SlowGuess/NeurIPS_2023_OCR huggingface dataset:
An illustrative example of applying the theorem is shown in Appendix B.1. Thm. 1 allows one to attribute spurious variations to latent variables influencing both  $X$  and  $Y$ . The key question is when such an attribution, as shown in Eq. 22, can be computed from observational data in practice (known as an identifiability problem [9]). In fact, when variables are added to the PA submodel in topological order, the attribution of variations to the latents  $U_{i}$  is identifiable, as we prove next:
Theorem 2 (Spurious decomposition identification in topological ordering). The quantity  $P(y \mid x^{U_{[i]}})$  can be computed from observational data using the expression
$$
P (y \mid x ^ {U _ {[ i ]}}) = \sum_ {z} P (y \mid z, x) P \left(z _ {- [ i ]} \mid z _ {[ i ]}, x\right) P \left(z _ {[ i ]}\right), \tag {23}
$$
rendering each term of decomposition in Eq. 22 identifiable from the observational distribution  $P(v)$ .
We discuss in Appendix B.2 why a decomposition that does not follow a topological order of the variables  $U_{i}$  is not identifiable.
4.2 Spurious Decompositions in Semi-Markovian Models
In the Markovian case, considered until now, there was a one-to-one correspondence between the observed confounders  $Z_{i}$  and their latent variables  $U_{i}$ . This, however, is no longer the case in Semi-Markovian models. In particular, it can happen that there exist exogenous variables  $U_{j}$  that induce common variations between  $X, Y$ , but affect more than one confounder  $Z_{i}$ . We are interested in  $U_{j} \subseteq U$  that have causal (directed) paths to both  $X, Y$ , described by the following definition:
Definition 3 (Trek). Let  $\mathcal{M}$  be an SCM corresponding to a Semi-Markovian model. Let  $\mathcal{G}$  be the causal diagram of  $\mathcal{M}$ . A trek  $\tau$  in  $\mathcal{G}$  (from  $X$  to  $Y$ ) is an ordered pair of causal paths  $(g_{l}, g_{r})$  with a common exogenous source  $U_{i} \in U$ . That is,  $g_{l}$  is a causal path  $U_{i} \to \dots \to X$  and  $g_{r}$  is a causal path  $U_{i} \to \dots \to Y$ . The common source  $U_{i}$  is called the top of the trek (ToT for short), denoted  $\text{top}(g_{l}, g_{r})$ . A trek is called spurious if  $g_{r}$  is a causal path from  $U_{i}$  to  $Y$  that is not intercepted by  $X$ .
When decomposing spurious effects, we are in fact interested in all the exogenous variables  $U_{i}$  that lie on top of a spurious trek between  $X$  and  $Y$ . It is precisely these exogenous variables that induce common variations between  $X$  and  $Y$ . Using any subset of the variables that are top of spurious treks, we define a set-specific notion of a spurious effect:
Definition 4 (Exogenous set-specific spurious effect). Let  $U_{sToT} \subseteq U$  be the subset of exogenous variables that lie on top of a spurious trek between  $X$  and  $Y$ . Suppose  $A, B \subseteq U_{sToT}$  are two nested subsets of  $U_{sToT}$ , that is  $A \subseteq B$ . We then define the exogenous experimental spurious effect with respect to sets  $A, B$  as
$$
E x p - S E _ {x} ^ {A, B} (y) = P \left(y \mid x ^ {A}\right) - P \left(y \mid x ^ {B}\right). \tag {24}
$$
Figure 5: Quantity  $\mathrm{Exp - SE}_x^{A,B}(y)$  as a graphical contrast. Dots indicate arbitrary observed confounders along the indicated pathway.
The above definition is analogous to Def. 2, but we are now fixing different subsets of the tops of spurious treks. Def. 2 supports partial abduction of exogenous variables that are not on top of a spurious trek, but we are seldom interested in these since they do not induce covariations of  $X, Y$ . The quantity  $\mathrm{Exp - SE}_x^{A,B}(y)$  is presented as a graphical contrast in Fig. 5. In particular, the set of tops of spurious treks  $U_{sToT}$  is partitioned into three parts  $(U_A, U_{B\backslash A}, U_{BC})$ . The causal diagram in the figure is informal, and the dots  $(\dots)$  represent arbitrary possible observed confounders that lie along indicated pathways. On the l.h.s. of the figure, the set  $U_A$  does not respond to the conditioning  $X = x$ , whereas  $U_{B\backslash A}, U_{BC}$  do. This is contrasted with the r.h.s., in which neither  $U_A$  nor  $U_{B\backslash A}$  respond to  $X = x$ , whereas  $U_{BC}$  still does respond to the  $X = x$  conditioning. The described contrast thus captures the spurious effect explained by the tops of spurious treks in  $U_{B\backslash A}$ .
Analogous to Thm. 1, we next state a variable-specific decomposition of the spurious effect, which is now with respect to exogenous variables that are top of spurious treks:
Theorem 3 (Semi-Markovian spurious decomposition). Let  $U_{sToT} = \{U_1, \dots, U_m\} \subseteq U$  be the subset of exogenous variables that lie on top of a spurious trek between  $X$  and  $Y$ . Let  $U_{[i]}$  denote the variables  $U_1, \dots, U_i$  ( $U_{[0]}$  denotes the empty set  $\emptyset$ ). The experimental spurious effect  $Exp - SE_x(y)$  can be decomposed into variable-specific contributions as follows:
$$
E x p - S E _ {x} (y) = \sum_ {i = 0} ^ {m - 1} \operatorname {E x p - S E} _ {x} ^ {U _ {[ i ]}, U _ {[ i + 1 ]}} (y) = \sum_ {i = 0} ^ {k - 1} P \left(y \mid x ^ {U _ {[ i ]}}\right) - P \left(y \mid x ^ {U _ {[ i + 1 ]}}\right). \tag {25}
$$
An example demonstrating the Semi-Markovian decomposition is given in Appendix B.3. We next discuss the question of identification. We begin by discussing how to annotate the exogenous variables given a Semi-Markovian causal diagram:
Definition 5 (Top of trek from the causal diagram). Let  $\mathcal{M}$  be a Semi-Markovian model and let  $\mathcal{G}$  be the associated causal diagram. A set of nodes fully connected with bidirected edges is called a clique. A maximal clique  $C_i$  is such that there is no clique  $C_i'$  such that  $C_i \subsetneq C_i'$ . The set of variables  $U_{sT_{oT}}$  can be constructed from the causal diagram in the following way:
(I) initialize  $U_{sToT} = \emptyset$
(II) for each maximal clique  $C_i$ , consider the associated exogenous variable  $U_{C_i}$  pointing to each node in the clique; if there exists a spurious trek between  $X$  and  $Y$  with a top in  $U_{C_i}$ , add  $U_{C_i}$  to  $U_{sToT}$ .
After defining the explicit construction of the set  $U_{sToT}$ , we define the notion of the anchor set:
Definition 6 (Anchor Set). Let  $U_1, \ldots, U_l \subseteq U$  be a subset of the exogenous variables. We define the anchor set  $AS(U_1, \ldots, U_l)$  of  $(U_1, \ldots, U_l)$  as the subset of observables  $V$  that are directly influenced by any of the  $U_i$ 's,
$$
A S \left(U _ {1}, \dots , U _ {l}\right) = \bigcup_ {i = 1} ^ {l} \operatorname {c h} \left(U _ {i}\right). \tag {26}
$$
Another important definition is that of anchor set exogenous ancestral closure:
SCM  $\mathcal{M}$
$$
\begin{array}{l} Z _ {1} \leftarrow B (0. 5) \\ Z _ {2} \leftarrow B \left(0. 4 + 0. 2 Z _ {1}\right) \\ Z _ {3} \leftarrow B \left(0. 3 + 0. 3 Z _ {1} Z _ {2}\right) \\ X \leftarrow B \left(0. 2 + \lambda_ {1} Z _ {1} + \lambda_ {2} Z _ {2} + \lambda_ {3} Z _ {3}\right) \\ Y \leftarrow B \left(0. 1 + 0. 2 X + \lambda_ {1} Z _ {1} + \lambda_ {2} Z _ {2} + \lambda_ {3} Z _ {3}\right) \\ \end{array}
$$
Causal Diagram  $\mathcal{G}$
Table 2: SCM and causal diagram for the Synthetic A example.
Definition 7 (Anchor Set Exogenous Ancestral Closure). Let  $U_s \subseteq U$  be a subset of the exogenous variables. Let  $AS(U_s)$  denote the anchor set of  $U_s$ , and let  $\mathrm{an}^{\mathrm{ex}}(AS(U_s))$  denote all exogenous variables that have a causal path to any variable in  $AS(U_s)$ .  $U_s$  is said to satisfy anchor set exogenous ancestral closure (ASEAC) if
$$
U _ {s} = \operatorname {a n} ^ {e x} \left(A S \left(U _ {s}\right)\right). \tag {27}
$$
Based on the above, we provide a sufficient condition for identification in the Semi-Markovian case: Theorem 4 (ID of variable spurious effects in Semi-Markovian models). Let  $U_s \subseteq U_{sToT}$ . The quantity  $P(y \mid x^{U_s})$  is identifiable from observational data  $P(V)$  if the following hold:
(i)  $X\notin AS(U_s),Y\notin AS(U_s)$
(ii)  $U_{s}$  satisfies anchor set exogenous ancestral closure,  $U_{s} = \mathrm{an}^{ex}(AS(U_{s}))$
Some instructive examples grounding Defs. 5-7 and Thm. 4 can be found in Appendix B.4. In words, the conditional expectation of  $Y$  given  $X$  in the partially abducted submodel w.r.t. a set  $U_{s}$  is identifiable whenever (i) neither  $X$  nor  $Y$  are elements of the anchor set of  $U_{s}$  and (ii) the set  $U_{s}$  satisfies the anchor set exogenous ancestral closure. Thm. 4 provides a sufficient, but not a necessary condition for identification. An additional discussion of the conditions is given in Appendix C. We hope to address in future work an algorithmic way for identifying spurious effects in full generality.
We now apply our framework to a synthetic example (called Synthetic A) with a known ground truth, summarized in Tab. 2 where the SCM  $\mathcal{M}$  and the causal diagram  $\mathcal{G}$  are given. The source code for the experiment can be found in our repository. For this example, we set the parameters  $\lambda_{1} = \lambda_{2} = \lambda_{3} = 0.2$ . We then vary each parameter  $\lambda_{i} \in [0,0.2]$  (while keeping the other two parameters fixed), which changes the value of the effect associated with latent variable  $U_{i}$ . The effects associated with each  $U_{i}, i \in \{1,2,3\}$  are computed based on the decomposition in Thm. 1:
$$
\operatorname {E x p} - \mathrm {S E} _ {x} ^ {U _ {1}} (y) := \operatorname {E x p} - \mathrm {S E} _ {x} ^ {\emptyset , U _ {1}} (y) = P (y \mid x ^ {\emptyset}) - P (y \mid x ^ {U _ {1}}) \tag {28}
$$
$$
\operatorname {E x p} - \mathbf {S E} _ {x} ^ {U _ {2}} (y) := \operatorname {E x p} - \mathbf {S E} _ {x} ^ {U _ {1}, \{U _ {1}, U _ {2} \}} (y) = P (y \mid x ^ {U _ {1}}) - P (y \mid x ^ {U _ {1}, U _ {2}}) \tag {29}
$$
$$
\operatorname {E x p} - \operatorname {S E} _ {x} ^ {U _ {3}} (y) := \operatorname {E x p} - \operatorname {S E} _ {x} ^ {\left\{U _ {1}, U _ {2} \right\}, \left\{U _ {1}, U _ {2}, U _ {3} \right\}} (y) = P \left(y \mid x ^ {U _ {1}, U _ {2}}\right) - P \left(y \mid x ^ {U _ {1}, U _ {2}, U _ {3}}\right). \tag {30}
$$
The key task is to compute the ground truth values of  $P(y \mid x^{U_{[i]}})$  for different values of  $i$ . According to Def. 1, we want to obtain the conditional distribution of  $Y$  given  $X = x$  but subject to not updating  $U_{[i]}$  according to the evidence  $X = x$ . Based on the true SCM, this can be done efficiently using rejection sampling as follows:
(1) Take  $N$  samples from the SCM  $\mathcal{M}$  in Tab. 2,
(2) For all samples  $k \in \{1, \dots, N\}$  with  $u^{(k)}$  such that
$$
X \left(u ^ {(k)}\right) \neq x, \tag {31}
$$
re-sample the part of the unit  $u^{(k)}$  that is not included in  $U_{[i]}$  (e.g., if  $U_{[i]} = \{U_1, U_2\}$ , latent  $u_1^{(k)}, u_2^{(k)}$  are not re-sampled but  $u_3^{(k)}$  is) and replace  $u^{(k)}$  with this new sample,
Figure 6: Experimental results on the Synthetic A example. Lines indicate the estimated values, dots the ground truth obtained from the SCM using rejection sampling, and the  $95\%$  confidence intervals are indicated with color. As expected, increasing the  $\lambda_{i}$  coefficient increases the spurious effect associated with the latent variable  $U_{i}$ .
(3) Evaluate the mechanisms  $\mathcal{F}$  of  $\mathcal{M}$  for all units  $u^{(k)}$ ,
(4) If there exists a sample  $k$  with  $X(u^{(k)})\neq x$  go back to Step (2),
(5) Return the mean of the  $Y$  variables  $\frac{1}{N}\sum_{k = 1}^{N}Y^{(k)}$ .
Notice that the described procedure gives us samples from the distribution  $P(y \mid x^{U_{[i]}})$ . The values of  $U_{[i]}$  are sampled only once and are not updated after the initial sampling. Other values in  $U$ , however, are sampled anew until their values are such that they are compatible with the evidence  $X = x$ . Therefore, the procedure guarantees that  $U_{[i]}$  do not respond to the evidence, whereas the complement of  $U_{[i]}$  does, allowing us to compute  $P(y \mid x^{U_{[i]}})$  and in turn the expressions in Eqs. 28-30. The effects are also estimated from observational data based on the identification expressions in Thm. 2. Fig. 6 demonstrates that the SCM-based ground truth matches the estimates based on Thm. 2.
In this paper, we introduced a general toolkit for decomposing spurious variations in causal models. In particular, we introduced a new primitive called partially abducted submodel (Def. 1), together with the procedure of partial abduction and prediction (Alg. 2). This procedure allows for new machinery for decomposing spurious variations in Markovian (Thm. 1) and Semi-Markovian (Thm. 3) models. Finally, we also developed sufficient conditions for identification of such spurious decompositions (Thms. 2, 4), and demonstrated the approach empirically (Sec. 5). The main limitation of our approach is the need for a fully-specified causal diagram, which may be challenging in practice. However, from a fully specified graph and the data, our tools for decomposing spurious effects give a fine-grained quantification of what the main confounders are. As is common in causal inference, the granularity of the obtained knowledge needs to be matched with the strength of the causal assumptions (in this case, specifying the causal diagram). Conversely, in the absence of such assumptions, fine-grained quantitative knowledge about these effects cannot be obtained in general [2], and we hypothesize that precise quantification of spurious effects is not attainable in the absence of a causal diagram. Finally, we discuss another technical solution that may alleviate some of the difficulty of causal modeling. Recently, cluster diagrams have been proposed [1], in which one can consider groups of confounders (instead of considering each confounder separately), and thus the specification of causal assumptions becomes less demanding (due to clustering, the number of nodes in the graph is smaller). However, causal decompositions as described in this paper can still be applied to cluster diagrams. This offers a way to choose a different level of granularity for settings where domain knowledge may not be specific enough to elicit a full causal diagram.
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Acknowledgements This research was supported in part by the NSF, ONR, AFOSR, DoE, Amazon, JP Morgan, and The Alfred P. Sloan Foundation. We would like to thank Inwoo Hwang for providing very useful comments on the paper during the NeurIPS conference.
Supplementary Material for A Causal Framework for Decomposing Spurious Variations
The source code for reproducing the experiments can be found in our code repository.
A Theorem and Proposition Proofs
A.1 Proof of Prop. 2
Proof. Note that TV and TE are defined as:
$$
\mathbf {T V} _ {x _ {0}, x _ {1}} (y) = P (y \mid x _ {1}) - P (y \mid x _ {0}) \tag {32}
$$
$$
\mathrm {T E} _ {x _ {0}, x _ {1}} (y) = P \left(y _ {x _ {1}}\right) - P \left(y _ {x _ {0}}\right). \tag {33}
$$
We can expand the TV measure in the following way:
$$
\begin{array}{l} \mathrm {T V} _ {x _ {0}, x _ {1}} (y) = P \left(y \mid x _ {1}\right) - P \left(y \mid x _ {0}\right) (34) \\ = P \left(y \mid x _ {1}\right) - P \left(y _ {x _ {1}}\right) + P \left(y _ {x _ {1}}\right) - P \left(y \mid x _ {0}\right) (35) \\ = \underbrace {P \left(y \mid x _ {1}\right) - P \left(y _ {x _ {1}}\right)} _ {\operatorname {E x p} - \operatorname {S E} _ {x _ {1}} (y)} + \underbrace {P \left(y _ {x _ {1}}\right) - P \left(y _ {x _ {0}}\right)} _ {\operatorname {T E} _ {x _ {0}, x _ {1}} (y)} + \underbrace {P \left(y _ {x _ {0}}\right) - P \left(y \mid x _ {0}\right)} _ {- \operatorname {E x p} - \operatorname {S E} _ {x _ {0}} (y)} (36) \\ = \mathrm {T E} _ {x _ {0}, x _ {1}} (y) + \operatorname {E x p - S E} _ {x _ {1}} (y) - \operatorname {E x p - S E} _ {x _ {0}} (y), (37) \\ \end{array}
$$
showing the required result.
A.2 Proof of Thm. 1
Proof. Note that
$$
\sum_ {i = 0} ^ {k - 1} \operatorname {E x p} - \operatorname {S E} _ {x} ^ {U _ {[ i ]}, U _ {[ i + 1 ]}} (y) = \sum_ {i = 0} ^ {k - 1} P \left(y \mid x ^ {U _ {[ i ]}}\right) - P \left(y \mid x ^ {U _ {[ i + 1 ]}}\right) \tag {38}
$$
is a telescoping sum, and thus we have that
$$
\begin{array}{l} \sum_ {i = 0} ^ {k - 1} \operatorname {E x p} - \operatorname {S E} _ {x} ^ {U _ {[ i ]}, U _ {[ i + 1 ]}} (y) = \sum_ {i = 0} ^ {k - 1} P \left(y \mid x ^ {U _ {[ i ]}}\right) - P \left(y \mid x ^ {U _ {[ i + 1 ]}}\right) (39) \\ = P (y \mid x ^ {\emptyset}) - P (y \mid x ^ {U _ {[ k ]}}) (40) \\ = P (y \mid x) - P \left(y _ {x}\right) (41) \\ = \operatorname {E x p} - \operatorname {S E} _ {x} (y), (42) \\ \end{array}
$$
completing the proof of the theorem.
A.3 Proof of Thm. 2
Proof. Notice that fixing a specific value for the variables  $(U_{1},\ldots ,U_{k}) = (u_{1},\ldots ,u_{k})$  also gives a unique value for the variables  $(Z_{1},\ldots ,Z_{k}) = (z_{1},\ldots ,z_{k})$ . Therefore, we can write
$$
\begin{array}{l} P \left(y \mid x ^ {U _ {[ i ]}}\right) = \sum_ {u _ {[ i ]}} P \left(u _ {[ i ]}\right) P \left(y \mid x, u _ {[ i ]}\right) (43) \\ = \sum_ {u _ {[ i ]}} P \left(u _ {[ i ]}\right) P \left(y \mid x, u _ {[ i ]}, z _ {[ i ]} \left(u _ {[ i ]}\right)\right) (44) \\ = \sum_ {z _ {[ i ]}} \sum_ {u _ {[ i ]}} P \left(u _ {[ i ]}\right) \mathbb {1} \left(Z _ {[ i ]} \left(u _ {[ i ]}\right) = z _ {[ i ]}\right) P (y \mid x, z _ {[ i ]}) (45) \\ = \sum_ {z _ {[ i ]}} P \left(z _ {[ i ]}\right) P \left(y \mid x, z _ {[ i ]}\right) (46) \\ = \sum_ {z} P (y \mid x, z) P \left(z _ {- [ i ]} \mid x, z _ {[ i ]}\right) P \left(z _ {[ i ]}\right). (47) \\ \end{array}
$$
The above proof makes use of the fact that the exogenous variables  $U_{i}$  are considered in the topological ordering in the decomposition in Eq. 22, since in this case a fixed value of  $u_{[i]}$  implies a fixed value of  $z_{[i]}$ . However, when considering decompositions that do not follow a topological ordering, this is not the case, and we lose the identifiability property of the corresponding effects, as shown in the example in Appendix B.2.
A.4 Proof of Thm. 3
Proof. The proof is analogous to the proof of Thm. 1, the only difference being that there is no longer a 1-to-1 of the latent variables  $U_{i}$  with the observed confounders  $Z_{i}$ . Rather, each  $U_{i}$  may correspond to one or more  $Z_{i}$  variables. However, we still have that
$$
\sum_ {i = 0} ^ {k - 1} \operatorname {E x p} - \operatorname {S E} _ {x} ^ {U _ {[ i ]}, U _ {[ i + 1 ]}} (y) = \sum_ {i = 0} ^ {k - 1} P \left(y \mid x ^ {U _ {[ i ]}}\right) - P \left(y \mid x ^ {U _ {[ i + 1 ]}}\right) \tag {48}
$$
$$
\begin{array}{l} \sum_ {i = 0} ^ {k - 1} \operatorname {E x p} - \operatorname {S E} _ {x} ^ {U _ {[ i ]}, U _ {[ i + 1 ]}} (y) = \sum_ {i = 0} ^ {k - 1} P \left(y \mid x ^ {U _ {[ i ]}}\right) - P \left(y \mid x ^ {U _ {[ i + 1 ]}}\right) (49) \\ = P \left(y \mid x ^ {\emptyset}\right) - P \left(y \mid x ^ {U _ {[ k ]}}\right) (50) \\ = P (y \mid x) - P \left(y _ {x}\right) (51) \\ = \operatorname {E x p} - \operatorname {S E} _ {x} (y), (52) \\ \end{array}
$$
A.5 Proof of Thm. 4
Proof. Let  $U_{PA}$  be the set of exogenous variables not updated according to evidence, and suppose that (i)  $X, Y \notin \mathrm{AS}(U_{PA})$ ; (ii)  $U_{PA} = \mathrm{an}^{\mathrm{ex}}(AS(U_{PA}))$ . Note that
$$
\begin{array}{l} P \left(y \mid x ^ {U _ {P A}}\right) \stackrel {\text {(d e f)}} {=} \sum_ {u _ {P A}} P \left(u _ {P A}\right) P \left(y \mid x, u _ {P A}\right) (53) \\ = \sum_ {u _ {P A}, z _ {A S}} P \left(u _ {P A}\right) P \left(y \mid x, u _ {P A}, z _ {A S}\right) P \left(z _ {A S} \mid x, u _ {P A}\right), (54) \\ \end{array}
$$
where  $Z_{AS}$  is the anchor set of  $U_{PA}$  and the second line follows from the law of total probability. Consider any exogenous ancestor of  $Z_{AS}$ , denoted by  $U_{z}$ . By condition (ii) of ancestral closure,  $U_{z}$  must be in  $U_{PA}$ . Therefore,  $U_{PA}$  contains all exogenous ancestors of  $Z_{AS}$ . Consequently, a fixed value of  $u_{PA}$  also implies a value of  $Z_{AS}$ , labeled  $z_{AS}$ . This means that
$$
P \left(z _ {A S} \mid x, u _ {P A}\right) = \mathbb {1} \left(Z _ {A S} \left(u _ {P A}\right) = z _ {A S}\right). \tag {55}
$$