SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Adam Li
Department of Computer Science, Columbia University
adam.li@columbia.edu
Amin Jaber
Synlico Inc.
amin.jaber@synlico.com
Elias Baireinboim
eb@cs.columbia.edu
A fundamental problem in many sciences is the learning of causal structure underlying a system, typically through observation and experimentation. Commonly, one even collects data across multiple domains, such as gene sequencing from different labs, or neural recordings from different species. Although there exist methods for learning the equivalence class of causal diagrams from observational and experimental data, they are meant to operate in a single domain. In this paper, we develop a fundamental approach to structure learning in non-Markovian systems (i.e. when there exist latent confounders) leveraging observational and interventional data collected from multiple domains. Specifically, we start by showing that learning from observational data in multiple domains is equivalent to learning from interventional data with unknown targets in a single domain. But there are also subtleties when considering observational and experimental data. Using causal invariances derived from do-calculus, we define a property called S-Markov that connects interventional distributions from multiple-domains to graphical criteria on a selection diagram. Leveraging the S-Markov property, we introduce a new constraint-based causal discovery algorithm, S-FCI, that can learn from observational and interventional data from different domains. We prove that the algorithm is sound and subsumes existing constraint-based causal discovery algorithms.
Causal discovery is the process of learning cause-and-effect relationships between variables in a given system, which is many times the final goal of the data scientist or a necessary step towards a more refined causal analysis [1, 2]. The learning process typically leverages constraints from data to infer the corresponding causal diagram. However, it is common that the data constraints do not uniquely identify the full diagram. Therefore, the target of analysis is often an equivalence class (EC) of causal diagrams that encodes constraints found in the data (implied by the underlying unknown causal system).
An EC encodes invariances in the form of graphical constraints, and thus is used to represent all causal diagrams that encode those constraints and invariances. Formal characterizations of ECs are
important to understand the output of a learning algorithm and how it relates to the underlying causal system the scientist aims to explain.
ECs are defined with respect to distributional invariances which are implied by the structure of the graph. For example, conditional independences (CI) are implied by d-separations in the causal graph. Hence, it is desirable to formally characterize the EC in the general setting where we have interventional data from multiple domains. A complete graphical characterization would enable i) an efficient representation of the distributional invariances in the data and ii) the ability to translate these data-invariances to graphical constraints (e.g. d-separation).
An early example of an EC when only observational data is available in a single domain is the Markov equivalence class (MEC). The MEC characterizes causal diagrams with the same set of d-separation statements over observed nodes [2-5]. Given interventional (i.e. experimental) data, one can reduce the size of the equivalence class [6, 7]. In the case of known intervention targets, the EC is known as the  $\mathcal{I}$ -MEC [7-9] and in the case of unknown targets, it is called the  $\Psi$ -MEC [6].
In prior research, domain-changes and interventions were treated similarly [10-14]. Nevertheless, various examples across scientific disciplines highlight their distinction (see Table 1). For instance, when extrapolating data-driven conclusions from bonobos to humans, consider Figure 1(b). Notably, the environment/domain, represented by the S-node pointing to  $X$ , illustrates differences in kidney function between the species. When applying a CRISPR intervention to a gene linked to kidney protein production  $(X)$ , researchers investigate the impact of medication  $(Y)$  on fluid balance in the body  $(Z)$ . This intervention is explicitly different from the kidney-function differences between bonobos and humans because the change-in-domain is there regardless of whether or not an intervention is made. This differentiation between interventions and domains holds significance, especially in causal discovery. By leveraging invariances across observational and interventional data from both bonobos and humans, one can learn additional causal relationships. Moreover, conflating these qualitatively distinct settings is generally invalid, as pointed out in transportability analysis [15]. Pearl and Baireinboim (2011) introduced clear semantics for S-nodes (environments), offering a unified representation in the form of a selection diagram.
In this paper, we investigate structure learning when mixtures of observational and interventional data (known and unknown targets) across multiple domains are available. The multi-domain setting has been analyzed from the lens of selection diagrams, where selection nodes (or S-nodes) encode distributional changes in the mechanisms, or exogenous variables due to a change in domain [16-18]. We will show in this paper a characterization of the EC for selection diagrams. Generalizing the structure learning setting to multiple domains requires a formal treatment because it is a common scenario in the sciences [19-31]; see Table 1 for an example of different settings and related literature). For example, in single-cell sequencing analysis, scientists are interested in analyzing the causal effects of proteins on one another. However, they may typically collect observational and/or experimental data from multiple labs (i.e. domains) and wish to combine them into one dataset. Also, scientists may collect observational and experimental data over multiple species in order to learn more about one specific species, or the relationships among species [25, 27, 32].
The celebrated FCI algorithm and its variants learn a partial ancestral graph (PAG), an MEC of causal diagrams, given purely observational data [1, 2, 33]. The  $\mathcal{I}$ -FCI (with known targets) and  $\Psi$ -FCI (with unknown targets) generalize these results to interventional data, and further reduce the size of the EC to an  $\mathcal{I}$ -PAG and  $\Psi$ -PAG, respectively [6, 7]. However, these algorithms operate in a single domain, or environment and do not account for combining known/unknown target interventions.
Various approaches have been proposed throughout the literature for causal discovery from multiple domains. The works in [10, 13, 34-38] assume Markovianity, a functional model (e.g. linearity) holds, and/or do not take into account arbitrary combinations of observational and interventional data with known and unknown targets. Alternatively, JCI pools data together and performs learning on the combined dataset [14]. Pooling data is an incomplete procedure when considering interventional data within a single domain let alone multiple domains [6][Appendix D.2].
In this paper, we take a principled approach to the multi-domain structure learning problem and formally characterize S-PAGs, the object of learning. This paper introduces the selection-diagram FCI algorithm (S-FCI) that learns from a mixture of observational and interventional data from multiple domains to construct an EC of selection diagrams, an S-PAG. Specifically, we contribute the following:
Domain	Obs.	Interv.	Property	FCI-variant	Related Lit.
K	U
1	✓	x	x	Markov [39]	[2, 33, 40–43]	[30, 31]
1	✓	✓	x	I-Markov [7, 44]	[7, 8, 44]	[22, 30]
1	✓	x	✓	Ψ-Markov [6]	[6, 13, 45]	[22, 27, 46]
k	✓	x	x	Ψ-Markov (Thm. 1)	[6] (Cor. 5)	[20, 21, 23, 24, 47, 48]
k	✓	✓	✓	S-Markov (Thm. 2)	S-FCI (Thm. 3)	[20–25, 28–31, 46–50]
Table 1: Summary of Markov property results, and related algorithms that learn the ancestral graph based on number of domains and types of interventional (interv.) data provided such as observational (obs.), and known  $(\mathcal{K})$  and unknown  $(\mathcal{U})$  targets. The last column indicates a brief survey of different fields in ecology, economics, genomics, neurosciences, neurology and medicine that attempt to answer questions at each level. The rows highlighted in "red" are new concepts.
1. Generalization of standard Markov properties - We introduce the S-Markov property, which extends and generalizes the normal Markov,  $\mathcal{I}$ -Markov, and  $\Psi$ -Markov properties to the setting of multiple domains with arbitrary mixtures of observational and interventional data with known and unknown targets.
2. Learning algorithm - We develop a sound learning algorithm for learning an equivalence class of selection diagrams with observational and/or interventional data across different domains. $^{1}$
2 Preliminaries and Notation
Uppercase letters  $(X)$  represent random variables, lowercase letters  $(x)$  signify assignments, and bold ones  $(\mathbf{X})$  indicate sets. The CI relation  $\mathbf{X}$  being independent of  $\mathbf{Y}$  given  $\mathbf{Z}$  is denoted as  $\mathbf{X} \perp \mathbf{Y}|\mathbf{Z}$ . The d-separation (or m-separation) of  $\mathbf{X}$  from  $\mathbf{Y}$  given  $\mathbf{Z}$  in graph  $G$  is expressed as  $(\mathbf{X} \perp \mathbf{Y}|\mathbf{Z})_G$ .  $G_{\overline{X}}$  depicts  $G$  with incoming edges to  $X$  removed, while  $G_{\underline{X}}$  omits all edges outgoing from  $X$ . Conventionally, every variable is d-separated from the empty set, denoted as  $(X \perp \{\})_G$ . Superscripts and subscripts will be dropped where feasible to simplify notation.
Causal Bayesian Network (CBN): Let  $P(\mathbf{V})$  be a probability distribution over a set of variables  $\mathbf{V}$ , and  $P_{\mathbf{x}}(\mathbf{V})$  denote the distribution resulting from the hard intervention  $do(\mathbf{X} = \mathbf{x})$ , which sets  $\mathbf{X} \subseteq \mathbf{V}$  to constants  $\mathbf{x}$ . Let  $\mathbf{P}^*$  denote the set of all interventional distributions  $P_{\mathbf{x}}(\mathbf{V})$ , for all  $\mathbf{X} \subseteq \mathbf{V}$ , including  $P(\mathbf{V})$ . A directed acyclic graph (DAG) over  $\mathbf{V}$  is said to be a causal Bayesian network compatible with  $\mathbf{P}^*$  if and only if, for all  $\mathbf{X} \subseteq \mathbf{V}$ ,  $P_{\mathbf{x}}(\mathbf{v}) = \prod_{\{i|V_i \notin \mathbf{X}\}} P(v_i|\mathbf{pa}_i)$ , for all  $\mathbf{v}$  consistent with  $\mathbf{x}$ , and where  $\mathbf{Pa}_i$  is the set of parents of  $V_i$  [41, 51, pp. 24]. Given that a subset of the variables are unmeasured or latent,  $G(\mathbf{V} \cup \mathbf{L}, \mathbf{E})$  will represent the causal graph where  $\mathbf{V}$  and  $\mathbf{L}$  denote the measured and latent variables, respectively, and  $\mathbf{E}$  denotes the edges. Following the convention in [41], for simplicity, a dashed bi-directed edge is used instead of the corresponding latent variables. CI relations can be read from the graph using a graphical criterion known as  $d$ -separation.
Soft Interventions: Under this type of interventions, the original conditional distributions of the intervened variables  $\mathbf{X}$  are replaced with new ones, without completely eliminating the causal effect of the parents. Accordingly, the interventional distribution  $P_{\mathbf{X}}(\mathbf{v})$  for  $\mathbf{X} \subseteq \mathbf{V}$  is such that  $P^{*}(X_{i}|Pa_{i}) \neq P(X_{i}|Pa_{i}), \forall X_{i} \in \mathbf{X}$ , and factorizes as follows:
$$
P _ {\mathbf {X}} (\mathbf {v}) = \sum_ {\mathbf {L}} \prod_ {\{i \mid X _ {i} \in \mathbf {X} \}} P ^ {*} (x _ {i} | \mathbf {p a} _ {i}) \prod_ {\{j \mid T _ {j} \notin \mathbf {X} \}} P (t _ {j} | \mathbf {p a} _ {j}) \tag {1}
$$
In this work, we assume no selection bias and solely consider soft interventions. In the presence of multiple domains, a selection diagram captures commonalities and differences between domains [16, 52, 53]. Represented as  $G_{S} = (\mathbf{V}\cup \mathbf{L}\cup \mathbf{S},\mathbf{E}\cup \mathbf{E}_{\mathbf{S}})$ , it extends a causal diagram by incorporating S-nodes and their edges.  $\binom{N}{2}$  S-nodes,  $S^{i,j}$ , indicate distribution changes across pairs among N domains, by pointing to nodes in V whose mechanism is altered between domains i and j. An example is shown in Figure 1(a), where the S-node is pointing to  $X$ , indicating that the distribution of X
<sup>1</sup>Our algorithm is implemented in open-source MIT-Licensed https://github.com/py-why/dodiscover.
changes, or that of the latent variable of  $\mathrm{X}$  is different across the two domains. Similarly, "F-nodes" are auxiliary nodes used in [1, 7, 54] to represent invariances with respect to interventions within the same domain. F-nodes in this paper when written as  $F_{X}^{i,j}$  means it intervenes on  $\mathrm{X}$  and compares distributions from domains i and j.  $F_{X}^{i}$  means it compares distributions within domain i. Unlike interventions, domain-shifts potentially alter latent variable distributions or functional relationships and persist irrespective of whether or not external intervention occurs. Distinguishing these concepts enables S-node learning, vital for transportability analysis on ancestral graphs. Appendix Section E.4 elaborates on our distinctions from previous work [11, 13, 14, 36].
Let  $\mathbf{S} = \{S^{1,2}, S^{1,3}, \ldots, S^{N-1,N}\}$  represent  $\binom{N}{2}$  S-nodes for distribution changes across domain pairs. When  $i = j$ ,  $S^{i,j} = \phi$ , indicating there is no S-node for a single domain.
Multi-domain setup The following objects are utilized repeatedly, and introduced here. Our notation borrows from [6] and the transportability literature [55].
1. Domains:  $\Pi = \{\Pi^1, \Pi^2, \dots, \Pi^N\}$  denotes a set of N domains.
2. Intervention targets:  $\Psi^{\Pi} = \langle \Psi_1^1, \Psi_2^1, \dots, \Psi_M^N \rangle$  is an ordered tuple of sets of intervention targets, with different sets of intervention targets occurring within each of the N domains for a total of M intervention target sets. We will denote  $\Psi^i$  as the intervention targets associated with domain i.
3. Distributions:  $\mathbf{P}^{\Pi} = \langle P_1^1, P_2^1, \dots, P_M^N \rangle$  is an ordered tuple of probability distributions that are available to learn from. Denote  $\mathbf{P}^i$  as the distributions associated with domain i. There is a one-to-one correspondence between  $\mathbf{P}$  and  $\Psi$ , such that  $P_j^i$  is the distribution associated with targets  $\Psi_j^i$  in domain i.
4. Known target indices:  $\mathcal{K}$  is a vector of 1's and 0's indicating which sets of interventions are known-targets.  $\mathcal{U} := 1 - \mathcal{K}$  represents therefore an index vector selecting the distributions and interventions with unknown targets.  $\mathbf{P}_{\mathcal{K}}$  and  $\Psi_{\mathcal{K}}$  denotes the set of distributions and intervention targets corresponding to the known target interventions.
5. Causal diagram:  $G = (\mathbf{V} \cup \mathbf{L}, \mathbf{E})$ , is a shared diagram over the  $\mathbf{N}$  domains.
6. Selection diagram:  $G_{S} = (\mathbf{V} \cup \mathbf{L} \cup \mathbf{S}, \mathbf{E} \cup \mathbf{E}_{\mathbf{S}})$ , extends  $\mathbf{G}$  with the corresponding S-nodes and their edges to represent each pair of domains. Let  $\mathbf{V}_{S^{i,j}}$  denote the set of nodes that S-node  $S^{i,j}$  points to and  $\mathbf{V}_{\mathbf{S}}$  as the set of children for all S-nodes of  $G_{S}$ .
$\mathbf{X}^i$  denotes the  $i$ th domain set of variables  $\mathbf{X}$ , and  $X_i \in \mathbf{X}$  indicates the  $i$ th variable within  $\mathbf{X}$ . When discussing intervention targets,  $X_j^{i,(k)}$  refers to the  $j$ th variable with the  $k$ th mechanism change in domain  $i$ . For instance,  $X^{i,(k)}, X^{i,(l)}$  represent two interventions with distinct mechanisms ( $k$  and  $l$ ) on variable  $\mathbf{X}$  in domain  $i$ .  $\{\}^i \in \Psi$  explicitly denotes the observational distribution for domain  $i$  and is by convention a "known-target". For concreteness, say  $\Pi = \{\Pi^1, \Pi^2, \Pi^3\}$  with  $\mathbf{P} = \langle P_1^1, P_2^1, P_3^1, P_1^3 \rangle$ ,  $\Psi = \langle \{\}^1, \{X^{(a)}\}^1, \{X, Y\}^1, \{\}^3 \rangle$ , and  $\mathcal{K} = [1, 1, 0, 1]$ . In words, there are three distributions available in domain  $1$ :  $P_1^1$  is observational,  $P_2^1$  is known-target on  $\mathbf{X}$  with a specific mechanism change and  $P_3^1$  is unknown-target that intervenes on  $\mathbf{X}$  and  $\mathbf{Y}$  simultaneously. In domain  $3$ ,  $P_1^3$  is observational. There are no distributions for domain  $2$ .
3 Multi-domain Markov Equivalence Class
Before designing a learning algorithm, we must characterize what can be learned from the given selection diagram. This section explores ECs in a multi-domain setting with arbitrary mixtures of observational and interventional data. The following assumptions are made throughout the paper.
Assumption 1 (Shared causal structure). We assume that each environment shares the same causal diagram. That is the S-nodes do not change the underlying causal diagram.
This means that the S-nodes do not represent structural changes such as when  $V_{i}$  has a different parent set across domains.
In the original selection diagram, each S-node points to a single node. Our adaptation simplifies it to a single S-node with multiple connections. Theoretical properties remain unaffected, as shown in the appendix.
<sup>3</sup>The assumption that there are no structural changes between domains can be relaxed in the context of inference, as specified in [16]. We do not explore this relaxation here in the context of structure learning.
Assumption 2 (Observational data is present across domains). We make the simplifying assumption that  $\{\} \in \Psi^i$ ,  $\forall i \in [N]$ , that is observational data is present in all domains.
This is a realistic assumption in many scientific applications highlighted in Table 1. Another assumption we make is that all soft interventions across domains are distinct.
Assumption 3 (Distinct interventions across domains). We assume all the interventions across different domains have unique mechanisms. That is, if  $X^{(m)} \in \Psi^i$  and  $X^{(n)} \in \Psi^j$ , where  $i \neq j$ , then  $m \neq n$ . In words,  $X$  has different mechanisms across the two distributions  $P_{\Psi^i}, P_{\Psi^j}$ .
This is a realistic assumption that precludes the possibility that any interventions that occur in different domains result in the same exact mechanism. For example, even if medication is given to humans and bonobos, it is unrealistic to expect the intervention has the same mechanism of action in each domain. Next, we define an important operation when comparing two different intervention sets.
Definition 3.1 (Symmetrical Difference Operator  $\Delta$  in Multiple Domains). For two domains  $i, j$  (possibly  $i = j$ ), given two sets of intervention targets,  $\Psi^i$  and  $\Psi^j$ , let  $\Psi^i \Delta \Psi^j$  denote the symmetrical difference set such that  $X \in \Psi^i \Delta \Psi^j$  if  $X^{(k)} \in \Psi^i$  and  $X^{(k)} \notin \Psi^j$  or vice versa.
This operation identifies the set of variables with unique interventional mechanisms across two intervention targets and also tracks the domain ids. For example, given  $\mathbf{I}^1 = \{X^1,Y,Z\}^1$  and  $\mathbf{J}^1 = \{X^2,Y\}^2$ , then  $\mathbf{I}^1\Delta \mathbf{J}^1 = \{X,Z\}^{1,2}$ . An implication of the above definition and Assumption 3 is that the symmetrical difference of two intervention target sets from two different domains is the union of all the variables in both sets since the mechanisms would be unique. For more details and discussion on the assumptions, see the Appendix.
3.1 Multi-distributional invariances: interventions and change-of-domain
This section elaborates on exactly what type of distributional invariances we characterize in the so called S-Markov EC.
When given only observational data, the celebrated FCI algorithm uses invariances of the form  $P(\mathbf{Y}|\mathbf{X},\mathbf{W}) = P(\mathbf{Y}|\mathbf{X})$  within the same probability distribution  $P(\mathbf{V})$  to characterize the Markov EC [2]. These invariances, or CI statements can be mapped to d-separation statements in the graphical model. The resulting learned object is the PAG, which represents the EC when only observational data is given within a single domain.
The works in [6-8, 44] build upon the Markov EC to characterize the so called interventional Markov EC, which uses distributional invariances of the form  $P_{\mathbf{W}}(\mathbf{Y}|\mathbf{X}) = P_{\mathbf{Z}}(\mathbf{Y}|\mathbf{X})$ . In words, these are conditional probabilities that remain invariant under different interventions. Importantly, this sort of invariance is markedly different from that of the CI statements, where only observational data is present, because one is now comparing probabilities across different distributions. These distributional invariances can be characterized graphically by the d-separation property when using an augmented graph with "F-nodes", which serve as graphical representations of the differences in distributions due to interventions. However, this body of work assumes that all the distributions, observational and interventional, are within the same domain.
In this work, we generalize this setting and consider an input set of distributions from (possibly) different domains to characterize the S-Markov EC. We consider distributional invariances of the form  $P_{\mathbf{W}}^{i}(\mathbf{Y}|\mathbf{X}) = P_{\mathbf{K}}^{j}(\mathbf{Y}|\mathbf{X})$  such that distributions could stem from different domains when  $i \neq j$ . Such an invariance implies that the conditional distribution of  $\mathbf{Y}|\mathbf{X}$  remains the same across domains  $i$  and  $j$  under interventions on  $\mathbf{W}$  and  $\mathbf{K}$ , respectively. Whenever  $i = j$ , these invariances reduce to the ones considered in the interventional Markov EC. From this perspective, it is clear that multi-domain invariances generalize the invariances analyzed in observational and interventional data in a single-domain.
3.2 S-Markov Property
Now, we are ready to generalize the previous Markov properties [2, 5-7, 41, 56] to the case when observational, and known/unknown-target interventional distributions in multiple domains are available.
<sup>4</sup>If one can collect experimental data in a domain, it is reasonable that they can also collect observational data. We discuss this further in the Appendix.
Figure 1: Example selection diagrams (a,b) and their respective augmented graphs (c,d). (c) corresponds to intervention set  $\Psi = \langle \{\}^1, \{\}^2, \{X\}^1 \rangle$  and (d) corresponds to intervention set  $\Psi = \langle \{\}^1, \{\}^2, \{Z\}^1 \rangle$ .
Definition 3.2 (S-Markov Property). Given the Multi-domain setup,  $\mathbf{P}$  satisfies the S-Markov property with respect to the pair  $\langle G_S,\Psi \rangle$  if the following holds for disjoint  $\mathbf{Y},\mathbf{W},\mathbf{Z}\subseteq \mathbf{V}$ :
1. For  $\Psi_j^i\in \Psi$  ..  $P_{j}^{i}(\mathbf{y}|\mathbf{w},\mathbf{z}) = P_{j}^{i}(\mathbf{y}|\mathbf{w})$  if  $(\mathbf{Y}\perp \mathbf{Z}|\mathbf{W},\mathbf{S})_{G_S}$
2. For  $\Psi_{m}^{i},\Psi_{l}^{j}\in \Psi$  ..  $P_{m}^{i}(\mathbf{y}|\mathbf{w}) = P_{l}^{j}(\mathbf{y}|\mathbf{w})$  if  $(\mathbf{Y}\perp \mathbf{K}|\mathbf{W}\backslash \mathbf{W}_{\mathbf{K}},\mathbf{S}\setminus \{S^{i,j}\})_{G_{S}}$ $\underline{\mathbf{W}_{\mathbf{K}},\overline{\mathbf{R}(\mathbf{W})}}$
where  $\mathbf{K} = (\Psi_m^i\Delta \Psi_l^j)\cup \{S^{i,j}\}$ ,  $\mathbf{W}_{\mathbf{K}} = \mathbf{W}\cap \mathbf{K}$ ,  $\mathbf{R} = \mathbf{K}\backslash \mathbf{W}_{\mathbf{K}}$  and  $\mathbf{R}(\mathbf{W})\subseteq \mathbf{R}$  are non-ancestors of  $\mathbf{W}$  in  $G_{S}$ .
Let  $S_{\mathcal{K}}^{\Pi}(G_S, \Psi)$  denote the set of distribution tuples that satisfy the S-Markov property with respect to  $\langle G_S, \Psi \rangle$  where  $\mathcal{K}$  denotes the known intervention targets.