(a)$ denote the potential
outcome for an individual under exposure assignment $A=a$. For example, $Y$
may reflect Harvard’s admissions decision, where $Y=1$ if Harvard denies an
applicant admissions, $A$ may be an indicator of Asian American identity, and
$X$ may include fixed demographic characteristics including age, sex, or
perhaps some measure of childhood socioeconomic position (SEP). Finally, we
posit the existence of potential outcomes with respect to Asian American
identity, where $Y(1)$ represents the admissions decision had the applicant
been racialized/identified as Asian American and $Y(0)$, the admissions
decision had the applicant been racialized/identified as white.222Whether and
how to define potential outcomes with respect to racial or other identity
categories is a topic of considerable controversy. Often, discussions of
racial discrimination in admissions or hiring appeal to “signals” of racial
identity, e.g., stereotyped associations with applicant names or other
signifiers that may be gleaned from a resume. This approach may also be
problematic (Kohler-Hausmann,, 2018), and related considerations have been
raised in the Harvard admissions case specifically Kohler-Hausmann, (2023).
For the sake of illustrating the proposed estimands, we will simply assume
these can be clearly defined in context. More broadly, we understand and
appreciate the challenges and critical perspectives on counterfactual
approaches to these questions. We can define the probability of causation as:

$\displaystyle\tau(x)$ $\displaystyle=P(Y(0)=0\mid Y(1)=1,X=x)$ (1)

This formalizes the probability that a Harvard applicant with covariates $X=x$
who would have been denied admission were they Asian American would have been
admitted were they not Asian American. We do not observe both potential
outcomes $Y(1)$ and $Y(0)$ in practice. We instead observe $n$ independent
copies of the data $O=(Y,A,X)$. We can nevertheless identify $\tau(x)$ in
observed data by making the following assumptions:

######  Assumption 1 (Y consistency).

$Y=AY(1)+(1-A)Y(0)$

######  Assumption 2 (A-Y ignorability).

$A\perp\\{Y(0),Y(1)\\}\mid X$

######  Assumption 3 (A positivity).

$P\\{\min_{a}P(A=a\mid X)\geq\epsilon\\}=1\qquad\epsilon>0$

######  Assumption 4 (Y positivity).

$P\\{P(Y=1\mid A=1,X)\geq\epsilon\\}=1\qquad\epsilon>0$

######  Assumption 5 (Y monotonicity).

$Y(1)\geq Y(0)$

Proposition 1 presents the identification result. For completeness, the proof
is provided in Appendix B.

######  Proposition 1.

Under Assumptions (1)-(5), $\tau(x)$ is identified in the observed data
distribution as,

$\displaystyle\tau(x)$ $\displaystyle=1-\frac{P(Y=1\mid A=0,X=x)}{P(Y=1\mid
A=1,X=x)}.$

######  Remark 1.

A more natural definition of the probability of causation conditions on the
observed outcomes and treatment assignment rather than the potential outcomes,
since the counterfactual quantities $Y(1)$ are unobserved for observations
where $A=0$ and are therefore not of practical interest. Specifically, we may
define the probability of causation as,

$\displaystyle\tilde{\tau}(x)$ $\displaystyle=P(Y(0)=0\mid Y=1,A=1,X=x).$ (2)

It is easy to see that this expression is equivalent to $\tau(x)$ under
assumptions (1)-(2). However, when defining the probability of causation as
$\tilde{\tau}(x)$ we can weaken the required identifying assumptions. For
example, in place of A-Y ignorability we only require that $A\perp Y(0)\mid
X$. We nevertheless introduce these estimands conditional on the
counterfactual quantities for conceptual clarity, noting throughout these
alternative, and arguably more intuitive, formulations.

##  3 Mediated probabilities of causation

To motivate the mediated probabilities of causation, consider the case where
we observe a binary causal descendant of the exposure, called $M$, that
mediates the effect of the exposure $A$ on the outcome $Y$, so that
$O=(X,A,M,Y)$. Moreover, we let $M=1$ denote some negative condition. For
example, in the Harvard discrimination case $M$ may indicate a personality
assessment. Figure 1 illustrates the assumed causal structure of the observed
data distribution distribution.

Figure 1: Assumed causal structureXAMY

We next define potential outcomes to illustrate how information on the
mediator adds complexity to this problem. We consider three specific
estimands: the total mediated probability of causation, the indirect
probability of causation, and the direct probability of causation. We motivate
and describe each quantity in greater detail below.

###  3.1 Total mediated probability of causation

Viewing the mediator as an “intermediate outcome” possibly affected by
exposure, we define a probability of causation on the stratum where $M(1)=1$
and outcome $Y(1)=Y(1,M(1))=1$ (noting that $Y(a)=Y(a,M(a))$ here and
throughout). In other words, we define a probability of causation where the
potential mediator under exposure and potential outcome under exposure are
both negative. We call this the “total mediated probability of causation.”
Mathematically, we can express this as:

$\displaystyle\delta(x)=P(Y(0)=0\mid Y(1)=1,M(1)=1,X=x)$ (3)

This estimand may be motivated by the question: given that an individual would
experience a negative outcome and negative mediator under an exposure (i.e.
$Y(1)=1$ and $M(1)=1$), what is the probability that the negative outcome
would not have occurred absent the exposure?

The quantity $\delta(x)$ as defined conditions on the potential rather than
observed outcomes. Just as we could define the probability of causation
conditional on only observed quantities, we can also define the total mediated
probability of causation quantity conditional on the observed outcome,
mediator, and exposure:

$\displaystyle\tilde{\delta}(x)=P(Y(0)=0\mid Y=1,M=1,A=1,X=x)$ (4)

This quantity expresses the probability that an individual who received
treatment and experienced a negative mediator and outcome under exposure would
not have experienced the negative outcome had they not been exposed. These two
quantities, $\delta(x)$ and $\tilde{\delta}(x)$, are equivalent under the
identifying assumptions that we outline in Section 3 below.

Finally, notice that the total mediated probability of causation is not
generally equal to the probability of causation. Instead, the probability of
causation can be expressed as a weighted average of the total mediated
probability of causation and an analogous term defined on the stratum where
$M(1)=0$. We can see this by iterating expectations for $\tau(x)$ over the
conditional distribution of $M(1)$:

$\displaystyle\tau(x)$ $\displaystyle=\sum_{m=0,1}P(Y(0)=0\mid
Y(1)=1,M(1)=m,X=x)$ $\displaystyle\times P(M(1)=m\mid Y(1)=1,X=x)$
$\displaystyle=\delta(x)\xi(x)+\delta^{\prime}(x)(1-\xi(x))$

where $\xi(x)=P(M(1)=m\mid Y(1)=1,X=x)$. This expression highlights the
relationship between these quantities, and shows that we should not in general
expect the probability of causation to equal the total mediated probability of
causation. We further discuss the term $\delta^{\prime}(x)$ and other possible
decompositions in Section 3.3 and in Appendix C.

###  3.2 Direct and indirect probabilities of causation

The total mediated probability of causation decomposes into the sum of what we
call the direct and indirect probabilities of causation. Specifically, we
iterate expectations of $\delta(x)$ over the conditional distribution of the
cross-world counterfactual quantity $Y(1,M(0))$ to obtain these quantities.

$\displaystyle\delta(x)$
$\displaystyle=\sum_{y=0,1}P(Y(1,M(0))=y,Y(0,M(0))=0\mid
Y(1,M(1))=1,M(1)=1,X=x)$ $\displaystyle=\psi(x)+\zeta(x)$

where

$\displaystyle\psi(x)=P(Y(1,M(0))=0,Y(0,M(0))=0\mid Y(1,M(1))=1,M(1)=1,X=x)$
(5)

and

$\displaystyle\zeta(x)$ $\displaystyle=P(Y(1,M(0))=1,Y(0,M(0))=0\mid
Y(1,M(1))=1,M(1)=1,X=x)$ (6)

$\psi(x)$ represents the probability of indirect causation, while $\zeta(x)$
is the direct probability of direct causation. We use these terms because
$\psi(x)$ represents the probability that the negative outcome was induced via
the $A\to M\to Y$ pathway, while the term $\zeta(x)$ reflects the probability
that the outcome was induced via the $A\to Y$ pathway. The total mediated
probability of causation is again simply the sum of these two quantities.

We can also define versions of these estimands that only condition on observed
variables. These are again equivalent to the quantities introduced above under
the identifying assumptions we outline below.

$\displaystyle\tilde{\psi}(x)$ $\displaystyle=P(Y(1,M(0))=0,Y(0,M(0))=0\mid
Y=1,M=1,A=1,X=x)$ (7) $\displaystyle\tilde{\zeta}(x)$
$\displaystyle=P(Y(1,M(0))=1,Y(0,M(0))=0\mid Y=1,M=1,A=1,X=x)$ (8)

Which estimand is of primary interest depends on the specific research
question, and possibly which pathways are legally or ethically impermissible.
For example, if the indirect channel is legally or ethically impermissible,
then we may wish to know the indirect probability of causation $\psi(x)$; if
the direct channel, then the direct probability of causation $\zeta(x)$. If
either pathway, and we observe a negative mediating event, then the total
mediated probability of causation $\delta(x)$. Finally, when all possible
pathways through which the exposure might induce the outcome are of interest,
then the probability of causation $\tau(x)$ may be the relevant causal
quantity.

###  3.3 Other possible quantities

A possible concern is that our proposed direct and indirect probabilities of
causation decompose the total mediated probability of causation rather than
the probability of causation itself. Mathematically, we could have instead
defined quantities similar to $\psi(x)$ and $\zeta(x)$ that do not condition
on the potential mediator $M(1)$, and that have the seemingly desirable
property of summing to the probability of causation instead of the total
mediated probability of causation. Moreover, these quantities would capture
information from the term $\delta^{\prime}(x)$ defined above, which
represented an analogous total mediated probability of causation term defined
on the $M(1)=0$ stratum.

We argue that these quantities have little practical interest: once we observe
the mediator, we know whether or not the negative condition occurred, and it
is thus natural to condition on this information. And if we observe no post-
treatment mediator, we show in Appendix C that we cannot generally identify
these probabilities, so that the question becomes strictly theoretic. Finally,
we may be in a setting where we observe a mediator, but the negative outcome
did not occur, so that the term $\delta^{\prime}(x)$ and an analogous
decomposition into direct and indirect pathways may seem desirable. However,
if the negative mediator did not occur under exposure, then, under the
monotonicity conditions we will require for causal identification, the outcome
can only have been induced via a direct pathway – that is, pathways not
through the mediator.333Intuitively, monotonicity rules out the case where the
exposure is protective of the mediator; that is, the mediating event would
have occurred absent exposure but did not occur due to the exposure. No
further decomposition of this term is necessary. In short, because most
conceivable scenarios where we would wish to estimate probabilities of direct
versus indirect causation involve the case where we observe a negative
mediating event under exposure, we argue that the total mediated probability
of causation and its decomposition are the most relevant quantities to
consider in practice. We nevertheless provide expressions for all of these
related quantities and identification results in Appendix C.

###  3.4 Identification

We can identify the mediated probabilities of causation in the observed data
assuming that for all values of $a,m$:

######  Assumption 6 (Y-M consistency).

$\displaystyle A=a,M=m$ $\displaystyle\implies Y=Y(am)$ $\displaystyle A=a$
$\displaystyle\implies M=M(a)$

######  Assumption 7 (Y-M monotonicity).

$\displaystyle Y(11)$ $\displaystyle\geq Y(10)\geq Y(00),$ $\displaystyle
Y(11)$ $\displaystyle\geq Y(01),$ $\displaystyle M(1)$ $\displaystyle\geq
M(0)$

######  Assumption 8 (A-YM ignorability).

$\displaystyle A\perp\\{Y(11),Y(10),Y(01),Y(00),M(1),M(0)\\}\mid X$

######  Assumption 9 (Cross-world ignorability).

$\displaystyle\\{Y(11),Y(10),Y(01),Y(00)\\}\perp\\{M(1),M(0)\\}\mid X$

######  Assumption 10 (A-M positivity).

$\displaystyle P\\{\min_{a,m}P(A=a,M=m\mid
X)\geq\epsilon\\}=1,\qquad\epsilon>0$

######  Assumption 11 (Y positivity).

$\displaystyle P\\{P(Y=1\mid A=1,M=1,X)\geq\epsilon\\}=1,\qquad\epsilon>0$

Assumption (6) precludes interference between individuals; that is, each
individual’s potential outcomes only depends on their own mediator and
exposure status. Assumption (7) implies that the mediator and exposure can
only work to induce the outcomes, and the exposure can only work to induce the
mediator, with one exception: the mediator may be protective against the
outcome in the absence of the exposure. In our working example, this precludes
a scenario where an individual, were they Asian American, would be accepted to
Harvard were they to receive a negative personality assessment but would be
rejected if they received a positive assessment. Assumption (8) implies that
the exposure is effectively randomized with the potential outcomes and
mediators given the covariates. Assumption (9) further implies that the
potential mediators are independent of the potential outcomes given
covariates: this is a strong assumption that is unenforceable even in a
randomized experiment. Assumption (10) implies that there is some value of
observing all exposure and mediator combinations for any covariate value;
finally, assumption (11) implies that there is a positive probability of
experiencing the outcome for any covariate value under exposure and the
mediator.

######  Remark 2.

We can weaken some of the above assumptions to identify $\psi(x)$ or
$\delta(x)$ alone; moreover, these assumptions are somewhat stronger than
strictly necessary to identify these three estimands. We provide slightly
weaker sets of identifying conditions for each individual estimand in Appendix
B.1.

######  Remark 3.

Substantially weaker assumptions to identify natural effects. For example,
assumption 7 is not required; assumption 10 is stronger than needed (see,
e.g., Vansteelandt and Daniel, (2017)); and no analogue of assumption 11 is
needed.444This is required here because the identified estimands involve risk-
ratios with this quantity in the denominator. Assumption 11 ensures these
quantities are well-defined. Finally, in place of assumption 9, one typically
assumes,

$\displaystyle Y(am)$ $\displaystyle\perp M\mid\\{X,A=a\\},\qquad a=0,1$ (9)
$\displaystyle Y(am)$ $\displaystyle\perp M(a^{\prime})\mid X,\qquad
m,a,a^{\prime}=0,1;a\neq a^{\prime}.$ (10)

Assumptions (6)-(8) imply equation 9, and assumption 9 alone implies equation
(10); however, the reverse is not true.

Theorem 1 provides identifying expressions for $\delta(x)$ and $\psi(x)$ in
terms of the observed data distribution. We define the observed data
quantities $\mu_{am}(x)=P(Y=1\mid A=a,M=m,X=x)$, and $\gamma_{a}(x)=P(M=1\mid
A=a,X=x)$.

######  Theorem 1 (Identification).

Under assumptions (7)-(11), $\psi(x)$, $\delta(x)$, and $\zeta(x)$ are
identified in the observed data as,

$\displaystyle\psi(x)$
$\displaystyle=\left[1-\frac{\mu_{10}(x)}{\mu_{11}(x)}\right]\left[1-\frac{\gamma_{0}(x)}{\gamma_{1}(x)}\right],$
(11) $\displaystyle\delta(x)$
$\displaystyle=\left[1-\frac{\mu_{00}(x)}{\mu_{11}(x)}\right]\left[1-\frac{\gamma_{0}(x)}{\gamma_{1}(x)}\right]+\left[1-\frac{\mu_{01}(x)}{\mu_{11}(x)}\right]\left[\frac{\gamma_{0}(x)}{\gamma_{1}(x)}\right],$
(12) $\displaystyle\zeta(x)$ $\displaystyle=\delta(x)-\psi(x).$ (13)

The identifying expressions in Theorem (1) have intuitive interpretations.
First, consider the expression for $\psi(x)$. Under our causal assumptions,
this expression is equivalent to 555This identity is implied by the Proof in
the Appendix.

$\displaystyle P(Y(10)=0\mid Y(11)=1,M(1)=1,X=x)P(M(0)=0\mid M(1)=1,X=x).$

The first quantity in this expression represents the total mediated
probability of causation with respect to a “controlled indirect effect.” In
contrast to natural effects, which allow the mediator to assume its natural
value under the relevant exposure condition, controlled (in)direct effects
conceive of deterministically setting both the exposure and mediator values.
This functional therefore represents the probability that a negative outcome
was induced by changing the mediator from $0$ to $1$ in the presence of an
exposure among the stratum where $Y(11)=1$ and $M(1)=1$. The second term is
simply the probability of causation with respect to the mediator; that is, it
is the probability of causation where we allow the mediator to play the role
of the outcome. The identifying expression for $\psi(x)$ is therefore
equivalent to the probability that the exposure induced the mediator times the
probability that the inducing the mediator caused the outcome in the presence
of the exposure (i.e. changing $Y(10)$ to $Y(11)$).

The identifying expression for $\delta(x)$ has a slightly more complex -
though still intuitive - explanation. In this case, we can show that this
expression is equivalent to,

$\displaystyle P(Y(00)=0\mid Y(11)=1,M(1)=1,M(0)=0,X=x)P(M(0)=0\mid
M(1)=1,X=x)$ $\displaystyle+P(Y(01)=0\mid
Y(11)=1,M(1)=1,M(0)=1,X=x)P(M(0)=1\mid M(1)=1,X=x).$

In other words, the total probability of causation is equal to the probability
that the exposure induced the mediator and either the mediator or the exposure
induced the outcome; plus the probability that the exposure did not induce the
mediator, but did induce the outcome in the presence of the mediator. These
represent all possible pathways from which $A$ might affect $Y$. Finally, to
arrive at the probability of indirect causation, we simply subtract from the
expression for the direct probability of causation from the expression for the
total mediated probability of causation. Intuitively, this removes the
particular case where the exposure induced the mediator and the mediator
induced the outcome, leaving only direct pathways from the exposure to the
outcome behind; that is, cases where the exposure alone induced the outcome
regardless of whether the exposure induced the mediator.

As a separate and perhaps useful point, the identification result for
$\psi(x)$ implies that a valid upper bound on the probability of indirect
causation is given by the probability of causation with respect to the
mediator. Intuitively, this follows because in the most extreme case the
probability of causation via a controlled indirect effect is one: in this case
it remains only to show that the mediator was induced by the exposure.
Conveniently, identification of this bound only requires assuming assumptions
(1)-(4) using $M$ and $M(a)$ in place of $Y$ and $Y(a)$. For example, in the
Harvard lawsuit, the probability of indirect causation of Asian American
identity via personality assessments is upper bounded by the probability that
Asian American identity caused an applicant’s negative personality assessment.
In practice, if the legal standard for some case depended on the probability
of indirect causation being less than some threshold, it would suffice to show
that the probability of causation with respect to the mediator is less than
this same threshold. Because the probability of causation requires far weaker
identifying assumptions than the probability of indirect causation, this bound
may be useful for some real-world settings.

###  3.5 Graphical intuition

To illustrate the relationships between the causal estimands we have discussed
– the average treatment effect, the probability of causation, and the mediated
probabilities of causation – we simulate data and display the implied
estimands.

Figure 2: Nuisance components

We first draw a single continuous covariate $X\stackrel{{\scriptstyle
iid}}{{\sim}}\mathcal{U}(0,1)$. We then generate propensity score, mediator,
and outcome models as polynomial functions of this variable, as illustrated in
Figure 2 (the precise specifications are available in Appendix D). All
functions are decreasing in $X$. For example, we can again think of these
variables as pertaining to the Harvard admissions case, where $X$ may
represent a measure of childhood SEP.

The simulated functions above suggest that the probability of rejection,
negative personality assessment, and Asian American identity are all
decreasing in the covariate. Moreover, the simulated functions give Asian
Americans have a higher probability of negative personality assessment than
corresponding non Asian Americans of similar childhood SEP ($P(M(1)=1\mid
x)>P(M(0)=1\mid x)$ for all $x$). However, the probability of rejection for
Asian Americans is lower compared to non Asian Americans for a fixed value of
the personality assessment and childhood SEP (that is, $P(Y(1m)=1\mid
x)<P(Y(0m)=1\mid x)$ for all $m,x$). Finally, we ensure that all monotonicity
assumptions hold when making Bernoulli draws of the potential outcomes and
mediators from these distributions.

Figure 3: Causal estimands

Figure 3 displays the implied causal estimands in our simulated dataset. The
first panel depicts the (conditional) average treatment effects. These
functions show that Asian American identity increases the probability of
rejection and receiving a negative personality assessment across the covariate
distribution. However, these functions do not address the question: given that
an applicant would be denied admission were they Asian American, would they be
denied application were they not Asian American? Or the question, given that
an applicant would receive a negative personality assessment were they Asian
American, would they not receive a negative personality assessment were they
not Asian American? These are the probabilities of causation with respect to
the outcome and the mediator, respectively, and are displayed in the second
panel. The values of both estimands are increasing in the covariate value.

Finally, the third panel depicts the mediated probabilities of causation:
these estimands condition on the subset of individuals who would receive a
negative personality assessment were they Asian American. The green line
represents the total probability of causation, while the purple and pink lines
represent the indirect and direct probabilities of causation, respectively.
Throughout the covariate distribution, the probability of indirect causation
is higher than the probability of direct causation, and for individuals with
values of $x$ approximately greater than $0.88$, the probability of indirect
causation is greater than 0.5. For these individuals, it is more likely than
they were they Asian American, they would be rejected from Harvard due to a
negative personality assessment that would have been positive had they not
been Asian American.666We again stress that this data is entirely simulated
and for illustrative purposes only. We are not asserting any factual claims
about the true data generating processes regarding Harvard application
admissions decisions.

##  4 Estimation

We propose estimating projections of the mediated probabilities of causation
onto parametric working models, following Cuellar and Kennedy, (2020). In
other words, we do not propose estimating the mediated probabilities of
causation directly, but instead target a summary measure. Figure 4 illustrates
this idea, displaying the true total and indirect probabilities of causation
as a function of $x$, denoted in the darker lines, versus their best fitting
linear approximation based on minimizing the squared error, denoted in the
lighter shades of the same color. Notably, the projections are almost never
exactly equal to the true quantities because the true functions are not
linear.777In practice, we could define a more complex projection that better
fits the true data generating process, though we do not do so to illustrate
the difference between these quantities. Nevertheless, these quantities are
generally close, illustrating that projections may be useful targets of
inference.

Figure 4: Projections

Targeting projections is also desirable from a statistical perspective
because, when the covariates are continuous, these estimands cannot generally
be estimated at root-n rates without making strong - and unrealistic -
parametric modeling assumptions. By contrast, we can obtain root-n consistent
and asymptotically normal estimates of projections of these parameters under
relatively weak conditions. We briefly review this approach and elaborate on
the properties of these estimators below, using a projection of $\psi(x)$ to
illustrate.

To be precise, we define a projection of $\psi(x)$ at a point $x$ as
$g(x;\beta)$ for some function $g$ parameterized by some finite-dimensional
parameter vector $\beta$, where $\beta$ is defined as the solution to equation
14)888While we define the projection with respect to the $L_{2}$ loss, we
could generalize this formula to minimize other loss functions instead.
Similarly, we could include pre-specified weights $w(X)$ into this
minimization that differentially weight the covariate space in the projection
(analogous to weighted least squares).,

$\displaystyle\beta=\arg\min_{\tilde{\beta}}\mathbb{E}[\\{\psi(X)-g(X;\tilde{\beta})\\}^{2}].$
(14)

We again stress that we need not assume $\psi(x)=g(x;\beta)$ for any $x$ for
the projection defined by $g(x;\beta)$ to be a valid and meaningful target of
statistical inference. This fact generally holds when using parametric models
to estimate unknown and possibly complex data generating mechanisms. Indeed,
one can often interpret model parameters as defining a projection of some true
process under no modeling assumptions (Angrist and Pischke,, 2009; Buja et
al.,, 2019).

To estimate this projection, we follow Cuellar and Kennedy, (2020) and propose
a “doubly-robust” estimation approach. Specifically, we propose using the
estimator $\hat{\beta}$ that satisfies the moment condition,

$\displaystyle\mathbb{P}_{n}\left[\frac{\partial
g(X;\hat{\beta})}{\partial\beta}w(X)\left(\varphi(O;\hat{\eta})-g(X;\hat{\beta})\right)\right]=0,$
(15)

where $\varphi(O;\hat{\eta})$ uses estimated “nuisance” functions $\hat{\eta}$
of $\eta=[\mu_{11},\mu_{10},P(A,M\mid X)]$999Note that we can recover
estimates of $(P(A=1\mid X),\gamma_{1}(X),\gamma_{0}(X))$ from estimates of
$P(A,M\mid X)$. Alternatively, we could separately estimate these functions to
obtain estimates of $P(A,M\mid X)$. plugged into the following expression,

$\displaystyle\varphi(O;\eta)=\varphi_{3}(O;\eta)-\varphi_{1}(O;\eta)-\varphi_{2}(O;\eta)+\psi(X;\eta),$

where

$\displaystyle\varphi_{1}(O;\eta)$
$\displaystyle=\frac{1}{\mu_{11}(X)}\left[\frac{A(1-M)\\{Y-\mu_{10}(X)\\}}{P(A=1,M=0\mid
X)}-\frac{\mu_{10}(X)}{\mu_{11}(X)}\frac{AM\\{Y-\mu_{11}(X)\\}}{P(A=1,M=1\mid
X)}\right],$ $\displaystyle\varphi_{2}(O;\eta)$
$\displaystyle=\frac{1}{\gamma_{1}(X)}\left[\frac{(1-A)(M-\gamma_{0}(X))}{P(A=0\mid
X)}-\frac{\gamma_{0}(X)}{\gamma_{1}(X)}\frac{A(M-\gamma_{1}(X))}{P(A=1\mid
X)}\right],$ $\displaystyle\varphi_{3}(Z;\eta)$
$\displaystyle=\varphi_{1}(Z;\eta)\left[\frac{\gamma_{0}(X)}{\gamma_{1}(X)}\right]+\varphi_{2}(Z;\eta)\left[\frac{\mu_{10}(X)}{\mu_{11}(O)}\right].$

Theorem 2 in Appendix A formalizes conditions where this estimator will yield
root-n consistent and asymptotically normal estimates of $\beta$ and therefore
$g(x;\beta)$.

A key benefit of this “doubly-robust” estimation strategy versus other
possible methods is that it allows for non-parametric estimation of the
nuisance functions while still obtaining parametric rates of convergence of
the estimated projection parameter under relatively mild conditions. At a high
level, this is possible because the error of doubly-robust estimators is a
function of the product of errors among the estimated nuisance functions. As a
result, the overall error of the estimator converges to zero more quickly than
each nuisance component. As long as certain error products between nuisance
components are estimated quickly enough (to be precise, are $o_{p}(n^{-1/2})$
in the $L_{2}$ norm), then the nuisance estimation can be ignored
asymptotically. In other words, the asymptotics of these estimators is as if
we regressed $\varphi(O;\eta)$ onto $g(X;\beta)$ directly, instead of the
estimated function $\varphi(O;\hat{\eta})$.

We refer to Cuellar and Kennedy, (2020) for a more thorough discussion of
using a non-parametric projection based estimation methods and (Kennedy et
al.,, 2021) for a more thorough review of doubly-robust estimators. We also
provide the analogous doubly-robust estimators for the projections of
$\delta(x)$ and $\zeta(x)$ in Appendix A.

##  5 Simulations

We conduct a simulation study to verify the expected theoretic performance of
our proposed estimators. In our first simulation study we examine the bias,
RMSE, and coverage rates for the proposed projection parameter while
simulating nuisance estimation error on the true data generating functions. In
particular, we draw 1,000 samples of size $n=1,000$ and add gaussian noise to
each nuisance parameter $\eta_{j}$ in the following way:

$\displaystyle\hat{\eta}_{j}(x)=\text{expit}(\text{logit}(\eta_{j}(x))+\mathcal{N}(C_{1j}n^{-\alpha},C_{2j}n^{-2\alpha}))$

for constants $C_{1}$ and $C_{2}$ that depend on the nuisance function and
$\alpha$ that defines the convergence rate. In other words, we simualted
nuisance parameters estimated at $\mathcal{O}_{p}(n^{-\alpha})$ rates in the
$L_{2}$ norm. We test $\alpha=0.3,0.1$, where the first parameterization is
fast enough to satisfy the rate conditions of Theorem 2, but the second does
not.101010This theorem states that a sufficient condition for root-n
consistency and asymptotic normality of this parameter estimate is that each
nuisance function be estimated at $o_{p}(n^{-1/4})$ rates. We then estimate a
linear projection of each mediated probability of causation (total, indirect,
and direct) at the value of $x=0.75$, as illustrated in Figure 3 above. Table
1 displays the results of this first simulation, illustrating minimal bias for
all estimands and showing that we are able to obtain approximately nominal
coverage rates for the projection parameters when $\alpha=0.3$, but not when
$\alpha=0.1$, as expected.

Table 1: Simulation results | $\alpha=0.3$ | $\alpha=0.1$
---|---|---
Estimand | Bias | RMSE | Coverage | Bias | RMSE | Coverage
Indirect | -0.01 | 0.05 | 0.96 | 0.23 | 0.30 | 0.80
Direct | 0.02 | 0.05 | 0.94 | -0.13 | 0.23 | 0.92
Total | 0.01 | 0.04 | 0.94 | 0.10 | 0.15 | 0.89

##  6 Conclusion

We have introduced a class of causal estimands that merge ideas from the
literature on mediation and the probability of causation that we call the
“mediated probabilities of causation.” We highlighted three quantities in
particular: the total mediated, indirect, and direct probabilities of
causation. We then provided a set of assumptions sufficient to identify these
estimands in observed data. While these assumptions are quite strong, we also
show that a far weaker set of assumptions suffices to identify upper bounds on
the indirect probability of causation. Finally, we propose estimating these
quantities using a doubly-robust projection based approach, following Cuellar
and Kennedy, (2020). In a set of simulation studies we verify that these
estimators can obtain root-n consistent and asymptotically normal estimates of
the projection parameters while allowing for relatively slow rates of
convergence on the nuisance estimates.

Although we used a stylized version of the Harvard admissions case as a
motivating example, the mediated probabilities of causation may not be of
primary interest in that case, since the actual arguments made have been
varied and
complicated111111https://www.supremecourt.gov/opinions/22pdf/20-1199_hgdj.pdf.
Indeed, the fundamental conceptual terms of this case have been challenged
(Kohler-Hausmann,, 2023). If in fact the plaintiffs would hypothesize a
particular mediating causal mechanism such as the one we describe, the burden
of proof would presumably fall on the plaintiff to provide evidence that this
mechanism is active. With the estimator we propose, this may be technically
feasible; however, we emphasize that the evidential burden may be steep, as
the identifying assumptions we describe may not be deemed credible in this
specific legal case setting. Nevertheless, we hope that it is useful to
formally explicate one possible quantity of interest in order to help
adjudicate between claims made without the aid of any statistical formalism,
and to clarify the strong assumptions required to identify these parameters
given observed data. Furthermore, the formalism may be useful outside of legal
contexts, as mentioned previously. For example, if medical providers or other
stakeholders hypothesize that a vaccinated patient’s negative outcome was
brought about by a negative mediating event - for example, an adverse reaction
to vaccination - then the indirect probability of causation may be of
substantive interest.

This paper also has technical limitations. For example, we only consider the
single mediator setting, and restrict that the mediators, exposure, and
outcome are all binary. Applied settings frequently have multiple possibly
continuous mediators with unknown causal orderings. Formalizing analogous
estimands in these more complicated – and realistic – settings would be a
useful area for future research. Additionally, our identification results
require strong unverifiable assumptions, including cross-world ignorability.
Whether it is possible to weaken these assumptions, or bound these estimands
under some assumed sensitivity parameters, would also be interesting to
investigate. Finally, our proposed estimation method targets a projection of
the causal estimands, rather than the causal estimands themselves. We could
instead use DR-learner (Kennedy,, 2020) to estimate the true mediated
probabilities of causation.121212At a high-level, this simply involves
replacing the parametric projection with a fully non-parametric model and
regressing the estimated quantity $\varphi(O;\hat{\eta})$ onto this quantity.
While this approach will not generally yield root-n consistent estimates, a
full articulation of this method applied to this setting may also be
worthwhile.

## References

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  * Buja et al., (2019) Buja, A., Brown, L., Berk, R., George, E., Pitkin, E., Traskin, M., Zhang, K., and Zhao, L. (2019).  Models as approximations i: Consequences illustrated with linear regression.  Statistical Science, 34(4):523–544.
  * Cuellar and Kennedy, (2020) Cuellar, M. and Kennedy, E. H. (2020).  A non-parametric projection-based estimator for the probability of causation, with application to water sanitation in kenya.  Journal of the Royal Statistical Society: Series A (Statistics in Society), 183(4):1793–1818.
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  * Kennedy, (2022) Kennedy, E. H. (2022).  Semiparametric doubly robust targeted double machine learning: a review.  arXiv preprint arXiv:2203.06469.
  * Kennedy et al., (2021) Kennedy, E. H., Balakrishnan, S., and Wasserman, L. (2021).  Semiparametric counterfactual density estimation.  arXiv preprint arXiv:2102.12034.
  * Kohler-Hausmann, (2018) Kohler-Hausmann, I. (2018).  Eddie murphy and the dangers of counterfactual causal thinking about detecting racial discrimination.  Nw. UL Rev., 113:1163.
  * Kohler-Hausmann, (2023) Kohler-Hausmann, I. (2023).  What does’ race neutral’admissions mean?  Yale Law School, Public Law Research Paper.
  * Lagakos and Mosteller, (1986) Lagakos, S. W. and Mosteller, F. (1986).  Assigned shares in compensation for radiation-related cancers.  Risk Analysis, 6(3):345–357.
  * Tian and Pearl, (2000) Tian, J. and Pearl, J. (2000).  Probabilities of causation: Bounds and identification.  Annals of Mathematics and Artificial Intelligence, 28(1):287–313.
  * VanderWeele and Robinson, (2014) VanderWeele, T. J. and Robinson, W. R. (2014).  On causal interpretation of race in regressions adjusting for confounding and mediating variables.  Epidemiology (Cambridge, Mass.), 25(4):473.
  * Vansteelandt and Daniel, (2017) Vansteelandt, S. and Daniel, R. M. (2017).  Interventional effects for mediation analysis with multiple mediators.  Epidemiology (Cambridge, Mass.), 28(2):258.

##  Appendix A Additional theoretic results

Proposition 2 formalizes the proposed influence function based estimator
discussed in Section 4.

######  Proposition 2.

Consider the moment condition for a fixed $\beta^{\star}$ suggested by
equation (14):

$\displaystyle\Psi(\beta^{\star})=\mathbb{E}[w(X)(\psi(X)-g(X;\beta^{\star}))]=0$

where $\psi(x)$ is given by equation (23). Under a non-parametric model, the
uncentered efficient influence curve for the moment condition
$\Psi(\beta^{\star})$ at any fixed $\beta^{\star}$ is given by

$\displaystyle\phi(Z;\beta^{\star},\eta)$ $\displaystyle=\frac{\partial
g(X;\beta^{\star})}{\partial\beta}w(X)\left(\varphi(O;\eta)-g(X;\beta^{\star})\right)$
(16)

where

$\displaystyle\varphi(O;\eta)=\varphi_{3}(O;\eta)-\varphi_{1}(O;\eta)-\varphi_{2}(O;\eta)+\psi(X;\eta)$

and

$\displaystyle\varphi_{1}(O;\eta)$
$\displaystyle=\frac{1}{\mu_{11}(X)}\left[\frac{A(1-M)\\{Y-\mu_{10}(X)\\}}{P(A=1,M=0\mid
X)}-\frac{\mu_{10}(X)}{\mu_{11}(X)}\frac{AM\\{Y-\mu_{11}(X)\\}}{P(A=1,M=1\mid
X)}\right]$ $\displaystyle\varphi_{2}(O;\eta)$
$\displaystyle=\frac{1}{\gamma_{1}(X)}\left[\frac{(1-A)(M-\gamma_{0}(X))}{P(A=0\mid
X)}-\frac{\gamma_{0}(X)}{\gamma_{1}(X)}\frac{A(M-\gamma_{1}(X))}{P(A=1\mid
X)}\right]$ $\displaystyle\varphi_{3}(O;\eta)$
$\displaystyle=\varphi_{1}(O;\eta)\left[\frac{\gamma_{0}(X)}{\gamma_{1}(X)}\right]+\varphi_{2}(O;\eta)\left[\frac{\mu_{10}(X)}{\mu_{11}(X)}\right]$

Proposition 3 provides an analogous estimator for the projection of
$\delta(x)$.

######  Proposition 3.

Consider the moment condition for a fixed $\alpha^{\star}$ suggested by
equation (14), where we assume the user-specified projection of $\delta(x)$
(see equation (11)), is given by the function $h(x;\alpha^{\star})$:

$\displaystyle\Psi_{tpc}(\alpha^{\star})=\mathbb{E}[w(X)(\delta(X)-h(X;\alpha^{\star}))]=0$

Under a non-parametric model, the uncentered efficient influence curve for the
moment condition $\Psi_{tpc}(\alpha^{\star})$ at any fixed $\alpha^{\star}$ is
given by

$\displaystyle\phi_{tpc}(O;\alpha^{\star},\eta)$ $\displaystyle=\frac{\partial
h(X;\alpha^{\star})}{\partial\alpha}w(X)\left(\varphi_{tpc}(O;\eta)-h(X;\alpha^{\star})\right)$
(17)

where

$\displaystyle\varphi_{tpc}(O;\eta)=\varphi_{tpc,4}(O;\eta)-\varphi_{tpc,5}(O;\eta)-\varphi_{tpc,1}(O;\eta)+\delta(X;\eta)$

and

$\displaystyle\varphi_{tpc,1}(O;\eta)$
$\displaystyle=\frac{1}{\mu_{11}(X)}\left[\frac{(1-A)(1-M)\\{Y-\mu_{00}(X)\\}}{P(A=0,M=0\mid
X)}-\frac{\mu_{00}(X)}{\mu_{11}(X)}\frac{AM\\{Y-\mu_{11}(X)\\}}{P(A=1,M=1\mid
X)}\right]$ $\displaystyle\varphi_{tpc,2}(O;\eta)$
$\displaystyle=\frac{1}{\mu_{11}(X)}\left[\frac{(1-A)M\\{Y-\mu_{01}(X)\\}}{P(A=0,M=1\mid
X)}-\frac{\mu_{01}(X)}{\mu_{11}(X)}\frac{AM\\{Y-\mu_{11}(X)\\}}{P(A=1,M=1\mid
X)}\right]$ $\displaystyle\varphi_{tpc,3}(O;\eta)$
$\displaystyle=\frac{1}{\gamma_{1}(X)}\left[\frac{(1-A)(M-\gamma_{0}(X))}{P(A=0\mid
X)}-\frac{\gamma_{0}(X)}{\gamma_{1}(X)}\frac{A(M-\gamma_{1}(X))}{P(A=1\mid
X)}\right]$ $\displaystyle\varphi_{tpc,4}(O;\eta)$
$\displaystyle=\varphi_{tpc,1}(O;\eta)\left[\frac{\gamma_{0}(X)}{\gamma_{1}(X)}\right]+\varphi_{tpc,3}(O;\eta)\left[\frac{\mu_{00}(X)}{\mu_{11}(X)}\right]$
$\displaystyle\varphi_{tpc,5}(O;\eta)$
$\displaystyle=\varphi_{tpc,3}(O;\eta)\left[\frac{\gamma_{0}(X)}{\gamma_{1}(X)}\right]+\varphi_{tpc,3}(O;\eta)\left[\frac{\mu_{01}(X)}{\mu_{11}(X)}\right]$

where $\eta=[\mu_{11},\mu_{00},\mu_{01},P(A,M\mid X)]$.

######  Remark 4.

To obtain an analogous result for the quantity $\zeta(x)$, one simply need
note that $\zeta(x)=\delta(x)-\psi(x)$: since the influence function of the
difference of two functionals is the difference of the influence functions,
the result immediately follows that the influence function for the
corresponding moment condition is given by:

$\displaystyle\phi_{pnde}(O;\rho^{\star},\eta)$ $\displaystyle=\frac{\partial
m(X;\rho^{\star})}{\partial\rho}w(X)\left(\varphi_{tpc}(O;\eta)-\varphi(O;\eta)-m(X;\rho^{\star})\right)$
(18)

where $m(x;\rho)$ defines the user-specified projection of $\zeta(x)$.

Theorem 2 establishes the conditions where the estimator proposed in equation
(15) is root-n consistent for $\psi$ and asymptotically normal.

######  Theorem 2.

Consider the moment condition $\mathbb{E}[\phi(O;\beta_{0},\eta_{0})]=0$
evaluated at the true parameters $(\beta_{0},\eta_{0})$. Now consider the
estimator $\hat{\beta}$ that satisfies
$\mathbb{P}_{n}[\phi(O;\hat{\beta},\hat{\eta})]=0$, where $\hat{\eta}$ is
estimated on an independent sample. Assume that:

  * •

The function class $\\{\phi(O;\beta,\eta):\beta\in\mathbb{R}^{p}\\}$ is
Donsker in $\beta$ for any fixed $\eta$

  * •

$\|\phi(O;\hat{\beta},\hat{\eta})-\phi(O;\beta_{0},\eta_{0})\|=o_{p}(1)$

  * •

The map $\beta\to\mathbb{P}[\phi(O;\beta,\eta)]$ is differentiable at
$\beta_{0}$ uniformly in $\eta$, with non-singular derivative matrix
$\frac{\partial}{\partial\beta}\mathbb{P}\\{\phi(O;\beta,\eta)\\}\mid_{\beta=\beta_{0}}=M(\beta_{0},\eta)$,
where

$M(\beta_{0},\hat{\eta})\overset{p}{\to}M(\beta_{0},\eta_{0})$

  * •

$\|\hat{\eta}-\eta\|=o_{p}(n^{-1/4})$

Then the proposed estimator attains the non-parametric efficiency bound and is
asymptotically normal with

$\displaystyle\sqrt{n}(\hat{\beta}-\beta_{0})\to^{d}\mathcal{N}(0,M^{-1}\mathbb{E}[\phi\phi^{\top}]M^{-\top})$
(19)

and, therefore, for any fixed value of $X=x$,

$\displaystyle\sqrt{n}(g(x;\hat{\beta})-g(x;\beta_{0}))\to^{d}\mathcal{N}\left(0,\left(\frac{\partial
g(x;\beta_{0})}{\partial\beta}\right)^{\top}M^{-1}\mathbb{E}[\phi\phi^{\top}]M^{-\top}\frac{\partial
g(x;\beta_{0})}{\partial\beta}\right).$ (20)

######  Remark 5.

Theorem 2 assumes that $\|\hat{\eta}-\eta\|=o_{p}(n^{-1/4})$. This condition
is sufficient, but not necessary, for this result to hold. The key requirement
is that the error of the influence function based estimator is a function of
the product of errors in the nuisance estimates. As long as these error
products are all $o_{p}(n^{-1/2})$, this result will hold.

######  Remark 6.

Analogous results to Theorem 2 for the projections of $\delta(x)$ and
$\zeta(x)$ can easily be derived: the difference comes in the precise form of
the second-order remainder terms, which we have omitted here, that are
implicitly assumed $o_{p}(n^{-1/2})$ under the assumption that
$\|\hat{\eta}-\eta\|=o_{p}(n^{-1/4})$. Additionally, further notice that we
have slightly abused notation to let $\eta$ to represent any of the nuisance
components in the corresponding influence functions; however, the specific
components differ slightly depending on the estimand (e.g. $\delta(x)$ depends
on $\mu_{01}(x)$ while $\psi(x)$ does not).

##  Appendix B Proofs

We divide the proofs into two subsections: first, proofs concerning
identification. Second, proofs concerning estimation. For the proofs on
identification, we first outline slightly weaker sets of assumptions (compared
to those outlined in Section 3) that are sufficient to identify each estimand.

###  B.1 Identification

####  B.1.1 Probability of causation

###### Proof of proposition (1).

$\displaystyle\tau(x)$ $\displaystyle=P(Y(0)=0\mid Y(1)=1,X=x)$
$\displaystyle=P(Y(0)=0\mid Y(1)=1,A=1,X=x)$ $\displaystyle=P(Y(0)=0\mid
Y=1,A=1,X=x)$ $\displaystyle=\frac{P(Y-Y(0)=1\mid A=1,X=x)}{P(Y=1\mid
A=1,X=x)}$ $\displaystyle=1-\frac{P(Y(0)=1\mid A=1,X=x)}{P(Y=1\mid A=1,X=x)}$
$\displaystyle=1-\frac{P(Y=1\mid A=0,X=x)}{P(Y=1\mid A=1,X=x)}$

where the first line follows by definition, the second by A-Y ignorability,
the third by Y consistency, the fourth by Bayes’ rule, the fifth by Y
monotonicity, and the final equality by A-Y ignorability and Y consistency. ∎

######  Remark 7.

The equivalence of $\tau(x)$ and $\tilde{\tau}(x)$ is implied in the proof,
following from the first three equalities.

####  B.1.2 Total probability of causation

We invoke assumptions (6) and (11). We further assume:

######  Assumption 7′ (Y-M monotonicity).

$\displaystyle M(1)$ $\displaystyle\geq M(0)$ $\displaystyle Y(11)$
$\displaystyle\geq Y(0m),\qquad m=0,1$

######  Assumption 8′ (A-YM ignorability).

$\displaystyle A$ $\displaystyle\perp\\{Y(11),Y(0m),M(1),M(0)\\}\mid X,\qquad
m=0,1$

######  Assumption 9′ (Cross-world ignorability).

$\displaystyle\\{M(0),M(1)\\}\perp Y(0m)\mid X,\qquad m=0,1$

######  Assumption 10′ (A-M positivity).

$\displaystyle P\\{P(A=0,M=m\mid X)$ $\displaystyle\geq\epsilon\\}=1,\qquad
m=0,1$ $\displaystyle P\\{P(A=1,M=1\mid X)$ $\displaystyle\geq\epsilon\\}=1$

###### Proof of total probability of causation identification.

We invoke assumptions (6) and (11), as well as assumptions (7′)-(10′) above.

$\displaystyle P(Y(0,M(0))=0\mid Y(1,M(1))=1,M(1)=1,x)$
$\displaystyle=\frac{P(Y(0,M(0))=0,Y(11)=1\mid M(1)=1,x)}{P(Y(11)=1\mid
M(1)=1,x)}$ (21) $\displaystyle=1-\frac{P(Y(0,M(0))=1\mid
M(1)=1,x)}{P(Y(11)=1\mid M(1)=1,x)}$ $\displaystyle=1-\frac{P(Y(0,M(0))=1\mid
M(1)=1,x)}{P(Y=1\mid A=1,M=1,x)}$ (22)

where the first equality holds by Bayes’ rule and consistency, the second by
Y-M monotonicity, and the third by A-YM ignorability and consistency. By the
law of iterated expectation and Y-M consistency we further obtain:

$\displaystyle P(Y(0,M(0))=1\mid M(1)=1,x)$
$\displaystyle=\sum_{m}P(Y(0m)=1\mid M(0)=m,M(1)=1,x)P(M(0)=m\mid M(1)=1,x)$
$\displaystyle=\sum_{m}P(Y(0m)=1\mid M(0)=m,M(1)=1,x)w_{m}(x)$

By A-YM ignorability and cross-world ignorability we further obtain that:

$\displaystyle P(Y(0m)=1\mid M(0)=m,M(1)=1,x)$ $\displaystyle=P(Y(0m)=1\mid
x)$ $\displaystyle=P(Y=1\mid A=0,M=m,x)$

Finally, Proposition 1 and the corresponding assumptions imply that:

$\displaystyle w_{0}(x)=1-\frac{P(M=1\mid A=0,x)}{P(M=1\mid A=1,x)}$

noting that $w_{1}(x)=1-w_{0}(x)$. Combining the expressions yields the
result. ∎

######  Remark 8.

The equivalence of $\tilde{\delta}(x)$ and $\delta(x)$, and therefore the
identification of $\tilde{\delta}(x)$, follows by noting that:

$\displaystyle P(Y(0,M(0))=0\mid Y=1,M=1,A=1,x)$
$\displaystyle=P(Y(0,M(0))=0\mid Y(11)=1,M(1)=1,A=1,x)$
$\displaystyle=P(Y(0,M(0))=0\mid Y(11)=1,M(1)=1,x)$
$\displaystyle=P(Y(0,M(0))=0\mid Y(1,M(1))=1,M(1)=1,x)$

where the first equality holds by consistency, the second by Y-AM
ignorability, and the final equality by consistency.

######  Remark 9.

In contrast to the positivity assumption in equation (10), we can allow that
$P(Y=1,M=0\mid x)=0$ for any value of $x$.

######  Remark 10.

Identification of the total probability of causation requires relatively
weaker monotonicity assumptions than stated in equation (7), allowing for
potentially negative direct and indirect effects, whereas assumption (7)
essentially restricts that these effects are non-negative. We discuss the
monotonicity conditions further in Remark 14.

####  B.1.3 Indirect probability of causation

######  Assumption 8⋆ (A-YM ignorability).

$\displaystyle A\perp\\{Y(11),Y(10),M(0),M(1)\\}\mid X,\qquad m=0,1$

######  Assumption 9⋆ (Cross-world ignorability).

$\displaystyle\\{M(0),M(1)\\}\perp\\{Y(10),Y(11)\\}\mid X,\qquad m=0,1$

######  Assumption 10⋆ (A-M positivity).

$\displaystyle P\\{P(A=1,M=m\mid X)\geq\epsilon\\}=1,\qquad m=0,1$
$\displaystyle P\\{P(A=0\mid X)\geq\epsilon\\}=1$

###### Proof of indirect probability of causation identification.

We invoke assumptions (6), (7), (11), and assumptions (8⋆)-(10⋆) above.

$\displaystyle P(Y(1,M(0))=0,Y(0,M(0))=0\mid Y(1,M(1))=1,M(1)=1,X=x)$
$\displaystyle=P(Y(1,M(0))=0\mid Y(11)=1,M(1)=1,X=x)$
$\displaystyle=\sum_{m}P(Y(1,M(0))=0\mid Y(11)=1,M(1)=1,M(0)=m,x)$
$\displaystyle\times P(M(0)=m\mid Y(11)=1,M(1)=1,x)$
$\displaystyle=\sum_{m}P(Y(1m)=0\mid Y(11)=1,M(1)=1,M(0)=m,x)P(M(0)=m\mid
M(1)=1,x)$ $\displaystyle=P(Y(10)=0\mid Y(11)=1,X=x)P(M(0)=0\mid M(1)=1,X=x)$
(23)

where the first equality follows by Y-M monotonicity and Y-M consistency, the
second by the law of iterated expectations, the third by cross-world
ignorability and Y-M consistency, and the fourth by the fact that
$P(Y(11)=0\mid Y(11)=1)=0$ and cross-world ignorability. Now consider the
first term:

$\displaystyle P(Y(10)=0\mid Y(11)=1,X=x)$
$\displaystyle=\frac{P(Y(10)=0,Y(11)=1\mid X=x)}{P(Y(11)=1\mid X=x)}$
$\displaystyle=\frac{P(Y(10)=0,Y(11)=1\mid X=x)}{P(Y(11)=1\mid A=1,M=1,X=x)}$
$\displaystyle=1-\frac{P(Y(10)=1\mid X=x)}{P(Y=1\mid A=1,M=1,X=x)}$
$\displaystyle=1-\frac{P(Y=1\mid A=1,M=0,X=x)}{P(Y=1\mid A=1,M=1,X=x)}$

where the first equality holds by Bayes’ rule, the second by A-YM and cross-
world ignorability, the third by Y-M monotonicity and Y-M consistency, and the
final equality by A-YM and cross-world ignorability and Y-M consistency. The
result follows from applying Proposition 1 to the second term (which is simply
the probability of causation with respect to the mediator) and multiplying the
expressions. ∎

######  Remark 11.

The equivalence of $\psi(x)$ and $\tilde{\psi}(x)$, and therefore the
identification of $\tilde{\psi}(x)$ can be seen by noting that:

$\displaystyle P(Y(1,M(0))=0,Y(0,M(0))=0\mid Y=1,M=1,A=1,X=x)$
$\displaystyle=P(Y(1,M(0))=0,Y(0,M(0))=0\mid Y(11)=1,M(1)=1,A=1,X=x)$
$\displaystyle=P(Y(1,M(0))=0\mid Y(11)=1,M(1)=1,X=x)$
$\displaystyle=P(Y(1,M(0))=0,Y(0,M(0))=0\mid Y(1,M(1))=1,M(1)=1,X=x)$

where the first equality uses consistency, the second monotonicity and A-YM
ignorability, and the final equality again uses consistency and monotonicity.

######  Remark 12.

The positivity requirement for the identification of $\psi(x)$ allows that for
any $x$, either $P(A=0,M=1\mid x)=0$ or $P(A=0,M=0\mid x)=0$ (but not both, as
we do require $P(A=0\mid x)\geq\epsilon$ for all $x$).

######  Remark 13.

Identification of $\zeta(x)$ follows directly from the proof of Theorem 1,
since $\delta(x)=\tau(x)+\zeta(x)$ by definition, though this result requires
the union of two sets of identifying assumptions outlined above.

######  Remark 14.

One assumption that we do not require for identification, but may nevertheless
be useful for interpretation, is a monotonicity condition that $Y(01)\geq
Y(00)$.

To see why, consider the case where for an individual $M(1)=1$ and $M(0)=0$;
that is, the exposure induces the mediator; and we have the following
potential outcomes: $Y(11)=1,Y(10)=1,Y(01)=0,Y(00)=1$. Under our definitions,
we would say that for this individual, there was no effect in total, no
indirect effect, and no direct effect. Yet in fact, for this individual, there
is an indirect effect that acts only in the absence of the exposure (a so-
called “pure” indirect effect), but a mediated interaction term that undoes
this effect in the presence of the exposure (VanderWeele and Robinson,, 2014).
While mathematically this does not present a problem, this scenario may be
conceptually challenging to allow for in applied examples. One can always
further assume the monotonicity condition that $Y(01)\geq Y(00)$ to rule these
out, if desired.

###  B.2 Estimation

###### Proof of Propositions 2, 3.

This proof follows directly from Cuellar and Kennedy, (2020) and applying the
chain-rule to the functionals $\mathbb{E}[\psi(X)]$ and
$\mathbb{E}[\delta(X)]$, treating $X$ as discrete (Kennedy,, 2022). For
example, for a fixed $\beta^{\star}$, we let
$h(X;\beta^{\star})=\frac{\partial g(X;\beta^{\star})}{\partial\beta}w(X)$:

$\displaystyle\text{EIF}[\mathbb{E}[h(X;\beta^{\star})(\psi(X)-g(X;\beta^{\star})))]]$
$\displaystyle=\text{EIF}[\sum_{x}h(x;\beta^{\star})(\psi(x)-g(x;\beta^{\star}))p(x)]$
$\displaystyle=\sum_{x}h(x;\beta^{\star})\text{EIF}(\psi(x))p(x)+\sum_{x}w(x)(\psi(x)-g(x;\beta^{\star}))\text{EIF}(p(x))$
$\displaystyle=\sum_{x}h(x;\beta^{\star})[\varphi_{3}(O;\eta)-\varphi_{1}(O;\eta)-\varphi_{2}(O;\eta)]+h(x;\beta^{\star})(\psi(x)-g(x;\beta^{\star}))[1(X=x)-p(x)]$
$\displaystyle=h(X;\beta^{\star})[\varphi_{3}(O;\eta)-\varphi_{1}(O;\eta)-\varphi_{2}(O;\eta)+\psi(X)-g(X;\beta^{\star})]-\Psi(\beta^{\star})$

where we again obtain $\text{EIF}[\psi(x)]$ by treating $x$ as discrete to
obtain:

$\displaystyle\text{EIF}[\psi(x)]$
$\displaystyle=\text{EIF}\left(1-\frac{\mu_{10}(x)}{\mu_{11}(x)}\right)\left(1-\frac{\gamma_{0}(x)}{\gamma_{1}(x)}\right)$
$\displaystyle=\text{EIF}\left[\frac{\mu_{10}(x)}{\mu_{11}(x)}\frac{\gamma_{0}(x)}{\gamma_{1}(x)}\right]-\text{EIF}\left[\frac{\mu_{10}(x)}{\mu_{11}(x)}\right]-\text{EIF}\left[\frac{\gamma_{0}(x)}{\gamma_{1}(x)}\right]$
$\displaystyle=\varphi_{3}(O;\eta)-\varphi_{1}(O;\eta)-\varphi_{2}(O;\eta)$

and applying the chain-rule to get each component part. To be precise, note
that:

$\displaystyle\text{EIF}\left[\frac{\mu_{10}(x)}{\mu_{11}(x)}\frac{\gamma_{0}(x)}{\gamma_{1}(x)}\right]$
$\displaystyle=\text{EIF}\left[\frac{\mu_{10}(x)}{\mu_{11}(x)}\right]\frac{\gamma_{0}(x)}{\gamma_{1}(x)}+\text{EIF}\left[\frac{\gamma_{0}(x)}{\gamma_{1}(x)}\right]\frac{\mu_{10}(x)}{\mu_{11}(x)}$
$\displaystyle=\varphi_{1}(O;\eta)\frac{\gamma_{0}(x)}{\gamma_{1}(x)}+\varphi_{2}(O;\eta)\frac{\mu_{10}(x)}{\mu_{11}(x)}$

Second, by the quotient rule we obtain:

$\displaystyle\varphi_{1}(O;\eta)$
$\displaystyle=\frac{1}{\mu_{11}(x)^{2}}(\text{EIF}[\mu_{10}(x)]\mu_{11}(x)-\text{EIF}[\mu_{11}(x)]\mu_{10}(x))$
$\displaystyle\varphi_{2}(O;\eta)$
$\displaystyle=\frac{1}{\gamma_{1}(x)^{2}}(\text{EIF}[\gamma_{0}(x)]\gamma_{1}(x)-\text{EIF}[\gamma_{1}(x)]\gamma_{0}(x))$

Finally, recall that $\text{EIF}[E[Y\mid X=x]]=\frac{1(X=x)}{P(X=x)}[Y-E[Y\mid
X=x]]$ (Kennedy,, 2022). It follows that for any $(a,m,x)$:

$\displaystyle\text{EIF}[\mu_{am}(x)]$
$\displaystyle=\frac{1(A=a,M=m,X=x)}{P(A=a,M=m,X=x)}(Y-\mu_{am}(X)),$
$\displaystyle\text{EIF}[\gamma_{a}(x)]$
$\displaystyle=\frac{1(A=a,X=x)}{P(A=a,X=x)}(M-\gamma_{a}(x))$

The result follows by combining expressions. The same logic can be used to
derive the influence function for the moment conditions
$\Psi_{ptc}(\alpha^{\star})$ and $\Psi_{pnde}(\rho^{\star})$, where for this
final derivation we can use the fact that since $\zeta(x)=\delta(x)-\psi(x)$,
then $\text{EIF}[\zeta(x)]=\text{EIF}[\delta(x)]-\text{EIF}[\psi(x)]$
(Kennedy,, 2022). To flesh this out slightly more, we sketch the derivation of
$\text{EIF}[\delta(x)]$ below:

$\displaystyle\text{EIF}[\delta(x)]$
$\displaystyle=\text{EIF}\left(\left[1-\frac{\mu_{00}(x)}{\mu_{11}(x)}\right]\left[1-\frac{\gamma_{0}(x)}{\gamma_{1}(x)}\right]+\left[1-\frac{\mu_{01}(x)}{\mu_{11}(x)}\right]\left[\frac{\gamma_{0}(x)}{\gamma_{1}(x)}\right]\right)$
$\displaystyle=\text{EIF}\left(\frac{\mu_{00}(x)}{\mu_{11}(x)}\frac{\gamma_{0}(x)}{\gamma_{1}(x)}-\frac{\mu_{10}(x)}{\mu_{11}(x)}\frac{\gamma_{0}(x)}{\gamma_{1}(x)}-\frac{\mu_{00}(x)}{\mu_{11}(x)}\right)$
$\displaystyle=\text{EIF}\left(\frac{\mu_{00}(x)}{\mu_{11}(x)}\right)\frac{\gamma_{0}(x)}{\gamma_{1}(x)}+\frac{\mu_{00}(x)}{\mu_{11}(x)}\text{EIF}\left(\frac{\gamma_{0}(x)}{\gamma_{1}(x)}\right)$
$\displaystyle-\text{EIF}\left(\frac{\mu_{10}(x)}{\mu_{11}(x)}\right)\frac{\gamma_{0}(x)}{\gamma_{1}(x)}-\text{EIF}\left(\frac{\gamma_{0}(x)}{\gamma_{1}(x)}\right)\frac{\mu_{10}(x)}{\mu_{11}(x)}-\text{EIF}\left(\frac{\mu_{00}(x)}{\mu_{11}(x)}\right)$

By the quotient rule, we know generally that:

$\displaystyle\text{EIF}\left(\frac{\mu_{am}(x)}{\mu_{a^{\prime}m^{\prime}}(x)}\right)$
$\displaystyle=\frac{1}{\mu_{a^{\prime}m^{\prime}}(x)^{2}}\left(\frac{I(A=a,M=m,X=x)}{P(A=a,M=m,X=x)}[Y-\mu_{am}(x)]\mu_{a^{\prime}m^{\prime}}(x)\right)$
$\displaystyle-\left(\frac{I(A=a^{\prime},M=m^{\prime},X=x)}{P(A=a^{\prime},M=m^{\prime},X=x)}[Y-\mu_{a^{\prime}m^{\prime}}(x)]\mu_{am}(x)\right)$

Applying this to the relevant quantities above and plugging the previously
derived result for
$\text{EIF}\left[\frac{\gamma_{0}(x)}{\gamma_{1}(x)}\right]$ gives the result.

∎

###### Proof of Theorem 2 (sketch).

The proof follows almost directly Lemma 3 in (Kennedy et al.,, 2021). To
formally complete the proof, we would also need to show that the quantity
$P[\phi(O;\beta_{0},\hat{\eta})-\phi(O;\beta_{0},\eta)]$ is second-order in
the nuisance estimation to derive sufficient conditions where this quantity is
$o_{p}(n^{-1/2})$, where $P[f(O)]=\mathbb{E}[f(O)\mid D_{0}^{n}]$ for a
training dataset $D_{0}^{n}$ used to estimate the nuisance parameters $\eta$.
The conditions for this to hold will then be satisfied when
$\|\hat{\eta}-\eta\|=o_{p}(n^{-1/4})$. ∎

##  Appendix C Probability of causation: other decompositions

As noted in Section 3, we can also decompose the probability of causation
decomposes into analogous terms that we described in our paper that do not
condition on $M(1)$. We call these the total probabilities of indirect and
direct causation and denote them $\alpha(x)$ and $\beta(x)$, respectively.
More formally, we can derive these quantities by applying the law of iterated
expectation the the probability of causation:

$\displaystyle P(Y(0)=0\mid Y(1)=1,x)$
$\displaystyle=P(Y(1,M(0))=0,Y(0,M(0))=0\mid Y(1,M(1))=1,x)$
$\displaystyle+P(Y(1,M(0))=1,Y(0,M(0))=0)\mid Y(1,M(1))=1,x)$
$\displaystyle=\alpha(x)+\beta(x)$

Proposition 4 shows that these expressions are identifiable in the observed
data distribution given $O=(X,A,M,Y)$.

######  Proposition 4 (Identification of total indirect and direct PCs).

Under assumptions (6)-(11), the total probabilities of indirect and direct
causation can be expressed as:

$\displaystyle\alpha(x)$
$\displaystyle=\left(1-\frac{\mu_{10}(x)}{\mu_{00}(x)}\right)\left(1-\frac{\gamma_{0}(x)}{\gamma_{1}(x)}\right)P(M=1\mid
A=1,Y=1,x)$ $\displaystyle\beta(x)$
$\displaystyle=\left(1-\frac{\mu_{0}(x)}{\mu_{1}(x)}\right)-\alpha(x)$

where $\mu_{a}(x)=P(Y=1\mid A=a,x)$.

######  Remark 15.

The identifying expression for $\alpha(x)$ is simply the identifying
expression for $\psi(x)$ scaled by the identifying expression for
$P(M(1)=1\mid Y(1)=1,x)$. This result is highly intuitive: under monotonicity,
an indirect effect can only occur on the stratum where $M(1)=1$: if $M(1)=0$,
an effect can only have been induced via the direct $A\to Y$ pathway.
Therefore, the total indirect probability of causation is simply a scaled
version of the indirect probability of causation. The total direct probability
of causation is then simply the probability of causation minus the total
indirect probability of causation.

######  Remark 16.

When we do not observe $M$, we cannot identify these expressions given the
observed data $(X,A,Y)$. On the other hand, once we observe $M$, we may able
to condition on this event, and so, as argued in the text, the expressions
$\alpha(x)$ and $\beta(x)$ have no obvious practical relevance.

######  Remark 17.

We can also define terms $\tilde{\alpha}(x)$ and $\tilde{\beta}(x)$ that
condition on the exposure and the observed rather than potential outcomes.
These quantities have the same identifying expression under our assumptions.

We also derived terms in Section 3 that were analogous to the total
probability of causation $\delta(x)$, but defined on the stratum where
$M(1)=0$ rather than the stratum where $M(1)=1$. We called this term
$\delta^{\prime}(x)$. We formally consider identification of this term;
however, we first note that by the law of iterated expectations, we can
similarly decompose this term into an analogous indirect and direct
probability of causation, $\psi^{\prime}(x)$ and $\zeta^{\prime}(x)$.

$\displaystyle\delta^{\prime}(x)$ $\displaystyle=P(Y(0)=0\mid
Y(1,M(1))=1,M(1)=0,x)$ $\displaystyle=P(Y(1,M(0))=0,Y(0,M(0))=0\mid
Y(1,M(1))=1,M(1)=0,X=x)$ $\displaystyle+P(Y(1,M(0))=1,Y(0,M(0))=0\mid
Y(1,M(1))=1,M(1)=0,X=x)$ $\displaystyle=\psi^{\prime}(x)+\zeta^{\prime}(x)$

Proposition 5 presents an identification result for these three terms. The
proofs of both propositions 4 and 5 are below.

######  Proposition 5 (Identification of
$\delta^{\prime}(x),\psi^{\prime}(x),\zeta^{\prime}(x)$).

Under assumptions (6)-(11), $\delta^{\prime}(x)$, $\psi^{\prime}(x)$, and
$\zeta^{\prime}(x)$ are identified by the following expressions:

$\displaystyle\zeta^{\prime}(x)$
$\displaystyle=\left(1-\frac{\mu_{10}(x)}{\mu_{11}(x)}\right)$
$\displaystyle\delta^{\prime}(x)$ $\displaystyle=\zeta^{\prime}(x)$
$\displaystyle\psi^{\prime}(x)$ $\displaystyle=0$

###### Proof of Proposition 4.

$\displaystyle\alpha(x)$
$\displaystyle=\sum_{m_{0},m_{1}=0,1}P(Y(1m_{0})=0,Y(0m_{0})=0\mid
M(1)=m_{1},M(0)=m_{0},Y(1m_{1})=1,x)$ $\displaystyle\times
P(M(0)=m_{0},M(1)=m_{1}\mid Y(1)=1,x)$
$\displaystyle=\sum_{m_{0},m_{1}=0,1}P(Y(1m_{0})=0\mid
M(1)=m_{1},M(0)=m_{0},Y(1m_{1})=1,x)$ $\displaystyle\times
P(M(0)=m_{0},M(1)=m_{1}\mid Y(1)=1,x)$
$\displaystyle=\sum_{m_{0},m_{1}=0,1}P(Y(1m_{0})=0\mid
Y(1m_{1})=1,x)P(M(0)=m_{0},M(1)=m_{1}\mid Y(1)=1,x)$
$\displaystyle=P(Y(10)=0\mid Y(11)=1,x)P(M(0)=0,M(1)=1\mid Y(1)=1,x)$
$\displaystyle=P(Y(10)=0\mid Y(11)=1,x)P(M(0)=0\mid
M(1)=1,Y(11)=1,x)P(M(1)=1\mid Y(1)=1,x)$
$\displaystyle=\underbrace{P(Y(10)=0\mid Y(11)=1,x)P(M(0)=0\mid
M(1)=1,x)}_{T_{1}}\underbrace{P(M(1)=1\mid Y(1)=1,x)}_{T_{2}}$

where the first equality follows by iterating expectations over the
conditional distributions of $M(1)$ and $M(0)$ and consistency, the second by
monotonicity, and the third by cross-world ignorability. The fourth equality
follows from the fact that when $m_{1}=m_{0}$, $P(Y(1m_{0})=0\mid
Y(1m_{1})=1)=0$, and by monotonicity, $P(M(1)=0,M(0)=1)=0$. The fifth equality
follows by definition, and the sixth by cross-world ignorability. The term
$T_{1}$ has previously been identified in the proof of identification of
$\psi(x)$; the final results holds by by applying A-YM ignorability and
consistency to $T_{2}$ (noting that we have used positivity assumptions
throughout). Identification of $\beta(x)$ follows from the fact that
$\alpha(x)+\beta(x)=\tau(x)$ and applying Proposition 1 to identify $\tau(x)$.
∎

###### Proof of Proposition 5.

$\displaystyle\psi^{\prime}(x)$ $\displaystyle=P(Y(1,M(0))=0,Y(0,M(0))=0\mid
Y(1,M(1))=1,M(1)=0,X=x)$ $\displaystyle=P(Y(1,M(0))=0,Y(0,M(0))=0\mid
Y(10)=1,M(1)=0,X=x)$ $\displaystyle=P(Y(10)=0,Y(00)=0\mid Y(10)=1,M(1)=0,X=x)$
$\displaystyle=0$

where the first equality follows by definition, the second by consistency, the
third by monotonicity ($M(1)=0\implies M(0)=0$) and consistency, and the final
equality by the fact that $P(Y(10)=0\mid Y(10)=1)=0$. Next, we consider
$\zeta^{\prime}(x)$:

$\displaystyle\zeta^{\prime}(x)$ $\displaystyle=P(Y(1,M(0))=1,Y(0,M(0))=0\mid
Y(1,M(1))=1,M(1)=0,X=x)$ $\displaystyle=P(Y(10)=1,Y(00)=0\mid
Y(10)=1,M(1)=0,X=x)$ $\displaystyle=P(Y(00)=0\mid Y(10)=1,M(1)=0,X=x)$
$\displaystyle=P(Y(00)=0\mid Y(10)=1,X=x)$
$\displaystyle=\left(1-\frac{\mu_{00}(x)}{\mu_{10}(x)}\right)$

where the first equality follows by definition, the second by consistency and
monotonicity, the third by the fact that $P(Y(10)=1\mid Y(10)=1,M(1)=0,x)=1$,
the fourth by cross-world ignorability, and the final follows from the proof
of Theorem 1. Since $\delta^{\prime}(x)=\zeta^{\prime}(x)+\psi^{\prime}(x)$,
the final result follows. ∎

######  Remark 18.

As noted both in Section 3 and Remark 15, the indirect probability of
causation on this stratum is equal to zero, essentially due to the
monotonicity requirements. As a result, the total probability of causation on
this stratum is equal to the direct probability of causation.

##  Appendix D Simulation specifications

We describe the data generating processes for the simulation studies. First,
for each simulation we make $n=1,000$ i.i.d. draws of a single covariate
$X\sim\mathcal{U}(0,1)$. Next, we generate the propensity scores, potential
mediators, and potential outcomes from the following functions:

$\displaystyle\pi(X)=\text{expit}(-1-0.5X)$
$\displaystyle\gamma_{0}(X)=0.25-0.1X-0.1X^{2}$
$\displaystyle\tilde{\gamma}_{1}(X)=0.65-0.1X-0.2X^{2}$
$\displaystyle\mu_{00}(X)=0.5-0.1X-0.2X^{2}$
$\displaystyle\mu_{01}(X)=0.3-0.1X-0.1X^{2}$
$\displaystyle\tilde{\mu}_{10}(X)=0.6-0.1X-0.1X^{2}$
$\displaystyle\tilde{\mu}_{11}(X)=0.4-0.1X-0.2X^{2}$

We then draw the treatment, potential mediators, and potential outcomes as
follows:

$\displaystyle A\mid X$ $\displaystyle\sim\text{Bern}(\pi(X))$ $\displaystyle
M(0)\mid X$ $\displaystyle\sim\text{Bern}(\gamma_{0}(X))$ $\displaystyle
M(1)\mid X,M(0)=0$ $\displaystyle\sim\text{Bern}(\tilde{\gamma}_{1}(X))$
$\displaystyle Y(00)\mid X$ $\displaystyle\sim\text{Bern}(\mu_{00}(X))$
$\displaystyle Y(01)\mid X$ $\displaystyle\sim\text{Bern}(\mu_{01}(X))$
$\displaystyle Y(10)\mid X,Y(00)=0$
$\displaystyle\sim\text{Bern}(\tilde{\mu}_{10}(X))$ $\displaystyle Y(11)\mid
X,Y(01)=0,Y(10)=0$ $\displaystyle\sim\text{Bern}(\tilde{\mu}_{11}(X))$

where we enforce the monotonicity constraints so that, for example,
$M(0)=1\implies M(1)=1$, $Y(00)=1\implies Y(10)=1,Y(01)=1,Y(11)=1$, and so
forth. This implies that:

$\displaystyle\gamma_{1}(X)$
$\displaystyle=\tilde{\gamma}_{1}(X)(1-\gamma_{0}(X))+\gamma_{0}(X)$
$\displaystyle\mu_{10}(X)$
$\displaystyle=\tilde{\mu}_{10}(X)(1-\mu_{00}(X))+\mu_{00}(X)$
$\displaystyle\mu_{11}(X)$
$\displaystyle=\tilde{\mu}_{11}(X)((1-\mu_{01}(X))(1-\tilde{\mu}_{10}(X))(1-\mu_{00}(X)))$
$\displaystyle+1-((1-\mu_{01}(X))(1-\tilde{\mu}_{10}(X))(1-\mu_{00}(X)))$

Finally, applying consistency we set:

$\displaystyle M$ $\displaystyle=AM(1)+(1-A)M(0)$ $\displaystyle Y$
$\displaystyle=AY(1)+(1-A)Y(0)$ $\displaystyle Y(1)$
$\displaystyle=Y(11)M(1)+Y(10)(1-M(1))$ $\displaystyle Y(0)$
$\displaystyle=Y(01)M(0)+Y(00)(1-M(0))$ $\displaystyle Y(1,M(0))$
$\displaystyle=Y(11)M(0)+Y(10)(1-M(0))$ $\displaystyle Y(0,M(1))$
$\displaystyle=Y(01)M(1)+Y(00)(1-M(1))$