Nested Markov Properties for Acyclic Directed Mixed Graphs

TL;DR

This paper introduces a new framework for understanding conditional independence and equality constraints in acyclic directed mixed graphs (ADMGs), extending DAG models and aiding causal effect identification.

Contribution

It develops a novel approach to characterize ADMG models using fixing operations, unifying conditional independences and equality constraints like the Verma constraint.

Findings

ADMG models can be characterized by new Markov properties and factorizations.

Verma constraints are interpretable as conditional independences in kernel objects.

The fixing operation simplifies the identification of causal effects in hidden variable models.

Abstract

Conditional independence models associated with directed acyclic graphs (DAGs) may be characterized in at least three different ways: via a factorization, the global Markov property (given by the d-separation criterion), and the local Markov property. Marginals of DAG models also imply equality constraints that are not conditional independences; the well-known ``Verma constraint'' is an example. Constraints of this type are used for testing edges, and in a computationally efficient marginalization scheme via variable elimination. We show that equality constraints like the ``Verma constraint'' can be viewed as conditional independences in kernel objects obtained from joint distributions via a fixing operation that generalizes conditioning and marginalization. We use these constraints to define, via ordered local and global Markov properties, and a factorization, a graphical model associated with acyclic directed mixed graphs (ADMGs). We prove that marginal distributions of DAG models lie in this model, and that a set of these constraints given by Tian provides an alternative definition of the model. Finally, we show that the fixing operation used to define the model leads to a particularly simple characterization of identifiable causal effects in hidden variable causal DAG models.

Equations

$$p(x_0,x_1,x_2,x_3,x_4)=p(x_0)p(x_1)p(x_2|x_0,x_1)p(x_3|x_1,x_2)p(x_4|x_0,x_3).$$

$$p^{\ast }(x_0,x_1,x_2,x_4|x_3)=p(x_0)p(x_1)p(x_2|x_0,x_1)p(x_4|x_0,x_3).$$

$$\sum _{x_2}p(x_4|x_1,x_2,x_3)p(x_2|x_1)\text{is a function of only }{x}_3\text{ and }{x}_4;$$

$$q_1(x_1)=p(x_1),q_{24}(x_2,x_4|x_1,x_3)=\frac{p(x_1,x_2,x_3,x_4)}{p(x_1)p(x_3|x_2,x_1)}.$$

$$q_4(x_4|x_3)$$

$$\mathcal{D}(\mathcal{G})\equiv \{D\mid D\text{ is a district in }\mathcal{G}\}$$

$$\sum _{x_V\in {\mathfrak{X}}_V}{q}_V(x_V|x_W)=1,\text{for all }{x}_W\in {\mathfrak{X}}_W.$$

$$q_V(x_A|x_W)\equiv \sum _{x_{V\setminus A}\in {\mathfrak{X}}_{V\setminus A}}{q}_V(x_V|x_W);q_V(x_{V\setminus A}|x_{W\cup A})\equiv \frac{q_V(x_V|x_W)}{q_V(x_A|x_W)}.$$

$$X_{A}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern 3.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{$\textstyle\perp$\hss}\mkern 3.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{$\scriptstyle\perp$\hss}\mkern 3.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern 3.0mu{\scriptscriptstyle\perp}}}X_{B}\mid X_{C}\quad[q_{V}]$$

$$q_V(x_A|x_B,x_C,x_{W\setminus (B\cup C)})=q_V(x_{V\cap (A\cup B\cup C)}|x_W)/q_V(x_{V\cap (B\cup C)}|x_W).$$

$$\displaystyle(X_{A}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern 3.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{$\textstyle\perp$\hss}\mkern 3.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{$\scriptstyle\perp$\hss}\mkern 3.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern 3.0mu{\scriptscriptstyle\perp}}}X_{B}\mid X_{C})\quad$$

$$\displaystyle(X_{A}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern 3.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{$\textstyle\perp$\hss}\mkern 3.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{$\scriptstyle\perp$\hss}\mkern 3.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern 3.0mu{\scriptscriptstyle\perp}}}X_{B}\mid X_{C\cup D})\land(X_{A}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern 3.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{$\textstyle\perp$\hss}\mkern 3.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{$\scriptstyle\perp$\hss}\mkern 3.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern 3.0mu{\scriptscriptstyle\perp}}}X_{D}\mid X_{C})\quad$$

$$q_{V\setminus H}^{\ast }(x_{V\setminus H}|x_H,x_W)=q_{V\setminus H}^{\ast }(x_R,x_T|x_H,x_W)$$

$$q_{V\setminus H}^{\ast }(x_R,x_T|x_H,x_W)$$

$$q_{V\setminus H}^{\ast }(x_R|x_H,x_T,x_W)$$

$$q_{V\setminus H}^{\ast }(x_T|x_H,x_W)$$

$$\displaystyle X_{A}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern 3.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{$\textstyle\perp$\hss}\mkern 3.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{$\scriptstyle\perp$\hss}\mkern 3.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern 3.0mu{\scriptscriptstyle\perp}}}X_{B}\mid X_{C}\quad[q_{V}]\quad\quad\Leftrightarrow\quad\quad X_{A}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern 3.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{$\textstyle\perp$\hss}\mkern 3.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{$\scriptstyle\perp$\hss}\mkern 3.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern 3.0mu{\scriptscriptstyle\perp}}}X_{B}\mid X_{C}\quad\bigl{[}q^{*}_{V\setminus H}\bigr{]}.$$

$$X_{A}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern 3.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{$\textstyle\perp$\hss}\mkern 3.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{$\scriptstyle\perp$\hss}\mkern 3.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern 3.0mu{\scriptscriptstyle\perp}}}X_{B}\mid X_{C}\quad[q_{V}]\quad\quad\Leftrightarrow\quad\quad X_{A}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern 3.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{$\textstyle\perp$\hss}\mkern 3.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{$\scriptstyle\perp$\hss}\mkern 3.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern 3.0mu{\scriptscriptstyle\perp}}}X_{B}\mid X_{C}\quad\bigl{[}q^{*}_{V\setminus H}\bigr{]}.$$

$$E^{\ast }\equiv E\cup \{w\leftrightarrow w^{\prime }\mid w,w^{\prime }\in W\}.$$

$$\displaystyle A\text{ is m-separated from }B\text{ given }C\text{ in }{\mathscr{G}}^{|W}\quad\implies\quad X_{A}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern 3.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{$\textstyle\perp$\hss}\mkern 3.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{$\scriptstyle\perp$\hss}\mkern 3.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern 3.0mu{\scriptscriptstyle\perp}}}X_{B}\mid X_{C}\quad[q_{V}].$$

$$\operatorname{mb}_{{\mathscr{G}}}({t})\equiv\operatorname{pa}_{{\mathscr{G}}}\bigl{(}\operatorname{dis}_{{\mathscr{G}}}(t)\bigr{)}\cup\bigl{(}\operatorname{dis}_{{\mathscr{G}}}(t)\setminus\{t\}\bigr{)}.$$

$$X_{t}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern 3.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{$\textstyle\perp$\hss}\mkern 3.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{$\scriptstyle\perp$\hss}\mkern 3.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern 3.0mu{\scriptscriptstyle\perp}}}X_{(V\cup W)\setminus(\operatorname{mb}_{{\mathscr{G}}}({t})\cup\{t\})}\mid X_{\operatorname{mb}_{{\mathscr{G}}}({t})}\quad[q_{V}].$$

$${\mathcal{P}}_{\mathrm{l}}^{\mathrm{c}}(\mathcal{G},\prec )\equiv$$

$$q_V(x_{V\cap A}|x_W)\text{ obeys the local Markov property for }$$

$$\displaystyle{\mathscr{G}}(V\cap A,W)\text{ at }{\max_{\prec}}(A)\Bigr{\}}.$$

$${\mathrm{mb}}_{\mathcal{G}}(t,A)\equiv {\mathrm{mb}}_{{\mathcal{G}}_A}(t).$$

$$\displaystyle\begin{split}{\mathcal{P}}^{\mathrm{c}}_{\mathrm{f}}({\mathscr{G}})\equiv{}&\biggl{\{}q_{V}\;\left|\;\,\text{for every ancestral set }A,\text{ there exist kernels }f_{D}^{A}(\cdot|\cdot)\right.\\ &\text{s.t. }q_{V}(x_{V\cap A}\,|\,x_{W})=\prod_{D\in{\mathcal{D}}({\mathscr{G}}_{A})}f_{D}^{A}(x_{D}\,|\,x_{\operatorname{pa}_{{\mathscr{G}}}(D)\setminus D})\biggr{\}}.\end{split}$$

$$f_D^A(x_D|x_{\mathrm{pa}(D)\setminus D})=\prod _{d\in D}{q}_V(x_d|x_{T_{\prec }^{(d,D)}}),$$

$$q_V(x_d|x_{T_{\prec }^{(d,D)}})=q_V(x_d|x_{A\cap {\mathrm{pre}}_{\mathcal{G},\prec }(d)},x_W).$$

$$\displaystyle\begin{split}{\mathcal{P}}^{\mathrm{c}}_{\mathrm{f}}({\mathscr{G}})={}&\biggl{\{}q_{V}\;\left|\;\,\text{for every ancestral set }A,\vphantom{f_{A}^{D}}\right.\\ &q_{V}(x_{V\cap A}\,|\,x_{W})=\prod_{D\in{\mathcal{D}}({\mathscr{G}}_{A})}q_{D}(x_{D}\,|\,x_{\operatorname{pa}(D)\setminus D})\biggr{\}},\end{split}$$

$${\mathbb{F}}({{\mathscr{G}}})\equiv\bigl{\{}v\in V\left|\;\operatorname{dis}_{{\mathscr{G}}}(v)\cap\operatorname{de}_{{\mathscr{G}}}(v)=\{v\}\right.\bigr{\}}.$$

$$\displaystyle X_{t}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern 3.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{$\textstyle\perp$\hss}\mkern 3.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{$\scriptstyle\perp$\hss}\mkern 3.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern 3.0mu{\scriptscriptstyle\perp}}}X_{\operatorname{nd}_{{\mathscr{G}}}(t)\setminus\operatorname{mb}_{{\mathscr{G}}}(t)}\,|\,X_{\operatorname{mb}_{{\mathscr{G}}}(t)}\quad[q_{V}]$$

Full text

We show that equality constraints like the “Verma constraint” can be viewed as conditional independences in kernel objects obtained from joint distributions via a fixing operation that generalizes conditioning and marginalization. We use these constraints to define, via ordered local and global Markov properties, and a factorization, a graphical model associated with acyclic directed mixed graphs (ADMGs). We prove that marginal distributions of DAG models lie in this model, and that a set of these constraints given by Tian provides an alternative definition of the model. Finally, we show that the fixing operation used to define the model leads to a particularly simple characterization of identifiable causal effects in hidden variable causal DAG models.

62H99,

60E05,

graphical models,

hidden variable models,

conditional independence,

causal inference,

1 Introduction

Graphical models provide a principled way to take advantage of independence constraints for probabilistic modeling, learning and inference, while giving an intuitive graphical description of qualitative features useful for these tasks. A popular graphical model represents a joint distribution by means of a directed acyclic graph (DAG), where each vertex in the graph corresponds to a random variable. The popularity of DAG models, also known as Bayesian network models, stems from their well-understood theory and from the fact that they admit an intuitive causal interpretation (under the assumption that there are no unmeasured common causes; see Spirtes et al. (1993)). An arrow from a variable $A$ to a variable $B$ in a DAG model can be interpreted, in a way that can be made precise, to mean that $A$ is a “direct cause” of $B$.

Starting from a causally interpreted DAG, the consequences of intervention in the system under study can be understood by modifying the graph via removing certain edges, and modifying the corresponding joint probability distribution via reweighting (Strotz and Wold (1960), Spirtes et al. (1993), Pearl (2000)). For example, the DAG in Figure 1(i) represents distributions that factorize as

[TABLE]

If the model is interpreted causally, an experiment to externally set the value of $X_{3}$ will break the dependence of $X_{3}$ on $X_{1}$ and $X_{2}$; however, the dependence of $X_{4}$ upon $X_{3}$ will be preserved; see Figure 1(ii). This is represented graphically by severing incoming edges to $3$, an operation some authors call “mutilation,” and probabilistically by removing the factor $p(x_{3}\,|\,x_{1},x_{2})$ from the factorization of $p$ to yield a new distribution:

[TABLE]

In certain contexts, we will later call these operations “fixing.”

The functional in (1) is sometimes called the g-formula (Robins, 1986), the manipulated distribution (Spirtes et al., 1993), or the truncated factorization (Pearl, 2000). The distribution $p^{*}(x_{0},x_{1},x_{2},x_{4}\,|\,x_{3})$ is commonly denoted by $p(x_{0},x_{1},x_{2},x_{4}\,|\,\operatorname{do}(x_{3}))$. In a causal model given by a DAG where all variables are observed, any interventional probability distribution can be identified by this method.

Often not all common causes are measured, or there is no way to know a priori whether this is the case. This motivates the study of DAG models containing latent variables (Allman et al., 2015), and the constraints they imply. Existing theoretical machinery based on DAGs can be applied to such settings, simply by treating the unobserved variables as missing data. However, this creates a number of problems that are particularly severe when the structure of the underlying DAG model with latents is unknown. First, there are, in general, an infinite number of DAG models with latent variables that imply the (independence) constraints holding in a given observed distribution. Second, assumptions concerning the state-space or distribution of latent variables may have a profound effect on the model. This is problematic if prior knowledge about latent variables is scarce.

An alternative approach considers a supermodel defined by taking a subset of the constraints implied by a DAG model with latent variables on the observed marginal distribution. More specifically, we consider models defined by equality constraints that are implied by the factorization of a DAG with latents, but that do not depend on assumptions regarding the state-space or distribution of the latent variables. Models defined by these constraints are naturally represented by mixed graphs, that is, graphs containing directed ($\rightarrow$) and bidirected ($\leftrightarrow$) edges, obtained from DAGs via a latent projection operation (Verma and Pearl, 1990); see the graph in Figure 3(i) for the latent projection of the DAG in Figure 1(i).

Much previous work (Wermuth et al., 1994; Richardson and Spirtes, 2002; Ali et al., 2009; Evans and Richardson, 2010; Wermuth, 2011; Sadeghi and Lauritzen, 2014) has defined Markov models for mixed graphs via independence constraints implied by hidden variable DAGs on the observed margin. It is well known, however, that DAG models with latent variables imply nonparametric constraints that are not conditional independence constraints. For example, consider the DAG shown in Figure 1(i), and take the vertex [math] as hidden. This DAG implies no conditional independence restrictions on the observed margin $p(x_{1},x_{2},x_{3},x_{4})$. This is because all vertex sets that d-separate pairs of observed variables—that is, the pairs $(x_{1},x_{4})$ and $(x_{2},x_{4})$—include the unobserved variable $x_{0}$. However, it may be shown that the $p(x_{1},x_{2},x_{3},x_{4})$ margin of any distribution $p(x_{0},x_{1},x_{2},x_{3},x_{4})$, which factorizes according to the DAG in Figure 1(i), obeys the constraint that

[TABLE]

see Verma and Pearl (1990); Robins (1986); Wermuth and Cox (2008). In Robins (1999), it is shown that this constraint is equivalent to the requirement that $X_{4}$ is independent of $X_{1}$ given $X_{3}$ in the distribution obtained from $p(x_{1},x_{2},x_{3},x_{4})$ after dividing by the conditional $p(x_{3}\,|\,x_{2},x_{1})$. Note that this is the same manipulation performed in (1), but the operation, which we later call “fixing,” is purely probabilistic and can be performed without requiring that the model has any causal interpretation. If we interpret the original DAG as causal, then (2) is an (identifiable) dormant independence constraint (Shpitser and Pearl, 2008; Shpitser et al., 2014).

Since, as we have seen, the DAG in Figure 1(i) implies no conditional independence restrictions on the joint $p(x_{1},x_{2},x_{3},x_{4})$, the set of distributions obeying these independence relations is (trivially) saturated. Consequently, a structure learning algorithm such as FCI (Spirtes et al., 1993) that learns a Markov equivalence class of DAGs with latent variables, under the assumption of faithfulness, will return a (maximally uninformative) unoriented complete graph. The assumption of “faithfulness” implies that if X_{A}\mathchoice{\mathrel{\hbox to0.0pt{\displaystyle\perp\hss}\mkern 3.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{\textstyle\perp\hss}\mkern 3.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptstyle\perp\hss}\mkern 3.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptscriptstyle\perp\hss}\mkern 3.0mu{\scriptscriptstyle\perp}}}X_{B}\mid X_{C} in the observed distribution then $A$ is d-separated from $B$ given $C$ in the underlying DAG with latent variables.

Indeed, as originally pointed out by Robins (1999), if we assume a certain generalization of faithfulness, if (2) holds, we may rule out the model in Figure 1(iii). More generally, Shpitser et al. (2009) used pairwise constraints of this form to test for the presence of certain directed edges (in the context of a specific graph). Further, Tian and Pearl (2002) presented a general algorithm for finding nonparametric constraints from DAGs with latent variables.

In this paper, we introduce a (statistical) model called the nested Markov model, defined by these nonparametric constraints, and associated with a mixed graph called an acyclic directed mixed graph (ADMG). We give equivalent characterizations of the model in terms of global and ordered local Markov properties, and a factorization. We next prove that the set of marginal distributions given by a latent DAG model is a submodel of the nested Markov model associated with the corresponding ADMG (obtained by latent projection). We show that our results lead to a particularly simple characterization of identifiable causal effects in hidden variable causal DAG models. Finally, we relate the nested Markov model to a set of constraints obtained by Tian and Pearl (2002).

1.1 Applications of the nested Markov model

Nested models are defined by constraints that correspond to the absence of direct effects (Robins and Wasserman, 1997; Robins, 1999). For example, consider the causal graph in Figure 1(i), where vertices $1$ and $3$ correspond to treatments $X_{1}$ and $X_{3}$, vertices $2$ and $4$ are responses $X_{2},X_{4}$ and vertex [math] is an unobserved confounding variable $X_{0}$. In this graph, $X_{1}$ has no direct effect on $X_{4}$, and as noted above, this implies (2). The nested model shown in Figure 3(i) is defined by this constraint (see Section 3).

Such restrictions have been increasingly exploited in the analysis of longitudinal observational data in medicine and public health. For example, consider the nested Markov constraint that a diagnostic or screening test (e.g., a mammogram) has no direct effect on a patient’s clinical outcome of interest (death from breast cancer), except through the effect of the test results on the choice of treatment (no treatment if the mammogram was negative and, if positive, biopsy possibly followed by breast surgery and/or chemotherapy). Several recent papers (Neugebauer et al., 2017; Caniglia et al., 2019; Kreif et al., 2021) have leveraged this no direct effect of screening constraint to construct novel highly efficient estimators of an optimal joint testing and treatment regime. What was surprising and, indeed, unprecedented is that, in actual medical studies, the new estimators have provided a 50-fold increase in efficiency (and thus a 50-fold reduction in the required sample size) compared to estimators that fail to leverage the no direct effect constraint (Caniglia et al., 2019).

Nested Markov constraints can also be used for structure learning. The review paper (Shpitser et al., 2014) includes all nested Markov equivalence classes over four variables; classes with nested constraints can be very small, so they are potentially extremely informative about causal structure. This reflects the fact that whereas conditional independences involve three sets, nested constraints are relations among more groups of variables (see Section 2.6). A general theory of nested Markov equivalence and search for ADMGs remains a subject for future work. However, distributions consistent with these constraints have already been observed in applications; see Bhattacharya et al. (2021).

The nested constraints also facilitate the construction of more computationally efficient marginalization methods in some causal graphs (Shpitser et al., 2011), for categorical variables. This is achieved via Theorem 48 and the parameterization of Evans and Richardson (2019).

This work is of separate interest to people constructing causal models for quantum mechanics, since it does not impose the inequalities implied by latent variable models (Navascués and Wolfe, 2020); see Example 47.

Evans (2018) shows that for categorical variables the nested Markov model is the closest approximation to the margin of a DAG model (ignoring inequalities), in that it incorporates all equality constraints, and in this sense is “complete.”

1.2 Overview of nested Markov models

We now outline our strategy for defining the nested Markov model in terms of ordinary conditional independence (in derived distributions) by analogy to a way of defining DAG models in terms of undirected graphical models. We give a specific example of a nested Markov model, outlining the key concepts, while providing references to the formal definitions within the paper.

A nested Markov model is represented by an ADMG, a mixed graph naturally derived from DAGs with latent variables via an operation called latent projection. Intuitively, the ADMG does not contain latent variables, but indicates the presence of such variables by the inclusion of bidirected ($\leftrightarrow$) edges. Earlier work (Richardson and Spirtes (2002); Richardson (2003)) established Markov properties for ordinary independence models defined by ADMGs. Such an independence model is defined by fewer constraints than the nested Markov model represented by the same ADMG, and hence the former is a supermodel of the latter. The global Markov property for these independence models simply corresponds to the natural extension of d-separation (Pearl, 1988) to ADMGs. This extension, which is sometimes called m-separation, consists of allowing colliders to involve bidirected edges. Latent projection is defined in Section A.3; ADMGs and m-separation in Sections 2.1 and A.2, respectively.

We also consider conditional ADMGs (CADMGs) where certain “fixed” vertices represent nonrandom variables that index distributions over the random variables. Such vertices are treated similarly to the so-called “strategy nodes” in influence diagrams (Dawid, 2002). The Markov property for CADMGs is a simple extension of m-separation that takes into account fixed nodes. CADMGs are defined formally in Section 2.3; the corresponding global Markov property for a CADMG is given in Section 2.8.1. Note that an ADMG is the special case of a CADMG with no fixed vertices.

CADMGs and their associated Markov models characterize the nested Markov model in much the same way that undirected graphs and their associated Markov models can be used to describe a DAG model. We first briefly review the characterization of DAGs via undirected models.

The global Markov property for DAGs may be obtained from the (union of the) Markov properties associated with undirected graphs derived from the DAG by the moralization operation ((Lauritzen, 1996), Theorem 3.27); the resulting property is equivalent to d-separation (Pearl, 1988). More precisely, the DAG Markov property corresponds to (the union of) the Markov properties associated with undirected graphs representing “ancestral” margins. Likewise, the set of distributions corresponding to the DAG is the intersection of the sets of distributions obeying the factorization properties associated with these undirected graphs; this is equivalent to the characterization that the joint distribution factors into the product of each variable given its parents in the graph. As an example, consider the DAG in Figure 2(i). Undirected graphs associated with some ancestral margins, and their factorizations, are shown in Figure 2(ii), (iii), and (iv).

Likewise, the set of distributions in the nested Markov model associated with an ADMG corresponds to the intersection of the sets of distributions obeying factorization properties encoded by specific CADMGs obtained from the original ADMG. However, whereas the undirected graphs corresponding to a DAG may be seen as representing specific (ancestral) margins, the CADMGs obtained from an ADMG represent “kernel” distributions obtained by sequentially applying a new “fixing” operation on distributions, one that generalizes conditioning and marginalizing. This fixing operation has a natural causal interpretation, as do the kernels that form CADMG factorizations of the nested Markov model. Specifically, in the context of a latent variable causal model whose projection is a given ADMG, kernels can be viewed as (identified) interventional distributions. Not all variables are fixable. From a causal perspective this is natural since in the presence of latent variables, not all interventional distributions are identifiable. The fixing operation and the set of fixable vertices are defined in Section 2.11.

As a specific example, consider the graph shown in Figure 3(i). In this ADMG, the vertex $3$ may be fixed to give the CADMG shown in Figure 3(ii), where the corresponding distribution and factorization are also shown, with

[TABLE]

Note that, although the original graph implied no conditional independences, the graph in Figure 3(ii) implies the independence X_{1}\mathchoice{\mathrel{\hbox to0.0pt{\displaystyle\perp\hss}\mkern 3.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{\textstyle\perp\hss}\mkern 3.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptstyle\perp\hss}\mkern 3.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptscriptstyle\perp\hss}\mkern 3.0mu{\scriptscriptstyle\perp}}}X_{4}\mid X_{3} via m-separation.

Whereas the undirected graphs associated with a DAG correspond to distributions obtained by specific marginalizations (namely those that remove vertices that have no children), CADMGs correspond to certain specific ordered sequences of fixing operations. Not all such sequences are allowed: in some cases a vertex may be fixable only after another vertex has already been fixed. The global nested Markov property corresponds to the (union of the) m-separation relations encoded in the CADMGs derived via allowed sequences of these fixing operations, which we call “valid.” Valid fixing sequences are defined in Section 2.13. These fixing sequences are closely related to a particular identification strategy for interventional distributions consisting of recursively applying the g-formula to an already identified intervention distribution to obtain the result of further interventions; this connection is explored further in Section 4.2.

Returning to the example, given the CADMG in Figure 3(ii), the vertex $1$ may be fixed to give the CADMG in Figure 3(iii). Further, given the CADMG in Figure 3(iii), we may fix $2$. The kernel in this graph is

[TABLE]

The quantity on the RHS of (3) is a function only of $x_{3}$ and $x_{4}$, and not $x_{1}$. This is precisely the constraint (2) implied by the original latent variable DAG model in Figure 1(i). This constraint characterizes the nested Markov model corresponding to the ADMG in Figure 3(i).

In the original ADMG in Figure 3(i), we could also have fixed $1$ and $4$. Had we chosen to fix $1$, we could then subsequently have fixed $3$, and would have arrived at the same CADMG and distribution as shown in Figure 3(iii). Thus, the operations of fixing $3$ and $1$ commute. However, this is not always the case: for example, $2$ may only be fixed after $3$.

Like the DAG model, the nested Markov model may be characterized via a factorization property, as well as by local and global Markov properties described in Section 3. In each case these properties are defined via the set of CADMGs that are “reached” via valid fixing sequences. Just as the factorization property of the DAG model leads naturally to a parameterization in terms of Markov factors, likewise the factorization property of the nested Markov model leads to a parameterization in terms of its factors in multivariate Gaussian and finite categorical cases (Shpitser et al., 2018; Evans and Richardson, 2019). The usual DAG parameters are such that the choice of value for one parameter $p(x_{v}\,|\,x_{\operatorname{pa}(v)})$ does not restrict the set of possible values for another, $p(x_{w}\,|\,x_{\operatorname{pa}(w)})$; in other words, they are variation independent, but this is not true for the nested Markov model.

The rest of the paper is organized as follows. In Section 2.1, we introduce graph theoretic preliminaries. In Section 2.2, we define conditional mixed graphs, which are used to construct our model. In Section 2.3, we introduce a generalization of conditional distributions which we call kernels, and in Section 2.4 we generalize the notion of conditional independence from standard distributions to these kernels. In Sections 2.5–2.7, we provide further properties of kernels, including a proof that conditional independence in kernels satisfies the semigraphoid axioms (Proposition 6). In Section 2.8 and Section 2.9, we give the Markov properties and factorization of kernels associated with CADMGs; we prove they are equivalent in Theorem 16 in Section 2.10. We define the graph-based fixing operation in Section 2.11, give some of its properties in Section 2.12, and define sets that can be reached by successive applications of this operation in Section 2.13.

We define the nested model in Section 3. Theorem 31 in Section 3.1 shows a key result, that graphs and kernels are invariant to the precise order of fixing in a sequence, as long as each fixing operation is valid. In Section 3.2, Section 3.3, and Section 3.4, respectively, we give definitions of the nested Markov model in terms of the global Markov property, the factorization, and the ordered local Markov property. Corollary 40 in Section 3.5 shows that the nested Markov model corresponding to any complete ADMG gives the saturated model. In Section 4.1, we show in Theorem 46, that any marginal distribution in a hidden variable DAG model lies in the nested Markov model associated with the latent projection. A simple characterization of identifiable causal effects in hidden variable causal DAG models, based on the fixing operation, is given in Theorem 48 of Section 4.3. Finally, a result stating that the nested Markov model may be defined by the set of constraints found by the algorithm in Tian and Pearl (2002) is given as Theorem 51 in Section 4.4. All proofs of claims are in the Supplementary Material (Richardson et al. (2023)).

2 Latent variable DAG models

We assume familiarity with standard graphical definitions relating to DAGs, latent projections, and mixed graphs. See Section A.1 in the Supplementary Material for a review.

2.1 Basic concepts

The motivation for introducing mixed graphs is twofold. First, by removing latent variables and replacing them with bidirected edges we simplify the representation. For example, to perform a search, instead of considering a potentially infinite class of DAGs with arbitrarily many latent variables, we need only consider a finite set of mixed graphs. Second, although the statistical models that we associate with mixed graphs capture many of the constraints implied by latent variable models, the resulting model will still, in general, be a superset of the set of distributions over the observables that are implied by the original DAG with latents. The use of a mixed graph to represent our model serves to emphasize that, in spite of this connection, the set of distributions we are constructing is nonetheless not a latent variable model (see Example 47).

We will make use of standard genealogic relations in graphs. The sets of parents, children, ancestors, descendants, and nondescendants of $a$ in ${\mathscr{G}}$ are written $\operatorname{pa}_{{\mathscr{G}}}(a)$, $\operatorname{ch}_{{\mathscr{G}}}(a)$, $\operatorname{an}_{{\mathscr{G}}}(a)$, $\operatorname{de}_{{\mathscr{G}}}(a)$, and $\operatorname{nd}_{{\mathscr{G}}}(a)$, respectively. By convention, every vertex is its own ancestor and descendant, so $\operatorname{an}_{{\mathscr{G}}}(a)\cap\operatorname{de}_{{\mathscr{G}}}(a)=\{a\}$. An ordering $\prec$ of nodes in ${\mathscr{G}}$ is said to be topological if for any pair of vertices $a,b$, if $a\prec b$, then $a\notin\operatorname{de}_{{\mathscr{G}}}(b)$; note that this implies that if $a\prec b$ then $a\neq b$. We define the set $\operatorname{pre}_{{\mathscr{G}},\prec}(b)\equiv\{a|a\prec b\}$. We apply these definitions disjunctively to sets, for example, $\operatorname{an}_{{\mathscr{G}}}(A)=\bigcup_{a\in A}\operatorname{an}_{{\mathscr{G}}}(a)$. A set of vertices $A$ in ${\mathscr{G}}$ is called ancestral if $\operatorname{an}_{{\mathscr{G}}}(a)\subseteq A$ whenever $a\in A$.

Given a DAG ${\mathscr{G}}$ with vertex set $V\mathbin{\dot{\cup}}L$ where $L$ is the set of latent variables, we associate with it a mixed graph, denoted ${\mathscr{G}}(V)$, via the standard operation of latent projection $\sigma_{L}$ (Pearl and Verma, 1991); see the Supplementary Material, Section A.3 for the definition (Richardson et al., 2023). Here and elsewhere, we use standard $\mathbin{\dot{\cup}}$ notation when we wish to emphasize that the sets in the union are disjoint.

The latent projection ${\mathscr{G}}(V)=\sigma_{L}({\mathscr{G}}(V\dot{\cup}L))$ represents the set of d-separation relations holding among the variables in $V$ in ${\mathscr{G}}$.

Proposition 1**.**

Let ${\mathscr{G}}$ be a DAG with vertex set $V\mathbin{\dot{\cup}}L$. For disjoint subsets $A,B,C\subseteq V$ (where $C$ may be empty), $A$ is d-separated from $B$ given $C$ in ${\mathscr{G}}$ if and only if $A$ is m-separated from $B$ given $C$ in ${\mathscr{G}}(V)$.

However, as we will see later, the latent projection ${\mathscr{G}}(V)$ captures much more than simply the d-separation relations holding in $V$. As suggested by Figures 1(i) and 3(i), ${\mathscr{G}}(V)$ also represents constraints such as (2), and further all those found by the algorithm in Tian and Pearl (2002), as well as others.

2.2 Conditional ADMGs

We define a conditional acyclic directed mixed graph(CADMG) ${\mathscr{G}}(V,W)$ to be an ADMG with two disjoint sets of vertices $V$ and $W$, which we call random and fixed, respectively. We require that for all $w\in W$, $\operatorname{pa}_{{\mathscr{G}}}(w)=\emptyset$ and there are no bidirected edges involving $w$.

Thus, in a CADMG ${\mathscr{G}}(V,W)$ there are no edges connecting vertices in $W$, and all edges connecting $w\in W$ and $v\in V$ take the form $w\to v$. Vertices in $V$ will represent random variables $X_{V}$, as in a standard graphical model. The rationale for excluding edges between vertices in $W$ or with arrowheads in $W$ is that the CADMG will represent objects where the vertices in $W$ correspond to (nonrandom) values that index distributions over $V$.

We note that CADMGs represent kernels that are not, in general, formed by standard conditioning from the original observed distribution. For example, a set of interventional distributions over $X_{V}$ indexed by possible interventions on $X_{W}$; note, however, that we do not require a causal interpretation of the graph. We also introduce operators $\mathbb{V}({{\mathscr{G}}})$ and $\mathbb{W}({{\mathscr{G}}})$ that return, respectively, the sets of random and fixed nodes associated with a CADMG ${\mathscr{G}}$. We will use circular nodes to indicate the random vertices $\mathbb{V}({{\mathscr{G}}})$, and square nodes to indicate the fixed vertices $\mathbb{W}({{\mathscr{G}}})$; see, for instance, the CADMGs in Figures 1(ii), 3(ii)–(iv) and 4(i). When the vertex sets are clear from context, we will abbreviate ${{\mathscr{G}}}(V,W)$ as ${\mathscr{G}}$.

We will apply the standard genealogical definitions to CADMGs. Conversely, since an ADMG ${\mathscr{G}}(V)$ may be seen as a CADMG in which $W=\emptyset$, all subsequent definitions for CADMGs will also apply to ADMGs.

The induced subgraph of a CADMG ${\mathscr{G}}(V,W)$ on a set $A$, denoted ${\mathscr{G}}_{A}$, is a CADMG with $\mathbb{V}({{\mathscr{G}}_{A}})=V\cap A$ and $\mathbb{W}({{\mathscr{G}}_{A}})=W\cap A$, and precisely those edges from ${\mathscr{G}}$ that have both endpoints in $A$. Note that in forming ${\mathscr{G}}_{A}$, the status of the vertices in $A$ with regard to whether they are in $V$ or $W$ is preserved.

Given a CADMG ${\mathscr{G}}(V\mathbin{\dot{\cup}}L,W)$ and a set of latent variables $L$ that are all random, we define the latent projection ${\mathscr{G}}(V,W)$ to be the CADMG obtained by applying the standard definition of latent projection for ADMGs (see the Supplementary Material, Section A.3).

Proposition 2**.**

In a CADMG ${\mathscr{G}}(V\cup L,W)$ if $V\cup W$ is ancestral, then ${{\mathscr{G}}}_{V\cup W}\equiv({\mathscr{G}}(V\cup L,W))_{V\cup W}={{\mathscr{G}}}(V,W)$.

Thus, the induced subgraph on an ancestral set $V\cup W$ is the same as the latent projection onto $V\cup W$.

Definition 3**.**

A set of vertices $C$ in a CADMG ${\mathscr{G}}$ is called bidirected-connected if for every pair of vertices $c,d\in C$ there is a bidirected path in ${\mathscr{G}}$ between $c$ and $d$ with every vertex on the path in $C$. A maximal bidirected-connected set of vertices in $\mathbb{V}({\mathscr{G}})$ is referred to as a district in ${\mathscr{G}}$. Let

[TABLE]

be the set of districts in ${\mathscr{G}}$. For $v\in\mathbb{V}({\mathscr{G}})$, let $\operatorname{dis}_{{\mathscr{G}}}(v)$ be the district containing $v$ in ${\mathscr{G}}$. We write $\operatorname{dis}_{A}(v)$ as a shorthand for $\operatorname{dis}_{{\mathscr{G}}_{A}}(v)$, the district of $v$ in the induced subgraph ${\mathscr{G}}_{A}$.

Tian and Pearl (2002) refer to districts in ADMGs as “c-components.” In models associated with ADMGs, districts are used to specify variable partitions that define terms in the factorization of observed random variables. For this reason, districts in CADMGs include only random vertices. In a DAG ${\mathscr{G}}(V)$, the districts ${\mathcal{D}}({\mathscr{G}})=\{\{v\}\mid v\in V\}$ are the singleton subsets of $V$, these are naturally associated with the usual Markov factors $p(x_{v}\,|\,x_{\operatorname{pa}_{{\mathscr{G}}}(v)})$.

2.3 Kernels

We consider collections of real-valued random variables $(X_{v})_{v\in V}$ taking values in $({\mathfrak{X}}_{v})_{v\in V}$. For $A\subseteq V$, we let ${\mathfrak{X}}_{A}\equiv\bigtimes_{v\in A}({\mathfrak{X}}_{v})$, and $X_{A}\equiv(X_{v})_{v\in A}$. Here, $v$ denotes a vertex and $X_{v}$ the corresponding random variable, likewise $A$ denotes a vertex set and $X_{A}$ is the vector $(X_{v}:v\in A)$.

Following Lauritzen (1996), p. 46, we define a kernel over $\mathfrak{X}_{V}$ and indexed by $\mathfrak{X}_{W}$ to be a nonnegative function $q_{V}(x_{V}\,|\,x_{W})$ satisfying

[TABLE]

We use the term “kernel” and write $q_{V}(\cdot|\cdot)$ (rather than $p(\cdot|\cdot)$) to emphasize that these functions, though they satisfy (4), and thus most properties of conditional densities, will not, in general, be formed via the usual operation of conditioning on the event $X_{W}=x_{W}$. Following standard conventions for densities, we will use $q_{V}(x_{V}\,|\,x_{W})$ to refer either to the function itself, or to its realization under specific values of $x_{V},x_{W}$, with the precise meaning being clear from context. To conform with standard notation for densities, we define for every $A\subseteq V$

[TABLE]

In general, for $R,S\subseteq V$ and $T\subset W$, the density $q_{V}(x_{R}\,|\,x_{S\cup T})$ may not be defined, since in the absence of a distribution over $X_{W}$, we cannot integrate out the variables $X_{W\setminus T}$ (though see Definition 4 and subsequent comments below). For disjoint $V_{1}\dot{\cup}V_{2}=V$ and $W_{1}\dot{\cup}W_{2}=W$, we will sometimes write $q_{V}(x_{V_{1}},x_{V_{2}}\,|\,x_{W_{1}},x_{W_{2}})$ to mean $q_{V}(x_{V_{1}\cup V_{2}}\,|\,x_{W_{1}\cup W_{2}})$.

2.4 Independence in kernels

We extend the notion of conditional independence to kernels over $\mathfrak{X}_{V}$ indexed by $\mathfrak{X}_{W}$. A rigorous treatment of conditional independence in settings where not all variables are random was given in Constantinou (2013).

Definition 4**.**

For disjoint subsets $A,B,C\subseteq V\cup W$, we define $X_{A}$ to be conditionally independent of $X_{B}$ given $X_{C}$ in a kernel $q_{V}$, written

[TABLE]

if either:

• (a)

$A\cap W=\emptyset$* and $q_{V}(x_{A}\,|\,x_{B},x_{C},x_{W\setminus(B\cup C)})$ is a function only of $x_{A}$ and $x_{C}$ (whenever this kernel is defined). In this case, we define $q_{V}(x_{A}\,|\,x_{C})\equiv q_{V}(x_{A}\,|\,x_{B},x_{C},x_{W\setminus(B\cup C)})$,*

or**

• (b)

$B\cap W=\emptyset$* and $q_{V}(x_{B}\,|\,x_{A},x_{C},x_{W\setminus(A\cup C)})$ is a function only of $x_{B}$ and $x_{C}$ (whenever this kernel is defined). In this case, we define $q_{V}(x_{B}\,|\,x_{C})\equiv q_{V}(x_{B}\,|\,x_{A},x_{C},x_{W\setminus(B\cup C)})$.*

See Figure 5 for an illustration of cases; it is necessary for either $A$ or $B$ not to intersect $W$ because there is no joint distribution over elements in $W$. This definition reduces to ordinary conditional independence in the case $W=\emptyset$. The condition that the kernel should be defined simply addresses the situation where the conditioning event has zero probability density. We remark that if $(A\cup B)\cap W=\emptyset$, then both definitions hold (or fail to hold) at the same time; the conditions become equivalent to saying that the kernel $q_{V}(x_{A},x_{B}\,|\,x_{C},x_{W\setminus C})$ should factorize into a piece that depends only upon $x_{A},x_{C}$ and a piece that depends only upon $x_{B},x_{C}$.

Note that the kernels appearing in (a) and (b) specify values for all of the variables $X_{W}$, and are defined via conditioning in (a margin of) the original kernel $q_{V}$. For example, in (a),

[TABLE]

Proposition 5**.**

In a kernel $q_{V}(x_{V}\,|\,x_{W})$, X_{A}\mathchoice{\mathrel{\hbox to0.0pt{\displaystyle\perp\hss}\mkern 3.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{\textstyle\perp\hss}\mkern 3.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptstyle\perp\hss}\mkern 3.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptscriptstyle\perp\hss}\mkern 3.0mu{\scriptscriptstyle\perp}}}X_{B}\mid X_{C} if and only if either X_{A}\mathchoice{\mathrel{\hbox to0.0pt{\displaystyle\perp\hss}\mkern 3.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{\textstyle\perp\hss}\mkern 3.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptstyle\perp\hss}\mkern 3.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptscriptstyle\perp\hss}\mkern 3.0mu{\scriptscriptstyle\perp}}}X_{B\cup(W\setminus C)}\mid X_{C} or X_{B}\mathchoice{\mathrel{\hbox to0.0pt{\displaystyle\perp\hss}\mkern 3.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{\textstyle\perp\hss}\mkern 3.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptstyle\perp\hss}\mkern 3.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptscriptstyle\perp\hss}\mkern 3.0mu{\scriptscriptstyle\perp}}}X_{A\cup(W\setminus C)}\mid X_{C}.

2.5 Semigraphoid axioms in kernels

Classical conditional independence constraints may logically imply other such constraints. Though no finite axiomatization of these connections is possible (Studený, 1992), deductive derivations of conditional independence constraints in DAGs can be restricted to the semigraphoid axioms of symmetry and the “chain rule” (Dawid, 1979), which we reproduce here:

[TABLE]

(The chain rule axiom is sometimes written as the three separate axioms of contraction, decomposition, and weak union.) We now show that, unsurprisingly, conditional independence constraints defined for kernels also obey these axioms. An additional set of axioms called separoids has been shown to apply to versions of conditional independence involving nonstochastic variables like $X_{W}$ (Constantinou, 2013).

Proposition 6**.**

The semigraphoid axioms are sound for kernel independence.

2.6 Constructing kernels

We will typically construct new kernels via the operation of dividing either a distribution $p(x_{V})$ by $p(x_{H}\,|\,x_{T})$ or an existing kernel $q_{V}(x_{V}\,|\,x_{W})$ by $q_{V}(x_{H}\,|\,x_{T\cup W})$, where $H\mathbin{\dot{\cup}}T\subseteq V$. For the results in the remainder of this section, we will consider a kernel $q_{V}(x_{V}\,|\,x_{W})$ where $V=R\mathbin{\dot{\cup}}H\mathbin{\dot{\cup}}T$, and a new object

[TABLE]

See Figure 6 for an illustration.

Lemma 7**.**

$q^{*}_{V\setminus H}(x_{R},x_{T}\,|\,x_{H},x_{W})$* is a kernel.*

Lemma 8**.**

For the kernel constructed in (6),

[TABLE]

*and hence *

[TABLE]

Note that if $R=\emptyset$, $q^{*}_{V\setminus H}(x_{V\setminus H}\,|\,x_{H},x_{W})=\sum_{x_{H}}q_{V}(x_{V}\,|\,x_{W})=q_{V}(x_{T}\,|\,x_{W})$.

Proposition 9** (Separation).**

For the kernel constructed in (6):

• (i)

(X_{H}\mathchoice{\mathrel{\hbox to0.0pt{\displaystyle\perp\hss}\mkern 3.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{\textstyle\perp\hss}\mkern 3.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptstyle\perp\hss}\mkern 3.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptscriptstyle\perp\hss}\mkern 3.0mu{\scriptscriptstyle\perp}}}X_{T}\mid X_{W})* holds in $q^{*}_{V\setminus H}(x_{V\setminus H}\,|\,x_{H},x_{W})$.*

• (ii)

If X_{V}\mathchoice{\mathrel{\hbox to0.0pt{\displaystyle\perp\hss}\mkern 3.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{\textstyle\perp\hss}\mkern 3.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptstyle\perp\hss}\mkern 3.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptscriptstyle\perp\hss}\mkern 3.0mu{\scriptscriptstyle\perp}}}X_{W\setminus W_{1}}\mid X_{W_{1}} in $q_{V}$, then X_{H\cup(W\setminus W_{1})}\mathchoice{\mathrel{\hbox to0.0pt{\displaystyle\perp\hss}\mkern 3.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{\textstyle\perp\hss}\mkern 3.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptstyle\perp\hss}\mkern 3.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptscriptstyle\perp\hss}\mkern 3.0mu{\scriptscriptstyle\perp}}}X_{T}\mid X_{W_{1}} in $q^{*}_{V\setminus H}$.

• (iii)

If $R=\emptyset$, X_{H}\mathchoice{\mathrel{\hbox to0.0pt{\displaystyle\perp\hss}\mkern 3.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{\textstyle\perp\hss}\mkern 3.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptstyle\perp\hss}\mkern 3.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptscriptstyle\perp\hss}\mkern 3.0mu{\scriptscriptstyle\perp}}}X_{V\setminus H}\,|\,X_{W} in $q^{*}_{V\setminus H}$.

2.7 Preservation of existing independences in a kernel

We now state two important properties that transfer conditional independence statements to a new kernel formed via the operation (6).

Proposition 10** (Ordering).**

Given disjoint sets $A,B,C$, where $C$ may be empty, if $A,B,C\subseteq T\cup W$, then

[TABLE]

This result, which follows directly from (9), states that, given any ordering of variables in which $W\cup T$ precedes $H$, which in turn precedes $R$, performing the operation in (6) on $X_{H}$ preserves conditional independence statements among variables that precede $H$ in the ordering; see Figure 7(i). Note that, by Definition 4, both independences in (10) imply either $A\subseteq T$ or $B\subseteq T$.

Proposition 11** (Modularity).**

Given disjoint sets $A,B,C$, where $C$ may be empty, if $A\subseteq R$ and $(B\cup C)\supseteq H\cup T$, then

[TABLE]

This result, which follows directly from (8), is illustrated in Figure 7(ii). In words, it says that if we express a probability distribution as a set of factors via the chain rule of probability, and a conditional independence statement can be stated exclusively in terms of one of the factors, then applying the operation in (6) such that another factor is dropped does not affect this conditional independence statement. In other words, “factors are modular.”

2.8 Markov properties for CADMGs

As described earlier, a CADMG ${\mathscr{G}}(V,W)$ represents the independence structure of a kernel $q_{V}(x_{V}\,|\,x_{W})$. We now introduce a number of Markov properties for a given kernel, the equivalence of which we will state in Section 2.10.

2.8.1 The CADMG global Markov property

The global Markov property for CADMGs may be derived from m-separation via the following simple construction.

Definition 12**.**

Given a CADMG ${\mathscr{G}}(V,W)$, we define ${\mathscr{G}}^{|W}$ to be a mixed graph with vertex set $V^{*}=V\cup W$, and edge set

[TABLE]

In words, the graph ${\mathscr{G}}^{|W}$ is formed from ${\mathscr{G}}$ by adding bidirected edges between all pairs of vertices $w,w^{\prime}\in W$, and then eliminating the distinction between vertices in $V$ and $W$; see Figure 4(ii) for an example.

Definition 13**.**

A kernel $q_{V}$ satisfies the global Markov property for a CADMG ${\mathscr{G}}(V,W)$ if for arbitrary disjoint sets $A,B,C$, ($C$ may be empty),

[TABLE]

We denote this set of kernels ${\mathcal{P}}^{\mathrm{c}}_{\mathrm{m}}({\mathscr{G}})$, where “c” and “m” denote CADMG and m-separation, respectively.

2.8.2 The CADMG local Markov property

The (ordered) local Markov property for a DAG states that each vertex $v$ is independent of vertices prior to $v$ (under a topological ordering) conditional on the parents of $v$. In a CADMG, the Markov blanket plays a role analogous to that of the set of parents.

If $t\in V$, then the Markov blanket of $t$ in ${\mathscr{G}}(V,W)$ is defined as

[TABLE]

Given a vertex $t\in V$ such that $\operatorname{ch}_{{\mathscr{G}}}(t)=\emptyset$, a kernel $q_{V}$ obeys the local Markov property for ${\mathscr{G}}$ at $t$ if

[TABLE]

If $\prec$ is a topological total ordering on the vertices in ${\mathscr{G}}$, then for a subset $A$ define $\max_{\prec}(A)$ to be the $\prec$-greatest vertex in $A$.

We define the set of kernels obeying the ordered local Markov property for the CADMG ${\mathscr{G}}(V,W)$ under the ordering $\prec$ as follows:

[TABLE]

We will make use of the following extension of the Markov blanket notation: for an ancestral set $A$ in a CADMG ${\mathscr{G}}$ and a vertex $t\in V\cap A$ such that $\operatorname{ch}_{{\mathscr{G}}}(t)\cap A=\emptyset$, let

[TABLE]

Proposition 14**.**

Given a CADMG ${\mathscr{G}}$, an ancestral set $A$ and a random vertex $t\in A$ such that $\operatorname{ch}_{{\mathscr{G}}}(t)\cap A=\emptyset$:

• (i)

$\operatorname{mb}_{{\mathscr{G}}}(t,A)\subseteq\operatorname{mb}_{{\mathscr{G}}}(t)\subseteq D\cup\operatorname{pa}_{{\mathscr{G}}}(D)$, where $t\in D\in{\mathcal{D}}({\mathscr{G}})$;

• (ii)

if $A^{*}$ is an ancestral set and $t\in A^{*}\subseteq A$, then $\operatorname{mb}_{{\mathscr{G}}}(t,A^{*})\subseteq\operatorname{mb}_{{\mathscr{G}}}(t,A)$.

2.8.3 The CADMG augmented Markov property

The following is the analog of moralization in DAGs for CADMGs. For a CADMG ${\mathscr{G}}(V,W)$, the augmented graph derived from ${\mathscr{G}}$, denoted $({\mathscr{G}})^{a}$, is an undirected graph with vertex set $V\cup W$ such that $c\text{ --- }d\text{ in }({\mathscr{G}})^{a}$ if and only if $c$ and $d$ are connected by a path in ${\mathscr{G}}^{|W}$ containing only colliders (or a single edge).

A kernel $q_{V}$ obeys the augmented Markov property for a CADMG ${\mathscr{G}}(V,W)$ if X_{A}\mathchoice{\mathrel{\hbox to0.0pt{\displaystyle\perp\hss}\mkern 3.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{\textstyle\perp\hss}\mkern 3.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptstyle\perp\hss}\mkern 3.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptscriptstyle\perp\hss}\mkern 3.0mu{\scriptscriptstyle\perp}}}X_{B}\mid X_{C} in $q_{V}$ whenever: for arbitrary disjoint sets $A,B,C$ ($C$ may be empty), every path in $({\mathscr{G}}_{\operatorname{an}(A\cup B\cup C)})^{a}$ from any $a\in A$ to any $b\in B$ passes through a vertex in $C$. We denote the set of such kernels by ${\mathcal{P}}^{\mathrm{c}}_{\mathrm{a}}({\mathscr{G}})$.

2.9 Tian factorization for CADMGs

The joint distribution under a DAG model may be factorized into univariate densities. In DAG models, these factors take the form $p(x_{a}\,|\,x_{\operatorname{pa}_{{\mathscr{G}}}(a)})$. This factor is a conditional distribution for a singleton variable $X_{a}$, given the set of variables corresponding to parents of $a$ in the graph. The factorization property may be generalized to CADMGs by requiring factorization of $q_{V}$ into kernels over districts.

We define the set of kernels that Tian factorize with respect to a CADMG:

[TABLE]

In the next lemma, we show that the terms $f_{D}^{A}(\cdot\,|\,\cdot)$ arising in (15) are equal to products of univariate conditional densities, that is, instances of the g-formula of Robins (1986), and thus $f_{D}^{A}(\cdot\,|\,\cdot)$ does not depend on $A$ other than through $D$.

Lemma 15**.**

Let ${\mathscr{G}}$ be a CADMG with topological ordering $\prec$. If $q_{V}\in{\mathcal{P}}^{\mathrm{c}}_{\mathrm{f}}({\mathscr{G}})$, then for every ancestral set $A$ and every $D\in{\mathcal{D}}({\mathscr{G}}_{A})$, we have

[TABLE]

where T^{(d,D)}_{\prec}\equiv\operatorname{mb}_{{\mathscr{G}}}(d,\operatorname{an}_{{\mathscr{G}}}(D)\cap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\operatorname{pre}_{{\mathscr{G}},\prec}(d)\cup\{d\})}), so that

[TABLE]

Conversely, if (17) holds for all $d\in A$ and ancestral sets $A$, and the $f_{D}^{A}$ functions are defined by (16), then $q_{V}\in{\mathcal{P}}^{\mathrm{c}}_{\mathrm{f}}({\mathscr{G}})$.

Note that by Proposition 14, $\operatorname{mb}_{{\mathscr{G}}}(d,\operatorname{an}_{{\mathscr{G}}}(D)\cap\operatorname{pre}_{{\mathscr{G}},\prec}(d))\subseteq D\cup\operatorname{pa}_{{\mathscr{G}}}(D)$. Lemma 15 has the following important consequence:

[TABLE]

where $q_{D}(x_{D}\,|\,x_{\operatorname{pa}(D)\setminus D})$ is defined via the right-hand side of (16) under any topological ordering. In a context where we refer to $q_{V}$ and $q_{D}$ for $D\in{\mathcal{D}}({\mathscr{G}})$, it is implicit that $q_{D}$ is derived from $q_{V}$ in this way. In Section 3 we will extend this notation to include other “reachable” sets.

2.10 Equivalence of factorizations and Markov properties for CADMGs

The above definitions describe the same set of kernels due to the following result.

Theorem 16**.**

${\mathcal{P}}^{\mathrm{c}}_{\mathrm{f}}({\mathscr{G}})={\mathcal{P}}^{\mathrm{c}}_{\mathrm{l}}({\mathscr{G}},\prec)={\mathcal{P}}^{\mathrm{c}}_{\mathrm{m}}({\mathscr{G}})={\mathcal{P}}^{\mathrm{c}}_{\mathrm{a}}({\mathscr{G}})$.

We thus use ${\mathcal{P}}^{\mathrm{c}}({\mathscr{G}})$ to denote the set of such kernels, and simply refer to a kernel $q_{V}\in{\mathcal{P}}^{\mathrm{c}}({\mathscr{G}})$ as being Markov with respect to a CADMG ${\mathscr{G}}$.

2.11 The fixing operation and fixable vertices

We now introduce a “fixing” operation on an ADMG or CADMG that has the effect of transforming a random vertex into a fixed vertex, thereby changing the graph. However, we define this operation only for a subset of the vertices in the graph, which we term the set of (potentially) fixable vertices.

Definition 17**.**

Given a CADMG ${{\mathscr{G}}}(V,W)$, the set of fixable vertices is

[TABLE]

In words, a vertex $v$ is fixable in ${\mathscr{G}}$ if there is no other vertex $t$ that is both a descendant of $v$ and in the same district as $v$ in ${\mathscr{G}}$. For details on the causal interpretation of fixing, see Section 4.

Proposition 18**.**

For any $v\in V$, we have $\operatorname{de}_{{\mathscr{G}}}(v)\cap\operatorname{dis}_{{\mathscr{G}}}(v)\cap{\mathbb{F}}({{\mathscr{G}}})\neq\emptyset$.

Thus, if a vertex in a district is not fixable then there is a descendant of the vertex within the district that is fixable. In particular, every district contains at least one vertex that is fixable.

Recall that $\operatorname{mb}_{{\mathscr{G}}}(t)$, defined in (11), is the set of vertices $v\in(V\setminus\{t\})\cup W$, which can be reached via paths of the form: $t\leftrightarrow\cdots\leftrightarrow v$ or $t\leftrightarrow\cdots\leftrightarrow\leftarrow v$ or $t\leftarrow v$. Suppose that $q_{V}$ is Markov with respect to the CADMG ${\mathscr{G}}$; if $\operatorname{ch}_{{\mathscr{G}}}(t)=\emptyset$, then (12) follows by the CADMG local Markov property. More generally, if $t$ is fixable then

[TABLE]

follows from m-separation in ${\mathscr{G}}^{|W}$; thus, in addition, X_{t}\mathchoice{\mathrel{\hbox to0.0pt{\displaystyle\perp\hss}\mkern 3.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{\textstyle\perp\hss}\mkern 3.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptstyle\perp\hss}\mkern 3.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptscriptstyle\perp\hss}\mkern 3.0mu{\scriptscriptstyle\perp}}}X_{W\setminus\operatorname{mb}_{{\mathscr{G}}}(t)}\,|\,X_{\operatorname{mb}_{{\mathscr{G}}}(t)} in $q_{V}$. These hold even if $\operatorname{ch}_{{\mathscr{G}}}(t)\neq\emptyset$.

Definition 19**.**

Given a CADMG ${{\mathscr{G}}}(V,W)$, and a kernel $q_{V}(x_{V}\,|\,x_{W})$, for every $r\in{\mathbb{F}}({{\mathscr{G}}})$, we associate a fixing transformation $\phi_{r}$ on the graph ${{\mathscr{G}}}$ defined as follows:

[TABLE]

where ${\mathscr{G}}^{*}(V\setminus\{r\},W\cup\{r\})$ has precisely the subset of edges in ${\mathscr{G}}(V,W)$ that do not have arrowheads at $r$. With slight abuse of notation, we define a fixing transformation $\phi_{r}$ on the pair $(q_{V}(x_{V}\,|\,x_{W}),{{\mathscr{G}}})$:

[TABLE]

Note that $\mathbb{V}(\phi_{r}({{\mathscr{G}}}))=\mathbb{V}({{\mathscr{G}}})\setminus\{r\}$ and $\mathbb{W}(\phi_{r}({{\mathscr{G}}}))=\mathbb{W}({{\mathscr{G}}})\cup\{r\}$, so that $\phi_{r}({{\mathscr{G}}})$ is a new CADMG in which the status of $r$ changes from random to fixed, while $\phi_{r}(q_{V};{{\mathscr{G}}})$ forms a new kernel, per Lemma 7. Although the CADMG $\phi_{r}({{\mathscr{G}}})$ is determined solely by the graph ${\mathscr{G}}$ given as input, the transformation $\phi_{r}(q_{V}(x_{V}\,|\,x_{W});{{\mathscr{G}}})$ on the kernel $q_{V}(x_{V}\,|\,x_{W})$ is a function of both the graph and the kernel itself.

Proposition 20**.**

If $q_{V}$ is Markov with respect to a CADMG ${{\mathscr{G}}}(V,W)$ and $\operatorname{ch}_{{\mathscr{G}}}(r)=\emptyset$ (and hence $r\in\mathbb{F}({\mathscr{G}})$), then

[TABLE]

Thus, if $\operatorname{ch}_{{\mathscr{G}}}(r)=\emptyset$, then $\phi_{r}$ simply marginalizes over $X_{r}$: the conditioning on $X_{r}$ in $\phi_{r}(q_{V}(x_{V}\,|\,x_{W});{\mathscr{G}})$ is vacuous in the sense that the resulting kernel does not depend on the value of $X_{r}$. Though it may at first appear unnatural, it greatly simplifies our subsequent analyses to unify marginalization and fixing in this way.

Proposition 21**.**

If $q_{V}$ is Markov w.r.t. ${{\mathscr{G}}}(V,W)$ and $r\in\mathbb{F}({\mathscr{G}})$, then

[TABLE]

Lemma 22**.**

If $r\in{\mathbb{F}}({{\mathscr{G}}})$, then ${\mathbb{F}}({{\mathscr{G}}})\setminus\{r\}\subseteq{\mathbb{F}}(\phi_{r}({{\mathscr{G}}}))$.

That is, any vertex $s$ that was fixable before $r$ was fixed is still fixable after $r$ has been fixed (with the obvious exception of $r$ itself). Thus, when fixing vertices, although our choices may be limited at various stages, we are never faced with a choice between fixing $r$ and $r^{\prime}$, whereby choosing $r$ precludes subsequently fixing $r^{\prime}$.

Proposition 23**.**

If ${\mathscr{G}}$ is a subgraph of ${{\mathscr{G}}}^{*}$ with the same random and fixed vertex sets, then ${\mathbb{F}}({{\mathscr{G}}}^{*})\subseteq{\mathbb{F}}({{\mathscr{G}}})$.

For conciseness, we use $D^{r}$ to denote $\operatorname{dis}_{{\mathscr{G}}}(r)$ when the graph ${\mathscr{G}}$ is clear from the context.

Proposition 24**.**

Let ${\mathscr{G}}$ be a CADMG, with $r\in{\mathbb{F}}({{\mathscr{G}}})$. If $r\in D^{r}\in{\mathcal{D}}({\mathscr{G}})$, then

[TABLE]

where the sets on the right are disjoint. Thus, if $D\in{\mathcal{D}}(\phi_{r}({{\mathscr{G}}}))$ then $D\subseteq D^{*}$ for some $D^{*}\in{\mathcal{D}}({{\mathscr{G}}})$; further if $D\neq D^{*}$, then $r\in D^{*}$.

In words, the set of districts in $\phi_{r}({\mathscr{G}})$, the graph obtained by fixing $r$, consists of the districts in ${\mathscr{G}}$ that do not contain $r$, together with new districts that are subsets of $D^{r}$, the district in ${\mathscr{G}}$ that contains $r$. The new districts are bidirected-connected subsets of $D^{r}$ after removing $r$.

2.12 Fixing and factorization

Proposition 25**.**

Take a CADMG ${\mathscr{G}}(V,W)$ with kernel $q_{V}\in{\mathcal{P}}^{\mathrm{c}}({\mathscr{G}})$ with associated district factorization:

[TABLE]

where the kernels $q_{D}(x_{D}\,|\,x_{\operatorname{pa}_{{\mathscr{G}}}(D)\setminus D})$ are defined via the right-hand side of (16). If $r\in{\mathbb{F}}({{\mathscr{G}}})$ and $D^{r}\in{\mathcal{D}}({\mathscr{G}})$ is the district containing $r$, then

[TABLE]

The proof is found in the Supplementary Material, Section A.6 in Richardson et al. (2023). In words, the result of a fixing operation is solely to marginalize the variable $X_{r}$ from the density $q_{D^{r}}$ associated with the district $D^{r}$ in which the vertex $r$ occurs, while leaving unchanged all of the other terms $q_{D}$ in the factorization.

2.13 Reachable graphs derived from an ADMG

A sequence of distinct vertices ${\mathbf{w}}$ is said to be valid in ${\mathscr{G}}$ if either ${\mathbf{w}}=\langle\rangle$, or ${\mathbf{w}}=\langle w_{1},w_{2},\ldots,w_{k}\rangle$, $w_{1}\in\mathbb{F}({\mathscr{G}})$ and $\langle w_{2},\ldots,w_{k}\rangle$ is valid in $\phi_{w_{1}}({\mathscr{G}})$. Given a valid ${\mathbf{w}}$, we define the fixing operator $\phi_{\mathbf{w}}({\mathscr{G}})$ on graphs, and the fixing operator $\phi_{\mathbf{w}}(q_{V};{\mathscr{G}})$ on kernels inductively as follows:

[TABLE]

Definition 26**.**

A CADMG ${{\mathscr{G}}}(V,W)$ is reachable from an ADMG ${{\mathscr{G}}}^{*}(V\cup W)$ if there exists a valid fixing sequence $\mathbf{w}$ of the vertices in $W$ such that ${\mathscr{G}}=\phi_{\mathbf{w}}({{\mathscr{G}}^{*}})$.

In words, a graph is reachable from ${\mathscr{G}}^{*}$ if there exists an ordering on vertices in $W$ such that the first element $w_{1}$ in the ordering may be fixed in ${\mathscr{G}}^{*}$, the second element $w_{2}$ in $\phi_{\langle w_{1}\rangle}({\mathscr{G}}^{*})$, the third element $w_{3}$ in $\phi_{\langle w_{1},w_{2}\rangle}({\mathscr{G}}^{*})$, and so on. Note that by definition ${\mathscr{G}}$ is reachable from itself. If a CADMG ${{\mathscr{G}}}(V,W)$ is reachable from ${{\mathscr{G}}}^{*}(V\cup W)$, we say that $V$ is reachable in ${\mathscr{G}}^{*}$.

A key result, which we will show later as Theorem 31, is that under the nested Markov model reachable CADMGs and their associated kernels are invariant with respect to any valid fixing sequence. It is not hard to see that if there are two valid fixing sequences ${\mathbf{w}}$ and ${\mathbf{u}}$ for $W$ then $\phi_{\mathbf{w}}({{\mathscr{G}}})=\phi_{{\mathbf{u}}}({{\mathscr{G}}})$. However, it requires more work to show that $\phi_{\mathbf{w}}(q_{V};{{\mathscr{G}}})=\phi_{{\mathbf{u}}}(q_{V};{{\mathscr{G}}})$.

3 Nested Markov models

In this section, we define a set of recursive Markov properties and a factorization, and show their equivalence, in Proposition 29 and Theorem 38. The models, which obey these properties, will be called “nested” Markov models. For the rest of this section, we will fix an ADMG ${\mathscr{G}}(V)$ and a density $p(x_{V})$.

Definition 27**.**

We say that a distribution $p(x_{V})$ is globally nested Markov with respect to ${\mathscr{G}}(V)$ if for all fixing sequences ${\mathbf{w}}$ valid in ${\mathscr{G}}$, the kernel $\phi_{{\mathbf{w}}}(p(x_{V});{\mathscr{G}})$ obeys the global Markov property for $\phi_{{\mathbf{w}}}({\mathscr{G}})$.

Note that the same graph may be reached by more than one sequence, that is, $\phi_{{\mathbf{w}}_{1}}({\mathscr{G}})=\phi_{{\mathbf{w}}_{2}}({\mathscr{G}})$ for two distinct valid sequences ${\mathbf{w}}_{1},{\mathbf{w}}_{2}$.

Definition 28**.**

We say that a distribution $p(x_{V})$ nested Markov factorizes with respect to ${\mathscr{G}}(V)$ if, for all valid fixing sequences ${\mathbf{w}}$ in ${\mathscr{G}}$, there exist kernels $\{f^{{\mathbf{w}}}_{D}(x_{D}\,|\,x_{\operatorname{pa}_{{\mathscr{G}}}(D)\setminus D}):D\in{\mathcal{D}}(\phi_{{\mathbf{w}}}({\mathscr{G}}))\}$ such that

[TABLE]

Proposition 29**.**

With respect to an ADMG ${\mathscr{G}}(V)$, a distribution $p(x_{V})$ is globally nested Markov if and only if it nested Markov factorizes.

3.1 Invariance to the order of fixing in an ADMG

In this section, we show that, given a distribution that obeys the nested Markov property with respect to an ADMG, any two valid fixing sequences that fix the same vertices will lead to the same reachable graph and kernel. For marginal distributions obtained from a hidden variable DAG, this claim follows by results in Tian and Pearl (2002) on identification of causal effects in hidden variable DAG models. However, for distributions which obey the nested Markov property for an ADMG, but which are not derived from any hidden variable DAG, the claim is far less obvious; see Example 47 below. For instance in the ADMG in Figure 8, the fixing sequence $\langle 4,3,1\rangle$, which leads to the kernel

[TABLE]

and the fixing sequence $\langle 3,4,1\rangle$, which leads to the kernel

[TABLE]

are both valid, and these two kernels are therefore the same, in the context of our model. This is not entirely obvious from inspecting these expressions. In addition, $q^{1}_{2,5}$ and $q^{2}_{2,5}$ are not functions of $x_{3}$ in our model; this is clear for $q^{1}_{2,5}$ since $x_{3}$ is summed out, but not so obvious for $q^{2}_{2,5}$.

Lemma 30**.**

Let ${\mathscr{G}}(V,W)$ be a CADMG with $r,s\in{\mathbb{F}}({\mathscr{G}})$ and let $q_{V}$ be a kernel Markov w.r.t. ${\mathscr{G}}$. Then

[TABLE]

In words, if we have a choice to fix two vertices in ${\mathscr{G}}$ then the order in which we do this does not affect the resulting graph, or kernel, provided that the original kernel is Markov with respect to ${\mathscr{G}}$.

Theorem 31**.**

Let ${p}(x_{V})$ be a distribution that is nested Markov with respect to an ADMG ${\mathscr{G}}$ (in either the sense of Definitions 27 or 28). Let ${\mathbf{u}},{\mathbf{w}}$ be different valid fixing sequences for the same set $W\subset V$. Then $\phi_{{\mathbf{u}}}({\mathscr{G}})=\phi_{{\mathbf{w}}}({\mathscr{G}})$ and

[TABLE]

Due to this theorem, our fixing operations $\phi_{{\mathbf{w}}}$, which were defined for a specific fixing sequence ${\mathbf{w}}$, can, under the model, be defined purely in terms of the set $W$ of nodes that were fixed; the order does not matter (provided that at least one valid fixing sequence exists). Consequently, we will subscript the fixing operator $\phi$ by a set rather than a sequence. That is, we write $\phi_{V\setminus R}({\mathscr{G}})$ and $\phi_{V\setminus R}(p(x_{V});{\mathscr{G}})$ to mean “apply the fixing operator $\phi_{\mathbf{w}}$, for any valid sequence ${\mathbf{w}}$ of elements in $V\setminus R$, to ${\mathscr{G}}$ and the pair $(p(x_{V});{\mathscr{G}})$, respectively.” For conciseness, and consistency with notation in Tian and Pearl (2002), we will also denote $\phi_{V\setminus R}({\mathscr{G}})$ by ${\mathscr{G}}[R]$.

Corollary 32**.**

If a distribution $p(x_{V})$ obeys the global nested Markov property for ${\mathscr{G}}(V)$, then for all reachable sets $R$, the kernel

[TABLE]

obeys the global Markov property for $\phi_{V\setminus R}({\mathscr{G}})$.

We will subsequently see that if we assume the existence of a latent variable DAG model (with observed variables $V\cup W$) that has latent projection ${\mathscr{G}}$, then if $W$ is fixable, the kernel $\phi_{W}(p(x_{V\cup W});{\mathscr{G}})$ can be interpreted as the intervention distribution $p(x_{V}\,|\,\operatorname{do}_{{\mathscr{G}}}(x_{W}))$; see Lemma D.3 in the Supplementary Material. In this context, a valid fixing sequence corresponds to a sequence of steps in the ID algorithm of Tian and Pearl (2002) that identify this intervention distribution; see Section 4.3. Consequently, were we to assume the existence of a DAG with latent variables, then the soundness of the ID algorithm would directly imply the equality (24). However, since we are not assuming such a DAG exists, $\phi_{W}(p(x_{V\cup W});{\mathscr{G}})$ may not correspond to an intervention distribution, and hence a separate proof is required; see Example 47 for an inequality constraint that is implied by the existence of a latent variable, but does not follow from the nested Markov property applied to the latent projection.

3.2 Intrinsic sets

We introduce the following two definitions that will prove useful.

Definition 33**.**

A set $C$ is intrinsic if it is a district in a reachable graph derived from ${\mathscr{G}}$. The set of intrinsic sets in an ADMG ${\mathscr{G}}$ is denoted by ${\mathcal{I({\mathscr{G}})}}$.

Definition 34**.**

For a set $R$ reachable in ${\mathscr{G}}$ and a vertex $v\in R$, with $\operatorname{ch}_{\phi_{V\setminus R}({\mathscr{G}})}(v)=\emptyset$, we define the Markov blanket of $v$ in $R$ to be

[TABLE]

Since every ancestral set $A$ is reachable in ${\mathscr{G}}$, this is a natural extension of our previous definition (14). The next two subsections give different characterizations of the nested Markov model.

3.3 The simplified nested factorization property

Theorem 35**.**

If $p(x_{V})$ nested Markov factorizes with respect to ${\mathscr{G}}$, then for every reachable $R$ in ${\mathscr{G}}$,

[TABLE]

Since all the sets $D$ quantified in the product are districts in a reachable graph derived from ${\mathscr{G}}$, it follows that in a nested Markov model every kernel corresponding to a reachable set can be constructed by combining kernels corresponding to intrinsic sets. We call (26) the simplified nested factorization.

3.4 The ordered local nested property

Notwithstanding Theorem 31, in general we may not know a priori that a distribution is Markov with respect to a graph. It is therefore helpful to have a canonical order in which variables should be fixed for the purposes of verifying that a distribution is in the nested Markov model.

Definition 36**.**

We define the intrinsic power DAG for an ADMG ${\mathscr{G}}$ and topological ordering to be the graph ${\mathfrak{I}}({\mathscr{G}})$ whose vertices are intrinsic sets ${\mathcal{I}}({\mathscr{G}})$, and such that there is an edge from $D$ to $C$ if one can obtain $C$ as the district of the maximal vertex in $D$ by fixing some other vertex in $D$.

To simplify the next definition, we set $\operatorname{fam}_{{\mathscr{G}}}(C):=C\cup\operatorname{pa}_{{\mathscr{G}}}(C)$.

Definition 37**.**

Let ${\mathscr{G}}$ be an ADMG with arbitrary topological order $\prec$; define $T_{v}=\{w:w\prec v\}\cup\{v\}$ as the initial segment for $v$. A distribution $p(V)$ is ordered local nested Markov with respect to $\prec$, if:

• (i)

for each $v\in V$, we have the independences

[TABLE]

where $C=\operatorname{dis}_{T_{v}}(v)$;

• (ii)

for every edge $D\rightarrow C$ in ${\mathfrak{I}}({\mathscr{G}})$ (obtained by fixing $w\in D\setminus\{v\}$) and $v$ being maximal under $\prec$ in $D$, we have

[TABLE]

where $q_{D}$ is the unique kernel resulting from some valid fixing sequence for $D$.

Note that in this definition, we assume that we have reached each node $D$ by traversals from a root node (i.e., a set $C$ in (i) for some $v$). In this way, crossing the edge $D\to C$ will (potentially) introduce a new conditional independence, as well as defining $q_{C}$.

Theorem 38**.**

$p(x_{V})$* is globally nested Markov with respect to ${\mathscr{G}}$ if and only if $p(x_{V})$ is ordered local nested Markov with respect to ${\mathscr{G}}$ for any topological ordering $\prec$.*

Proposition 29 and Theorem 38 thus yield three equivalent ways of defining the set ${\mathcal{P}}^{\mathrm{n}}({{\mathscr{G}}})$ of distributions in the nested Markov model of ${{\mathscr{G}}}$.

3.5 Nested Markov models for complete graphs are saturated

It is known that any distribution is Markov relative to a complete DAG or ADMG (i.e., a graph with at least one edge between every pair of vertices). This also holds in the nested Markov case.

Theorem 39**.**

Let ${\mathscr{G}}$ be an ADMG. The model ${\mathcal{P}}^{\mathrm{n}}({\mathscr{G}})$ is saturated if and only if for every valid fixing sequence $r_{1},\ldots,r_{k}$, and every $i,j\in 1,\ldots,k$, either $r_{i}\in\operatorname{mb}_{{\mathscr{G}}^{(j)}}(r_{j})$ or $r_{j}\in\operatorname{mb}_{{\mathscr{G}}^{(i)}}(r_{i})$. Here, ${\mathscr{G}}^{(1)}\equiv{\mathscr{G}}$ and ${\mathscr{G}}^{(\ell+1)}\equiv\phi_{r_{\ell}}({\mathscr{G}}^{(\ell)})$.

Corollary 40**.**

Let ${\mathscr{G}}$ be a complete ADMG; then ${\mathcal{P}}^{\mathrm{n}}({\mathscr{G}})$ is saturated.

4 Connections with causal inference

The fixing operation is closely related to the identification of intervention distributions.

4.1 Latent variable DAG models are in the nested Markov model

We first show that if $p(x_{L\cup V})$ is Markov relative to a DAG ${\mathscr{G}}(L\cup V)$, then $p(x_{V})$ is in the nested Markov model of ${\mathscr{G}}(V)$. We extend the latent projection operation in the natural way from ADMGs to CADMGs, and denote the operation of creating a latent projection ${\mathscr{G}}(V,W)$ of a CADMG ${\mathscr{G}}(L\cup V,W)$ onto the subset $V$ as $\sigma_{L}$. That is, $\sigma_{L}({\mathscr{G}}(L\cup V,W))={\mathscr{G}}(V,W)$, and ${\mathscr{G}}(V,W)^{|W}$ encodes the m-separation relations holding among $V\cup W$ in ${\mathscr{G}}(L\cup V,W)^{|W}$. Precise statements appear in the Supplementary Material, Section A.3.

We will call a CADMG, which does not contain bidirected arrows, a conditional acyclic directed graph (CDAG). It is a corollary of the definition of ${\mathcal{P}}^{\mathrm{c}}_{\mathrm{f}}$ that if ${\mathscr{G}}(V,W)$ is a CDAG, then $q_{V}(x_{V}\,|\,x_{W})\in{\mathcal{P}}^{\mathrm{c}}_{\mathrm{f}}({\mathscr{G}})$ if

[TABLE]

Lemma 41**.**

Let ${\mathscr{G}}$ be a DAG with a vertex set $V$. Then every nonempty subset $S$ of $V$ is reachable, and if $p(x_{V})$ is Markov with respect to ${\mathscr{G}}$,

[TABLE]

In other words, $\phi_{V\setminus S}(p(x_{V});{\mathscr{G}})\in{\mathcal{P}}^{\mathrm{c}}(\phi_{V\setminus S}({\mathscr{G}}))$.

For a DAG ${\mathscr{G}}$, let ${\mathcal{P}}^{\mathrm{d}}({\mathscr{G}})$ denote the set of distributions Markov with respect to ${\mathscr{G}}$ (see the Supplementary Material, Section A.1, for details).

Corollary 42**.**

For a DAG ${\mathscr{G}}$, ${\mathcal{P}}^{\mathrm{d}}({\mathscr{G}})={\mathcal{P}}^{\mathrm{n}}({\mathscr{G}})$.

That ${\mathcal{P}}^{\mathrm{d}}({\mathscr{G}})\subseteq{\mathcal{P}}^{\mathrm{n}}({\mathscr{G}})$ is a special case of Theorem 46 below when $L$ is empty.

Lemma 43**.**

Let ${\mathscr{G}}(L\cup V,W)$ be a CDAG. Assume $v\in V$ is fixable in ${\mathscr{G}}(V,W)=\sigma_{L}({\mathscr{G}}(L\cup V,W))$. Then $\sigma_{L}(\phi_{v}({\mathscr{G}}(L\cup V,W)))=\phi_{v}(\sigma_{L}({\mathscr{G}}(L\cup V,W)))$. That is, the commutative diagram in Figure 9(a)* holds.*

Note that $v\in V$ is fixable in ${\mathscr{G}}(V,W)$ by assumption, while the fact that every element of $V$ is fixable in ${\mathscr{G}}(L\cup V,W)$ follows since ${\mathscr{G}}(L\cup V,W)$ is a CDAG, and has no bidirected edges.

In fact, an inspection of the proof of this lemma (found in the Supplementary Material) shows it does not rely on the vertex $v$ being fixable in ${{\mathscr{G}}}(V,W)$, only on the specific way edges are removed by $\phi$. In Lemma 45, we give a more general version of this result, that is useful for deriving properties of causal models, which we discuss later in Section 4.2.

Given a DAG ${\mathscr{G}}(V\dot{\cup}L)$, there is always a well-defined intervention distribution associated with any vertex $v\in V$; however, if $v$ is not fixable in ${\mathscr{G}}(V)$, then this distribution may not be identifiable from $p(x_{V})$; see (29) below. We use $\phi^{*}_{v}$ to denote the graphical operation that corresponds to this intervention, and note that this applies to all vertices in ${\mathscr{G}}(V)$, as opposed to $\phi_{v}$, which presupposes identifiability from $p(x_{V})$, and hence applies only to elements in $\mathbb{F}({\mathscr{G}}(V))$. The resulting graph $\phi_{v}^{*}({\mathscr{G}}(V))$, without distinguishing fixed and random vertices, was denoted ${\mathscr{G}}_{\overline{v}}$ by Pearl (2000). We use $\phi^{*}_{v}$ to make the connection to fixing more explicit.

For any $r,s\in V$, $\phi^{*}_{\langle r,s\rangle}({\mathscr{G}})=\phi^{*}_{\langle s,r\rangle}({\mathscr{G}})$, and so for any $Z\subseteq V$, we define $\phi^{*}_{Z}({\mathscr{G}})$ inductively under any order as in Lemma 30.

Corollary 44**.**

Let ${\mathscr{G}}(L\cup V,W)$ be a CDAG. Then for any $v\in V$, $\sigma_{L}(\phi^{*}_{v}({\mathscr{G}}(L\cup V,W)))=\phi^{*}_{v}(\sigma_{L}({\mathscr{G}}(L\cup V,W)))$.

Corollary 44 is essentially equivalent to Proposition 8 of Evans (2016).

Lemma 45**.**

Let ${\mathscr{G}}(L\cup V,W)$ be a CDAG, and assume $q_{L\cup V}(x_{L\cup V}\,|\,x_{W})\in{\mathcal{P}}^{\mathrm{c}}_{\mathrm{f}}({\mathscr{G}}(L\cup V,W))$. Assume $v\in V$ is fixable in ${\mathscr{G}}(V,W)=\sigma_{L}({\mathscr{G}}(L\cup V,W))$. Then

[TABLE]

That is, the commutative diagram in Figure 9(b)* holds.*

Robins (1986) proves a similar result that he calls the “collapse of the g-formula.” We now have enough to prove the main result of this section.

Theorem 46**.**

Let ${\mathscr{G}}(V\cup L)$ be a DAG. Then

[TABLE]

Thus, the constraints implied by the nested model for the latent projection of ${\mathscr{G}}(V\cup L)$ also hold in the hidden variable CDAG model. Note that the converse is not true in general. There are distributions $p(x_{V})\in{\mathcal{P}}^{\mathrm{n}}({\mathscr{G}}(V))$ for which there exists no joint distribution $p(x_{V\cup L})\in{\mathcal{P}}^{\mathrm{d}}({\mathscr{G}}(V\cup L))$. This is because marginals of hidden variable DAGs may induce additional inequality constraints, which are not satisfied by all elements of the associated nested Markov model. The best known of these are the instrumental inequalities of Pearl (1995), which were generalized by Bonet (2001), Evans (2012) and Kédagni and Mourifié (2020). Building on the above result, Evans (2018) showed further that the discrete latent variable DAG model does not imply any additional equality constraints not implied by the discrete nested Markov model.

Example 47**.**

Consider variables $X_{0},\ldots,X_{4}$ under a distribution, which is Markov with respect to the graph in Figure 1(i). Then the marginal distribution over $X_{1},\ldots,X_{4}$ satisfies the nested Markov property with respect to the graph in Figure 3(i). However, if the observed variables are binary (and regardless of the state-space of $X_{0}$) their marginal distribution also satisfies the following inequality constraints not implied by the nested Markov property:

[TABLE]

for each $x_{1},x_{3}\in\{0,1\}$; here, for example, $0_{2}$ is a shorthand for the event $\{X_{2}=0\}$. These are related to the CHSH inequalities of Clauser et al. (1969), and are sometimes referred to as Bell inequalities after Bell (1964).

4.2 Causal model of a DAG

The statistical model of a DAG ${\mathscr{G}}$ with vertices $V$, described earlier, is a set of distributions $p(x_{V})$ defined by the factorization (A.1) formulated on ${\mathscr{G}}$. We define a causal model of a DAG ${\mathscr{G}}$ by a set of similar factorizations that yield joint distributions under an intervention operation, which corresponds to setting values of variables from outside the system.

Specifically, for a DAG ${\mathscr{G}}$ with vertices $V$, and any $A\subseteq V$, the causal model for ${\mathscr{G}}$ defines the kernel resulting from intervention on $x_{A}$ to be

[TABLE]

This is known as the g-formula, truncated factorization or manipulated distribution. Note that since for any DAG ${\mathscr{G}}$, ${\mathcal{P}}^{\mathrm{d}}({\mathscr{G}})={\mathcal{P}}^{\mathrm{n}}({\mathscr{G}})$, we have

[TABLE]

We will drop the ${\mathscr{G}}$ subscript from $\operatorname{do}(\cdot)$ when the graph is obvious. Thus, (30) provides a causal interpretation of the fixing operation in a DAG.

In this light the earlier propositions relating to the preservation of independence in kernels may also be reinterpreted. Specifically, Proposition 9 (Separation) states that a variable that is intervened on no longer depends on its direct causes. Proposition 10 (Ordering) states that interventions in the future cannot causally affect the past. While Proposition 11 (Modularity) says that in the setting given, intervening and conditioning are interchangeable.

4.3 Reformulation of the ID algorithm via fixing

Identification of causal effects is a more difficult problem in causal DAGs where some variables are unobserved. In particular, not every distribution $p(x_{Y}\mid\operatorname{do}(x_{A}))$ is identified in every ${\mathscr{G}}(L\cup V)$. Given a causal model ${\mathscr{G}}(L\cup V)$, the goal is to find an identifying functional for $p(x_{Y}\mid\operatorname{do}(x_{A}))$ in terms of the observed marginal distribution $p(x_{V})$ or to show that no such functional exists.

The problem may be formalized by considering a latent projection ADMG ${\mathscr{G}}(V)$ in place of the original causal DAG with hidden variables, ${\mathscr{G}}(L\cup V)$. A well-known “folklore” result in causal inference states that any two hidden variable DAGs ${\mathscr{G}}^{1}(L^{1}\cup V)$ and ${\mathscr{G}}^{2}(L^{2}\cup V)$ with the same latent projection ${\mathscr{G}}(V)$ will share all identifying functionals, and so there is no loss of generality in reasoning on ${\mathscr{G}}(V)$. We prove this folklore result later in this section as Corollary 49.

The ID algorithm, introduced by Tian and Pearl (2002) solves the identification problem; the algorithm, which applies to ADMGs, was proved to be complete by Shpitser and Pearl (2006) and also independently by Huang and Valtorta (2006). Here, “complete” means that whenever the algorithm fails to find an expression for $p(x_{Y}\mid\operatorname{do}(x_{A}))$ in terms of $p(x_{V})$ in the causal model given by ${\mathscr{G}}(L\cup V)$, no other algorithm is able to do so without making more assumptions. In this section, we use the fixing operation to give a simple constructive characterization (via an algorithm) of all causal effects identifiable by the ID algorithm, and thus of all causal effects identifiable in any hidden variable causal DAG ${\mathscr{G}}(L\cup V)$. We can view this characterization as using the fixing operation to simplify the ID algorithm from its original two-page formulation down to the single formula (31).

Theorem 48**.**

Let ${\mathscr{G}}(L\cup V)$ be a causal DAG with latent projection ${\mathscr{G}}(V)$. For $A\mathbin{\dot{\cup}}Y\subseteq V$, let $Y^{*}=\operatorname{an}_{{\mathscr{G}}(V)_{V\setminus A}}(Y)$. Then if ${\mathcal{D}}({\mathscr{G}}(V)_{Y^{*}})\subseteq{\mathcal{I}}({\mathscr{G}}(V))$,

[TABLE]

If not, there exists $D\in{\mathcal{D}}({\mathscr{G}}(V)_{Y^{*}})$ that is not intrinsic in ${\mathscr{G}}(V)$, and $p(x_{Y}\mid\operatorname{do}_{{\mathscr{G}}(L\cup V)}(x_{A}))$ is not identifiable in ${\mathscr{G}}(L\cup V)$.

Note that $Y^{*}$ is the set of vertices $v\in V\setminus A$ for which, for some $y\in Y$, there is a directed path $v\rightarrow\cdots\rightarrow y$, with no vertex on the path in $A$; such paths are called “proper causal paths” in Perković et al. (2015). Also note that since, by Theorem 46, X_{D}\mathchoice{\mathrel{\hbox to0.0pt{\displaystyle\perp\hss}\mkern 3.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{\textstyle\perp\hss}\mkern 3.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptstyle\perp\hss}\mkern 3.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptscriptstyle\perp\hss}\mkern 3.0mu{\scriptscriptstyle\perp}}}X_{V\setminus(D\cup\operatorname{pa}_{{\mathscr{G}}(V)}(D))}\mid X_{\operatorname{pa}_{{\mathscr{G}}(V)}(D)\setminus D} in $\phi_{V\setminus D}(p(x_{V});{\mathscr{G}}(V))$, it follows that $\phi_{V\setminus D}(p(x_{V});{\mathscr{G}}(V))$ is a function solely of $x_{D}$ and $x_{\operatorname{pa}_{{\mathscr{G}}(V)}(D)\setminus D}$. Thus, the product on the RHS of (31) is a function of the “bound” variables $x_{Y^{*}\setminus Y}$ present in the sum and (a subset of) the “input” variables on the LHS, $x_{Y}$, $x_{A^{*}}$ where $A^{*}=A\cap\bigcup_{D\in\mathcal{D}({\mathscr{G}}(V)_{Y*})}\operatorname{pa}_{{\mathscr{G}}(V)}(D)$.

Corollary 49**.**

Let ${\mathscr{G}}^{1}(L^{1}\cup V)$ and ${\mathscr{G}}^{2}(L^{2}\cup V)$ be two causal DAGs, with the same latent projection, so ${\mathscr{G}}^{1}(V)={\mathscr{G}}^{2}(V)$. Then for any $A\dot{\cup}Y\subseteq V$:

• (i)

$p(Y\mid\operatorname{do}_{{\mathscr{G}}^{1}}(A))$* is identified if and only if $p(Y\mid\operatorname{do}_{{\mathscr{G}}^{2}}(A))$ is identified;*

• (ii)

if $p(Y\mid\operatorname{do}_{{\mathscr{G}}^{1}}(A))$ is identified, then $p(Y\mid\operatorname{do}_{{\mathscr{G}}^{1}}(A))=p(Y\mid\operatorname{do}_{{\mathscr{G}}^{2}}(A))$.

Example 50**.**

Given some hidden variable DAG ${\mathscr{G}}(V\cup L)$, where $V=\{x_{1},\ldots,x_{4}\}$ with latent projection ${\mathscr{G}}(V)$ given by the ADMG in Figure 3(i), consider the problem of identifying $p(x_{4}\mid\operatorname{do}_{{\mathscr{G}}}(x_{2}))$. Mapping this problem to the notation of Theorem 48, we have $Y=\{4\}$, $A=\{2\}$, $Y^{*}=\{4,3,1\}$. The districts of ${{\mathscr{G}}}_{Y^{*}}$ are $\{4\}$, $\{3\}$ and $\{1\}$. In fact, these three sets are intrinsic in ${\mathscr{G}}$, and thus a fixing sequence exists for each corresponding kernel:

[TABLE]

[TABLE]

[TABLE]

The last step here follows because $q_{24}(x_{2},x_{4}\,|\,x_{1},x_{3})=p(x_{2}\,|\,x_{1})p(x_{4}\,|\,x_{1},x_{2},x_{3})$, and $q_{24}(x_{2}\,|\,x_{1},x_{3},x_{4})=q_{24}(x_{2},x_{4}\,|\,x_{1},x_{3})/(\sum_{x_{2}}q_{24}(x_{2},x_{4}\,|\,x_{1},x_{3}))$. Combining these kernels as in Theorem 48 yields the same identifying functional as the one obtained by the ID algorithm applied to ${\mathscr{G}}$:

[TABLE]

where we relabel $x_{2}$ as $x^{\prime}_{2}$ in the last kernel to avoid confusion between free and summation quantifier-captured versions of the variable $x_{2}$ in the final expression.

4.4 Connections with Tian’s constraint algorithm

An algorithm for enumerating constraints on kernels in marginals of DAG models was given in Tian and Pearl (2002). Tian’s algorithm may be viewed as implementing fixing for both graphs and kernels, with three important differences from our formalization. First, unlike CADMGs, subgraphs obtained by fixing in Tian and Pearl (2002) do not show fixed nodes explicitly. This makes it difficult to graphically represent constraints that may involve fixed nodes. Second, the kernel objects obtained in intermediate steps of the algorithm, called “q-factors” and written as $Q[V]$ where $V$ is the set of nodes not yet fixed, do not explicitly show the dependence on nodes already fixed. This makes it hard to explicitly write down restrictions on q-factors, since these restrictions often state that dependence on some already fixed nodes does not exist. Third, there is no unified fixing operation on kernels, instead the algorithm in Tian and Pearl (2002) alternates between steps corresponding to the application of the g-formula (division by a conditional density), and steps corresponding to marginalization.

For a given DAG ${\mathscr{G}}(V\cup L)$ and a density $p(x_{V\cup L})$ Markov relative to ${\mathscr{G}}(V\cup L)$, a subset of observable nodes $V$, and a topological order $\prec$ on ${\mathscr{G}}$, Tian’s constraint algorithm gives a list of constraints of the form “a kernel corresponding to a q-factor $Q[C]$ obtained by some set of applications of the g-formula and marginalization on $p(x_{V})$ does not functionally depend on a set $X_{D}$, for some $D\subseteq V$.”

The algorithm in Tian and Pearl (2002) was developed in the context of hidden variable DAG models only. We have reformulated this algorithm, using the language of CADMGs and kernels, to yield an algorithmic specification of a set of generalized independence constraints implied by the larger nested model, which does not rely on the existence of an underlying hidden variable DAG; see Algorithm 1 in the Supplementary Material, Section E. As noted before, this is not a trivial reformulation, since the issue of invariance to choice of valid fixing sequences arises if we do not assume the existence of an underlying hidden variable DAG. We now state a key result relating this algorithm and the nested Markov model.

Theorem 51**.**

For an ADMG ${\mathscr{G}}(V)$, let ${\mathcal{P}}_{\mathrm{t}}({\mathscr{G}},V,\prec)$ be the set of densities $p(x_{V})$ in which the list of constraints found by Algorithm 1 holds. Then ${\mathcal{P}}_{\mathrm{t}}({\mathscr{G}},V,\prec)={\mathcal{P}}^{\mathrm{n}}({\mathscr{G}}(V))$.

Thus, the set of constraints given by Algorithm 1 implicitly defines the nested Markov model.

4.5 Connections with r-factorization

Shpitser et al. (2011) used constraints in causal DAG models with latent variables to construct a variable elimination (VE) algorithm for evaluating causal queries $p(x_{Y}\,|\,\operatorname{do}(x_{A}))$ in a computationally efficient manner. This algorithm used an older definition called the “r-factorization property.” The nested Markov model r-factorizes, which implies that the VE algorithm applies to these models as well.

Theorem 52**.**

If $p(x_{V})\in{\mathcal{P}}^{\mathrm{n}}({\mathscr{G}}(V))$, then $p(x_{V})$ r-factorizes with respect to ${\mathscr{G}}$ and $\{\phi_{V\setminus C}(p(x_{V});{\mathscr{G}})\mid C\in{\mathcal{I}}({\mathscr{G}})\}$.

5 Summary

We have introduced a novel statistical model defined by the equality constraints holding in marginals of DAG models, such as the Verma constraint. Though this model represents constraints found in marginal distributions, it does not itself model latent variables explicitly; indeed the existence of a latent variable may imply additional inequality constraints not captured by our model; see Example 47. We call this model the nested Markov model, and it is represented by means of an acyclic directed mixed graph (ADMG). Our model is “nested” because it is recursively defined. Specifically, just as a DAG model links a graph and a distribution, the nested model links sets of graphs derived from the original ADMG by a graphical fixing operation with sets of kernels obtained from the original distribution by an analogous fixing operation. This operation unifies certain marginalizations, conditioning operations and applications of the g-formula. Central to our model definition is the fact that any two valid sequences of fixing operations that fix the same set of nodes give the same result. We have characterized our model via Markov properties and a factorization. We have also shown a close connection between our model and a constraint enumeration algorithm for marginals of causal DAG models given in Tian and Pearl (2002), and used the fixing operation to characterize all identifiable causal effects in hidden variable DAG models using a one line formula (31).

Acknowledgements

We thank Zhongyi Hu for pointing out an error in the proof of Proposition 6, and the Associate Editor and referees for their helpful comments. The authors completed work on this paper while visiting the American Institute for Mathematics and the Simons Institute, Berkeley, California. The first author was supported in part by ONR Grants N00014-19-1-2446 and N00014-15-1-2672, and NIH Grant R01 AI032475.

The third author was supported in part by ONR Grant N00014-19-1-2446, and NIH Grant R01 AI032475.

The fourth author was supported in part by ONR Grant N00014-21-1-2820, NSF Grants 2040804 and 1942239, and NIH Grant R01 AI127271-01A1.