SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Dennis Frauen, Valentyn Melnychuk & Stefan Feuerriegel
{frouen,melnychuk,feuerriegel}@lmu.de
Causal inference from observational data is crucial for many disciplines such as medicine and economics. However, sharp bounds for causal effects under relaxations of the unconfoundedness assumption (causal sensitivity analysis) are subject to ongoing research. So far, works with sharp bounds are restricted to fairly simple settings (e.g., a single binary treatment). In this paper, we propose a unified framework for causal sensitivity analysis under unobserved confounding in various settings. For this, we propose a flexible generalization of the marginal sensitivity model (MSM) and then derive sharp bounds for a large class of causal effects. This includes (conditional) average treatment effects, effects for mediation analysis and path analysis, and distributional effects. Furthermore, our sensitivity model is applicable to discrete, continuous, and time-varying treatments. It allows us to interpret the partial identification problem under unobserved confounding as a distribution shift in the latent confounders while evaluating the causal effect of interest. In the special case of a single binary treatment, our bounds for (conditional) average treatment effects coincide with recent optimality results for causal sensitivity analysis. Finally, we propose a scalable algorithm to estimate our sharp bounds from observational data.
Causal effects are crucial for decision-making in many disciplines, such as marketing [71], medicine [77], and economics [1]. For example, physicians need to know the treatment effects to personalize medical care, and governments are interested in the causal effects of policies on infection rates during a pandemic. In many such applications, randomized experiments are costly or infeasible, because of which causal effects must be estimated from observational data [59].
Estimating causal effects from observational data may lead to bias due to the existence of confounders, i.e., variables that affect both treatment and outcome [55]. A remedy is to observe all confounders and thus to assume unconfoundedness (e.g., as in [16, 63, 64]). However, in many practical applications, the assumption of unconfoundedness is violated. For example, electronic health records do not capture a patient's ethnic background, which is a known confounder in medicine [52]. In such cases, the causal effect is not identified from observational data, and unbiased estimation is thus impossible [27].
A popular way to perform causal inference in the presence of unobserved confounders is causal sensitivity analysis. Causal sensitivity analysis aims to derive bounds on the causal effect of interest under relaxations of the unconfoundedness assumption. Here, the strength of unobserved confounding is typically controlled by some sensitivity parameter  $\Gamma$ , which also determines the tightness of the bounds. In practice, one chooses  $\Gamma$  through domain knowledge [14, 33, 82] or data-driven heuristics [29]. Then, one derives that the causal effect of interest lies in some informative region. For example,
a suitable  $\Gamma$  may enable - despite unobserved confounding - to infer the sign of the treatment effect, which is often sufficient for consequential decision-making [35].
A common model for causal sensitivity analysis is the marginal sensitivity model (MSM) [31, 35, 36, 67]. A benefit of the MSM is that it does not impose any kind of parametric assumptions on the data-generating process. However, the standard MSM is only applicable in settings where the treatment is a single binary variable. Different extensions have been proposed for continuous treatments [10, 32, 45] and for time-varying treatments and confounders [10]. However, these works are restricted to specific settings, while a unified framework for causal sensitivity analysis is still missing.
Contributions: In this paper, we propose a generalized marginal sensitivity model (GMSM). Our GMSM provides a unified framework for causal sensitivity analysis under unobserved confounding in various settings with multiple discrete, continuous, and time-varying treatments. Crucially, our GMSM includes existing models, such as those in [67] and [32], as special cases. As a result, our GMSM enables a unified approach to deriving sharp bounds for a large class of
Figure 1: Three examples of causal inference settings where our GMSM and bounds are applicable.  $M_{1}$ ,  $M_{2}$  are mediators,  $Y$  is the outcome, and  $\mathbf{X}$  and  $\mathbf{A}$  are (multiple) discrete, continuous, or time-varying covariates and treatments. Variables  $\mathbf{U}_W$  are unobserved confounders between  $\mathbf{A}$  and  $W \in \{M_1, M_2, Y\}$ .
causal effects. To do so, we bound a distribution shift in the unobserved confounders while performing an intervention on the treatments. As a result, we obtain sharp bounds for various causal effects, including (conditional) average treatment effects but also effects for mediation analysis and path analysis (see Fig. 1) and for distributional effects, which have not been studied under an MSM-type sensitivity analysis previously. We also show that, for binary treatments, our bounds coincide with recent optimality results for (conditional) average treatment effects under the MSM. Finally, we propose a scalable algorithm to estimate our sharp bounds from observational data and perform extensive computational experiments to show the validity of our bounds empirically.
In the following, we review related work on causal sensitivity analysis. For a more general review of partial identification and treatment effect estimation under unconfoundedness, we refer to Appendix A.
Causal sensitivity analysis: Causal sensitivity analysis dates back to a study from 1959 showing that unobserved confounders cannot explain away the causal effect of smoking on cancer [14]. This was formalized by introducing sensitivity models that yield bounds on the causal effect of interest under some restriction on the amount of confounding. Previous works introduced a variety of sensitivity models that make different assumptions about the data-generating process and the confounding mechanism. Examples include sensitivity models based on parametric assumptions [30, 62], difference between potential outcome distributions [60, 70], and Rosenbaum's sensitivity model that uses randomization tests [61].
Marginal sensitivity model (MSM): The marginal sensitivity model (MSM) [67] is a common model for sensitivity analysis aimed at settings with binary treatments. Many methods were proposed to estimate bounds from observational data under the MSM. Examples of such methods include linear fractional programs [82] and machine learning such as including kernel-based methods [35] and deep neural networks [31]. Recently, Dorn et al. [18] and Jin et al. [33] showed that estimates from the previous methods are too conservative bounds and, as a remedy, derived closed-form solutions for sharp bounds under the MSM. This was also extended to semiparametric inference [19, 53]. However, all previous methods are limited to binary treatments and (conditional) average treatment effects.
Different extensions for the MSM have been proposed. Jesson et al. [32], Bonvini et al. [10], and Marmarelis et al. [45] developed extensions for continuous treatments. Furthermore, Bonvini et al. [10] also extended the MSM to time-varying treatments and confounders. However, both works are not applicable to settings beyond the estimation of (conditional) average treatment effects, such as
$^{1}$ Code is available at https://github.com/DennisFrauen/SharpCausalSensitivity.
mediation analysis and path analysis or distributional effects. In contrast, we propose a generalized marginal sensitivity model that is compatible with binary, continuous, and time-varying treatments for a variety of causal effects.
Research gap: To the best of our knowledge, no existing causal sensitivity analysis based on the MSM provides a unified framework for deriving bounds for binary, continuous, and time-varying treatments. Furthermore, MSM-based causal sensitivity analysis is restricted to (conditional) average treatment effects and is thus not applicable for complex settings such as effects for mediation analysis and path analysis or distributional effects.
3 Generalized marginal sensitivity model (GMSM)
We first formally define a general setting for causal sensitivity analysis that includes mediation and path analysis (Sec. 3.1 and Sec. 3.2). We then propose our generalized marginal sensitivity model (GMSM) in Sec. 3.3 and compare it with existing sensitivity models from the literature.
Notation: We write random variables as capital letters  $X$  and their realizations in lowercase  $x$ . Bold letters  $\mathbf{X}$  or  $\mathbf{x}$  represent (random) vectors. If  $\mathbf{X} = (X_1, \ldots, X_\ell)$  is a sequence of random variables of length  $\ell$ , we denote  $\bar{\mathbf{X}}_k = (X_1, \ldots, X_k)$  for  $1 \leq k \leq \ell$ . We denote probability distributions over  $X$  as  $\mathbb{P}^X$  where required. The probability mass function for a discrete  $X$  is denoted as  $\mathbb{P}(x) = \mathbb{P}(X = x)$ . If  $X$  is continuous,  $\mathbb{P}(x)$  is the probability density function. We denote  $\mathbb{P}(\cdot)$  as the corresponding probability mass/density function not evaluated at a specific  $x$ . Similarly, we write conditional probability mass functions/density functions as  $\mathbb{P}(y | x) = \mathbb{P}(Y = y | X = x)$  and conditional expectations as  $\mathbb{E}[Y | x] = \mathbb{E}[Y | X = x] = \int y \mathbb{P}(y | x) \, \mathrm{d}y$ . We denote  $\mathbb{P}(y | do(X = x))$  as the probability mass function/density function of  $Y$  after performing the do-intervention  $do(X = x)$  [55]. Finally, we define  $\mathbf{X}_Y = p a(Y) \cap \mathbf{X}$ , where  $p a(Y)$  denotes the parents of  $Y$  in a given causal graph.
3.1 Problem setup for generalized causal sensitivity analysis
We formalize causal sensitivity analysis based on Pearl's structural causal model framework [55].
Definition 1. A structural causal model (SCM)  $\mathcal{M}$  is a tuple  $(\mathbf{V},\mathbf{U},\mathcal{F},\mathbb{P}^{\mathbf{U}})$ , where  $\mathbf{V} = (V_{1},\ldots ,V_{k})$  are observed endogenous variables,  $\mathbf{U}$  are unobserved exogenous variables determined outside of the model,  $\mathcal{F} = \{f_{V_1},\dots,f_{V_k}\}$  is a set of functions so that each  $f_{V_i}$  maps a set of parents  $pa(V_i)\subseteq \mathbf{V}\cup \mathbf{U}$  to  $V_{i}$ , and  $\mathbb{P}^{\mathbf{U}}$  is a probability distribution on  $\mathbf{U}$ .
Every SCM  $\mathcal{M}$  induces unique directed graph $^2$ $\mathcal{G}_{\mathcal{M}}$  on  $\mathbf{V} \cup \mathbf{U}$  by drawing a directed edge from  $V_1$  to  $V_2$  if  $V_1 \in pa(V_2)$ . In this paper, we assume that  $\mathcal{G}_{\mathcal{M}}$  is acyclic, i.e., does not contain any directed cycle. Then,  $\mathcal{M}$  induces a unique joint probability distribution  $\mathbb{P}^{\mathbf{V} \cup \mathbf{U}}$  on  $\mathbf{V} \cup \mathbf{U}$ . Furthermore,  $\mathcal{M}$  induces unique interventional distributions  $\mathbb{P}^{\mathbf{V} \cup \mathbf{U}}_{do(\mathbf{A} = \mathbf{a})}$  when performing the do-intervention  $do(\mathbf{A} = \mathbf{a})$  for some observed treatments  $\mathbf{A} \subseteq \mathbf{V}$  [4, 55].
We consider settings where we observe four distinct types of endogenous variables  $\mathbf{V} = \{\mathbf{X},\mathbf{A},\bar{\mathbf{M}}_{\ell},Y\}$ : observed confounders  $\mathbf{X}\in \mathcal{X}\subseteq \mathbb{R}^{d_x}$ , treatments  $\mathbf{A}\in \mathcal{A}\subseteq \mathbb{R}^{d_a}$ , discrete mediators  $\bar{\mathbf{M}}_{\ell} = (M_1,\dots ,M_\ell)$  with  $M_{i}\in \mathbb{N}$  for  $1\leq i\leq \ell$ , and an outcome  $Y\in \mathbb{R}$ . In a medical setting,  $\mathbf{X}$  might be patient characteristics (gender, age, medical history, etc.),  $\mathbf{A}$  a medical treatment,  $\bar{\mathbf{M}}_{\ell}$  a change in diet, and  $Y$  a variable indicating a health outcome. Possible causal graphs are shown in Fig. 1. We assume w.l.o.g. that  $(M_1,\ldots ,M_\ell)$  are ordered causally, i.e.,  $\bar{\mathbf{M}}_{i - 1}$  are parents of  $M_{i}$  for each  $i\in \{2,\dots ,\ell \}$ . Given treatment interventions  $\bar{\mathbf{a}}_{\ell +1} = (\mathbf{a}_1,\dots ,\mathbf{a}_{\ell +1})$ , we are interested in causal effects of the form
$$
Q (\mathbf {x}, \bar {\mathbf {a}} _ {\ell + 1}, \mathcal {M}) = \sum_ {\bar {\mathbf {m}} _ {\ell} \in \mathrm {N} ^ {\ell}} \mathcal {D} \left(\mathbb {P} ^ {Y} (\cdot | \mathbf {x}, \bar {\mathbf {m}} _ {\ell}, d o (\mathbf {A} = \mathbf {a} _ {\ell + 1})\right) \prod_ {i = 1} ^ {\ell} \mathbb {P} (m _ {i} | \mathbf {x}, \bar {\mathbf {m}} _ {i - 1}, d o (\mathbf {A} = \mathbf {a} _ {i})), \tag {1}
$$
where  $\mathcal{D}$  is some functional that maps the density  $\mathbb{P}^Y (\cdot |\mathbf{x},\mathbf{m},do(\mathbf{A} = \mathbf{a}_{\ell +1}))$  to a scalar value and we sum over all possible realizations  $\bar{\mathbf{m}}_{\ell}$  of  $\bar{\mathbf{M}}_{\ell}$ . We are also interested in averaged causal effects
<sup>2</sup>Note that we consider graphs over both observed and unobserved variables. In the causal inference literature, these are also called augmented causal graphs[9].
$\int_{\mathcal{X}}Q(\mathbf{x},\bar{\mathbf{a}}_{\ell +1},\mathcal{M})\mathbb{P}(\mathbf{x})\mathrm{d}\mathbf{x}$  or differences (Appendix D). The effect  $Q(\mathbf{x},\bar{\mathbf{a}}_{\ell +1},\mathcal{M})$  generalizes many common effects across different causal inference settings.
Example 1 (CATE,  $\ell = 0$ ). If  $\mathcal{D} = \mathbb{E}[\cdot]$  is the expectation functional, Eq. (1) reduces to  $Q(\mathbf{x}, \mathbf{a}, \mathcal{M}) = \mathbb{E}[Y \mid \mathbf{x}, \text{do}(\mathbf{A} = \mathbf{a})]$ . When  $\mathbf{A}$  is continuous, this is known as the conditional dose-response function [7, 32]. For binary treatments  $\mathbf{A} \in \{0, 1\}$ , the query  $Q(\mathbf{x}, 1, \mathcal{M}) - Q(\mathbf{x}, 0, \mathcal{M})$  is known as the conditional average treatment effect (CATE) [16, 78], and its averaged version as the average treatment effect (ATE) [64, 69].
Example 2 (Mediation analysis,  $\ell = 1$ ). If  $\mathcal{D} = \mathbb{E}[\cdot]$  and  $\bar{\mathbf{M}}_1 = M$  is a single mediator, Eq. (1) reduces to  $Q(\mathbf{x},\bar{\mathbf{a}}_1,\mathcal{M}) = \sum_m\mathbb{E}\left[Y\mid \mathbf{x},m,do(\mathbf{A} = \mathbf{a}_2)\right]\mathbb{P}(m\mid \mathbf{x},do(\mathbf{A} = \mathbf{a}_1))$ . If  $A\in \{0,1\}$  is binary and the  $M - Y$  relationship is unconfounded, we obtain the following causal effects studied in mediation analysis [56]:  $Q(\mathbf{x},(\mathbf{a}_1 = 0,\mathbf{a}_2 = 1),\mathcal{M}) - Q(\mathbf{x},(\mathbf{a}_1 = 0,\mathbf{a}_2 = 0),\mathcal{M})$  is the (conditional) natural direct effect (NDE), and  $Q(\mathbf{x},(\mathbf{a}_1 = 1,\mathbf{a}_2 = 0),\mathcal{M}) - Q(\mathbf{x},(\mathbf{a}_1 = 0,\mathbf{a}_2 = 0),\mathcal{M})$  is the (conditional) natural indirect effect (NIE).
In general, Eq. (1) includes so-called path-specific effects if the relationship between mediators and outcome is unconfounded [2, 15]. For example, by setting  $\mathbf{a}_1 = 1$  and  $\mathbf{a}_k = 0$  for all  $2 \leq k \leq \ell + 1$ , we obtain the indirect effect that is passed through the mediator sequence  $(M_1, \ldots, M_\ell)$ . Path-specific effects are important in various applications, including algorithmic fairness, where the aim is to mitigate effects through paths that are considered unfair [50, 51]. Furthermore, we can set  $\mathcal{D}$  to a quantile instead of using the mean in all the examples above. This results in distributional versions of CATE, NDE, NIE, and path-specific effects. For example, if the outcome distribution is skewed or contains outliers, practitioners might prefer the median or other quantiles over the mean due to their robustness properties [17].
If all confounders between treatment and mediators and outcome were observed, we could (under additional assumptions) identify the causal effect  $Q(\mathbf{x}, \bar{\mathbf{a}}_{\ell + 1}, \mathcal{M})$  from the observational distribution  $\mathbb{P}^{\mathbf{V}}$  on  $\mathbf{V}$  by replacing all do-operations with conditional probabilities according to the backdoor criterion [55]. However, under unobserved confounding, identifiability is not possible because SCMs  $\mathcal{M}$  and  $\mathcal{M}'$  exist, which induce the same observational distribution  $\mathbb{P}^{\mathbf{V}}$ , but which result in different causal effects  $Q(\mathbf{x}, \bar{\mathbf{a}}_{\ell + 1}, \mathcal{M}) \neq Q(\mathbf{x}, \bar{\mathbf{a}}_{\ell + 1}, \mathcal{M}')$  [54, 74].
3.2 Causal sensitivity analysis
In the following, we formalize causal sensitivity analysis as maximizing/minimizing the causal effect  $Q(\mathbf{x}, \bar{\mathbf{a}}_{\ell + 1}, \mathcal{M})$  over all SCMs  $\mathcal{M}$  that are compatible with a predefined sensitivity model. Similar approaches have been used for partial identification with instrumental variables [40, 54] and testing identifiability of counterfactuals [74, 75], where all SCMs are considered that are compatible with the observational data. However, a sensitivity model must additionally restrict the joint distribution of both observed and unobserved variables in order to allow for informative bounds on the causal effect.
Definition 2 (Sensitivity model). Let  $\mathbb{P}^{\mathbf{V}}$  denote the distribution of the observed variables  $\mathbf{V} = (\mathbf{X},\mathbf{A},\bar{\mathbf{M}}_{\ell},Y)$ . A sensitivity model  $S$  is a tuple  $(\mathbf{U},\mathcal{P})$ , where  $\mathbf{U} = (\mathbf{U}_W)_W$  are unobserved confounders between  $\mathbf{A}$  and  $W \in \{M_1,\dots,M_\ell,Y\}$ , respectively (see Fig. 1) and a family  $\mathcal{P}$  of joint probability distributions on  $\mathbf{V} \cup \mathbf{U}$ , such that, for all  $\mathbb{P} \in \mathcal{P}$ , it holds that  $\int \mathbb{P}(\mathbf{v},\mathbf{u})\mathrm{d}\mathbf{u} = \mathbb{P}^{\mathbf{V}}(\mathbf{v})$ . We denote the set of all SCMs  $\mathcal{M}$  compatible with  $S$  (i.e., that respect the causal graph, induce a distribution  $\mathbb{P} \in \mathcal{P}$ , and do not contain additional confounders) as  $\mathcal{C}(S)$  (see Appendix E).
Our definition of sensitivity models excludes unobserved confounding between mediators and the outcome. This ensures that the causal effect from Eq. (1) can be interpreted as a path-specific effect [2, 56]. We refer to Sec. 6 for a detailed discussion.
Using the definitions above, we can define causal sensitivity analysis as the following partial identification problem: we aim to obtain bounds  $Q^{-}(\mathbf{x}, \bar{\mathbf{a}}_{\ell + 1}, \mathcal{S}) \leq Q^{+}(\mathbf{x}, \bar{\mathbf{a}}_{\ell + 1}, \mathcal{S})$  so that
$$
Q ^ {+} (\mathbf {x}, \bar {\mathbf {a}} _ {\ell + 1}, \mathcal {S}) = \sup  _ {\mathcal {M} \in \mathcal {C} (\mathcal {S})} Q (\mathbf {x}, \bar {\mathbf {a}} _ {\ell + 1}, \mathcal {M}) \text {a n d} Q ^ {-} (\mathbf {x}, \bar {\mathbf {a}} _ {\ell + 1}, \mathcal {S}) = \inf  _ {\mathcal {M} \in \mathcal {C} (\mathcal {S})} Q (\mathbf {x}, \bar {\mathbf {a}} _ {\ell + 1}, \mathcal {M}). \tag {2}
$$
$Q^{+}(\mathbf{x},\bar{\mathbf{a}}_{\ell +1},\mathcal{S})$  is the maximal causal effect that can be achieved by any SCM that is compatible with the sensitivity mode  $\mathcal{S}$  (and vice versa for  $Q^{-}(\mathbf{x},\bar{\mathbf{a}}_{\ell +1},\mathcal{S}))$ . Hence, if the sensitivity model is valid, i.e., contains the ground-truth distribution over observed variables and unobserved confounders, we know that the ground-truth causal effect must be contained in the interval  $[Q^{-}(\mathbf{x},\bar{\mathbf{a}}_{\ell +1},\mathcal{S}),Q^{+}(\mathbf{x},\bar{\mathbf{a}}_{\ell +1},\mathcal{S})]$ . Bounds for average causal effects and effect differences follow immediately (see Appendix D).
3.3 Generalized marginal sensitivity model (GMSM)
We introduce now the GMSM. We begin by providing the general definition and then show that this extends various marginal sensitivity models from existing literature.
Definition 3 (GMSM). The generalized marginal sensitivity model (GMSM) is a sensitivity model  $(\mathbf{U},\mathcal{P})$ , where  $\mathcal{P}$  contains all  $\mathbb{P}$  that satisfy the following sensitivity constraint: For each  $W\in$ $\{M_1,\ldots ,M_\ell ,Y\}$ , there exist bounds  $s_W^{-}(\mathbf{a},\mathbf{x})\leq 1\leq s_W^{+}(\mathbf{a},\mathbf{x})$ , so that, for all  $\mathbf{u}_W$ ,  $\mathbf{x}$ , and  $\mathbf{a}$
$$
s _ {W} ^ {-} (\mathbf {a}, \mathbf {x}) \leq \frac {\mathbb {P} \left(\mathbf {U} _ {W} = \mathbf {u} _ {W} \mid \mathbf {x} , \mathbf {a}\right)}{\mathbb {P} \left(\mathbf {U} _ {W} = \mathbf {u} _ {W} \mid \mathbf {x} , d o (\mathbf {A} = \mathbf {a})\right)} \leq s _ {W} ^ {+} (\mathbf {a}, \mathbf {x}). \tag {3}
$$
The GMSM bounds the distribution shift in the unobserved confounders  $\mathbf{U}_W$  when performing the intervention  $do(\mathbf{A} = \mathbf{a})$  instead of conditioning on  $\mathbf{A} = \mathbf{a}$ . This is a restriction on the strength of the effect the unobserved confounders  $\mathbf{U}_W$  can have on the treatment  $\mathbf{A}$ . If  $\mathbf{U}_W$  has no effect on  $\mathbf{A}$ , Eq. (3) holds with  $s_W^-(\mathbf{a}, \mathbf{x}) = s_W^+(\mathbf{a}, \mathbf{x}) = 1$ . Hence, the further  $s_W^-(\mathbf{a}, \mathbf{x})$  and  $s_W^+(\mathbf{a}, \mathbf{x})$  deviate from 1, the larger is the effect from  $\mathbf{U}_W$  on  $\mathbf{A}$  that the GMSM allows for. We will often use a weighted GMSM, which expresses the bounds in terms of a sensitivity parameter.
Definition 4 (Weighted GMSM). A weighted GMSM is a GMSM where  $s_W^- (\mathbf{a},\mathbf{x})$  and  $s_W^+ (\mathbf{a},\mathbf{x})$  can be written as  $s_W^- (\mathbf{a},\mathbf{x}) = \frac{1}{(1 - \Gamma_W)q_W(\mathbf{a},\mathbf{x}) + \Gamma_W}$  and  $s_W^+ (\mathbf{a},\mathbf{x}) = \frac{1}{(1 - \Gamma_W^{-1})q_W(\mathbf{a},\mathbf{x}) + \Gamma_W^{-1}}$  for a sensitivity parameter  $\Gamma_W \geq 1$  and a weight function  $q_{W}(\mathbf{a},\mathbf{x}) \in [0,1]$  for all  $\mathbf{x}$  and  $\mathbf{a}$ .
In a weighted GMSM, the sensitivity parameter  $\Gamma_W$  captures the overall restriction on the unobserved confounding strength across individuals. If  $\Gamma_W = 1$ , no unobserved confounding is allowed, and unconfoundedness holds. For  $\Gamma_W \to \infty$ , the restriction is relaxed completely, and arbitrary confounding strength is allowed. The weight function  $q_W(\mathbf{a}, \mathbf{x}) \in [0,1]$  offers several advantages. It allows to further restrict confounding for individuals with treatments  $\mathbf{a}$  and covariates  $\mathbf{x}$ . As  $q_W(\mathbf{a}, \mathbf{x}) \to 1$ , confounding is more strongly restricted until unconfoundedness is reached. This is helpful in applications where prior knowledge about the confounding structure is available. As an example, consider a medical setting where we want to estimate the effect of new drug treatments on the risk of developing a certain disease, but we suspect that the data is confounded by the individual's genetic risk for the disease, which is not measured in the observational data. However, we also might know that genetic risk for the disease is not relevant for a specific combination of age and gender.