Causal inference with imperfect instrumental variables

1 Introduction

Inferring causal relations from data is a central goal in any empirical science. Yet, in spite of its importance, causality has remained a thorny issue. Misled by the commonplace sentence stating that “correlation does not imply causation,” causal inference persists in the view of many as a noble but practically impossible task. Contrary to that, however, the surge and development of the causality theory [1,2] has proven formal conditions under which cause and effect relations can be extracted.

Consider the simplest and fundamental question of deciding whether observed correlations between two variables, A (also known as treatment) and B (also known as effect), are due to some direct causal influence of the first over the second, or due to a common cause, a third, potentially latent (nonobservable) variable Λ . Both causal models are observationally equivalent, meaning that both models can generate the same set of possible correlations observed between the target variables. Notwithstanding, causal conclusions can be reached if, instead of passively observing the events, we perform interventions [1,3,4]. In particular, interventions on A put this variable under the experimenter’s control, turning it independent of any latent common cause. If after the intervention, one still observes correlations between A and B , then it is possible to unambiguously conclude that A is a cause of B . Interventions, however, are often unavailable for various practical, fundamental, or ethical issues.

An elegant way to circumvent such issues is the instrumental variables [1,5,6, 7,8,9, 10,11]. If a proper instrument X , correlated with A but statistically independent of Λ , can be found, then the causal effect of A over B can be estimated even in the absence of interventions or structural equations. Nevertheless, since the instrumental conditions depend on an unobservable variable, identifying an instrument seems to be a matter of judgment that cannot be supported solely by the data. To cope with that, instrumental inequalities have been introduced [12,13,14], constraints that should be respected by any experiment in compliance with the instrumental assumptions. Thus, the violation of instrumental inequality is an explicit proof that one does not have a proper instrument. Does that mean, however, that no causal inference at all can be made if an instrumental inequality is violated? Or can we still rely on that instrument, even though imperfect, to infer causal relations?

Motivated by these questions, we analyze in detail a generalization of the instrumental causal structure, where we drop the assumption that one has a perfect instrument. More specifically, we relax the assumption that the instrumental variable X should be independent of the latent factor Λ . Considering the case where A and B are dichotomic and X is also discrete, we derive new instrumental inequalities that take explicitly into account the correlation between X and Λ . We also generalize the bounds on the average causal effect (ACE) [1,3,4], maximized over all possible realizations of A and B , for more general instruments. Although non-standard in the literature, this last maximization comes naturally from the idea of taking the highest possible causal influence from A into B .

Finally, we make a connection with the field of quantum foundations, where violation of instrumental inequalities can appear without relaxing the measurement independence assumption [15,16, 17,18]. Using our results, we establish the minimal measurement dependence needed in the classical instrumental scenario to explain such violations and, as a result, we analyze the robustness of instrumental tests as witnesses of nonclassical behavior.

The article is organized as follows. In Section 2, we discuss how instrumental variables can be employed to put lower bounds on the cause and effect relations between two variables. In Section 3, we discuss the violations of independence assumption and how modified instrumental inequalities and causal bounds on ACE can be derived to take that into account. In Section 4, we discuss quantum violations of the modified inequalities. In Section 5, we discuss our findings and point out interesting questions for future research.

Notations: Throughout the article, we denote random variables by capital letters A , B , X , and Λ , as well as Λ X , Λ A , Λ B . Without loss of generally, we consider these random variables taking values in the set of nonnegative integers Z 0 + . Probability of an event E is denoted as p ( E ) . We use a common shorthand notation p ( a ) = p ( A = a ) to denote the probability of A taking value a . Similar shorthand notation is used for conditional probabilities, e.g., p ( a ∣ x ) = p ( A = a ∣ X = x ) , and interventions, e.g., p ( b ∣ do ( a ) ) = p ( B = b ∣ do ( A = a ) ) . We use one exception to this rule for probabilities of the form p ( i , j ∣ k ) and p ( l ∣ do ( m ) ) , which should be read as p ( A = i , B = j ∣ X = k ) and p ( B = l ∣ do ( A = m ) ) , respectively, for any i , j , k , l , m ∈ Z 0 + . We also use the common notation [ n ] = { 0 , 1 , … , n − 1 } .

2 Instrumental variables, instrumental inequalities, and causal bounds

Before getting into details and illustrating the power of an instrumental variable as a causal inference tool, we discuss a simple linear structural model, b = β a + λ , where β can be understood as the strength of the causal influence of A over B and Λ is a latent factor that might affect both A and B . By introducing the instrumental variable X and assuming its statistical independence from Λ , one can infer the causal strength β . For that aim, it is enough to multiply both sides of the structural equation by x and compute the observed correlations, defined as corr ( A , B ) = ⟨ A , B ⟩ / ⟨ A ⟩ ⟨ B ⟩ where ⟨ A , B ⟩ = ∑ a , b ( a ⋅ b ) p ( a , b ) is the expectation value of A and B . By doing that, we obtain that β = corr ( X , B ) /corr ( X , A ) . It is worth highlighting that the linear structural model is not a requirement. As will be detailed below, under certain assumptions, instrumental variables provide a general tool for estimating causal influences, even in the absence of structural equations.

More formally, an instrumental variable X has only a direct causal influence over A and should be independent of any latent factors acting as a common cause for variables A and B . The latter assumption is known by various names such as the independence assumption [8], ignorable treatment assignment [6], no confounding for the effect of X on B [9], and in the literature of quantum foundations is termed as the measurement independence assumption [15,16,17, 18,19], an issue of crucial relevance for the violation of Bell inequalities [20,21]. Furthermore, even though X and B might be correlated, those correlations can only be mediated by A , that is, there is no direct causal influence of X on B , the so-called exchangeability assumption or exclusion restriction [10,11]. See Figure 1(a) for a directed acyclic graph (DAG) description of the instrumental scenario. Altogether, any observed distribution p ( a , b , x ) compatible with these instrumental conditions should then be decomposable as

Typically, instead of looking at the joint distribution p ( a , b , x ) , one rather considers the conditional distribution p ( a , b ∣ x ) that, under the same causal assumptions, can be decomposed as

Figure 1

Causal graphs describing the instrumental scenario and its relaxations. Circular nodes correspond to observed variables, and rectangular ones are latent. Directed edges represent the causal links. (a) The instrumental scenario: the controlled variable X is completely independent of a latent variable Λ . (b) A relaxed instrumental scenario, where the exchangeability assumption (also known as exclusion restriction) is relaxed; in other words, there is a direct causal influence of X over B . We are not focusing on this relaxation. (c) A relaxed instrumental scenario, where the independence assumption is relaxed. Differently from (a), there is a causal link from Λ to X . Consequently, we no longer assume that the instrumental variable X and the common cause Λ are independent, that is, p ( x , λ ) ≠ p ( x ) p ( λ ) . We are focusing on this relaxation.

The set of probability distributions of the form in equation (2) is bounded in the space of all possible distributions p ( a , b ∣ x ) . These bounds are given by the so-called instrumental inequalities [12,13,14]. For the simplest case of dichotomic variables there is only one type of instrumental inequalities, which we will call Pearl’s inequality [12] and can be summarized as follows:

Importantly, the instrumental variable can be used for causal inference even in the absence of structural models, something typical in the context of quantum information and refereed there as the device-independent framework [22,23]. In particular, simply from the observed data distribution p ( a , b ∣ x ) one can infer the effect of interventions on the variable A and thus obtain a lower bound on the ACE A → B defined as

in which

and do ( a ) represents the intervention over the variable A . It is worth highlighting that the maximization over a , a ′ , b , as we used in our definition of ACE, is not common in the literature, where it is typically defined as E [ B ∣ do ( A = 1 ) ] − E [ B ∣ do ( A = 0 ) ] , where E [ ⋅ ] denotes expectation with respect to the probability distribution p ( b ∣ do ( a ) ) . Our aim is to identify a causal influence of one random variable over the other, hence, it is natural to consider the maximum ACE that could occur over possible values of a , a ′ , and b . The resulting definition is not linear, but piecewise linear (e.g., absolute value for the case of binary A and B). However, by fixing the relation between the do-probabilities that enter the ACE definition, e.g., p ( 0 ∣ do ( 0 ) ) ≥ p ( 0 ∣ do ( 1 ) ) , and treating every such case separately, we can effectively linearize the ACE function. Also, since we consider a theoretical study with no particular meaning given to outcomes “0” or “1,” there is always freedom of relabeling the outcomes.

As shown in ref. [3], for the case of binary random variables A , B , and X the value of the ACE in equation (4) can be lower-bounded as

The bound above is particularly relevant because it shows that the effect of interventions can be inferred simply from the observational data. Thus, instrumental variables offer a central tool for situation where interventions are not possible.

The bound in equation (6) is one of the eight expressions given in ref. [3], which provide nontrivial lower bounds on ACE A → B . The three of these eight bounds can be obtained by relabeling the one in equation (6). The remaining four are

and the ones obtained by relabeling of the values of X , A , and B . However, these four inequalities are not interesting for us, since they continue to hold for any causal structure (including the ones with imperfect instruments). To say it otherwise, these inequalities cannot be violated even if the latent variables (possibly quantum) and the instrumental variables are perfectly correlated.

For the case of more general random variables (not only binary), one can obtain a system of linear inequalities of the form

where C i = ∑ c i a , b , x p ( a , b ∣ x ) are linear expressions of the probabilities with c a , b , x ∈ R and the maximum is taken over all such expressions. These lower bounds C i can be found using the tools of linear programming [24]. We refer to this type of bounds on the ACE as causal bounds. The causal bound in equation (6) we denote as C 1 . Other bounds studied in this work are given in Section 3.4.

In this work, we focus on the case where the variables A and B are binary, i.e., taking values a , b ∈ { 0 , 1 } but the instrumental variable can take more values (we resort to an arbitrary set of values when discussing the instrumental inequalities and the following two cases x ∈ { 0 , 1 } and x ∈ { 0 , 1 , 2 } , when referring to the problem of causal bounds). At the same time, the methods developed in this article are applicable to the general case where all the random variables take values in arbitrary finite sets. We remark, however, that there are frameworks different from the linear programming approach, we pursue here. For instance, refs [25,26,27] employ Artstein’s theorem [28] from random set theory to bound continuous functionals of potential outcomes from the literature on treatment effects and offer a valuable tool for when analytical bounds are hard to obtain, as would be the case for variables with high cardinality or even continuous.

3 Relaxing the independence assumption

For the causal bounds such as in equation (6) to hold, one has to guarantee that the instrumental causal assumptions are fulfilled. If any instrumental inequality such as in equation (3) is violated by the observed data distribution p ( a , b ∣ x ) , then one can unambiguously conclude that at least one of the instrumental assumptions does not hold. Such a violation can have two distinct roots. As shown in refs [23,29, 30,31], even if one imposes the instrumental causal structure to a quantum experiment, still some instrumental inequalities can be violated. This can be seen as a stronger version of Bell’s theorem [20], showing that correlations mediated via quantum entanglement can fail to have a description in terms of standard causal models. The second kind of mechanism, purely classical, and the one we mainly focus on in this article, is the failure of causal assumptions.

For instance, the violation of an instrumental inequality could be motivated by a direct causal influence of X over B , a violation of the exchangeability assumption (exclusion restriction) shown in Figure 1(b), a scenario for which particular cases have been analyzed in ref. [23] and for which the LP framework we develop can be readily applied. Here, as shown in Figure 1(c) we focus on the violation of the independence assumption. Differently from the typical scenario, we no longer assume that the instrumental variable X and the common source Λ are independent, that is, p ( x , λ ) ≠ p ( x ) p ( λ ) .

In order to facilitate our analysis, we focus on the DAG including an additional causal link between a latent variable Λ and instrument X (Figure 1(c)). In this case, the observed probability distribution decomposes as

In the following, we argue that without loss of generality, we can consider the response functions p ( a ∣ λ , x ) , p ( b ∣ λ , a ) , and p ( x ∣ λ ) to be deterministic. To see this, consider rewriting p ( a ∣ λ , x ) using a uniformly distributed auxiliary variable ξ a taking values in [ 0 , 1 ] in the following way:

where p ˜ ( a ∣ λ , x , ξ a ) = 1 , if ξ a ≤ p ( a ∣ λ , x ) and 0, otherwise. Replacing the original response function with p ˜ ( a ∣ λ , x , ξ a ) leads to the same observed distribution. We do the same for the other two response functions introducing auxiliary variables ξ b and ξ x . The collection of the newly introduced auxiliary variables can be considered as a part of the global variable Λ , resulting in the decomposition involving only deterministic conditional probability distributions. Finally, even though for this argument we made the enlarged confounding variable Λ continuous, it is clear that for the case of discrete observed variables A , B , and X there is only a finite number of possible combination of deterministic assignments, which means that considering Λ discrete is not a loss of generality. A formal proof of the above argument can be found in refs [32,33] for a related causal scenario of Bell test [20].

It is convenient to treat a realization of Λ as a vector ( λ x , λ a , λ b ) , where λ x ∈ [ m x ] , λ a takes its values in [ k a m x ] and λ b ∈ [ k b k a ] . In this relaxed case, any distribution p ( a , b ∣ x ) factorizes as follows:

where we use the same notation p ( ⋅ ) for different response functions in order to avoid cumbersome expressions. As argued above, the conditional probabilities p ( a ∣ x , λ a ) and p ( b ∣ a , λ b ) , and p ( x ∣ λ x ) can be taken to be deterministic leading to,

where f i ( ⋅ ) and g i ( ⋅ ) denote deterministic functions, specified by λ a and λ b , respectively, and we took p ( x ∣ λ x ) = δ x , λ x .

In analogy to ref. [17], we use a common measure of dependence between X and Λ for the instrumental scenario given by

Crucially to our subsequent analysis, we cast it as the l 1 -norm of the following vector:

where q λ x , λ a , λ b = p ( λ x , λ a , λ b ) and for the canonical basis { e i , j , k } i , j , k in R m x k a m x k b k a , we have a matrix M ,

Before stating our results, we note that in the literature there are a number of studies dealing with the estimation of causal effects when some assumptions are relaxed. A standard tool is that of the potential outcomes model. Considering a binary treatment A, this model implies that

where B 0 and B 1 are the potential (nonobserved) outcomes. It is well known that under the assumption of unconfoundedness, implying the conditional independencies A ⊥ ⊥ B 1 ∣ W and A ⊥ ⊥ B 0 ∣ W , where W is a vector of observed covariates, it is possible to estimate the ACE. Understanding the sensitivity of causal inference to partial conditional independence is a central problem addressed in a number of studies [34,35, 36,37,38, 39,40]. Note, however, that the independence assumption we consider here is native of the instrumental DAG and different of that of the potential outcomes model. Our work is thus complementary to those relaxing unconfoundedness in the potential outcomes model.

Note that the scenario with of a direct cause from X to A is equivalent to the scenario in which X and A are correlated due to confounding variables. This can be verified by noting that the set of probability distributions compatible with the instrumental scenario with the arrow X → A , described by the following causal model

is the same set as that of the model where X ← Δ → A , since

where in the second step we have simply chosen p ( δ ∣ x ) = δ x , δ . Note, however, that the same reasoning does not necessarily apply in the case where we permit the violation of the independence assumption. In this case, one needs to distinguish two situations (Figure 2). In the case when the confounding factor Δ is directly influenced by Λ , the analysis of this article can be applied (with a slight modification). As for the second case, in which the confounding factors Δ and Λ are independent, a different method would need to be developed, since the independence condition p ( Δ = i , Λ = j ) = p ( Δ = i ) p ( Λ = j ) cannot be directly incorporated into a linear program (LP). At the same time, ignoring the independence of Δ and Λ (which are latent factors) would always lead to a valid, but perhaps nontight linear bounds.

Figure 2

Causal graphs describing a noncausal instrumental variable, where the correlations between X and A are mediated by another latent factor Δ and its relaxations. (a) Relaxation of the independence assumption where Λ has some causal influence over Δ . (b) The independence assumption is relaxed by a direct influence of Λ over X .

3.2 Dependencies in the simplest instrumental scenario

Building on the results of this section, here we investigate the minimal required dependence for a fixed violation of instrumental inequalities and bounds on ACE in the simplest instrumental scenario when all the observed random variables are binary. For the both types of inequalities, namely instrumental inequality in equation (3) and causal bound in equation (6), we give exact solutions to the corresponding LPs in equation (20).

Lemma 2

For the instrumental scenario with binary observed random variables X , A , B and a latent variable Λ , the minimal dependence required to explain a violation of the instrumental inequality by α is ℳ X : Λ = 4 p ( X = 0 ) p ( X = 1 ) α .

See Appendix A for the proof. We conclude that for a given violation α , the uniformly distributed instrumental variable X requires the highest dependence. The reverse also holds true: if the instrumental variable is uniformly random, a given dependence will permit the lowest amount of violation. Our result implies that even though we do not have direct empirical access to the common source between A and B , from observational data p ( a , b ∣ x ) alone we can lower-bound the amount of dependence ℳ X : Λ present in a given experiment.

Next we investigate how the violation of the lower bound on ACE as in equation (6) translates to the required measurement dependence.

Lemma 3

For the instrumental scenario with binary observed random variables X , A , B and a latent variable Λ , the minimal measurement dependence required to explain a violation of the lower bound on ACE as in equation (6) by α is ℳ X : Λ = 4 p ( X = 0 ) p ( X = 1 ) 2 − p ( X = 0 ) α .

See Appendix B for the proof. Until now, we asked a question which degree of measurement dependence is required to explain violation of a linear inequality (e.g., instrumental inequalities or causal bounds) and we gave an analytical solution for the simplest scenario with binary observed variables. One can, however, ask the reverse question of how the linear inequalities change in the simplest instrumental scenario, if some level of measurement dependence is present in a given setup. This is the inverse problem to the one considered in this section. Since both of these problems aim at estimating the same dependency, they have the same solution, namely the piecewise linear dependence in Observation 1. As a result, we can derive adapted linear inequalities (e.g., binary instrumental inequalities and causal bounds) that accounts for the dependence between X and Λ , explicitly.

Corollary 3.1

Given a linear inequality valid for the simplest instrumental scenario, K inst ≥ 0 , the adapted linear inequality in terms of the measurement dependency is

(21) K inst + ℳ X : Λ u ≥ 0 ,

where u is the optimization parameter of the dual LP in equation (20).

The above corollary shows that one can still infer cause and effect relations even with nonperfect instruments. Also note that in case of independence, ℳ X : Λ = 0 , we directly recover the inequalities valid in instrumental scenario (Pearl’s inequality in equation (3) and the causal bound in equation (6)).

For a more general case, when the instrumental variable can take more than two values, adapting a linear inequality valid for the perfect instrumental scenario is also possible. However, it is a more involving task as the minimal measurement dependence does not have to be linear in the observed violation, as pointed out in Observation 1. We give numerical treatment for this problem in Section 3.4 and in Figure 3.

Figure 3

Measurement dependence ℳ X : Λ for violations α of instrumental inequality I 2 (left), and causal bounds C 2 (center) and C 3 (right). The maximal violation of each inequality attainable in quantum theory is marked by Q max (Section 4).

4 Quantum violations of instrumental tests

We saw that the instrumental and causal bounds can be violated if a certain amount of measurement dependence is present between an instrumental variable and a common cause Λ . However, a violation is also possible if we do not assume any relaxation on the instrumental scenario, but instead we consider the case, when the unobserved common cause can be a quantum state [23,48,49]. Since Bell’s theorem [20], it is known that classical causal models are unable to explain the correlations observed by measurements on entangled quantum systems. In other terms, if we impose a given causal structure to a quantum experiment, the probability distribution obtained by employing the quantum rules cannot be explained if we assume a classical causal model where each node, including the confounder, corresponds to a random variable.

More precisely, the postulates of quantum theory impose that (i) the state of a physical system is described by a positive, trace-1 linear operator, known as the density matrix ρ , acting on a Hilbert space, (ii) an observed outcome is the result of a measurement, characterized by a positive operator-valued measure (POVM), which is a set of linear operators M i respecting the completeness relation ∑ i M i = 1 ( 1 being the identity operator in the Hilbert space), (iii) the probability of a given outcome is given by the Born’s rule p ( i ) = Tr ( M i ρ ) , (iv) a composite quantum system is modeled by a tensor product of Hilbert spaces. This set of rules implies a well-defined description of the theory where the probability of all outcomes of an experiment are positive and sum up to one. Applying the quantum postulates to the instrumental scenario considered here, all three observable variables, X , A , and B , are still classical random variables, but instead of the latent variable Λ , we have a latent quantum state ρ A B of a composite system acting on a tensor product of Hilbert spaces. This type of quantum causal model produces observable correlations given by

where ρ A B is a quantum state of two subsystems, represented by the density matrix acting on the tensor product of two Hilbert spaces ℋ A ⊗ ℋ B , { M a x } a is a POVM acting on the first subsystem (Hilbert space ℋ A ) and describes a measurement depending on the choice x with the outcome a . Similarly, { N b a } b is a POVM acting on the second subsystem (Hilbert space ℋ B ) and describes a measurement depending on the choice a (which is the measurement outcome obtained on the first subsystem) with the outcome b .

Remember that the bounds such as instrumental inequalities and causal bounds derived in refs [3,11,13] assume that the latent node is a classical random variable. Consequently, they can all be systematically obtained using linear programming (for the case of discrete observed random variables). On the other hand, once we allow quantum common causes and use Born’s rule for probabilities, the description of the set of all possible observed distributions p ( a , b ∣ x ) becomes a much more tedious problem [23,48]. In particular, it is not, in general, possible to give a description of the observed probabilities in the form of equations (2) and (5). Since quantum theory encapsulates the classical probability theory as a special case, the former set is known to be strictly larger. Consequently, we cannot expect that the derived inequalities, which are the results of the linear programming, also hold for quantum common causes. Valid bounds for quantum common causes can be obtained using more general convex optimizations tools, such as infinite hierarchies of semidefinite programs [54] or trace inequalities [48]. Finding such valid bounds for quantum common causes is, however, not the aim of this section. Instead, we compare here the violations of classical bounds due to quantum common causes and due to relaxations of the independence assumption. More precisely, we establish the amount of the minimal measurement dependence needed to explain quantum effects in classical causal models.

Interestingly, in the case of the simplest instrumental scenario, the statistics obtained from a latent quantum state cannot violate the instrumental inequalities in equation (3) and no further analysis on minimal measurement dependency is required. Thus, in the simplest instrumental scenario, inequalities can be violated only if causal assumptions are relaxed.

This is no longer the case for the causal bounds on ACE in the simplest instrumental scenario. Refs [23,48] show that in the case of binary observed variables, the causal bound in equation (6) can be violated if the common cause is a quantum state (without any measurement dependence). In order to demonstrate such a violation, in a full analogy with the classical case, one can define (see ref. [47]) interventions on a classical random variable A , which is an outcome of a quantum measurement, as

The identity operator 1 in the above formula represents the fact that the outcome of the measurement on the first subsystem, generating A is discarded, and instead the value a is supplied to the measurement that produces B . Alternatively, it can be understood as simply “doing nothing” to the first subsystem. The observed quantum average causal effect (qACE) is then given by the difference,

Ref. [48] considered the following quantum causal model. The common cause is given by a pure two-qubit entangled state (rank-1 density operator with both Hilbert spaces being two-dimensional complex vector spaces C 2 ). Such states can be represented as ρ A B = ∣ ψ ⟩ ⟨ ψ ∣ , with ∣ ψ ⟩ = sin α ∣ 0 ⟩ ⊗ ∣ 0 ⟩ + cos α ∣ 1 ⟩ ⊗ ∣ 1 ⟩ , where ∣ 0 ⟩ , ∣ 1 ⟩ ∈ C 2 are basis vectors in C 2 in the so-called Dirac notation. Note that a quantum state is entangled if it cannot be expressed as a tensor product of the states of the individual subsystems, which is the case for 0 < α ≤ π / 4 in the considered example. We say that the state is maximally entangled if sin α = cos α = 1 / 2 . Entanglement is the central phenomenon in quantum mechanics and its applications, and it has only very recently been analyzed in the context of quantum causal discovery [23,48,55,56].

In ref. [48] it was shown that for all 0 < α ≤ π / 4 , and for appropriately chosen quantum measurements { M a x } a , { N b a } b , the causal bound in equation (6) can be violated. The degree of violation depends on the parameter α , which governs the amount of entanglement present in the common cause. The maximal possible violation was numerically obtained (and was verified by the hierarchy of semidefinite programs [54]) to be 3 − 2 2 ≈ 0.1716 .

We use the results of the previous sections to explain the maximal known quantum violation in the classical instrumental scenario with the relaxed measurement independence assumption: the amount of minimum measurement dependency in the classical instrumental causal structure must at least be ℳ X : Λ = 4 p ( X = 0 ) p ( X = 1 ) 2 − p ( X = 0 ) ( 3 − 2 2 ) . This quantity is maximal for P ( X = 0 ) = ( 2 − 2 ) , and equals ℳ X : Λ = ( 68 − 48 2 ) ≈ 0.1177 . Note that we are indeed imposing the instrumental causal structure to the quantum experiment. What the violation of instrumental inequality shows is that the only way to find a classical explanation (a causal model described by random variables) for the observed probability distribution is if some of the assumptions in the instrumental scenario (in this the independence assumption) are relaxed. We see that given a DAG, the quantum description of it will always lead to a larger (or at least equal) set of compatible correlations with the DAG.

In case of more general instrumental scenario, where X can take more than two values, I 2 ≤ 0 and I 3 ≤ 0 can be violated by quantum states and measurements [23], both by the maximally entangled state (i.e., when sin α = 1 2 ) with the amount of 1 2 − 1 2 ≈ 0.2071 and 2 − 1 ≈ 0.4142 , respectively. See Figure 3 (left) for the relation between the minimal required measurement dependence in classical instrumental scenario and the amount of violation of I 2 ≤ 0 for various probability distributions of the instrumental variable. In particular, for uniformly distributed instrumental variables the minimal measurement dependence required to explain the quantum violation is ℳ X : Λ = 1 3 ( 2 − 1 ) ≈ 0.1381 . The minimal measurement dependence needed to explain the maximal quantum violation of I 3 ≤ 0 for the uniformly distributed instrumental variables is ℳ X : Λ = 1 2 − 1 2 ≈ 0.2071 .

Finally, we consider the quantum violation of the causal bounds for the instrument that takes three values. The inequality ACE A → B ≥ C 2 can be violated by the maximally entangled state with the amount of 1 2 − 1 2 ≈ 0.2071 and the inequality ACE A → B ≥ C 3 with the amount of 2 − 1 ≈ 0.4142 . In order to explain these violations, the amount of minimum measurement dependency in the classical instrumental causal structure depends on a probability distribution of the instrumental random variable. See Figure 3 (center) and (right) for the particular examples of such distributions. We highlight that even though the violations of causal bounds match with the violations of instrumental inequalities, these quantities are of a very different nature. In particular, the violation of causal bounds required both interventional and observational probability distributions, while the violation of instrumental inequalities relies solely on observational data.

5 Discussion

Instrumental variables offer ways to estimate causal influence even under confounding effects and without the need for interventions. Strikingly, as discovered in ref. [3], one can infer the effect of interventions, without resorting to any structural equations, simply from observational data obtained with the help of an instrument. As already recognized long ago [57], however, “the real difficulty in practice of course is actually finding variables to play the role of instruments.” Since the potential correlation of the instrument with any latent variables is in principle unobservable, it might seem that the exogeneity of a given instrument is a matter of trust and intuition rather than a fact supported by the data.

Motivated by this fundamental problem, the data from an instrumental test [11,12, 13,14] can be employed to benchmark the amount of dependence the instrument can have with a confounding variable. More precisely, we quantify such correlations via a l 1 -norm, measuring by how much the instrumental variable fails to be exogenous. The violation of an instrumental inequality allows us then to put lower bounds on this dependence. In turn, we derive bounds for the ACE [1] taking into account that some level of dependence, lower bounded by the violation of instrumental inequality, is present. That is, we turn the causal bounds in a reliable tool even if the instrument is not really exogenous.

Relying on an LP description, we obtain fully analytical results for the simplest instrumental scenario where all variables are binary. We study a more general case of trinary instrumental variable numerically using our linear programming technique. In parallel, we also derived new bounds for the ACE (equation (25)), which to the best of our knowledge are new to the literature. We also consider applications of our generalized instrumental inequalities and causal bounds to consider the problem of measurement independence (also known as “free-will”) in the foundations of quantum physics.

It is interesting to compare the lower bounds on the dependence between Λ and X obtained either from the instrumental inequality or the causal bound violations, when the observational data include both the distribution p ( a , b ∣ x ) and the estimated value of the ACE. The required amount of dependence between instrumental variables and confounders that would explain such violations would then be at least the maximum among the tested inequalities. However, more generally, one would need to solve the LP described in Section 3 with all the tested inequalities included as constraints. The required amount of the relaxation of the independence assumption also provides a way to compare different inequalities in their bounding power in regard to the considered causal structure.

It is worth noting that the effect of imperfect instruments has previously been considered [6,58,59, 60,61]. Those works, however, relied on a number of additional assumptions, for instance, the study in [59] was limited to regression bivariate models, while here our results are free of any structural equations and valid for any causal mechanisms between the variables. Even though we have focused on the case where treatment and effect variables are binary, the LP framework we propose can also be extended to variables assuming any discrete number of values (limited, of course, by the computational complexity of the problem). Another interesting question for future research is to understand whether similar results may hold for the case of continuous variables or consider statistical tests as in ref. [62], a direction that we hope might be triggered by our results.