Monotone Missing Data: A Blessing and a Curse
Authors: Santtu Tikka
Juha Karvanen
Venue: Accepted by TMLR ()

Abstract: Monotone missingness is commonly encountered in practice when a missing measurement compels another measurement to be missing. Because of the simpler missing data pattern, monotone missing data is often viewed as beneficial from the perspective of practical data analysis. However, in graphical missing data models, monotonicity has implications for the identifiability of the full law, i.e., the joint distribution of actual variables and response indicators. In the general nonmonotone case, the full law is known to be nonparametrically identifiable if and only if specific graphical structures are not present. We show that while monotonicity may enable the identification of the full law despite some of these structures, it also prevents the identification in certain cases that are identifiable without monotonicity. The results emphasize the importance of proper treatment of monotone missingness in the analysis of incomplete data.

Published in Transactions on Machine Learning Research (11/2025)
Monotone Missing Data: A Blessing and a Curse
Santtu Tikka
santtu.likaa
Deportment of Mothematics and Statistics University of Jyvaskyla, Finland
Juha Karvanen
juha. kareanen@jyu.j
Department of Mathematics and Statistics University of Jyvaskyla, Finland
Reviewed on OpenReview: https: // openreview. net/ pormforum? id- kVthdlAVks
Abstract
Monotone missingness is commonly encountered in practice when a missing measurement compels another measurement to be missing. Because of the simpler missing data pattern, monotone missing data often viewed as beneficial from the perspective of practical data analysis. However, in graphical missing data models, monotonicity has implications for the identifiability of the full law, ie., the joint distribution of actual variables and response indicators. In the general nommonotone case, the full law is known to be nonparametrically identifiable if and only specific graphical structures are not present. We show that while monotonicity may enable the identification of the full law despite some these structures, also prevents the identification in certain cases that are identifiable without monotonicity. The results emphasize the importance of proper treatment of monotone missingness the analysis of incomplete data.
1 Introduction
Missing data is ubiquitous across all fields of scientific study and it has the potential to severely impact the results of statistical analyses. In particular, monotone missingness occurs when missing measurement implies that another measurement must also be missing. In longiludinal sludies, monotone missing dala pattern is encountered when dropout is permanent, meaning that subjects do not return to the study after missing one measurement. For instance, in a clinical trial where response variables Y1, Y2 and Y3 are mcasured at three consecutive monthly visits, a subject who drops out after the first visit will have both Y2 and Ya missing. Monotone missing data also arise there are logical constraints or technical restrictions between the measurements. For instance, if the information on the number of children (of a person) is missing, the information on the children's ages will be missing as well.
The missing data literature has been moving from the classical characterization: missing completely at random (MCAR), missing at random (MAR) and missing not at random (MNAR) (Rubin) 1976 Little & Rubin 2002) towards more sophisticated assumptions which are often expressed using graphical models (Daniel
et al. 2012 Mohan et al. 2013: Mohan & Pearl 2014b Karvanen 2015 One the major goals related nonparametric missing data models has been to characterize the set of missing data distributions that are identifiable as functionals of the observed data distribution. Several graphical criteria and identifiability algorithms have been developed for this purpose (e.g.. Tian| 2017 Bhattacharya al. 2020 Mohan & Pearl 2021 Guo et al. 2023). including a sound and complete graphical criterion by Nabi et al. (2020). As result. of these efforts, it is known when the joint distribution variables and response indicators is identifiable in nonparametric MNAR models with nonmonotone missingness.
In contrast, identifiability under monotone missingness is far less studied despite the prevalence of monotone missing data mechanisms in real-world scenarios. Identification strategies for monotone missingness usually consider MCAR MAR settings, and identifiability is achieved by imposing identifying restrictions, such

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available case missing value restriction Molenberghs et al. 1998), neighboring-case missing value restriction (Thijs 2002), and specific donor-based identification
the complete case missing value restriction Little 1993).
restrictions (Chen & Sadinle 2019). Identifying restrictions have also been developed for specific MNAR
settings (c.g.. Kenward et al. 2003 Tang et al. 2003
Inference under monotone missingness is often viewed as a simpler problem than scenarios involving non- monotone missingness. For instance, [Sikov (2018) states that "The advantage of the monotone missingness condition is that it considerably simplifies the analysis of the data." It turns out, that identification under monotone missingness is far from simple and it is not subset of general nonparametric identification under missing data, but distinct problem that only partially overlaps the general problem. The presence of monotonic relationships in the missingness mechanism implies that the probability of some combinations of values of the response indicators is exactly zero. This reduces the number of parameters in the missingness mechanism to be identified, leading to new identification results. On the other hand, monotonicity violates the positivity assumptions that are explicitly or implicitly needed in many identificatinn results with nonmonotone missing data rendering them inapplicable under monotone missingness (Nabi et al. 2024
In this paper, we consider missing data models represented by directed acyelic graphs (DAGs) and scenarios where the assumption of a monotonic relationship between response indicators enables us to identify distri- butions of interest that would otherwise be nonidentifiable, and the converse, where the same assumption renders otherwise identifiable distributions nonidentifiable. To the best of our knowledge, there are no previous graphical criteria or algorithms for determining identifiability or nonidentifiability of the missing data distribution under monotone missing data in nonparametric MNAR settings for missing data DAGs.
The rest of the paper is organized as follows. Scction introduces the notation and the relevant definitions. Section 3 considers missing data models and monotone missingness. Section | discusses identifiability in missing data models and the applicability of previous identifiability results for nonmonotone missingness under monotone missingness. Sections 5 and G present new results for identifiability and nonidentifiability under monotone missingness, respectively. Section concludes the paper with a discussion.
2 Notations and Definitions
We use capital letters to denote random variables or vertices, and small letters to denote the values or value assignments of random variables. We use bold letters to denote sets and vectors of random variables, vertices, or values.
A directed graph G is a pair (V, E) where V is the vertex set and E is the set of directed edges (i.e., pairs (V, V2), Vi * Vg, Vi, Vj E V). An edge from Vú to V, is also denoted by V, + Vj. A directed graph over a set of vertices V is denoted by G(V). If the edge (V,, Vú) exists in G, we say that Vú2 is a parent of VG and Vj is a child of V4. The set of parents of a vertex V; in G is denoted by (V,) and the set of children is denoted by (Vi), respectively. If the graph G is clearly determined by the context, we will simply write (V.) and ch(V,) for clarity.
A directed path is a sequence of distinct edges (E,) such that E, = (V,, Vi+1) and each vertex V may have at most one incoming and one outgoing edge in the sequence. A directed path where the first and the last vertex are the same is called a cycle. If a directed path cxists in g from V; to Vj, then Vú is an ancestor of V, and Vú is a descendant of V, The set of ancestors of vertex VÚ2 in G is denoted by an;(V) and the set of descendants by deg (V,), respectively (omitting again the graph if the context is clearly defined). A directed acyclic graph (DAG) is directed graph that contains no cycles.
We consider DAGs whose vertices represent random variables. For simplicity, we will denote both vertices and the associated random variables using the same symbols. A statistical model of a DAG G(V) is set of distributions that factorize according to the structure of the DAG as follows
PVV)
II p(V] (V)().)), V.Ev
(1)
Whenever a joint distribution compatible with a DAG, the conditional independence constraints of the distribution can be derived from the DAG using the d-separation criterion (Pearl 1995 2009). Tn general,
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we do not make distinction between d-separation statements (such as X is d-separated from Y given Z) from conditional independence statements (such as X is independent of Y given Z). However, we will also consider models that contain deterministic relationships between the variables of interest, meaning that d-separation will not imply conditional independence in all instances for such models. Thus, we will only denote conditional independence constraints as X Y Z and explicitly explain when such statements are not implied by the DAG due to deterministic relationships.
3 Missing Data Models
Missing data models are sets of distributions over a set of random variables V where can be partitioned into four distinct sets: the set of fully observed variables O, the set of partially observed variables X the set of observed proxy variables X, and the set of response indicators R (sometimes referred to as missingness indicators or simply indicators). Each partially observed variable X(1) E (1)) has corresponding observed proxy and & response indicator that have the following deterministic relationship
X(I)) if Rx =1 1, NA
X
(2)
Rx 00 0,
where NA (not available) denotes a missing value. In other words, Rx = means that the true value of the variable (1) was observed, and Rx- = 0 indicates that it is missing. For set Y S x()) of partially observed variables, we denote the corresponding set of response indicators by Ry defined as Ry = Uy ].r.
By 2 we can factorize the joint distribution of V as
(VV) = p(O, X, (1) R) -
where the nondeterministic term (O,X()) R) is the full law which can be further partitioned into two terms: the larget law p(O, x and the missingness mechanism p(R(O, x(1).. Finally, the information available under missing data is represented by the observed data law p(O, X, R). This distribution is the one we actually have access to. We note that the term "missingness mechanism" " is sometimes used to refer to the set of response indicators R or its members instead of their conditional distribution. Similarly, the target law is sometimes simply referred to as the joint distribution (see e.g., |Mohan et al. 2013 Mohan & Pearl 201 2a), but to avoid ambiguity, we use the terms "target law" and "full law" to distinguish between the two joint distributions (1)) and P(O, (O))(()) R), respectively.
Missing data models can be represented by missing dala DAGs (m-DAGs). A DAG G is missing data DAG if it has the following properties: the vertex set of G is OUXUX() UR; for each X E X, pay(X) (x Rx} and (X) 0; and for each Rx E R, de n (O U ( 0. A missing data model associated with a missing data DAG G is the set of joint distributions p(O, X, ( R) that factorize as
X, R)
II Xex
p(X (
II
p(V| (V))))(V)).
(3)
VEOUXOOU
Conditional independence constraints of (O) analogously to DAGs.
(I) R) can be determined via d-separation in m-DAGs
The missingness mechanism of a missing data model may contain monotonic relationships between response indicators, meaning that missingness in one variable always renders another variable to be missing well. We define this property follows.
Definition 1. The missingness mechanism of a missing data model associated with an m-DAG g(V) is locally monotone with respect to (Rz, Rw) if the edge Ri + Rw exists in G and p(R = r|0, x = 0 for all value assignments r to R where rz = O and rw 1. Furthermore, we say that such a value assignment r to R violates monotonicity.
As a shorthand notation, we will denote the assumption that the missingness mechanism locally monotone with respect to (Rz, Rw) Rz 2 Rw- Graphically, we denote the same assumption as Rz 2 RR
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Rz= 2 Rw. Furthermore, we say that missing data model is monotone if the missingness mechanism is locally monolone with respect to at least one pair of response indicators.
We note that the concept of monotone missingness is used to refer to different properties of the missingnoss mechanism in literature. For example, Cui et al. (2117) use the term "monotone missing data mechanism" to describe that the missingness probability is a monotonic function of set of covariates and a response variable. Miao et al. (2016) define a "monotone missing mechanism" in a similar way. In this paper, we will only consider monotonicity in accordance with Definition
4 Identifiability in Missing Data Models
In this section, we revisit some identifiability results for nonmonotone missingness and discuss their applicability under monotone missingness. We begin by defining the notion of identifiability, which is also referred to recoverabiliity in missing data models (Mohan et al. 2013: Mohan & Pearl 2014a
Definition 2. Given missing data model M associated with m-DAG G(V), estimand or probabilistic query Q is said to be identifiable if Q can be expressed in terms of the observed data distribution p(X, O, R), that is, if Q1 = Q2 for every pair of distributions P1,P2 E M such that (X, O, R) = P2 (X. O, R).
The query of interest Q for nonparametric identification in missing data models is typically either the target law p(O, x the full law p(O. effects can also be considered (Shpitser et al. 2015). Because the component Ixxex |[X|xxx Xxxx)(1)) in (3) is deterministic, we can ignore in all identifiabilit bility considerations. For the same reason, we also omit the observed proxy variables and the corresponding deterministic edges related to them from all figures.
or some functional of them, but other quantities such as causal
Typically, definitions of identifiability also include positivity assumption such as p(X - x, O o, R R) >0 Tian 2017], which we omit. from our definition because our goal is to consider identifiability not only under nonmonotone missingness but also under monotone missingness, where some events have zero probability. This mcans that we must consider a different positivity assumptions depending on whether the missingness mechanism is assumed to be locally monotone or not. Thus, for the remainder of the paper, we will assume that p(X = x, O = o, R- = r) > for all value assignments in scenarios where monotonicity is not assumed, and for the scenarios where monotonic relationships between response indicators are present, we assumee that p(X= x, O o, R = r) > for only those value assignments r R that do not violate monotonicity. In simpler terms, we will assume positive probability for events whose probability is not zero due to deterministic relationships between response indicators.
The missingness mechanism p( x is key component for the identification of both the target law and the full law due to the following identities:
(()) (1), R=
p(R = 100, X())
(0,x R 1)
O (1) RR) pPR= = (10, (m) p( (R
where R = 1 means that all response indicators have a value assignment of 1. In other words, identifiability of the target law equivalent, to the identifiability of p(R = 1,0, (()) and identifiability of the full law is oquivalent to the identifiability of the missingness mechanism as the numerator p(O, ( R = 1) always identified from the observed data distribution. This is why methods for identification in missing data models often target the missingness mechanism instead of the full law or target law directly. These identities also hold for locally monotone missingness mechanisms under the corresponding positivity assumption.
There are Iwo important graphical structures related to nonidentifiability in missing data DAGs. The first is a self-censoring edge where a partially observed variable is a parent of its corresponding response indicator, i.c. x()) + Rx (also referred to as "self-masking,, e.g., Mohan et al. 2018). The second is a colluder where a partially observed variable and its response indicators are the parents of a response indicator of another partially observed variable, i.e. (1) - Ry +- Rx. Self-censoring edges and colluders are the only structures
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that render the full law nonparametrically nonidentifiable in missing data DAGs [hhattacharya al. 2020
Nabi et al. 2020). Self-censoring also renders the target law non-identifiable (Mohan et al.
2013). We say
that response indicator Ry is colluded if a colluder structure x + Ry + Rx is present in the m-DAG.
Building on the seminal works of Mohan et al. 2013) and Mohan & Pearl 20144a b). Nabi et al. (2020) provided
a sound and complete criterion for full law identifiability under a general nouparametric setting without
assumptions of monotonicity. Importantly, when no colluders or self-censoring edges are present, this criterion
provides identifying functional that relies the odds ratio (OR) parameterization of the missingness
mechanism (Yun Chen 2006). Denote R_* = R R3, R (RI...., RRRI...,.Rk.Ih and R {Rs+1,... RR).
Now we can write
1 p(R 0
II
(Rs R= II OR (R,, R RA* R 1, 1,0,"(),
(4)
Z k=1
k=
where
[RR Ry = 1, 1, O, 0, PRR = = p(Rx =
OR(RR, R Ry 1, 1,0,
1| R = Rx O, x()) pRR, R 1,, x())
and Z is the normalizing constant. However, this identifying functional is no longer valid if the missingness mechanism is locally monotone as this implies that some of the terms in the denominator of the OR terms will be zero. This will occur even if we restrict our attention to value assignments that do not violate monotonicity, because the term p(R, = R = 1, RR O, x will be zero for some k as R& is conditioned on all other response indicators, and the normalizing term evaluates this probability for all value assignments of R In contrast, an carlier identifiability result by Bhattacharya et al. (2020) based on propensity scores can be applied under monotone missingness, as it only involves terms related to response indicators of the form RRa = pag( R())) [ (R.))R)) in a denominator term, meaning that only nonzero probabilities (assuming positivity) are considered in the denominator of the identifying functional.
There are also methods for target law identification that can be applied under monotone missingness. The propensity score approach of BBhatacharya et al. (2020) is equally valid for target law identification. The MID algorithm by (Shpitser et. al. (015)) also remains applicable under monotone missingness as it attempts to identify p(Rg- pag
(Rx) )) (pag (R)) for each response indicator Rg. A graphical criterion by Mohan et al. 2013) states that if no partially observed variable is an ancestor of its own response indicator, then the target law is identifiable, and its expression is
p(X (1) (,0)=
II VEEXO
p(Vilpag(V)) (x))
The conditional independence restrictions implied by the non-ancestrality assumption also hold monotonicity is assumed.
Outside of graphical missing data models, the impact of functional relationships identification has been studied in the context of causal inference by Chen & Darwiche (2024). In their approach, some variables are assumed functionally dependent (i.e,, deterministic) on their parents and subsequently removed via functional elimination. Unfortunately, this approach does not directly generalize to the setting of full law identifiability and monotone missingness, as monotonicity does not impose a true deterministic relationship between response indicators.
2020 or the While identification strategies such as the propensity score based method of Bhattacharya et al.
MID algorithm Sf et al. (015)) remain valid under monotone missingness, their scope is simultaneously limited by the standard positivity assumption. Intuitively, because p(X = x, O - o. R r) 0 for values assignments that violate positivity, the missingness mechanism has fewer parameters (and consequently the full law), and thus identification easier in a sense when the missingness model is monotone. Importantly, the crucial colluder structure does not always prohibit identification under monotone missingness. In the following sections, we will focus on these special instances where monotonicity makes otherwise nonidentifiable quantities identifiable, and the converse, where monotonicity makes otherwise identifiable quantities nonident nonidentifiable.
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X(
y())
Rx
>
Ry
Figure 1: An example m-DAG where the monotonicity assumption enables the full law to be identified.
5 Identifiability Gained under Monotonicity
A locally monotone missingness mechanism can enable us to identify the full law in scenarios where iden- tifiability could not be achieved otherwise. As an example, we consider the graph of Figure [ where the monotonicity assumption allows us to identify the full law. The full law is not identifiable without this assumption because of the colluder x - Ry + Rx (Bhattacharya et al. 2020).
As practical example of Figure 1 consider an intervention program to improve the physical condition of the participants. Variable (1) stands for the result of physical test at the beginming of the intervention and y()) represents the result of the same test after the intervention. The pre-interventional physical condition x affects both the post-interventional condition y()) and the participant's 's decision Ry to complete the intervention and the final test. The monotonicity Rx 2 Ry occurs because the measurement of X is a prerequisite for participating in the intervention.
We begin by considering the identifiability of the full law under the assumption of monotonicity, meaning that that Rx 2 Ry. This means that
Rr p(((), y(), Rx By =
from which we obtain that
7(x , Rx ,0,, 0, = Rx = Rx = 0)
and consequently
(xx0),(1)) Rx = 0, R 0)
=p(X(1, y(1, Rx 0)
-p(r() x(),, Rx = 0)p(x(1) Rx = 0)p(Rx =0)
=P(Y(') x(),, Rx 1, Ry 1)p(x(1)) (1) Rxx 1)p(Rx 0)
= =P(Y XX, Rx = 1, Ry = 1)p(X Rx 1)p(Rx 0)
where we used the following facts
(1) IL Rx, ()) 1L Rx (x(
(()) 11 (Ry, Rx (1)
Thus (1X(1), (L)), Rx = 0, Ry = 0) is identifiable. The term ((X(1), (1),, Rx = 1, =1) is directly identifiable from the complete cases. It remains to show that ((x (()) Rx = 1, Ry = 0) is also identifiable. To show this, we can write
P(X(), Y(), Rx = 1, = =0)
=p(Y()) x(1), Rx R, 0))(x(1), RRx R, 0)
-p(Y(1) x11) Rx = 1, = =1)p(x(1) 11x = 1, = 0)
-P(Y X, Rx = Ry = 1)p(X, Rx = 1, Ry 0)
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Y(), Rx = 11 =0) is identifiable By combining the above cases, we conclude that the full law p(x(1), Y()), Rx. Ry) is identifiable in the m-DAG of Figure [ under the assumption of monotonicity.
where again used the fact that Y()) 1 {Rx, Ry) |x))) Thus p(x() ((X())(0)
It evident that we cannot leverage monotonicity when self-censoring edges are present for identifiability purposes. However, as the previous example shows, monotonicity can be beneficial when colluders are present. We present several generalizations of the previous example in the form of graphical criteria and show how identifiability be regained when the m-DAG contains colluders. We defer all proofs in Sections 5 and to Appendices A and B respectively. As starting point, the following definition characterizes multiple simultancous colluders affecting the same response indicator.
Definition 3. A pair (CO), Ryy is a marimal colluder in an m-DAG 9 i contains the edges c. + Ry and Rc. + Ry for all Cf E C() and there does not erist zO) E x CO) such that G contains the edges z + Ry and Rz + Ry.
For convenience, among members of Rc. In other words, min Re is 0 if at least one response indicator in Re is 0 and otherwise its value is 1, i.., when all response indicators in Re have the value 1. The following result is immediate.
use the notation min Rc to denote the random variable that takes the smallest value
Theorem 1 Let G be an m-DAG with marimal colluder (CO2, If min Re 2 Ry then p(Ry| pag Ry)) min Ro=o" i.e,, p(Ry| pag(By)) under the value assignment min Rc = 0, is identifiable, and
By 1 pag (Ry )) min Re=0 0,
p Ry 0| pag (Ry) )) min Rc=0
1.
Theorem essentially states that we can always identify the conditional distribution of Ry for those value assignments of the relevant response indicators where the monotonicity is violated. Thus it remains to consider value assignments that do not violate monotonicity, i.e., the case with 11
When Ry has parents that include partially observed variables whose response indicators are not parents of Ry, we can identify the conditional distribution of Ry if Ry is conditionally independent of the respective response indicators of the partially observed variables that are parents of Ry.
Theorem
Let G be an m-DAG with a maximal colluder (CO), Ry). If min Rc 2 Ry and
Ry $IR' R' pag(Ry).
where R' R Rpag (By) \ (Ry) then p(Br) pay( )) Re=1 is identifiabl, and Ro=1 Ro=1= p(Ry (Ry ),R) RC=1,R==
p(Ry pag (Ry))
In other words, R' is set of response indicators for those partially observed variables that are parents of Ry but that are not themselves parents of Ry.
Figure 2a illustrates a scenario considered by Theorem 2 In this m-DAG (X(1), Ry) is a maximal colluder such that Rx 2 Ry, and we have that Ry R' (Ry which in this case translates Ry Rz (1),, z(1, 20,,,xx))))))))))) Rx. Now, we can write
(( 20), Rx = 1)= p(By|x( 2(0, Rx = 1, R2 = 1),
where the right-hand side is identifiable from the observed data distribution.
Another way to identify the conditional distribution of Ry is instead identify the conditional distribution of the other partially observed variables that are parents of Ry given Ry and the other parents of Ry, but whose response indicators are not parents of Ry.
Theorem 3 Let be an m-DAG with maximal colluder (CU),B)), If min Re Ry and
Z 11 Rz Ry U (pag(Rr)) Z)
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X
Y(I)
2
X
Y
2
2
Rx
Ry
Rz
Rx
Ry
Rz
(a)
(b)
X(I)
((1)
z(1)
W
X(I
Y(I)
2(1)
W(I)
Rx
Ry
Rz
Rx
Ry
Rz
Rw
(c)
(d)
X(I)
Y(I)
Z(()
W(I)
Rx
Ry
Rz
Rw
(e)
Figure 2: Example m-DAGs for Theorems 2 where the conditional distribution of Ry is identifiable when the missingness mechanism is locally monotone with respect to (Rx,Ry) but not otherwise.
where z(0 (20)= x()) z E paggRyy) Rz & (ay) then p(Ry| pag(Ry)) m==1 identifiable, and
p(Ry| (Ry)) RC=1
(1) Rzz Ry, pag (Ry) Z( )(Ry, (alpp/aa.aaaayy/)))) (By) (Z())
ERY (()) Rz, Ry, pag (Ry) Z (())p((s pag (Ry(RR)))))))a))a Z
Re=1,
An example use case of Theorem 3 is shown in Figure 2b In contrast to Figure 2a there is now an addit ional edge from Ry to Rz, which means that we can no longer use Theorem 2 to identify the conditional distribution of Ry. Again, (X(1), Ry) is a maximal colluder such that Rx 2 Ry, but now we have that ( Rz Ry U (Pag(Ry) (() which reads as ((1) Rz Rx, Ry, (1) in this instance. By Theorem we can write
p(Br (1)) (()), Rx 1)
p(Z Ry, Rx = 1, ) Rx = 1, 1, E Ry Pl (Z()) Rz = Ry, Rx 1, (p)) Rx X()
Another strategy is to identify the conditional distribution of Ry is analogous to the previous but also employs fully observed proxy variable.
Theorem Let G be m-DAG with marimal colluder (CO), Ry). If min Rc Ry and there exists W c (0) (ay(R))) such that
( 1 Rz WW Ry U (pag(Rr)) (1)
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where z =(Z E x(1) z paggRy)) Rz $ pag(Ry)), then p(Ry| pag(Ry)) no=1 is identifiable, and
p(rr| | pag(Rr)) Re=1
Ewp(Z W, By, påg Ryy) \ za, Rz)p( W, pag (By) Z()
Ew.y p(Z W, Ry, pag (Ry) Z R2)p((W)WWW W. Ry, pag (Ry) \ Z(1)
Rc=1,
Figure 2c depicts a scenario where Theorems and 3 cannot be applied to identify the conditional distribution of Ry, but Theorem applics. The required conditional independence ( Rz WURy Upag )
corresponds to Z(1) Rz W, Rx, Ry, which holds in this m- m-DAG. We can write
pl p( (11, RR 1)
Σ p(( R2 Ry Rx = 1, ,,,,,,,,)xxx) W. Rx =1, Ry),
W
and we have that
(( z(1),
R Rxx =
Ewp(Z() R2 Ry, Rx 1, ()x(O) W, Rx R,
Ryy, W p( (2(20) = 1, Rl, Rx = 1, w)p W, = Ry)
If a fully observed proxy variable required by Theorem f does not exist, it may stiill be possible to use a partially observed variable instead. However, in this case we must also be able to identify the conditional distribution of this partially observed variable, thus we employ another assumption of conditional independence, as outlined by the next theorem.
Theorem 5
Let G be m-DAG with maximal colluder (C(), Ry). If min Rc 2 Ry and there erists pag(By)) such that
W(1) C (X(T)
(T)
2(2) I1 Rz U (Rw (Py) W()) U Ry U (pag \ 2()), and
w(1) Rw \ (Ry) (Ry)
U (pas )))
where Z = (Z(1) E (()) z(1) pag Ry), ) Rz & (ay(BB)), )), then (Ry pag (x))) RC=1 is identifiable, and
Ewar Q
p(Ry pag Ry )) RC=1
Σ Ewo, Q
R2=
where
Q (1 ((I) Rz, Rw, Ry, pag ( Rxy) x p(((()) Rw, Ry, pag (Ry) (z U Rw))p(R (aappp(yy,aa(((0pp (Ry) ())
(Z()) U Rw))
As the conditions of Theorem 5 are rather complicated, we will illustrate the theorem using two examples where W(1) is singleton (WT)). The first example considers the case where Rw E pay(Ry) and the corresponding m-DAG is shown in Figure 2d In this scenario, the first required conditional independence is 2 1 Rz W(2), X Rx, Rw, Ry which holds in the m-DAG, and the second condition is trivially satisfied. This allows us to write
p(Z (1), Ry, Rw 1, =1)
Σ ((Z()) Ry, R2 1, way
1, Rw 1, Rx =1,((), ((())(Ry,WW)) Rx 1, R, =1),
and we have that
(1) (1)), Rw 1, Rx = 1)
Σ (1) Ry = 1, Rw = Rx =1 |,
W())
Σ P(Z() Ry, R, Rx =1, X(), ( ((), X0),x =1 Rw =1) W(,
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Intuitively, Theorems 4 and
operate almost identically when Rw C (aagRyy) because the partially observed variables W essentially act as an observed proxy in Theorem in such cases. In the second example sccnario depicted in Figure 2e have that Rw & paa(Ry)) Now, the required conditional independence restrictions are
z(4) L RRz // ,1), Ry, w LL Rw (x Rx,
which hold in the m-DAG. We can write
p( x(1, Ry, Rx 1)
Σ p(Z(1) Ry, Rr, R.w 1, Rx 1,x(),(())
wa
x xP(W(1) ())) ,Rx ,Rx Rw 1)p(By, x x Rx 1)
and thus obtain p p(2r)X()) 2 Rw Rx 1) as in the first example.
It is important to keep in mind the functional relationships between response indicators when considering d-soparation statements under monotone missing data. For example, in Figure 3a it is the case that y()) Ry Rx if there no monotonic relationship between Rx and Ry irrespective of the values of Rx and Ry. However, if the monotonic relationship is present, the conditional independence only holds only if Rx- = because only then is Ry true random variable. We note that in Theorems 25 we only use the conditional independence statements implied by d-separation under the value assignment of min Rc 1, thus avoiding false implications of conditional independence due to the functional relationships arising from monotone missingness. However, if there are other monotonic relationships present in the missing data model than those related to the colluder structure, then it may be the case that the required conditional independence properties no longer hold even implied by d-separation due to the functional ronshiss induced by monotonicity. One might also consider D-scparation (Gegger et al., 1990 that takes into account deterministic variables. However, D-separation may also lead to false conclusions about conditional independence under monotonicity, monotonicity only enforces determinism for subset of value assignments of the response indicators.
As a tool for full law ifeetiii,ation, Theorems 5 should be applied to identify the conditional distributions of response indicators that are colluded. If all conditional distributions of colluded response indicators can be identified in this way, we can attempt to identify the remaining conditional distributions of response indicators using existing methods (c.g.. Bhattacharya al.) 2020
6 Identifiability Lost under Monotonicity
Monotonicity is not always beneficial for identification tasks in missing data models. As example, we consider the m-DAG of Figure 3a Without assumptions of monotonicity, the full law is identifiable as the m-DAG does not contain sell-censoring edges or colluders. The identifying formula can be derived as follows:
p(X() y(), y(,,,,,,) Rx,Ry) Ry)
- Rx Ry ( Ry )p( Ry)
=P(X(")(X(") Rx 1, Ry (1) Rx Ry 1)p((xxxXX))) Ry)
=P(XIY, Rx 11 -1) Rxx By 1)p( Rr)
(5)
where we used the facts that X(I) 1 {Rx, Ry YO)) and Y(I) IL Ry Rx. However, if monotonicity is assumed, i.e. Rx 2 Ry, the response indicators Rx and Ry become functionally dependent, and the conditional independence y() 1 &Ry| Rx that is critical for obtaining the identifying functional no longer holds. The nonidentifiability construction provided in Appendix [[ that neither the full law nor even the marginal distributions p(x()) and p(Y are identifiable under the monotonicity constraint. This example has been considered previously by e.g., Mohan 2022
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X Y()
x
X2
Rx
Ry
Rx.
Rx2
> Rxs
> Rxx
(a)
(b)
Figure 3: Example m-DAGs where the monotonicity assumption (Rx 2 Ry in (a) and Rx, 2 Rx, 2 Rx,
Rx, in (b)) renders the full law nonidentifibll.
As practical example of Figure 3a consider an cpidemiological study on hobbies and cognitive abilities in elderly people. At the first stage of the study, the participants are asked to fill out a questionnaire on their hobbies and activities during the past ten years (variable X(").. Those who returned the questionnaire are then invited to participate in measurements of cognitive abilities (variable y Restricting these measurements those who returned the questionnaire implies the monotonic relationship Rx 2 Ry. is reasonable assume that past hobbies and activities have effect on the current cognitive abilities (edge x + (()), Cognitive abilities may affect the ability and motivation to fill out the questionnaire (edge y() + Rx) but past hobbies and activitics do not have direct effect on the filling of the questionnaire (the absence of self-censoring for ) Neither past hobbies and activities nor cognitive abilities have a direct effect on whether cognitive abilities are mcasured or not because this part of the study is not self-administratedd (absence of edges ( + Ry and (() + Ry).
The monotonicity assumption Rx Ry means that the cognitive abilities can be measured only from those individuals who returned the questionnaire. This breaks identifiability because the observed data do not provide information the relation of cognitive abilities (1) and the decision to return the questionnaire Rx. Regarding this means that we are not able to identify ((Y Rx = 0, Ry = 1) from p(X, Y, Rx Ry).
It is well known that self-censoring odges, i.c., odges of the form ( + Rx render the corresponding marginal distribution p(X(1) nonidentifiable (Mohan et al. 2013 Intuitively, monotonicity can induce analogous structures which we call self-censoring paths in an m-DAG due to the functional dependency between the response indicators, i.e., a path from a partially observed variable to its own response indicator via other response indicators, all of which have monotonic relationship. An example of a self-censoring path is presented in Figure 3b A self-censoring path renders the marginal distribution of the corresponding partially observed variable nonidentifiable under monotonicity (and consequently the target law and the full law). This notion is formalized by the following theorem.
Theorem 6. Let G be an m-DAG that contains the edge x(1) - Rx, and the edges Rxy-1 - Rx, for all 2.....k l (a self-censoring path). Tf Rx 2 Rx, for all j = 2...., k, then p( ((x()) is not identifiable.
The following corollary is immediate.
If m-DAG G contains self- relf-censoring path where the response indicators have monotonic relationship, then neither the full law nor the target law is identifiabl.
Corollary 1.
In addition, if there is true dependency between the partially observed variables whose response indicators are part of the self-censoring path, it may not be possible to identify even the marginal distributions of these variables, as demonstrated in Appendix
7 Discussion
We considered monotone missing data and its implications on nonparametric full law identifiability from a graphical modeling perspective. Specifically, we showed that the colluder structure is not always a detriment to identification under missing data, and conversely, how the self-censoring path structure emerges
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barrier to identification, analogously self-censoring. These findings emphasize that monotone missing data must be properly accounted for both in identification and est estimation, and not merely dismissed as special instance of missing data.
In the present work, we focused on missing data models associated with DAGs. In other words, we assumed that. no hidden variables are present. Missing data. models with hidden variables are often represented via acyelic directed mixed graphs (ADMG) where bidirected edges are used to denote the effects of hidden confounders (Richardson et al., 2023). We note however, that our results directly generalize to such models because Theorems 5 only make use of d-separation properties of m- m- DAGs to derive conditional independence properties, which can be accomplished analogously via m-separation in missing data ADMGs (under value assignments of response indicators where monotonicity is not violated). Similarly, the construction used to prove Theorem G is also valid for missing data ADMGs.
The presented work leaves some avenues for potential future research. It may be possible to devise an identifiability algorithm a graphical criterion for monotone missingness similar those for nommonotone missingness. However, this approach would face several challenges. First, the deterministic relationships implied by locally monotone missingness mechanism make it impractical to leverage d-separation for conditional independence constraints. Furthermore, the deterministic relationships are dependent on the particular value assignment. to the response indicators. Second, while the result. by Nabi et al. (2020) provides a complete characterization of full law identifiability under nonmonotone missingness, the result relies on the OR factorization in (4) meaning that it does not directly translate to monotone settings. Similarly, there is currently no complete characterization of target law identifiability even under nonmonotone missingness.
The question remains whether the self-censoring path is the only new structure under monotone missing data that renders the full law (and the larget law) nonparametrically nonidentifiable. We hypothesize that this is the case. It also seems that monotonicity is only beneficial for full law identification and not for target law identification. Intuitively, this would seem to be true as identification of the target law does not necossitate identification of the entire missingness mechanism, but only the fully observed portion where all response indicators have a value assignment of one, and monotonicity does not have an impact on positivity.
Acknowledgement
This work was supported by the Research Council of Finland under grant number 368935.
References
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A Proofs for Section (Identifiability Gained under Monotonicity)
Theorem 1 Let g be an m-DAG with a morimal colluder (C),Ry). If min Rc 2 Ry then pl Ry| pag RR)) min Re=o iee, pl Bay (nn(Rr)) under the value assignment min Rc = 0, identifiable, and
pfy pag(Ry )) min Re=0 O, pfy 0| pag (fr )) min Re=0 1.
Proof. The claim follows directly from Definition as the missingness mechanism is locally monotone with respect to each pair (Re, Ry), C()) E C
Theorem Let be an m-DAG with a marimal colluder (C Ry). If min Re Ry and
Ry 1 R' (ay(Ry);
where R' pag (By)nxx)) \ (agg(Ry), then pl (By| pag( (Ry )) Rt=1 identifiable, and p(Ry pag (Ry (Ry)(R))) RC=1 p(Ry pag(Ry) R') Rc 11--1
Proof. If n ( = 0, the claim is immediate because pag( is fully observed. Suppose now that ( ( * 0. Then, by the assumed conditional independence, we have that
p(Ry) (ay))))) Re=1 p(Ry pag) Re=1,/==1'
where the right-han hand side is a function of the observed data distribution.
Theorem
3 Let g be an m DAG with mazimal colluder (C()), Ry). If min Rc 2 Ry and
Z LL Rz Ry U (pag(IBr) Z)
where z(1 (2()) x(1) z E (ag (Ry) Rz & pag }} then p(Ry| pag(Ry)) nc=1 is identifiable,
and
p(Rr (agg(Rr))) RC=1
(1) Riz, Ry, pag (Ry) (Z))) (( (agpp(ayyaaaaaaaaaaa (By) ((1)
Exy ny (1 Rz, Ry, pag (Ry) Z ( p(Ry, pag (Ry) Z()) Re=1,
Proof. If (1) = 0, the claim follows by observing that p(Ry pag((Ry)) R==1 is directly identifiable from the observed data distribution, as the rinht-hand side may only contain observed proxies, response indicators, and partially observed variables whose response indicators are also parents of Ry.
Suppose now that (1) * 0. Then, by the assumed conditional independence, have that
(1) ( Ry, pay
Ryy) (ay)
(())
Re=1
-z( Ry, Pa (yy) (()) Ry, PaR, z())
Re=1
-x((11 Rzz Ry, pag (Ry) Z p( Ry, pag Ryy) (Rx) (
Be=l,Bzan
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where both terms the last line identifiable from the observed data distribution. Thus p(Z(), Ry. pag (Ry) ( Re=1 is identifiable, and we can write
p(Ry| pag(Ry Rc=1
pl Ry pag (Ry))
= Eny By pl Ry, pag Ry))
Mc=1
p(Z(), Ry, pag Ryy )
En, p(((1,, Ry, p Ryy) \ Z
Rc=1
((Z Rz, Ry, pag (Ry) Z )p(yy (Ry (Ry) ((1)
ERy By (( Rz, Ry, pag RRy) Z ( pag (Ry) 2
R.
and the claim follows. Note that despite the monotonicity min Rc 2 Ry: the denominator in the identifying functional is always positive because of the value assignment Rc = 1.
Theorem 4 Let G be m-DAG with maximal colluder (C(1) Ry). If min Rc 2 Ry and there erists W W (0 pag (Ry)) such that
Z Rz W By U (pag(Rr)) z
where ((1 =2 E x()) 2() E (ag( Rz & pag(Hyy),, then p(Rr I pag(Rr)) Rc=1 is identifiable, and
p(Br) (ag(Br))) RC=1
Ewp(Z() W, Ry, pag (Ry) Z(), Rr)pp((pW W, Ry, pag (Ry) 2
= Ew.Ry O(Z W, Ry, pag (Ry) Z R2)p W. Ry, (Ry) (Ry) ((()
Rc Rg=1,8z=1
Proof. By the assumed conditional independence, have that
((Z Ry, pag (Ry) (
Ro=1
Σ
p (Z (1, W, Pay (By) ())
(ay(Ba)
W
Re=1
Σ (1) W. Ry, pag z (1 (W, Ry, (By) ( 22
Rc=1
Σ (2 W, By pag (Rr) 2() (Z)p(W, pag (By) ())
W
Re=1,
thus the claim follows by using the same reasoning as in the proof of Theorem 3
Let G be an m- m-DAG with a maximal colluder (C(1) Ry). If min Re 2 Ry and there erists W c (X (Ry( such that
Theorem
( IL Rz U (Rw (Ba)) w() U Ry U (pag (22),, and
w(1) Rw \ (Pg(Ry) (Ry) U (pag(Riv) z()).
where Z = (2 (() ( pag (Ry). Rz & pag(Ry)), }), then p(Ry pag(Ry)) nc=1 is identifiable, and
Ewas Q
p(Rr| (ar(( Re=l Σ Eway Q
Ro
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where
Q -((()) W())) Rz, Rw, By, pag (By) (Z() U Rw))
x xp(((") Rw, Ry, Ray (Ry) (Z U (w))p(Ry,, BW)))((y 2
Proof. Let C( (W x | w ww., Rw pag(Ry)). By the assumed conditional
independence properties, have that
p(Z Hy, (oyypag())) )
Rb=1
Σ (( w Ry, pag (Ry) | ) W
RD=1
Σ (2 w()) Ry, pag (rr) 2 )p( (w()) Rr, (ag((r)) 2
W()
x p( Ry, pag (Ry) \ 2
RD
Σ (Z (1) W(2), Rz, Rw, Ry, (Rr)) \ (Z u Pww)
x Pl (()) Rw: Ry, pag Ry) (z (Z Rw)))((),,)).m Pay (Ry Z(1)
Re=1,8y=1,
thus the claim follows by using the same reasoning again in the proof of Theorem
B Proofs for Section (Identifiability Lost under Monotonicity)
Thcorem Let G be on m-DAG that contains the edge x( Rx, and the edges Rx Rx, for all 22, ...... lo (a self-censoring path). Rx Rx, for all j 2, ...... k, then p( p(x(1)) is not identifiable.
Proof. Without loss of generality, we may assume that O 0. If k: 1 the claim immediate due to the self- censoring edge X₁ + Rx,- Suppose now that k 1. We construct two models (parametrizations) such that the obscrved data laws agree but the full laws disagroe between the models. We denote = (x(") Rx,...., Rx, and let W (x() UR) \ Z. We let the effects of variables in W on those Z be null effects and vice versa for all value assignments. Further, we assume that following probability is constant and the same in both models
II p(V/ pag(VV)) V.W
Next, we define the parametrizations along the self-censoring path in both models (we assume that x is binary):
P. Pı (X 01 pag ()() =
8. (X (1) 0l pag (x( =1
$1,0,1 Pı (Rx, 0 pag Rxx) )) 1)x(!0= =7,
81,0,2 (2 0l pag (Rx,) )) x 1 1-7
11,11 = (Rx, = 0l pag Rx xx = -
$1,12 = (2 = 0 pag (Rx, xx =%,
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where y can be chosen freely long 7 * 0.5, and
$2.0, = pı (Rxx, = 0l pag ( Rx, = 1,
$2, P2(Rx; 0l pag (xx,)) nx,=1 0.5,
:
83.03 = pi(Rx, = 0) (ex(Bx))) Rx 1,
By.. = P2(Rxx = 0 pag Rx, Rxc-1=1 = 0.5,
for 1,2. First, must ensure that the observed data laws agrec. To simplify the exposition, define the following functions for the models:
f ,X,....;Txx)
= II PV(V (V)(V V.EZ
x 1) Rxx =r Rxy
1-10 (1) Pos & (1 30,) 8t (1
II
k
kax, R
rxy
(1 BRAxx (.
10c 1
j=2
We can simplify these functions by writing them separalely for the two models
72
zy (1) 0,rx, =
(1 (1-99 7)2
2 (1) 1, O,
f1(0)) (1) Px.. 20(r2....7k)
7(1 7))
rx, = 1, 2 TX21125 --
0
otherwise
?2
Fk (1) 1, = 0,
(1 7)2
(1) 0, = O,
(1) f2(0k) ,xx,).., 2c(r2....7))
7(1 -1 7)
1,TX, 2 2x21111 = -....,,
otherwise
where e(r2, e(r,...., nk) Ej TX; Denote fr = rc) fe. We can now express the full law as follows for both models
=x, = c(rx, - xx Waf (o TX.)...
We have the following system of equations for the observed data law, where the following functions must. have the same value for i 1 and i for all value assignments X =x
(x(x)
=
Σ
Px = -X RT).
(z),1) ((0,0), (x st. (x )E
if if, NA if X, NA
Because the factors and c(.) agrec between the models, it suffices to focus on the functions f.. If Xx * NA, then it must. be that X, + NA also, and it is easy to see that the value of the functions hs agree as = f+)=))()=
again (7))=((==)) (or equal to zero) for all value assignments. When X, = NA and X, * NA, we have 7) (or equal to zero) for all value assignments, thus all terms the sum agree between the models. When X NA and X₃ NA the terms in the sum can be split into pairs such that each pair has term with the factor 72 and term with the factor - = (corresponding to eit X 0
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X5 =1 depending the model). Thus the expression of h, is symmetric with respect to y and (1 (1-9) -)).
Thus h1 (x) = h2(x) for all value assignments.
Next, we consider the marginal distribution of x in both models
(x( mx)
E Σ px
Σ
Σ 2 of
aff (x :x.... ,x.....
x(
We may again ignore the factors and c(-) and focus on the following expression
Σ Ei .x.. "... ,x.. x
Without loss of generality, we assume that the number of partially observed variables is l and that all other
partially observed variables are also binary. When at least of Rx..-... Rx, is not zero, the functions f always output the value 0 for value assignments that violate monotonicity and the value >(1 7) otherwise.
When all of ,x,...., Rxx are equal to O, then fz = 0) = 7² and fz(=1"" = 0, = (1 - 7)2. There
are 2h-1 ways lo choose the values of x x and h value assignments R. that do not violate monotonicity where at least partially observed variable is not missing. Thus we have that
Σ
Et .x.
x
2k-1
Σ
( .'......, Txx)
Nx
22 + - -7)
= 1, (1)
= 0
k
(1 - 7)- + ky(1 7) 1, B
(1) 1,
2
7)- k7(1 -7) 2 (1) O,
+ -7)
2, T
(1) 1.
Because we can freely choose the value of y (as long as 7 * 0), there are infinitely many models that agree on the observed data law, but disagree on the marginal distribution of (1)
C Nonidentifiability construction for a bivariate self-censoring path
We consider a scenario with locally monotone missingness mechanism with respect to (Rx, Ry) and a self-censoring path where there also a true dependency between ( and y()) Thus we assume that a d and c of in the parametrization below.
X
(1))
Rx
Ry
YY
X
Y
XOO
YOU P( (
fX
p
(OU)
Rx
RY
play Rx)
X
b
1
1 1-
U
d
1
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Rx
Ry
X
YOU
P(XU), ytu, R.x. Rly
M(,, V.X,YByy Rx, Ry)
abef
1
abei p11
a11 b)ef
0
a(1 b)ef = P10
(1 ofdof Pot
(1 - a1 d)ef
0
(1 - a0 d)ef
abo(1
a () Ø
albe (1 - b1
PONA
NA
(1 a)do(1
(1-a)]de+11- (1- PINA
(1 a1 d)e(1
0
ab(1 - e)
(1 o)[(1 d)(1 -) c)
6)
NA
+ +[(11 - 6) - e) -(1-))
(1 a)
a)(1 di1
PNANA
The probabilities pı1, Pio, Poi, poo, PONA + PINA, and PNANA are observed and they sum to 1. Note that PoNA and PINA cannot be specified separately because they are functionally dependent on other probabilities. We use shortcut notations "n P11 and " Pio/Puo- When paramcter is fixed, the other parameters can be solved from the observed probabilities as follows:
70 - I-a
70 - 71
P11 7
(1 a>oo'1 071
Tu 1-a
d
7/ - 7
P10(70 71)
070 (1 (1-a))m a0'''
P11 P1o + Po1 Poo
f P11 + P1o + Pui + Pno + PINA + PONA
It remains to check that the solutions for b, c, d, e, f are between 0 and 1. This directly holds for parameter f that does not depend on parameter a. The constraints b < 1 and 0 < d < induce necessary condition min(20, 71) a/(1 a) < max( 71).
In order to present specific construction for nonidentifiability, consider the observed probabilities P11 1/5, P1o = 1/10, Po1 = 1/10, Pon = 1/5, PINA = 1 1/20 and PONA = 1/10. We obtain 71 = 2 and ?o = 1/2, which implies 1/3 < a < 2/3. Investigation of the constraints < c < and 0 < e < reduces this interval to 9/20 < a < 11/20. This means that we may define two models by picking two different values of a from this interval and then solve the values of the other parameters. The two models will agree on all observed distributions but differ by unobserved distributions. As illustration consider the models Mi with 7/15 R 0.47 and M2 with 8/15 Rs 0.53.
Parameter
M1
M2
a
7/15
8/15
12/21
3/4
c
15/16
5/8
1/4
9/21
5/8
15/16
f
4/5
4/5
The table below shows the probabilities of the full law under Mi and M2. It can be seen that the distributions differ when Rx = 0 and Ry = and are otherwise the same. This means the full laws differ between the models Mj and M2 while the observed data law are the same. By summing the probabilities over Rx and Ry can be confirmed that also the target laws and the marginal distributions P(X()) and P(Y()) differ between the models Mi and M2-
20

---

Published in Transactions on Machine Learning Research (11/2025)
Rx
Ry
X
yuu
p(X!), yuu, Rx, Ry)
Mi
abef
al1 bjef
(1 def
(] d)of
aba(1 - f)
b)e(1
(1 a dd f
a) d)e(1 0
0
0
0
ab(1 c)
a(1 6) c)
(1 a)d(1 c)
130
a1 d1
21


Reviews:
[{"review_id": "xmntNbLede", "reviewer": "Reviewer_RXd2", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "I find the contributions of this work to be both novel and valuable. However, I have a few comments aimed primarily at improving the writing for greater clarity.\n\nAbstract: The abstract would benefit from being framed in more general terms. Avoid using specific technical terminology such as “colluders,” which may not be familiar to a broad audience. Instead, emphasize the central idea of monotonicity—briefly explain what it means, why it is important, and how it motivates the proposed work. A short, intuitive explanation in plain language would make the abstract more engaging and accessible\n\nThe concept of monotonic missingness should be explained more clearly, ideally with a simple illustrative example involving three or four variables. Providing this early in the paper (e.g., on page 1) would help readers grasp the intuition behind the concept before delving into formal definitions.\n\nLocal monotone is mentioned but is there a notion of global monotone as well? May be worth stating somewhere.\n\nSome terminology appears to deviate from established usage in the literature, which may confuse readers familiar with prior frameworks. For instance, the term missingness mechanism is used here to denote a distribution, whereas in Pearl’s framework it refers to the R variables. Similarly, the target law in this work corresponds to the joint distribution in Pearl’s terminology. It would be helpful to clearly state these distinctions to avoid ambiguity.\n\nDefinition 1 could be made more precise. In Rx-->Ry, here X is used as a variable, but previously X also denotes a set of proxies, which may cause confusion. Consider using W or another symbol instead. Explain what \"<\" indicates in the definition. In the last sentence of the definition, specify which assignment the phrase “such a value assignment” refers to.\n\nIn the paragraph following Definition 1, the authors mention two ways of graphical representation. Since only one is used consistently throughout the paper, it may suffice to describe just that one.\n\nAgain about notations, self-censoring is called as self-masking in Pearl's work. See https://www.ijcai.org/proceedings/2018/705 Acknowledging this alternative terminology would align the paper with existing literature.\n\nWhen citing Nabi et al. for the non-identifiability of the full law, please also acknowledge that these results build upon earlier findings by Mohan et al. (2013) and Mohan & Pearl (2014). Specifically, Mohan et al. (2013) established the non-identifiability of the joint distribution under self-masking, and Mohan & Pearl (2014) showed non-identifiability when a path exists from Z to Rz where all intermediate nodes are colliders (later termed colluders).\n\nSome of the cases discussed here have been examined in https://ftp.cs.ucla.edu/pub/stat_ser/mohan-ch34-acm-2021.pdf, although without taking monotonicity into account.\n\nOverall, I find this work to be a valuable contribution and support its acceptance. I do not anticipate the need for further review from my side, unless the authors strongly disagree with any of the comments provided and wish to contest/discuss them.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The primary contribution is the analysis of identification under monotonicity. Interestingly, monotonicity can enable identification of distributions that are otherwise nonidentifiable under general (nonmonotone) missing data. Conversely, monotonicity can destroy identifiability for models that would otherwise be identifiable.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "The requested changes are noted above in the section on strengths and weaknesses. The key point is to ensure that acknowledgement and credit are given where they are due.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "VyvWou2vpv", "reviewer": "Reviewer_31LF", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- Positivity constraints in missing data are understudied and the authors present interesting (and to my knowledge, novel)  identification and non-identification results for monotone missingness\n\nWeaknesses:\n\n- The authors point out that d-separation is not sufficient to determine all independence/dependence relations when there is functional determinism in the graph due to monotonicity. However, the authors do not make sufficiently clear whether D-separation (uppercase D rather than lowercase) is also not suitable for this purpose --- https://ftp.cs.ucla.edu/pub/stat_ser/r116.pdf. If it is suitable, it would be great to present the results in terms of D-separation statements (and perhaps simplify and strengthen the existing results presented in the paper). In particular, it would be interesting to draw connections to this paper --- Identifying Causal Effects Under Functional Dependencies -- Chen and Darwiche (https://www.mdpi.com/1099-4300/26/12/1061). It presents extensions of the do-calculus to account for functional determinism and uses D-separation for this purpose. Their results might apply to monotone missingness as well\n\n- The extent to which the identification results easily generalize to imputation is unclear; this contribution feels ad hoc and underdeveloped in the current version of the paper. In particular, the statements of the form \"The identifiability of the full law in the case of Figure 1 implies that multiple imputation can be used to estimate the target law p(X(1), Y (1))\" are not sufficiently rigorous. Identification alone is not sufficient---an imputation procedure that accounts for the manner in which the model is identified is also important. E.g., Applying MICE, which assume the data are MAR, probably would not directly work for models that are MNAR---if that is the implication being made, this should be supported with theory supporting why this is the case. If the authors mean that a sequential imputation procedure like the one in Karvanen and Tikka (2024) might be applied to certain identified graphs that have a sequential identification structure, this is also not sufficiently clear. In my understanding, that procedure assumed the absence of colluders, so would still need some modification when they are present. Other MNAR imputation procedures for graphical models of missing data, such as [Ren et al (2021)](https://journals.sagepub.com/doi/pdf/10.1177/09622802231188520?casa_token=O1x-eLPYMM8AAAAA:pg_49g_YG87H1Cd66C4ir4SLzupdkFEKxSlj1X64JCFCa6slosApFOZ6P8q1Im860hzgDPtHLl3ivK4) and [Phung et al (2025)](https://www.arxiv.org/pdf/2507.16107), also assume no colluders.\n\nTo summarize the above, I agree with the claim that imputation is possible when the model is fully identified, but the authors do not make it sufficiently rigorous/clear, what the imputation procedure should be. In my opinion, this detracts from the genuinely interesting identification results presented in the paper.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "- The authors propose identification and non-identification results for graphical models of missing data with monotone missing data. These restrictions are relatively understudied in the current graphical modeling literature on missing data. The author's identification results are interesting and, to my knowledge, novel\n- There is a proposed connection between the identifiability result and imputation. This connection, at least in the current presentation of the paper, is much less clear to me and perhaps underdeveloped", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please see the Weakness section above --- I'd recommend making some connections to D-separation and addressing the comments regarding connections to imputation---maybe it's too ambitious to have a completely general procedure, but it would be good to be explicit about the procedure for at least some of the graphs/class of graphs discussed here\n\nMinor typos:\n\n-  Furthermore, we say that a such a value --> remove the first \"a\"\n\n- \"Second, there is no sound and complete identifiability algorithm for nonmonotone missingness which could serve as a starting point, as the result by Nabi et al. (2020) relies on the OR factorization in (4) which does not directly translate to monotone settings\". This sentence is a little unclear. I think you mean to say there is no sound and complete result for nonmonotone missingness with functional dependence?\n\n\nEdit:\n\nAcknowledging that my comments have been sufficiently addressed and updating my review accordingly.", "broader_impact_concerns": "No concerns", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "eqV18VYVYi", "reviewer": "Reviewer_x5Ds", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths\n------------\n- The paper is well written and motivated (though even with that, the paper is still a dry read)\n- Monotone missingness is important and discussing the identifiability possible in this case is important\n- It is quite surprising that some cases are actually not identifiable when monotonicity is present!\n\nWeaknesses\n-------------------\n- The paper is hard to read for somebody who is not familiar with the literature on identification in graphical misisng data models. Unfortunately I do not see a clear fix for this.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors consider identification of monotone missingness mechanism in graphical models. They show that in some graphs identifiability can be gained compared to the non-monotone case, while, somewhat surprisingly, there can also be a loss of identifiability.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- This is just a suggestion, but it might make sense to take the two nice examples used in the beginnings of Section 4 and 5 and to put them in a separate motivation section before Sections 4 and 5.\n- The notation p(X,O,R) > 0 and the surrounding discussion on page 4 is very confusing to me (though again this seems to be something coming out of this specific literature). First, what does \"p(X,O,R) > 0\" mean? (with random variables). Second, it is clear that adding the assumption of monotonicity may induce (extreme) dependence, which simply reduces the support of (X,O,R), but how exactly is this a problem for positivity? My understanding is that this comes later on page 5, where it is explained why an earlier result cannot be used and has to do with the fact that we are interested in R_k=1 specifically. However, this all might be explained better.\n- Related to this in the example at the end of page 10/beginning of page 11, can you explain (ideally in the example of cognitivie abilities), how the monotonicity exactly breaks identifiability?\n- The notation of min R_C seems to be somewhat hidden at the end of page 7. As this seems quite important, it might make sense to put it somewhere more prominently, or at least define it before the result.\n- End of page 6: There is an \"also\" too much", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)