SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Liang Wendong  $^{1,2}$  Armin Kekić  $^{1}$  Julius von Kügelgen  $^{1,3}$  Simon Buchholz  $^{1}$
Michel Besserve<sup>1</sup> Luigi Gresele*<sup>1</sup> Bernhard Schölkopf*<sup>1</sup>
$^{1}$  Max Planck Institute for Intelligent Systems, Tübingen, Germany  
 $^{2}$  ENS Paris-Saclay, Gif-sur-Yvette, France  $^{3}$  University of Cambridge, United Kingdom {wendong.liang, armin.kekic, jvk, simon.buchholz}@tue.mpg.de {besserve, luigi.gresele, bs}@tue.mpg.de
Independent Component Analysis (ICA) aims to recover independent latent variables from observed mixtures thereof. Causal Representation Learning (CRL) aims instead to infer causally related (thus often statistically dependent) latent variables, together with the unknown graph encoding their causal relationships. We introduce an intermediate problem termed Causal Component Analysis (CauCA). CauCA can be viewed as a generalization of ICA, modelling the causal dependence among the latent components, and as a special case of CRL. In contrast to CRL, it presupposes knowledge of the causal graph, focusing solely on learning the unmixing function and the causal mechanisms. Any impossibility results regarding the recovery of the ground truth in CauCA also apply for CRL, while possibility results may serve as a stepping stone for extensions to CRL. We characterize CauCA identifiability from multiple datasets generated through different types of interventions on the latent causal variables. As a corollary, this interventional perspective also leads to new identifiability results for nonlinear ICA—a special case of CauCA with an empty graph—requiring strictly fewer datasets than previous results. We introduce a likelihood-based approach using normalizing flows to estimate both the unmixing function and the causal mechanisms, and demonstrate its effectiveness through extensive synthetic experiments in the CauCA and ICA setting.
Independent Component Analysis (ICA) [7] is a principled approach to representation learning, which aims to recover independent latent variables, or sources, from observed mixtures thereof. Whether this is possible depends on the identifiability of the model [32, 57]: this characterizes assumptions under which a learned representation provably recovers (or disentangles) the latent variables, up to some well-specified ambiguities [22, 58]. A key result shows that, when nonlinear mixtures of the latent components are observed, the model is non-identifiable based on independent and identically distributed (i.i.d.) samples from the generative process [8, 19]. Consequently, a learned model may explain the data equally well as the ground truth, even if the corresponding representation is strongly entangled, rendering the recovery of the original latent variables fundamentally impossible.
Identifiability can be recovered under deviations from the i.i.d. assumption, e.g., in the form of temporal autocorrelation [14, 18] or spatial dependence [15] among the latent components; auxiliary variables which render the sources conditionally independent [17, 21, 25, 26]; or additional, noisy views [11]. An alternative path is to restrict the class of mixing functions [4, 13, 63].
*Shared last author. Code available at https://github.com/akekic/causal-component-analysis.
Figure 1: Causal Component Analysis (CauCA). We posit that observed variables  $\mathbf{X}$  are generated through a nonlinear mapping  $\mathbf{f}$ , applied to unobserved latent variables  $\mathbf{Z}$  which are causally related. The causal structure  $G$  of the latent variables is assumed to be known, while the causal mechanisms  $\mathbb{P}_i(Z_i\mid \mathbf{Z}_{\mathrm{pa}(i)})$  and the nonlinear mixing function are unknown and to be estimated. (Known or observed quantities are highlighted in red.) CauCA assumes access to multiple datasets  $\mathcal{D}_k$  that result from stochastic interventions on the latent variables.
Despite appealing identifiability guarantees for ICA, the independence assumption can be limiting, since interesting factors of variation in real-world data are often statistically, or causally, dependent [51]. This motivates Causal Representation Learning (CRL) [45], which aims instead to infer causally related latent variables, together with a causal graph encoding their causal relationships. This is challenging if both the graph and the unmixing are unknown. Identifiability results in CRL therefore require strong assumptions such as counterfactual data [1, 3, 9, 37, 54], temporal structure [30, 33], a parametric family of latent distributions [5, 6, 30, 47, 59, 60], graph sparsity [28-30], pairs of interventions and genericity [55] or restrictions on the mixing function class [2, 49, 52]. It has been argued that knowing either the graph or the unmixing might help better recover the other, giving rise to a chicken-and-egg problem in CRL [3].
We introduce an intermediate problem between ICA and CRL which we call Causal Component Analysis (CauCA), see Fig. 1 for an overview. CauCA can be viewed as a generalization of ICA that models causal connections (and thus statistical dependence) among the latent components through a causal Bayesian network [41]. It can also be viewed as a special case of CRL that presupposes knowledge of the causal graph, and focuses on learning the unmixing function and causal mechanisms.
Since CauCA is solving the CRL problem with partial ground truth information, it is strictly easier than CRL. This implies that impossibility results for CauCA also apply for CRL. Possibility results for CauCA, on the other hand, while not automatically generalizing to CRL, can nevertheless serve as stepping stones, highlighting potential avenues for achieving corresponding results in CRL. Note also that there are only finitely many possible directed acyclic graphs for a fixed number of nodes, but the space of spurious solutions in representation learning (e.g., in nonlinear ICA) is typically infinite. By solving CauCA problems, we can therefore gain insights into the minimal assumptions required for addressing CRL problems. CauCA may be applicable to scenarios in which domain knowledge can be used to specify a causal graph for the latent components. For instance, in computer vision applications, the image generation process can often be modelled based on a fixed graph [44, 50].
Structure and Contributions. We start by recapitulating preliminaries on causal Bayesian networks and interventions in § 2. Next, we introduce Causal Component Analysis (CauCA) in § 3. Our primary focus lies in characterizing the identifiability of CauCA from multiple datasets generated through various types of interventions on the latent causal variables (§ 4). Importantly, all our results are applicable to the nonlinear and nonparametric case. The interventional perspective we take exploits the modularity of the causal relationships (i.e., the possibility to change one of them without affecting the others)—a concept that was not previously leveraged in works on nonlinear ICA. This leads to extensions of existing results that require strictly fewer datasets to achieve the same level of identifiability. We introduce and investigate an estimation procedure for CauCA in § 5, and conclude with a summary of related work (§ 6) and a discussion (§ 7). We highlight the following main contributions:
- We derive sufficient and necessary conditions for identifiability of CauCA from different types of interventions (Thm. 4.2, Prop. 4.3, Thm. 4.5).
- We prove additional results for the special case with an empty graph, which corresponds to a novel ICA model with interventions on the latent variables (Prop. 4.6, Prop. 4.7, Corollary 4.8, Prop. 4.9).
- We show in synthetic experiments in both the CauCA and ICA settings that our normalizing flow-based estimation procedure effectively recovers the latent causal components (§ 5).
Notation. We use  $\mathbb{P}$  to denote a probability distribution, with density function  $p$ . Uppercase letters  $X,Y,Z$  denote unidimensional and bold uppercase  $\mathbf{X},\mathbf{Y},\mathbf{Z}$  denote multidimensional random variables. Lowercase letters  $x,y,z$  denote scalars in  $\mathbb{R}$  and  $\mathbf{x},\mathbf{y},\mathbf{z}$  denote vectors in  $\mathbb{R}^d$ . We use  $[i,j]$  to denote the integers from  $i$  to  $j$ , and  $[d]$  denotes the natural numbers from 1 to  $d$ . We use common graphical notation, see App. A for details. The ancestors of  $i$  in a graph are the nodes  $j$  in  $G$  such that there is a directed path from  $j$  to  $i$ , and they are denoted by  $\mathrm{anc}(i)$ . The closure of the parents (resp. ancestors) of  $i$  is defined as  $\overline{\mathrm{pa}}(i) := \mathrm{pa}(i) \cup \{i\}$  (resp.  $\overline{\mathrm{anc}}(i) := \mathrm{anc}(i) \cup \{i\}$ ).
A key definition connecting directed acyclic graphs (DAGs) and probabilistic models is the following.
Definition 2.1 (Distribution Markov relative to a DAG [41]). A joint probability distribution  $\mathbb{P}$  is Markov relative to a DAG  $G$  if it admits the factorization  $\mathbb{P}(Z_1, \ldots, Z_d) = \prod_{i=1}^d \mathbb{P}_i(Z_i | \mathbf{Z}_{pa(i)})$ .
Defn. 2.1 is a key assumption in directed graphical models, where a distribution being Markovian relative to a graph implies that the graph encodes specific independences within the distribution, which can be exploited for efficient computation or data storage [43, §6.5].
Causal Bayesian networks and interventions. Causal systems induce multiple distributions corresponding to different interventions. Causal Bayesian networks [CBNs; 41] can be used to represent how these interventional distributions are related. In a CBN with associated graph  $G$ , arrows signify causal links among variables, and the conditional probabilities  $\mathbb{P}_i\left(Z_i \mid \mathbf{Z}_{\mathrm{pa}(i)}\right)$  in the corresponding Markov factorization are called causal mechanisms.
Interventions are modelled in CBNs by replacing a subset  $\tau_{k} \subseteq V(G)$  of the causal mechanisms by new, intervened mechanisms  $\{\widetilde{\mathbb{P}}_j(Z_j \mid \mathbf{Z}_{\mathrm{pa}^k(j)})\}_{j \in \tau_k}$ , while all other causal mechanisms are left unchanged. Here,  $\mathrm{pa}^k(j)$  denotes the parents of  $Z_{j}$  in the post-intervention graph in the interventional regime  $k$  and  $\tau_{k}$  the intervention targets. We will omit the superscript  $k$  when the parent set is unchanged and assume that interventions do not add new parents,  $\mathrm{pa}^k(j) \subseteq \mathrm{pa}(j)$ . Unless  $\widetilde{\mathbb{P}}_j$  is a point mass, we call the intervention stochastic or soft. Further, we say that  $\widetilde{\mathbb{P}}_j$  is a perfect intervention if the dependence of the  $j$ -th variable from its parents is removed ( $\mathrm{pa}^k(j) = \emptyset$ ), corresponding to deleting all arrows pointing to  $i$ , sometimes also referred to as graph surgery [48]. An imperfect intervention is one for which  $\mathrm{pa}^k(j) \neq \emptyset$ . We summarise this in the following definition.
Definition 2.2 (CBN). A causal Bayesian network (CBN) consists of a graph  $G$ , a collection of causal mechanisms  $\{\mathbb{P}_i(Z_i \mid \mathbf{Z}_{pa(i)})\}_{i \in [d]}$ , and a collection of interventions  $\{\{\widetilde{\mathbb{P}}_j^k(Z_j \mid \mathbf{Z}_{pa^k(j)})\}_{j \in \tau_k}\}_{k \in [K]}$  across  $K$  interventional regimes. The joint probability for interventional regime  $k$  is given by:
$$
\mathbb {P} ^ {k} (\mathbf {Z}) := \left\{ \begin{array}{l l} \prod_ {i = 1} ^ {d} \mathbb {P} _ {i} \left(Z _ {i} \mid \mathbf {Z} _ {p a (i)}\right) & k = 0 \\ \prod_ {j \in \tau_ {k}} \widetilde {\mathbb {P}} _ {j} ^ {k} \left(Z _ {j} \mid \mathbf {Z} _ {p a ^ {k} (j)}\right) \prod_ {i \notin \tau_ {k}} \mathbb {P} _ {i} \left(Z _ {i} \mid \mathbf {Z} _ {p a (i)}\right) & \forall k \in [ K ] \end{array} \right. \tag {1}
$$
where  $\mathbb{P}^0$  is the unintervened, or observational, distribution, and  $\mathbb{P}^k$  are interventional distributions. Remark 2.3. The joint probabilities  $\mathbb{P}^k$  in (1) are uniquely factorized into causal mechanisms according to  $G$ . We therefore use the equivalent notation  $(G, (\mathbb{P}^k, \tau_k)_{k \in [0, K]})$ , where  $\mathbb{P}^k$  is defined as in (1).
The main object of our study is a latent variable model termed latent causal Bayesian network (CBN).
Definition 3.1 (Latent CBN). A latent CBN is a tuple  $(G, \mathbf{f}, (\mathbb{P}^k, \tau_k)_{k \in [0, K]})$ , where  $\mathbf{f}: \mathbb{R}^d \to \mathbb{R}^d$  is a diffeomorphism (i.e., invertible with both  $\mathbf{f}$  and  $\mathbf{f}^{-1}$  differentiable).
2The term can also be used in structural causal models to denote deterministic functions of endogenous and exogenous variables in assignments, see [43, Def. 3.1]. A central idea in causality [41, 43] is that causal mechanisms are modular or independent, i.e., it is possible to modify some without affecting the others: after an intervention, typically only a subset of the causal mechanisms change.
<sup>3</sup>A special case of perfect interventions are hard interventions, where  $\widetilde{\mathbb{P}}_j$  corresponds to a Dirac distribution:  $\mathbb{P}(\mathbf{Z} \mid \mathrm{do}(Z_j = z_j)) = \delta_{Z_j = z_j} \prod_{i \neq j} \mathbb{P}_i(Z_i \mid \mathbf{Z}_{\mathrm{pa}(i)})$ .
Data-generating process for Causal Component Analysis (CauCA). In CauCA, we assume that we are given multiple datasets  $\{\mathcal{D}_k\}_{k\in [0,K]}$  generated by a latent CBN  $(G,\mathbf{f},(\mathbb{P}^k,\tau_k)_{k\in [0,K]})$
$$
\mathcal {D} _ {k} := \left(\tau_ {k}, \left\{\mathbf {x} ^ {(n, k)} \right\} _ {n = 1} ^ {N _ {k}}\right), \quad \text {w i t h} \quad \mathbf {x} ^ {(n, k)} = \mathbf {f} \left(\mathbf {z} ^ {(n, k)}\right) \quad \text {a n d} \quad \mathbf {z} ^ {(n, k) \stackrel {\text {i . i . d .}} {\sim}} \mathbb {P} ^ {k}, \tag {2}
$$
where  $N_{k}$  denotes the sample size for interventional regime  $k$ , see Fig. 1 for an illustration. The graph  $G$  is assumed to be known. Further, we assume that the intervention targets  $\tau_{k}$  are observed, see § 7 for further discussion. Both the mixing function  $\mathbf{f}$  and the latent distributions  $\mathbb{P}^k$  in (2) are unknown.
The problem we aim to address is the following: given only the graph  $G$  and the datasets  $\mathcal{D}_k$  in (2), can we learn to invert the mixing function  $\mathbf{f}$  and thus recover the latent variables  $\mathbf{z}$ ? Whether this is possible, and up to what ambiguities, depends on the identifiability of CauCA.
Definition 3.2 (Identifiability of CauCA). A model class for CauCA is a tuple  $(G, \mathcal{F}, \mathcal{P}_G)$ , where  $\mathcal{F}$  is a class of functions and  $\mathcal{P}_G$  is a class of joint distributions Markov relative to  $G$ . A latent CBN  $(G, \mathbf{f}, (\mathbb{P}^k, \tau_k)_{k \in [0, K]})$  is said to be in  $(G, \mathcal{F}, \mathcal{P}_G)$  if  $\mathbf{f} \in \mathcal{F}$  and  $\mathbb{P}^k \in \mathcal{P}_G$  for all  $k \in [0, K]$ . We say  $(G, \mathcal{F}, \mathcal{P}_G)$  has known intervention targets if all its elements share the same  $G$  and  $(\tau_k)_{k \in [0, K]}$ .
We say that  $\text{CauCA}$  in  $(G, \mathcal{F}, \mathcal{P}_G)$  is identifiable up to  $\mathcal{S}$  (a set of functions called "indeterminacy set") if for any two latent CBNs  $(G, \mathbf{f}, (\mathbb{P}^k, \tau_k)_{k \in [0, K]})$  and  $(G', \mathbf{f}', (\mathbb{Q}^k, \tau_k)_{k \in [0, K]})$ , the equality of pushforward  $\mathbf{f}_* \mathbb{P}^k = \mathbf{f}_*' \mathbb{Q}^k \forall k \in [0, K]$  implies that  $\exists \mathbf{h} \in S$  s.t.  $\mathbf{h} = \mathbf{f}'^{-1} \circ \mathbf{f}$  on the support of  $\mathbb{P}$ .
We justify the definition of known intervention targets and generalize them to a more flexible scenario in App. E. Defn. 3.2 is inspired by the identifiability definition of ICA in [4, Def. 1]. Intuitively, it states that, if two models in  $(G,\mathcal{F},\mathcal{P}_G)$  give rise to the same distribution, then they are equal up to ambiguities specified by  $\mathcal{S}$ . Consequently, when attempting to invert  $\mathbf{f}$  based on the data in (2), the inversion can only be achieved up to those ambiguities.
In the following, we choose  $\mathcal{F}$  to be the class of all  $\mathcal{C}^1$ -diffeomorphisms  $\mathbb{R}^d \to \mathbb{R}^d$ , denoted  $\mathcal{C}^1(\mathbb{R}^d)$ , and suppose the distributions in  $\mathcal{P}_G$  are absolutely continuous with full support in  $\mathbb{R}^d$ , with the density  $p^k$  differentiable.
A first question is what ambiguities are unavoidable by construction in CauCA, similar to scaling and permutation in ICA [22, § 3.1]. The following Lemma characterizes this.
Lemma 3.3. For any  $(G,\mathbf{f},(\mathbb{P}^k,\tau_k)_{k\in \llbracket 0,K\rrbracket})$  in  $(G,\mathcal{F},\mathcal{P}_G)$ , and for any  $\mathbf{h}\in S_{scaling}$  with
$$
\mathcal {S} _ {\text {s c a l i n g}} := \left\{\mathbf {h}: \mathbb {R} ^ {d} \rightarrow \mathbb {R} ^ {d} \mid \mathbf {h} (\mathbf {z}) = \left(h _ {1} \left(z _ {1}\right), \dots , h _ {d} \left(z _ {d}\right)\right), h _ {i} \text {i s a d i f f e o m o r p h i s m i n} \mathbb {R} \right\}, \tag {3}
$$
there exists a  $(G,\mathbf{f}\circ \mathbf{h},(\mathbb{Q}^k,\tau_k)_{k\in [[0,K]]})$  in  $(G,\mathcal{F},\mathcal{P}_G)$  s.t.  $\mathbf{f}_{*}\mathbb{P}^{k} = (\mathbf{f}\circ \mathbf{h})_{*}\mathbb{Q}^{k}$  for all  $k\in [[0,K]]$
Lemma 3.3 states that, as in nonlinear ICA, the ambiguity up to element-wise nonlinear scaling is also unresolvable in CauCA. However, unlike in nonlinear ICA, there is no permutation ambiguity in CauCA: this is a consequence of the assumption of known intervention targets. The next question is under which conditions we can achieve identifiability up to (3), and when the ambiguity set is larger.
In this section, we investigate the identifiability of CauCA. We first study the general case (§ 4.1), and then consider the special case of ICA in which the graph is empty (§ 4.2).
4.1 Identifiability of CauCA
Single-node interventions. We start by characterizing the identifiability of  $\mathrm{CauCA}$  based on single-node interventions. For datasets  $\mathcal{D}_k$  defined as in (2), every  $k > 0$  corresponds to interventions on a single variable: i.e.,  $\forall k > 0, |\tau_k| = 1$ . This is the setting depicted in Fig. 1, where each interventional dataset is generated by intervening on a single latent variable. The following assumption will play a key role in our proofs.
Assumption 4.1 (Interventional discrepancy). Given  $k \in [K]$ , let  $p_{\tau_k}$  denote the causal mechanism of  $z_{\tau_k}$ . We say that a stochastic intervention  $\tilde{p}_{\tau_k}$  satisfies interventional discrepancy if
$$
\frac {\partial \left(\ln p _ {\tau_ {k}}\right)}{\partial z _ {\tau_ {k}}} \left(z _ {\tau_ {k}} \mid \mathbf {z} _ {p a (\tau_ {k})}\right) \neq \frac {\partial \left(\ln \widetilde {p} _ {\tau_ {k}}\right)}{\partial z _ {\tau_ {k}}} \left(z _ {\tau_ {k}} \mid \mathbf {z} _ {p a ^ {k} (\tau_ {k})}\right) \quad \text {a l m o s t e v e r y w h e r e (a . e .)}. \tag {4}
$$
Figure 2: Violation of the Interventional Discrepancy Assumption. The shown distributions constitute a counterexample to identifiability that violates Asm. 4.1 and thus allows for spurious solutions, see App. C for technical details. (Left) Visualisation of the joint distributions of two independent latent components  $z_{1}$  and  $z_{2}$  after no intervention (red), and interventions on  $z_{1}$  (green) and  $z_{2}$  (blue). As can be seen, each distribution reaches the same plateau on some rectangular interval of the domain, coinciding within the red square. (Center/Right) Within the red square where all distributions agree, it is possible to apply a measure preserving automorphism which leaves all distributions unchanged, but non-trivially mixes the latents. The right plot shows a distance-dependent rotation around the centre of the black circle, whereas the middle plot show a reference identity transformation.
Asm. 4.1 can be applied to imperfect and perfect interventions alike (in the latter case the conditioning on the RHS disappears). Intuitively, Asm. 4.1 requires that the stochastic intervention is sufficiently different from the causal mechanism, formally expressed as the requirement that the partial derivative over  $z_{i}$  of the ratio between  $p_{i}$  and  $\tilde{p}_{i}$  is nonzero a.e. One case in which Asm. 4.1 is violated is when  $\partial p_{i} / \partial z_{i}$  and  $\partial \tilde{p}_i / \partial z_i$  are both zero on the same open subset of their support. In Fig. 2 (Left), we provide an example of such a violation (see App. C for its construction), and apply a measure-preserving automorphism within the area where the two distributions agree (see Fig. 2 (Right)).
We can now state our main result for CauCA with single-node interventions.
Theorem 4.2. For CauCA in  $(G,\mathcal{F},\mathcal{P}_G)$
(i) Suppose for each node in  $[d]$ , there is one (perfect or imperfect) stochastic intervention that satisfies Asm. 4.1. Then CauCA in  $(G, \mathcal{F}, \mathcal{P}_G)$  is identifiable up to
$$
\mathcal {S} _ {\overline {{G}}} = \left\{\mathbf {h}: \mathbb {R} ^ {d} \rightarrow \mathbb {R} ^ {d} | \mathbf {h} (\mathbf {z}) = \left(h _ {i} \left(\mathbf {z} _ {\overline {{a n c}} (i)}\right)\right) _ {i \in [ d ]}, \mathbf {h} i s \mathcal {C} ^ {1} - d i f f e o m o r p h i s m \right\}. \tag {5}
$$
(ii) Suppose for each node  $i$  in  $[d]$ , there is one perfect stochastic intervention that satisfies Asm. 4.1. Then CauCA in  $(G, \mathcal{F}, \mathcal{P}_G)$  is identifiable up to  $\mathcal{S}_{\mathrm{scaling}}$ .
Thm. 4.2 (i) states that for single-node stochastic interventions, perfect or imperfect, we can achieve identifiability up to an indeterminacy set where each reconstructed variable can at most be a mixture of ground truth variables corresponding to nodes in the closure of the ancestors of  $i$ . While this ambiguity set is larger than the one in eq. (3), it is still a non-trivial reduction in ambiguity with respect to the spurious solutions which could be generated without Asm. 4.1. A related result in [49, Thm. 1] shows that for linear mixing, linear latent SCM and unknown graph,  $d$  interventions are sufficient and necessary for recovering  $\overline{G}$  (the transitive closure of the ground truth graph  $G$ ) and the latent variables up to elementwise reparametrizations. Thm. 4.2 (i) instead proves that  $d$  interventions are sufficient for identifiability up to mixing of variables corresponding to the coordinates in  $\overline{G}$  for arbitrary  $\mathcal{C}^1$ -diffeomorphisms  $\mathcal{F}$ , non-parametric  $\mathcal{P}_G$  and known graph.
Thm. 4.2 (ii) shows that if we further constrain the set of interventions to perfect single-node, stochastic interventions only, then we can achieve a much stronger identifiability—i.e., identifiability up to scaling, which as discussed in § 3 is the best one we can hope to achieve in our problem setting without further assumptions. In short, the unintervened distribution together with one single-node, stochastic perfect intervention per node is sufficient to give us the strongest achievable identifiability in our considered setting. In App. D, we also discuss identifiability when only imperfect stochastic interventions are available. In short, with a higher number of imperfect interventions, the ambiguity in Thm. 4.2 (i) can be further constrained to the closure of parents, instead of the closure of ancestors.
Thm. 4.2 (ii) shows that  $d$  datasets generated by single-node interventions on the latent causal variables are sufficient for identifiability up to  $S_{\mathrm{scaling}}$ . We additionally prove below that  $d$  interventional datasets are necessary—i.e., for CauCA, and for any nonlinear causal representation learning problem,  $d - 1$  single-node interventions are not sufficient for identifiability up to  $S_{\mathrm{scaling}}$ .
Proposition 4.3. Given a DAG  $G$ , with  $d - 1$  perfect stochastic single node interventions on distinct targets, if the remaining unintervened node has any parent in  $G$ ,  $(G, \mathcal{F}, P_G)$  is not identifiable up to