SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Example 3 (Connected components of the factual level sets). Here, we construct the function with the level sets consisting of multiple connected components. For that, we extend Example 2 with the Box-Müller transformation. We define a so-called oscillating Box-Müller transformation
$$
f _ {Y} \left(U _ {Y ^ {1}}, U _ {Y ^ {2}}\right) = \sqrt {- 2 \log \left(U _ {Y ^ {1}}\right)} \cos \left(2 ^ {- \lceil \log_ {2} \left(U _ {Y ^ {2}}\right) \rceil} \pi U _ {Y ^ {2}}\right). \tag {51}
$$
We plot the level set for  $y = -0.5$  in Fig. 9, namely  $E(y) = \{u_{Y} \in [0,1]^{2} : f_{Y}(u_{Y^{1}}, u_{Y^{2}}) = y\}$ . Here, we see that the level sets consist of an infinite number of connected components with an infinite total length.
Figure 9: Level sets with multiple connected components  $E(y) = \{u_{Y} \in [0,1]^{2} : f_{Y}(u_{Y^{1}}, u_{Y^{2}}) = y\}$  for oscillating Box-Müller transformation (see Example 3).
Example 4 (Curvature of level sets). Let us consider three SCMs of class  $\mathfrak{B}(C^2, 2)$ , which satisfy the Assumption  $\kappa$  with  $\kappa = 50$ . Then, the curvature of the level sets is properly defined for the function  $f_{Y}$  (see Eq. (11)). Fig. 10 provides a heatmap with the absolute curvature,  $|\kappa_1(u_Y)|$ , for the counterfactual level sets,  $\{E(y, a): y \in \mathcal{V}_a\}$ . Here, all three SCMs are instances of our APID.
Figure 10: Curvature of the counterfactual level sets  $\{E(y,a):y\in \mathcal{Y}_a\}$  in blue for three SCMs, satisfying the Assumption  $\kappa$ . Factual level set for the quantile of the distribution,  $E(\mathbb{F}_{a'}^{-1}(0.5),a')$  is given for reference in orange. As can be seen, "bent" level sets have larger curvature (right) than "straight" (left). Here,  $\kappa = \max_{y\in \mathcal{Y}_a,u_Y\in E(a,y)}|\kappa_1(u_Y)|$  amounts to 50.
Example 5 ( $\mathcal{M}_{\mathrm{tri}}$ ). There exist BGMs in class  $\mathfrak{B}(C^1, 1)$ , for which the curvature of level sets are not properly defined at certain points. For example, an SCM  $\mathcal{M}_{\mathrm{tri}}$ , which induces triangular distributions, like in Example 1. For this example,  $\mathcal{M}_{\mathrm{tri}}$  will have the following function for  $Y$ :
$$
\mathcal {M} _ {\text {t r i}}: f _ {Y} (A, U _ {Y}) = \left\{ \begin{array}{l l} A \sqrt {2 U _ {Y}} + (A - 1) (\sqrt {2 U _ {Y}} - 1), & \text {i f} U _ {Y} \in [ 0. 0, 0. 5 ], \\ A (2 - \sqrt {2 (U _ {Y} + 1)}) + (A - 1) (1 - \sqrt {2 (U _ {Y} + 1)}), & \text {i f} U _ {Y} \in (0. 5, 1. 0 ]. \end{array} \right. \tag {52}
$$
Example 6 ( $\mathcal{M}_{\mathrm{flat}}$ ). Here, we provide examples of SCMs in the class  $\mathfrak{B}(C^\infty, d)$ , for which the bundles of the level sets coincide for both treatments, i.e.,  $\{E(y, a) : y \in \mathcal{Y}_a\} = \{E(y, a') : y \in \mathcal{Y}_{a'}\}$ , and all the level sets are flat hypersurfaces ( $\kappa = 0$ ). For example, SCMs with the following functions for  $Y$ :
$$
\mathcal {M} _ {\text {f l a t}}: f _ {Y} (A, U _ {Y}) = g \left(A, w _ {1} U _ {Y ^ {1}} + \dots + w _ {d} U _ {Y ^ {d}}\right), \tag {53}
$$
where  $g(a, \cdot)$  is an invertible function in class  $C^\infty$  and  $w_1, \ldots, w_d$  are coefficients from the linear combination. After a reparametrization, it is always possible to find an equivalent BGM with the function:
$$
g \left(A, w _ {1} U _ {Y ^ {1}} + \dots + w _ {d} U _ {Y ^ {d}}\right) = g (A, \tilde {U} _ {Y}) = g \left(A, \mathbb {F} _ {\tilde {U} _ {Y}} ^ {- 1} \left(U _ {Y ^ {1}}\right)\right), \tag {54}
$$
where  $\mathbb{F}_{\tilde{U}_Y^{-1}}(\cdot)$  is an inverse CDF of  $\tilde{U}_Y = w_1U_{Y^1} + \dots +w_dU_{Y^d}$ . The smoothness of the inverse CDF then defines the smoothness of the BGM.
Example 7 ( $\mathcal{M}_{\mathrm{curv}}$ ). This example is the generalization of the Example 6. Here, we also use the same bundles of the level sets for both treatments, but allow for the curvature of  $\kappa > 0$ :
$$
\mathcal {M} _ {\text {c u r v}}: f _ {Y} (A, U _ {Y}) = g (A, h (U _ {Y})), \tag {55}
$$
where  $g(a, \cdot)$  is an invertible function in class  $C^\infty$ , and  $h(\cdot): [0,1]^d \to \mathbb{R}$  is a function in class  $C^\infty$  with bounded by  $\kappa$  curvature of the level sets. In this case, we also can reparametrize the latent space with a non-linear transformation and find an equivalent BGM:
$$
g (A, h \left(U _ {Y}\right)) = g (A, \tilde {U} _ {Y}) = g (A, \mathbb {F} _ {\tilde {U} _ {Y}} ^ {- 1} \left(U _ {Y ^ {1}}\right)), \tag {56}
$$
where  $\mathbb{F}_{\tilde{U}_Y}^{-1}(\cdot)$  is an inverse CDF of  $\tilde{U}_Y = h(U_Y)$ . The smoothness of the inverse CDF then analogously defines the smoothness of the BGM.
Example 8  $(\mathcal{M}_{\mathrm{perp}})$ . We can construct an SCM of class  $\mathfrak{B}(C^\infty, 2)$ , so that the level sets are all straight lines ( $\kappa = 0$ ) and perpendicular to each other for different  $a \in \{0,1\}$ , e.g.,
$$
\mathcal {M} _ {\text {p e r p}}: f _ {Y} \left(A, U _ {Y ^ {1}}, U _ {Y ^ {2}}\right) = A g _ {1} \left(U _ {Y ^ {1}}\right) + (A - 1) g _ {2} \left(U _ {Y ^ {2}}\right), \tag {57}
$$
where  $g_{1}(\cdot), g_{2}(\cdot)$  are invertible functions in class  $C^\infty$ . In this case, the ECOU for  $y' \in \mathcal{V}_{a'}$  always evaluates to the conditional expectation
$$
\begin{array}{l} Q _ {0 \rightarrow 1} ^ {\mathcal {M} _ {\text {p e r p}}} \left(y ^ {\prime}\right) = \frac {1}{\mathbb {P} ^ {\mathcal {M} _ {\text {p e r p}}} \left(Y = y ^ {\prime} \mid A = 0\right)} \int_ {\left\{u _ {Y 2} = g _ {2} ^ {- 1} \left(y ^ {\prime}\right)\right\}} \frac {g _ {1} \left(u _ {Y ^ {1}}\right)}{\left| \nabla_ {u _ {Y ^ {2}}} g _ {2} \left(u _ {Y ^ {2}}\right)\right|} d \mathcal {H} ^ {1} \left(u _ {Y}\right) (58) \\ = \frac {\mathbb {P} ^ {\mathcal {M} _ {\text {p e r p}}} \left(Y = g _ {2} \left(g _ {2} ^ {- 1} \left(y ^ {\prime}\right)\right) \mid A = 0\right)}{\mathbb {P} ^ {\mathcal {M} _ {\text {p e r p}}} \left(Y = y ^ {\prime} \mid A = 0\right)} \int_ {0} ^ {1} g _ {1} \left(u _ {Y ^ {1}}\right) \mathrm {d} u _ {Y ^ {1}} = \mathbb {E} ^ {\mathcal {M} _ {\text {p e r p}}} \left(Y \mid A = 1\right). (59) \\ \end{array}
$$
Similarly, the ECOT is  $\mathbb{E}^{\mathcal{M}_{\mathrm{perp}}}(Y\mid A = 0)$ . This result is, in general, different from the result of BGMs, which, e.g., would yield  $Q_{a^{\prime}\to a}^{\mathcal{M}}(y^{\prime}) = \mathbb{F}_{a}(0.5)$  for  $y^\prime = \mathbb{F}_{a'}^{-1}(0.5)$ . Thus, point identification is not guaranteed with  $\kappa = 0$ .
Uniformity of the latent noise. Classes of SCMs  $\mathfrak{B}(C^k,d)$  assume independent uniform latent noise variables, i.e.,  $U_{Y}\sim \mathrm{Unif}(0,1)^{d}$ . This assumption does not restrict the distribution of the latent noise. Namely, for a bivariate SCM  $\mathcal{M}_{\mathrm{non - unif}}$  with  $d$ -dimensional non-uniform continuous latent noise, it is always possible to find an equivalent<sup>7</sup> SCM  $\mathcal{M}_{\mathrm{unif}}$  of class  $\mathfrak{B}(C^k,d)$ . This follows from the change of variables formula and the inverse probability transformation [25, 116]:
$$
\mathcal {M} _ {\text {n o n - u n i f}}: Y = \tilde {f} _ {Y} (A, \tilde {U} _ {Y}), \quad \tilde {U} _ {Y} \sim \mathbb {P} (\tilde {U} _ {Y}); \tilde {f} _ {Y} (a, \cdot) \in C ^ {k}, \tag {60}
$$
$$
\Longleftrightarrow \mathcal {M} _ {\text {u n i f}}: Y = \tilde {f} _ {Y} (A, T (U _ {Y})) = f _ {Y} (A, U _ {Y}), \quad U _ {Y} \sim \operatorname {U n i f} (0, 1) ^ {d}; f _ {Y} (a, \cdot) \in C ^ {\min  (1, k)}, \tag {61}
$$
where  $\tilde{U}_Y = T(U_Y)$ , and  $T(\cdot):(0,1)^d\to \mathbb{R}^d$  is a diffeomorphism (bijective  $C^1$  transformation) and a solution to the following equation:
$$
\underbrace {\mathbb {P} \left(\tilde {U} _ {Y} = \tilde {u} _ {Y}\right)} _ {\in C ^ {0}} = \left| \det  \underbrace {\left(\frac {\mathrm {d} T ^ {- 1} \left(\tilde {u} _ {Y}\right)}{\mathrm {d} \tilde {u} _ {Y}}\right)} _ {\in C ^ {0}} \right|. \tag {62}
$$
For further details on the existence and explicit construction of  $T(\cdot)$ , see [25]. Thus,  $f_{Y}(a,\cdot)$  is of class  $C^{\min(1,k)}$ , as the composition of  $T(\cdot)$  and  $\tilde{f}_{Y}(a,\cdot)$ .
In our CSM we require the functions of the SCMs to be of class  $C^2$ . In this case,  $\mathfrak{B}(C^2,d)$  also includes all the SCMs with non-uniform continuous latent noise with densities of class  $C^1$ . This also follows from the change of variables theorem, see Eq. (62).
More general classes of functions, e. g., all measurable functions, or more general classes of distributions, e. g., distributions with atoms, fall out of the scope of this paper, even though we can find bijective (but non-continuous) transformations between probability spaces of the same cardinality (see the isomorphism of Polish spaces, Theorem 9.2.2 in [12]).
$$
\mathbb {P} ^ {\mathcal {M}} (Y = y \mid a) = \int_ {E (y, a)} \frac {1}{\left\| \nabla_ {u _ {Y}} f _ {Y} (a , u _ {Y}) \right\| _ {2}} \mathrm {d} \mathcal {H} ^ {d - 1} \left(u _ {Y}\right), \tag {1}
$$
where  $E(y, a)$  is a level set (preimage) of  $y$ , i.e.,  $E(y, a) = \{u_{Y} \in [0,1]^{d} : f_{Y}(a, u_{Y}) = y\}$ , and  $\mathcal{H}^{d-1}(u_{Y})$  is the Hausdorff measure (see Appendix B for the definition).
Proof. Lemma is a result of the coarea formula (see Eq. (16)):
$$
\begin{array}{l} \mathbb {P} ^ {\mathcal {M}} (Y = y \mid a) = \int_ {[ 0, 1 ] ^ {d}} \mathbb {P} ^ {\mathcal {M}} (Y = y, U _ {Y} = u _ {Y} \mid a) \mathrm {d} u _ {Y} (63) \\ \stackrel {(*)} {=} \int_ {[ 0, 1 ] ^ {d}} \mathbb {P} ^ {\mathcal {M}} (Y = y \mid u _ {Y}, a) \mathbb {P} ^ {\mathcal {M}} \left(U _ {Y} = u _ {Y}\right) \mathrm {d} u _ {Y} (64) \\ = \int_ {[ 0, 1 ] ^ {d}} \delta \left(f _ {Y} (a, u _ {Y}) - y\right) 1 \mathrm {d} u _ {Y} = \int_ {E (y, a)} \frac {1}{\| \nabla_ {u _ {Y}} f _ {Y} (a , u _ {Y}) \| _ {2}} \mathrm {d} \mathcal {H} ^ {d - 1} \left(u _ {Y}\right), (65) \\ \end{array}
$$
where  $(\ast)$  holds, as  $U_{Y}$  is independent of  $A$ . In a special case, when we set  $d = 1$ , we obtain a change of variables formula
$$
\mathbb {P} ^ {\mathcal {M}} (Y = y \mid a) = \int_ {E (y, a)} \frac {1}{| \nabla_ {u _ {Y}} f _ {Y} (a , u _ {Y}) |} d \mathcal {H} ^ {0} (u _ {Y}) = \sum_ {u _ {Y} \in E (y, a)} | \nabla_ {u _ {Y}} f _ {Y} (a, u _ {Y}) | ^ {- 1}. \tag {66}
$$
Equivalence of the SCMs is defined as the almost surely equality of all the functions in two SCMs [13].
Corollary 1 (Identifiable functions (BGMs)[93]). The function  $f_{Y}(a, u_{Y})$  can be identified (up to a sign) given the observational distribution  $\mathbb{P}(Y \mid a)$ , induced by some SCM  $\mathcal{M}$  of class  $\mathfrak{B}(C^1, 1)$  if it is strictly monotonous in  $u_{Y}$ . In this case, the function is
$$
f _ {Y} (a, u _ {Y}) = \mathbb {F} _ {a} ^ {- 1} (\pm u _ {Y} \mp 0. 5 + 0. 5), \tag {67}
$$
where  $\mathbb{F}_a^{-1}$  is an inverse CDF of the observational distribution, and the sign switch corresponds to the strictly monotonically increasing and decreasing functions.
Proof. As  $d = 1$ , we can apply the change of variables formula from Eq. (66):
$$
\mathbb {P} (Y = y \mid a) = \sum_ {u _ {Y} \in E (y, a)} | \nabla_ {u _ {Y}} f _ {Y} (a, u _ {Y}) | ^ {- 1} = | \nabla_ {u _ {Y}} f _ {Y} (a, u _ {Y}) | ^ {- 1}, \quad u _ {Y} = f _ {Y} ^ {- 1} (a, y), \tag {68}
$$
where  $f_{Y}^{-1}(a,y)$  is an inverse function with respect to  $u_{Y}$  (it is properly defined, as the function is strictly monotonous). Thus, to find  $f_{Y}$ , we have to solve the following differential equation [116]:
$$
\mathbb {P} (Y = f _ {Y} (a, u _ {Y}) \mid a) = | \nabla_ {u _ {Y}} f _ {Y} (a, u _ {Y}) | ^ {- 1}. \tag {69}
$$
By the properties of derivatives, this equation is equivalent to
$$
\mathbb {P} (Y = y \mid a) = | \nabla_ {y} f _ {Y} ^ {- 1} (a, y) |. \tag {70}
$$
Since the latter holds for every  $y$ , we can integrate it from  $-\infty$  to  $f_{Y}(a, u_{Y})$ :
$$
\begin{array}{l} \int_ {- \infty} ^ {f _ {Y} (a, u _ {Y})} \mathbb {P} (Y = y \mid a) \mathrm {d} y = \int_ {- \infty} ^ {f _ {Y} (a, u _ {Y})} | \nabla_ {y} f _ {Y} ^ {- 1} (a, y) | \mathrm {d} y (71) \\ = \left\{ \begin{array}{l l} \int_ {f _ {Y} ^ {- 1} (a, - \infty)} ^ {u _ {Y}} \mathrm {d} t = \int_ {0} ^ {u _ {Y}} \mathrm {d} t = u _ {Y}, & f _ {Y} \text {i s s t r i c t l y i n c r e a s i n g ,} \\ - \int_ {f _ {Y} ^ {- 1} (a, - \infty)} ^ {u _ {Y}} \mathrm {d} t = \int_ {u _ {Y}} ^ {1} \mathrm {d} t = 1 - u _ {Y}, & f _ {Y} \text {i s s t r i c t l y d e c r e a s i n g .} \end{array} \right. (72) \\ \end{array}
$$
Therefore,  $\mathbb{F}_a(f_Y(a,u_Y)) = \pm u_Y\mp 0.5 + 0.5$
Corollary 2 (Critical points of  $f_{Y}(a,\cdot)$ ). Lemma 1 also implies that critical points of  $f_{Y}(a,u_{Y})$ , i.e.,  $\{u_{Y}\in [0,1]^{d}:\| \nabla_{u_{Y}}f_{Y}(a,u_{Y})\|_{2} = 0\}$ , are mapped onto points  $y$  with infinite density. Therefore, the assumption, that the function is regular, namely  $\nabla_{u_Y}f_Y(a,u_Y) > 0$ , is equivalent to the assumption of the continuous observational density with the finite density.
where  $E(y', a')$  is a (factual) level set of  $y'$ , i.e.,  $E(y', a') = \{u_{Y} \in [0,1]^{d} : f_{Y}(a', u_{Y}) = y'\}$  and  $a' \neq a$ .
Proof. Both counterfactual queries can be inferred with the abduction-action-prediction procedure.
(1) Abduction infers the posterior distribution of the latent noise variables, conditioned on the evidence:
$$
\begin{array}{l} \mathbb {P} ^ {\mathcal {M}} \left(U _ {Y} = u _ {Y} \mid a ^ {\prime}, y ^ {\prime}\right) = \frac {\mathbb {P} ^ {\mathcal {M}} \left(U _ {Y} = u _ {Y} , A = a ^ {\prime} , Y = y ^ {\prime}\right)}{\mathbb {P} ^ {\mathcal {M}} \left(A = a ^ {\prime} , Y = y ^ {\prime}\right)} \stackrel {*} {=} \frac {\mathbb {P} ^ {\mathcal {M}} \left(Y = y ^ {\prime} \mid a ^ {\prime} , u _ {Y}\right) \mathbb {P} ^ {\mathcal {M}} \left(U _ {Y} = u _ {Y}\right)}{\mathbb {P} ^ {\mathcal {M}} \left(Y = y ^ {\prime} \mid a ^ {\prime}\right)} (73) \\ = \frac {\delta \left(f _ {Y} \left(a ^ {\prime} , u _ {Y}\right) - y ^ {\prime}\right)}{\mathbb {P} ^ {\mathcal {M}} \left(Y = y ^ {\prime} \mid a ^ {\prime}\right)}, (74) \\ \end{array}
$$
where  $(*)$  holds, as  $U_{Y}$  is independent of  $A$ .
(2)-(3) Action and prediction are then pushing forward the posterior distribution with the counterfactual function. For example, the density of the counterfactual outcome distribution of the [un]treated is
$$
\begin{array}{l} \mathbb {P} ^ {\mathcal {M}} \left(Y _ {a} = y \mid a ^ {\prime}, y ^ {\prime}\right) = \int_ {[ 0, 1 ] ^ {d}} \mathbb {P} ^ {\mathcal {M}} \left(Y _ {a} = y, U _ {Y} = u _ {Y} \mid a ^ {\prime}, y ^ {\prime}\right) \mathrm {d} u _ {Y} (75) \\ = \int_ {[ 0, 1 ] ^ {d}} \mathbb {P} ^ {\mathcal {M}} \left(Y _ {a} = y \mid a ^ {\prime}, y ^ {\prime}, u _ {Y}\right) \mathbb {P} ^ {\mathcal {M}} \left(U _ {Y} = u _ {Y} \mid a ^ {\prime}, y ^ {\prime}\right) d u _ {Y} (76) \\ \stackrel {(*)} {=} \frac {1}{\mathbb {P} ^ {\mathcal {M}} (Y = y ^ {\prime} \mid a ^ {\prime})} \int_ {[ 0, 1 ] ^ {d}} \delta \left(f _ {Y} \left(a, u _ {Y}\right) - y\right) \delta \left(f _ {Y} \left(a ^ {\prime}, u _ {Y}\right) - y ^ {\prime}\right) d u _ {Y} (77) \\ = \frac {1}{\mathbb {P} ^ {\mathcal {M}} \left(Y = y ^ {\prime} \mid a ^ {\prime}\right)} \int_ {E \left(y ^ {\prime}, a ^ {\prime}\right)} \frac {\delta \left(f _ {Y} \left(a , u _ {Y}\right) - y\right)}{\| \nabla_ {u _ {Y}} f _ {Y} \left(a ^ {\prime} , u _ {Y}\right) \| _ {2}} d \mathcal {H} ^ {d - 1} \left(u _ {Y}\right), (78) \\ \end{array}
$$
where  $(\ast)$  holds due to the independence of  $Y$  from  $Y^{\prime}$  and  $A^\prime$ , conditional on  $U_{Y}$  (see the parallel worlds network in Fig. 1). The inference of the ECOU [ECOT] is analogous
$$
\begin{array}{l} Q _ {a ^ {\prime} \rightarrow a} ^ {\mathcal {M}} \left(y ^ {\prime}\right) = \mathbb {E} ^ {\mathcal {M}} \left(Y _ {a} \mid a ^ {\prime}, y ^ {\prime}\right) = \int_ {\mathcal {Y} _ {a}} \int_ {[ 0, 1 ] ^ {d}} \mathbb {P} ^ {\mathcal {M}} \left(Y _ {a} = y, U _ {Y} = u _ {Y} \mid a ^ {\prime}, y ^ {\prime}\right) \mathrm {d} u _ {Y} \mathrm {d} y (79) \\ = \int_ {[ 0, 1 ] ^ {d}} \mathbb {E} ^ {\mathcal {M}} \left(Y _ {a} \mid a ^ {\prime}, y ^ {\prime}, u _ {Y}\right) \mathbb {P} ^ {\mathcal {M}} \left(U _ {Y} = u _ {Y} \mid a ^ {\prime}, y ^ {\prime}\right) \mathrm {d} u _ {Y} (80) \\ \stackrel {(*)} {=} \frac {1}{\mathbb {P} ^ {\mathcal {M}} (Y = y ^ {\prime} \mid a ^ {\prime})} \int_ {[ 0, 1 ] ^ {d}} f _ {Y} \left(a, u _ {Y}\right) \delta \left(f _ {Y} \left(a ^ {\prime}, u _ {Y}\right) - y ^ {\prime}\right) \mathrm {d} u _ {Y} (81) \\ = \frac {1}{\mathbb {P} ^ {\mathcal {M}} \left(Y = y ^ {\prime} \mid a ^ {\prime}\right)} \int_ {E \left(y ^ {\prime}, a ^ {\prime}\right)} \frac {f _ {Y} \left(a , u _ {Y}\right)}{\left\| \nabla_ {u _ {Y}} f _ {Y} \left(a ^ {\prime} , u _ {Y}\right) \right\| _ {2}} d \mathcal {H} ^ {d - 1} \left(u _ {Y}\right), (82) \\ \end{array}
$$
where  $(\ast)$  holds due to the independence of  $Y$  from  $Y^{\prime}$  and  $A^\prime$ , conditional on  $U_{Y}$  (see the parallel worlds network in Fig. 1), and due to the law of the unconscious statistician.
Corollary 3 (Identifiable counterfactuals (BGMs bounds)[93]). The ECOU [ECOT] can be identified (up to a sign) given the observational distribution  $\mathbb{P}(Y\mid a)$ , induced by some SCM  $\mathcal{M}$  of class  $\mathfrak{B}(C^1,1)$  if it is strictly monotonous in  $u_{Y}$ . In this case, the ECOU [ECOT] is
$$
Q _ {a ^ {\prime} \rightarrow a} ^ {\mathcal {M}} \left(y ^ {\prime}\right) = \mathbb {F} _ {a} ^ {- 1} \left(\pm \mathbb {F} _ {a ^ {\prime}} \left(y ^ {\prime}\right) \mp 0. 5 + 0. 5\right). \tag {83}
$$
Proof. The corollary is a result of the Lemma 2 and the Corollary 1:
$$
\begin{array}{l} Q _ {a ^ {\prime} \rightarrow a} ^ {\mathcal {M}} \left(y ^ {\prime}\right) = \frac {1}{\mathbb {P} ^ {\mathcal {M}} \left(Y = y ^ {\prime} \mid a ^ {\prime}\right)} \int_ {E \left(y ^ {\prime}, a ^ {\prime}\right)} \frac {f _ {Y} \left(a , u _ {Y}\right)}{\left| \nabla_ {u _ {Y}} f _ {Y} \left(a ^ {\prime} , u _ {Y}\right)\right|} \mathrm {d} \mathcal {H} ^ {0} \left(u _ {Y}\right) (84) \\ = \left| \nabla_ {u _ {Y}} f _ {Y} \left(a ^ {\prime}, u _ {Y}\right) \right| \frac {f _ {Y} \left(a , u _ {Y}\right)}{\left| \nabla_ {u _ {Y}} f _ {Y} \left(a ^ {\prime} , u _ {Y}\right) \right|} = f _ {Y} (a, u _ {Y}), \quad u _ {Y} = f _ {Y} ^ {- 1} \left(a ^ {\prime}, y ^ {\prime}\right), (85) \\ \end{array}
$$
Therefore, the ECOU [ECOT] is
$$
Q _ {a ^ {\prime} \rightarrow a} ^ {\mathcal {M}} \left(y ^ {\prime}\right) = \mathbb {F} _ {a} ^ {- 1} \left(\pm u _ {Y} \mp 0. 5 + 0. 5\right) = \mathbb {F} _ {a} ^ {- 1} \left(\pm \mathbb {F} _ {a ^ {\prime}} \left(y ^ {\prime}\right) \mp 0. 5 + 0. 5\right). \tag {86}
$$
Theorem 1 (Non-informative bounds of ECOU (ECOT)). Let the continuous observational distribution  $\mathbb{P}(Y\mid a)$  be induced by some SCM of class  $\mathfrak{B}(C^\infty ,d)$ . Let  $\mathbb{P}(Y\mid a)$  have a compact support  $\mathcal{V}_a = [l_a,u_a]$  and be of finite density  $\mathbb{P}(Y = y\mid a) < + \infty$ . Then, the ignorance interval for the partial identification of the ECOU [ECOT] of class  $\mathfrak{B}(C^\infty ,d)$ ,  $d\geq 2$ , has non-informative bounds:  $\overline{Q_{a^{\prime}\to a}(y^{\prime})} = l_{a}$  and  $\overline{Q_{a^{\prime}\to a}(y^{\prime})} = u_{a}$ .
Proof. Without the loss of generality, let us consider the lower bound of the ECOU [ECOT], namely,
$$
\underline {{Q}} _ {a ^ {\prime} \rightarrow a} (y ^ {\prime}) = \inf  _ {\mathcal {M} \in \mathfrak {B} \left(C ^ {k}, d\right)} Q _ {a ^ {\prime} \rightarrow a} ^ {\mathcal {M}} \left(y ^ {\prime}\right) \quad \text {s . t .} \forall a \in \{0, 1 \}: \mathbb {P} (Y \mid a) = \mathbb {P} ^ {\mathcal {M}} (Y \mid a). \tag {87}
$$
The proof then proceeds in two steps. First, we prove the statement of the theorem for  $d = 2$ , i.e., when latent noise is two-dimensional. Then, we extend it to arbitrary dimensionality.
Step 1 ( $d = 2$ ). Lemma 2 suggests that to minimize the ECOU [ECOT], we have to either increase the proportion of the length (one-dimensional volume) of the factual level set, which intersects the bundle of counterfactual level sets, or change the counterfactual functions, by "bending" the bundle of counterfactual level sets around a factual level set. Here, we focus on the second case, which is sufficient to construct an SCM with non-informative bounds.
Formally, we can construct a sequence of SCMs  $\{\mathcal{M}_{\mathrm{non - inf}}^{\varepsilon (y)}:y\in \mathcal{Y}_a,0 <   \varepsilon (y) <   1\}$  of class  $\mathfrak{B}(C^\infty ,2)$ , for which  $Q_{a^{\prime}\to a}^{\mathcal{M}^{(\varepsilon (y))}}(y^{\prime})$  gets arbitrarily close to  $l_{a}$  as  $y\rightarrow l_{a}$  and  $\varepsilon (y)\rightarrow 0$
For all  $\{\mathcal{M}_{\mathrm{non - inf}}^{\varepsilon (y)}:y\in \mathcal{V}_a,0 <   \varepsilon (y) <   1\}$ , we choose the same factual function
$$
f _ {Y} \left(a ^ {\prime}, U _ {Y ^ {1}}, U _ {Y ^ {2}}\right) = \mathbb {F} _ {a ^ {\prime}} ^ {- 1} \left(U _ {Y ^ {2}}\right), \tag {88}
$$
where  $\mathbb{F}_{a'}^{-1}(\cdot)$  is the inverse CDF of the observational distribution  $\mathbb{P}(Y \mid a')$ . This is always possible as a result of the Corollaries 1 and 2. Hence, all the level sets of the factual function are horizontal straight lines of length 1 (see Fig. 4), due to  $E(y', a') = \{\mathbb{F}_{a'}^{-1}(u_{Y^2}) = y'\}$ .
Now, we construct the counterfactual functions in the following way. For a fixed  $\varepsilon(y)$ , we choose the level sets of the function  $f_{Y}(a,\cdot),\{E^{\varepsilon(y)}(y,a),y\in \mathcal{V}_{a}\}$ , so that they satisfy the following properties. (1) Each  $E^{\varepsilon(y)}(y,a)$  intersects the factual level set  $E(y',a')$  at a certain point once and only once and ends at the boundary of the boundaries of the unit square. (2) Each  $E^{\varepsilon(y)}(y,a)$  splits the unit square on two parts, an interior with the area  $\mathbb{F}_a(y)$  and an exterior, with the area  $1 - \mathbb{F}_a(y)$  respectively (where  $\mathbb{F}_a(\cdot)$  is the CDF of the counterfactual distribution). This property ensures that the induced CDF coincides with the observational CDF at every point, namely,  $\forall y\in \mathcal{V}_a:\mathbb{F}_a^{\mathcal{M}_{\mathrm{non - inf}}^{\varepsilon(y)}}(y) = \mathbb{F}_a(y)$ . (3) All the level sets have the nested structure, namely, all the points of the interior of  $E^{\varepsilon(y)}(y,a)$  are mapped to  $\underline{y} < y$ , and points of the exterior, to  $\overline{y} > y$ . Additionally, for some fixed  $y\in \mathcal{V}_a$ , we set  $\varepsilon = \varepsilon(y)$  as the proportion of the factual level set,  $E(y',a')$ , fully contained in the exterior of the counterfactual level set,  $E(y,a)$ . Thus, the interior of the counterfactual level set covers  $1 - \varepsilon(y)$  of the factual level set.
Therefore, the ECOU [ECOT] for  $\mathcal{M}_{\mathrm{non - inf}}^{\varepsilon (y)}$  for some fixed  $y\in \mathcal{V}_a$  is
$$
Q _ {a ^ {\prime} \rightarrow a} ^ {\mathcal {M} _ {\text {n o n - i n f}} ^ {\varepsilon (y)}} \left(y ^ {\prime}\right) = \left(1 - \varepsilon (y)\right) \underline {{y}} + \varepsilon (y) \bar {y}, \quad \underline {{y}} \in [ l _ {a}, y ], \bar {y} \in [ y, u _ {a} ], \tag {89}
$$
which follows from the mean value theorem for integrals, since, for every  $y \in \mathcal{V}_a$ , we can choose  $\varepsilon(y)$  arbitrarily close to zero, and, thus
$$
\inf  _ {y \in \mathcal {Y} _ {a}, 0 <   \varepsilon (y) <   1} Q _ {a ^ {\prime} \rightarrow a} ^ {\mathcal {M} _ {\text {n o n - i n f}} ^ {\varepsilon (y)}} \left(y ^ {\prime}\right) = \inf  _ {y \in \mathcal {Y} _ {a}, 0 <   \varepsilon (y) <   1} \left(1 - \varepsilon (y)\right) \underline {{y}} + \varepsilon (y) \bar {y} = l _ {a}. \tag {90}
$$
Step 2 ( $d > 2$ ). The construction of the factual and counterfactual level sets extends straightforwardly to the higher dimensional case. In this case, the factual level sets are straight hyperplanes and the counterfactual level sets are hypercylinders, "bent" around the factual hyperplanes.