SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Abhin Shah
abhin@mit.edu
mkocaoglu@purdue.edu
Causal effect estimation from data typically requires assumptions about the cause-effect relations either explicitly in the form of a causal graph structure within the Pearlian framework, or implicitly in terms of (conditional) independence statements between counterfactual variables within the potential outcomes framework. When the treatment variable and the outcome variable are confounded, front-door adjustment is an important special case where, given the graph, causal effect of the treatment on the target can be estimated using post-treatment variables. However, the exact formula for front-door adjustment depends on the structure of the graph, which is difficult to learn in practice. In this work, we provide testable conditional independence statements to compute the causal effect using front-door-like adjustment without knowing the graph under limited structural side information. We show that our method is applicable in scenarios where knowing the Markov equivalence class is not sufficient for causal effect estimation. We demonstrate the effectiveness of our method on a class of random graphs as well as real causal fairness benchmarks.
Causal effect estimation is at the center of numerous scientific, societal, and medical questions [Nabi et al., 2019, Castro et al., 2020]. The  $do(\cdot)$  operator of Pearl represents the effect of an experiment on a causal system. For example, the probability distribution of a target variable  $y$  after setting a treatment  $t$  to  $t$  is represented by  $\mathbb{P}(y|do(t = t))$  and is known as an interventional distribution. Learning this distribution for any realization  $t = t^1$  is what causal effect estimation entails. This distribution is different from the conditional distribution  $\mathbb{P}(y|t = t)$  as there may be unobserved confounders between treatment and outcome that cannot be controlled for.
A causal graph, often depicted as a directed acyclic graph, captures the cause-and-effect relationships between variables and explains the causal system under consideration. A semi-Markovian causal model represents a causal model that includes unobserved variables influencing multiple observed variables [Verma and Pearl, 1990, Acharya et al., 2018]. In a semi-Markovian graph, directed edges between observed variables represent causal relationships, while bi-directed edges between observed variables represent unobserved common confounding (see Figure 1). Given any semi-Markovian
<sup>1</sup>Depending on the context, causal effect estimation sometimes refers to the estimating the difference of assigning  $t = 1$  vs.  $t = 0$  on the target variable  $y$ , e.g.,  $\mathbb{E}[y|do(t = 1)] - \mathbb{E}[y|do(t = 0)]$ . This quantity is computable if we can identify  $\mathbb{P}(y|do(t = t))$  for  $t = \{0,1\}$ .
Figure 1: Representative graphs for back-door adjustment (left) and front-door adjustment (right).
graph, complete identification algorithms for causal effect estimation are known. For example, if  $\mathbb{P}(y|do(t = t))$  is uniquely determined by the observational distribution and the causal graph, the algorithm by Shpitser and Pearl [2006] utilizes the graph to derive an estimand, i.e., the functional form mapping the observational distribution to the interventional distribution.
Certain special cases of estimands have found widespread use across several domains. One such special case is the back-door adjustment [Pearl, 1993] shown in Figure 1(left). The back-door adjustment utilizes the pre-treatment variable  $\mathbf{z}$  (that blocks back-door paths) to control for unobserved confounder as follows:
$$
\mathbb {P} (y | d o (t = t)) = \sum_ {\boldsymbol {z}} \mathbb {P} (y | t = t, \boldsymbol {z} = \boldsymbol {z}) \mathbb {P} (\boldsymbol {z} = \boldsymbol {z}), \tag {1}
$$
where the do-calculus rules of Pearl [1995] are used to convert interventional distributions into observational distributions by leveraging the graph structure. However, the back-door adjustment is often inapplicable, e.g., in the presence of an unobserved confounder between  $t$  and  $y$ . Surprisingly, in such scenarios, it is sometimes possible to find the causal effect using the front-door adjustment [Pearl, 1995] shown in Figure 1(right). Utilizing the front-door variable  $\mathbf{z}$ , the front-door adjustment estimates the causal effect from observational distributions using the following formula (which is also obtained through the do-calculus rules and the graph structure):
$$
\mathbb {P} (y | d o (t = t)) = \sum_ {\boldsymbol {z}} \left(\sum_ {t ^ {\prime}} \mathbb {P} (y | t = t ^ {\prime}, \boldsymbol {z} = \boldsymbol {z}) \mathbb {P} (t = t ^ {\prime})\right) \mathbb {P} (\boldsymbol {z} = \boldsymbol {z} | t = t) \tag {2}
$$
Recently, front-door adjustment has gained popularity in analyzing real-world data [Glynn and Kashin, 2017, Bellemare et al., 2019, Hünermund and Bairenboim, 2019] due to its ability to utilize post-treatment variables to estimate effects even in the presence of confounding between  $t$  and  $y$ . However, in general, front-door adjustment also relies on knowing the causal graph, which may not always be feasible, especially in domains with many variables.
An alternative approach uses observational data to infer a Markov equivalence class, which is a collection of causal graphs that encode the same conditional independence relations [Spirtes et al., 2000]. A line of work [Perkovic et al., 2018, Jaber et al., 2019] provide identification algorithms for causal effect estimation from partial ancestral graphs (PAGs) [Zhang, 2008], a prominent representation of the Markov equivalence class, whenever every causal graph in the collection shares the same causal effect estimand. However, learning PAGs from data is challenging in practice due to the sequential nature of their learning algorithms, which can propagate errors between tests [Strobl et al., 2019a]. Further, to the best of our knowledge, there is no existing algorithm that can incorporate side information, such as known post-treatment variables, into PAG structure learning.
In this work, we ask the following question: Can the causal effect be estimated with a testable criteria on observational data by utilizing some structural side information without knowing the graph?
Recent research has developed such testable criteria to enable back-door adjustment without knowing the full causal graph [Entner et al., 2013, Cheng et al., 2020, Gultchin et al., 2020, Shah et al., 2022]. These approaches leverage structural side information, such as a known and observed parent of the treatment variable  $t$ . However, no such results have been established for enabling front-door adjustment. We address this gap by focusing on the case of unobserved confounding between  $t$  and  $y$ , where back-door adjustment is inapplicable. Traditionally, this scenario has been addressed by leveraging the presence of an instrumental variable [Mogstad and Torgovitsky, 2018] or performing sensitivity analysis [Veitch and Zaveri, 2020], both of which provide only bounds in the non-parametric case. In contrast, we achieve identifiability by utilizing structural side information.
Contributions. We propose a method for estimating causal effects without requiring the knowledge of causal graph in the presence of unobserved confounding between treatment and outcome. Our approach utilizes front-door-like adjustments based on post-treatment variables and relies on conditional independence statements that can be directly tested from observational data. We
require one structural side information which can be obtained from an expert and is less demanding than specifying the entire causal graph. We illustrate that our framework provides identifiability in random ensembles where existing PAG-based methods are not applicable. Further, we illustrate the practical application of our approach to causal fairness analysis by estimating the total effect of a sensitive attribute on an outcome variable using the German credit data with fewer structural assumptions. The source code of our implementation is available at https://github.com/abhin-shah/FD-adjustment-with-limited-graph.
Effect estimation from causal graphs/Markov equivalence Class: The problem of estimating interventional distributions with the knowledge of the semi-Markovian model has been studied extensively in the literature, with important contributions such as Tian and Pearl [2002] and Shpitser and Pearl [2006]. Perkovic et al. [2018] presented a complete and sound algorithm for identifying valid adjustments from PAGs. Going beyond valid adjustments, Jaber et al. [2019] proposed a complete and sound algorithm for identifying causal effect from PAGs. However, our method can recover the causal effect in scenarios where these algorithms are inapplicable.
Effect estimation via front-door adjustment with causal graph: Several recent works have contributed to a better understanding of the statistical properties of front-door estimation [Kuroki, 2000, Kuroki and Cai, 2012, Glynn and Kashin, 2018, Gupta et al., 2021], proposed robust generalizations [Hunermund and Bareinboim, 2019, Fulcher et al., 2020], and developed procedures to enumerate all possible front-door adjustment sets [Jeong et al., 2022, Wienobst et al., 2022]. However, all of these require knowing the underlying causal graph. By contrast, Bhattacharya and Nabi [2022] verified the front-door criterion without knowing the causal graph using Verma constraint-based methodology. While their method was limited to a small set of graphs, ours leverages conditional independence, making it applicable to a much broader class of graphs. We note that they're applicable in different settings, depending on what the analyst knows about the problem.
Notations. For a sequence of realizations  $r_1, \dots, r_n$ , we define  $\mathbf{r} \triangleq \{r_1, \dots, r_n\}$ . For a sequence of random variables  $r_1, \dots, r_n$ , we define  $\mathbf{r} \triangleq \{r_1, \dots, r_n\}$ . Let  $\mathbb{1}$  denote the indicator function.
Semi-Markovian Model and Effect Estimation. We consider a causal effect estimation task where  $\mathbf{x}$  represents the set of observed features,  $t$  represents the observed treatment variable, and  $y$  represents the observed outcome variable. We denote the set of all observed variables jointly by  $\mathcal{V} \triangleq \{\mathbf{x}, t, y\}$ . Let  $\mathcal{U}$  denote the set of unobserved features that could be correlated with the observed variables.
We assume  $\mathcal{W} \triangleq \mathcal{V} \cup \mathcal{U}$  follows a semi-Markovian causal model [Tian and Pearl, 2002] as below.
Definition 1. A semi-Markovian causal model (SMCM)  $\mathcal{M}$  is specified as follows:
1.  $\mathcal{G}$  is a directed acyclic graph (DAG) over the set of vertices  $\mathcal{W}$  such that each element of the set  $\mathcal{U}$  has no parents.
2.  $\forall v\in \mathcal{V}$  , let  $\pi^{(o)}(v)\subseteq \mathcal{V}$  and  $\pi^{(u)}(v)\subseteq \mathcal{U}$  denote the set of parent of  $v$  in  $\mathcal{V}$  and  $\mathcal{U}$  , respectively.
3.  $\mathbb{P}(\mathbf{u})$  is the unobserved joint distribution over the unobserved features.
4. The observational distribution is given by  $\mathbb{P}(\mathbf{v}) = \mathbb{E}_{\mathbf{u}}\Big[\prod_{v\in \mathcal{V}}\mathbb{P}\big(v|\pi^{(o)}(v),\pi^{(u)}(v)\big)\Big]$ .
5. The interventional distribution when the variables  $\mathbf{r} \subset \mathcal{V}$  are set to a fixed value  $\mathbf{r}$  is given by
$$
\mathbb {P} (\mathbf {v} | d o (\mathbf {r} = \mathbf {r})) = \mathbb {1} _ {\mathbf {r} = \mathbf {r}} \cdot \mathbb {E} _ {\mathbf {u}} \left[ \prod_ {v \in \mathcal {V} \backslash r} \mathbb {P} (v | \pi^ {(o)} (v), \pi^ {(u)} (v)) \right]. \tag {3}
$$
6. For any  $v_1, v_2 \in \mathcal{V}$ , if  $\pi^{(u)}(v_1) \cap \pi^{(u)}(v_2) \neq \emptyset$ , then  $v_1$  and  $v_2$  have a bi-directed edge in  $\mathcal{G}$ .
In this work, we are interested in the causal effect of  $t$  on  $y$ , i.e.,  $\mathbb{P}(y|do(t = t))$ . We define this formally by marginalizing all variables except  $y$  in the interventional distribution in (3).
Definition 2. The causal effect of  $t$  (when forced to a value  $t$ ) on  $y$  is given by:
$$
\mathbb {P} (y | d o (t = t)) = \sum_ {\mathbf {v} \backslash \{y \}} \mathbb {P} \left(\mathbf {v} \backslash \{y \}, y | d o (t = t)\right). \tag {4}
$$
Next, we define the notion of average treatment effect for a binary treatment  $t$ .
Definition 3. The average treatment effect (ATE) of a binary treatment  $t$  on outcome  $y$  is given by  $ATE = \mathbb{E}[y|do(t = 1)] - \mathbb{E}[y|do(t = 0)]$ .
Next, we define when the causal effect (Definition 2) is said to be identifiable from the observational distribution and the causal graph.
Definition 4. (Causal effect identifiability) Given an observational distribution  $\mathbb{P}(\mathbf{v})$  and a causal graph  $\mathcal{G}$ , the causal effect  $\mathbb{P}(y|do(t = t))$  is identifiable if it is identical for every semi-Markovian Causal model with (a) same graph  $\mathcal{G}$  and (b) same observational distribution  $\mathbb{P}(\mathbf{v})$ .
In a causal graph  $\mathcal{G}$ , a path is an ordered sequence of distinct nodes where each node is connected to the next in the sequence by an edge. A path starting at node  $w_{1}$  and ending at node  $w_{2}$  in  $\mathcal{G}$  is blocked by a set  $\mathbf{w} \subset \mathcal{W} \setminus \{w_{1}, w_{2}\}$  if there exists  $w \in \mathbf{w}$  such that (a)  $w$  is not a collider or (b)  $w$  is a collider and neither  $w$  nor any of its descendants is in  $\mathbf{w}$ . Further,  $w_{1}$  and  $w_{2}$  are said to be  $d$ -separated by  $\mathbf{w}$  in  $\mathcal{G}$  if  $\mathbf{w}$  blocks every path between  $w_{1}$  and  $w_{2}$  in  $\mathcal{G}$ . Let  $w_{1} \perp_{d} w_{2} | \mathbf{w}$  denote that  $w_{1}$  and  $w_{2}$  are  $d$ -separated by  $\mathbf{w}$  in  $\mathcal{G}$ . Similarly, let  $w_{1} \perp_{p} w_{2} | \mathbf{w}$  denote that  $w_{1}$  and  $w_{2}$  are conditionally independent given  $\mathbf{w}$ . We assume causal faithfulness, i.e., any conditional independence  $w_{1} \perp_{p} w_{2} | \mathbf{w}$  implies a  $d$ -separation relation  $w_{1} \perp_{d} w_{2} | \mathbf{w}$  in the causal graph  $\mathcal{G}$ .
2.1 Adjustment using pre-treatment variables
It is common in causal effect estimation to consider pre-treatment variables, i.e., variables that occur before the treatment in the causal ordering, and identify sets of variables that are valid adjustments. Specifically, a set  $\mathbf{z} \subset \mathcal{V}$  forms a valid adjustment if the causal effect can be written as  $\mathbb{P}(y|do(t = t)) = \sum_{\mathbf{z}} \mathbb{P}(y|t = t, \mathbf{z} = \mathbf{z}) \mathbb{P}(\mathbf{z} = \mathbf{z})$ . In other words, a valid adjustment  $\mathbf{z}$  averages an estimate of  $y$  regressed on  $t$  and  $\mathbf{z}$  with respect to the marginal distribution of  $\mathbf{z}$ , A popular criterion to find valid adjustments is to find a set  $\mathbf{z} \subset \mathcal{V}$  that satisfies the back-door criterion [Pearl, 2009]. Formally, a set  $\mathbf{z}$  satisfies the back-door criterion if (a) it blocks all back-door paths, i.e., paths between  $t$  and  $y$  that have an arrow pointing at  $t$  and (b) no element of  $\mathbf{z}$  is a descendant of  $t$ . While, in general, back-door sets can be found with the knowledge of the causal graph, recent works (see the survey Cheng et al. [2022]) have proposed testable criteria for identifying back-door sets with some causal side information, without requiring the entire graph.
2.2 Adjustment using post-treatment variables
While back-door adjustment is widely used, there are scenarios where no back-door set exists, e.g., when there is an unobserved confounder between  $t$  and  $y$ . If no back-door set can be found from the pre-treatment variables, Pearlian theory can be used to identify post-treatment variables, i.e., the variables that occur after the treatment in the causal ordering, to obtain a front-door adjustment.
Definition 5 (Front-door criterion). A set  $\mathbf{z} \subset \mathcal{V}$  satisfies the front-door criterion with respect to  $t$  and  $y$  if (a)  $\mathbf{z}$  intercepts all directed paths from  $t$  to  $y$  (b) all back-door paths between  $t$  and  $\mathbf{z}$  are blocked, and (c) all back-door paths between  $\mathbf{z}$  and  $y$  are blocked by  $t$ .
If a set  $\mathbf{z}$  satisfies the front-door criterion, then the causal effect can be written as
$$
\mathbb {P} (y | d o (t = t)) = \sum_ {\boldsymbol {z}} \left(\sum_ {t ^ {\prime}} \mathbb {P} (y | t = t ^ {\prime}, \boldsymbol {z} = \boldsymbol {z}) \mathbb {P} (t = t ^ {\prime})\right) \mathbb {P} (\boldsymbol {z} = \boldsymbol {z} | t = t). \tag {5}
$$
Intuitively, front-door adjustment estimates the causal effect of  $t$  on  $y$  as a composition of two effects: (a) the effect of  $t$  on  $\mathbf{z}$  and (b) the effect of  $\mathbf{z}$  on  $y$ . However, one still needs the knowledge of the causal graph  $\mathcal{G}$  to find a set satisfying the front-door criterion.
Inspired by the progress in finding back-door sets without knowing the entire causal graph, we ask: Can testable conditions be derived to identify front-door-like sets using only partial structural information about post-treatment variables? To that end, we consider the following side information.
Assumption 1. The outcome  $y$  is a descendant of the treatment  $t$ .
Assumption 2. There is an unobserved confounder between the outcome  $y$  and the treatment  $t$ .
Assumption 3. b, the set of all children of the treatment  $t$ , is observed and known.
Assumption 1 is a fundamental assumption in most causal inference works, as it forms the basis for estimating non-trivial causal effects. Without it, the causal effect would be zero. Assumption 2 rules out the existence of sets that satisfy the back-door criteria, necessitating a different way of estimating the causal effect. Assumption 3 captures our side information by requiring every children of the treatment to be known and observed. To contrast, the side information in data-driven works on back-door adjustment requires a parent of the treatment to be known and observed [Shah et al., 2022].
Our assumptions imply that  $\mathbf{b}$  intercepts all the directed paths from  $t$  to  $y$ . Given this, it is natural to ask whether  $\mathbf{b}$  satisfies the front-door criterion (Definition 5). We note that, in general, this is not true. We illustrate this via Figure 2 where we provide a causal graph  $\mathcal{G}^{toy}$  satisfying our assumptions. However,  $\mathbf{b}$  is not a valid front-door set in  $\mathcal{G}^{toy}$  as the back-door path between  $\mathbf{b}$  and  $y$  via  $\mathbf{z}^{(i)}$  is not blocked by  $t$ . Therefore, estimating the causal effect by assuming  $\mathbf{b}$  is a front-door set might not always give an unbiased estimate. In the next section, we leverage the given side information and provide testable conditions to identify front-door-like sets.
Figure 2: The graph  $\mathcal{G}^{toy}$  satisfying Assumptions 1 to 3 where  $u_{i}$  are unobserved.
3 Front-door Adjustment Beyond Markov Equivalence
In this section, we provide our main results, an algorithm for ATE estimation, and discuss the relationship to PAG-based methods. Our main results use observational criteria for causal effect estimation under Assumptions 1 to 3 using post treatment variables.
3.1 Causal effect estimation using post-treatment variables
First, we state a conditional independence statement implying causal identifiability. Then, we provide additional conditional independence statements resulting in a unique formula for effect estimation.
Causal identifiability (Definition 4) implies that the causal effect is uniquely determined given an observational distribution  $\mathbb{P}(\mathcal{V})$  and the corresponding causal graph  $\mathcal{G}$ . We now show that satisfying a conditional independence statement (which can be tested solely from observational data, without requiring the graph  $\mathcal{G}$ ) guarantees identifiability. We provide a proof in Appendix D.
Theorem 3.1 (Causal Identifiability). Suppose Assumptions 1 to 3 hold. If there exists a set  $\mathbf{z} \subseteq \mathcal{V} \setminus \{t, \mathbf{b}, y\}$  such that  $\mathbf{b} \perp_{d} y|t, \mathbf{z}$ , then the causal effect of  $t$  on  $y$  is identifiable from observational data without the knowledge of the underlying causal graph  $\mathcal{G}$ .
While the above result leads to identifiability, it does not provide a formula to compute the causal effect. In fact, the conditional independence  $\mathbf{b} \perp_{d} y|t$ ,  $\mathbf{z}$  alone is insufficient to establish a unique formula, and different causal graphs lead to different formula. To illustrate this, we provide two SMCMs where Assumptions 1 to 3 and  $\mathbf{b} \perp_{d} y|t$ ,  $\mathbf{z}$  hold, i.e., causal effect is identifiable from observational data via Theorem 3.1, but the formula is different. First, consider the SMCM in Figure 3(top) with  $\mathbf{z} = (z_1, z_2)$  where causal effect is given by following formula (derived in Appendix D):
$$
\mathbb {P} (y | d o (t = t)) = \sum_ {z _ {1}, z _ {2}, \boldsymbol {b}} \left(\sum_ {t ^ {\prime}} \mathbb {P} (y | z _ {1}, z _ {2}, t ^ {\prime}) \mathbb {P} \left(t ^ {\prime} | z _ {1}\right)\right) \times \mathbb {P} \left(z _ {2} | \boldsymbol {b}\right) \mathbb {P} (\boldsymbol {b} | t, z _ {1}) \mathbb {P} \left(z _ {1}\right). \tag {6}
$$
Next, consider the SMCM in Figure 3(bottom) with  $\mathbf{z} = z_{2}$  where the causal effect is given by the front-door adjustment formula in (5) as  $\mathbf{z}$  satisfies the front-door criterion. It remains to explicitly show that the formula in (6) is different from (5). To this end, we create a synthetic structural equation model (SEM) respecting the graph in Figure 3(top) and show that the formula in (5) gives a non-zero ATE error. In our SEM, the unobserved variable has a uniform distribution over [1, 2]. Each observed variable except  $t$  is a sum of  $(i)$  a linear combination of its parents with
Figure 3: SMCMs on (top) & (bottom) satisfy  $\mathbf{b} \perp_{d} y|t$ ,  $\mathbf{z}$  but have different causal effect estimation formulae.
coefficients drawn from uniform distribution over  $[1,2]$  and  $(ii)$  a zero-mean Gaussian noise. The treatment variable is binarized by applying a standard logistic model to a linear combination of its parents with coefficients drawn as before. The ATE error averaged over 50 runs with 50000 samples in each run is  $0.3842 \pm 0.0207$ . See more experimental details in Appendix G.
Next, we provide two additional conditional independence statements that imply a unique formula for causal effect estimation. Our result is a generalized front-door with a formula identical to (5) as if  $\mathbf{z}$  were a traditional front-door set. We also offer an alternative formula by utilizing a specific partition of  $\mathbf{z}$  obtained from the conditional independence statements. We provide a proof in Appendix E.
Theorem 3.2 (A generalized front-door condition). Suppose Assumptions 1 to 3 hold. Let  $\mathbf{z} \subseteq \mathcal{V} \setminus \{t, \mathbf{b}, y\}$  be a set satisfying
$$
\mathbf {b} \perp_ {d} y | t, \mathbf {z}, \tag {7}
$$
such that  $\mathbf{z}$  can be decomposed into  $\mathbf{z}^{(o)}\subseteq \mathbf{z}$  and  $\mathbf{z}^{(i)} = \mathbf{z}\setminus \mathbf{z}^{(o)}$  with
$$
(i) \mathbf {z} ^ {(i)} \perp_ {d} t \quad a n d \quad (i i) \mathbf {z} ^ {(o)} \perp_ {d} t | \mathbf {b}, \mathbf {z} ^ {(i)}. \tag {8}
$$
Then,  $\mathbf{z}$  and  $\mathbf{s} \triangleq (\mathbf{b}, \mathbf{z}^{(i)})$  are generalized front-doors, and the causal effect of  $t$  on  $y$  can be obtained using any of the following equivalent formulae:
$$
\mathbb {P} (y | d o (t = t)) = \sum_ {\boldsymbol {z}} \left(\sum_ {t ^ {\prime}} \mathbb {P} (y | \boldsymbol {z}, t ^ {\prime}) \mathbb {P} \left(t ^ {\prime}\right)\right) \mathbb {P} (\boldsymbol {z} | t). \tag {9}
$$