SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Ultimately, we summarize related works on both point and partial identification for two layers of causality, namely interventional and counterfactual, in Table 1.
Table 1: Overview of methods for point and partial identification, assuming a known causal diagram. Relevant methods are highlighted in yellow.
Layer	M/SM	Symbolic identifiability	Point identification methods	Discrete outcomes	Partial identification methods Continuous outcomes
Informative bounds	Non-informative bounds
L2Interventional	M	Always via back-door criterion [8]	Deep generative models [74, 144]	—	—	—
SM	Do-calculus & rules of probability [56, 79, 120]	Potential outcomes framework [14, 28, 52, 90, 111, 130]; binary IV [39, 58, 131]; proxy variables [85, 92]	Partially observed back-/front-door variables [81]; canonical SCM [136]	No-assumptions bound [88]; MSM [15, 35, 40, 41, 60, 61, 62, 98, 126]; outcome sensitivity models [15, 106]; con-founding functions [11, 17, 109]; noisy proxy variables [49]; IV [48, 55, 68, 145]; clustered DAGs [99]	MSM [40, 41, 89]; ATD [5]
L3Counterfactual	M	Parallel worlds net-works [3, 121], counterfactual unnesting theorem [27]	Deep generative models [21, 29, 70, 100, 113, 114, 117, 118]; Markovian BGMs [59, 66, 93, 94, 123, 148]; transport-based counterfactuals [32, 96]	PN, PS, PNS [6, 80, 82, 103, 127]; response func-tions framework / canon-ical partitioning [6, 97, 112, 137, 141, 142, 143, 147]; causal marginal problem [46, 115]; deep twin networks [129]	Variance of the treatment effects [2]; CDF of the joint potential outcomes distribu-tion [37]; conservative coun-terfactual effects [4]	CSM (this paper)
SM	ETT [119]; path-specific effects [122, 146]; deep generative mod-els [31, 84, 134, 135, 140]; semi-Markovian BGMs [93]	Future work (out of the scope of this paper)	Future work (see discussion in Appendix E); ANMs with hidden confounding [69]
Legend:
M/SM: Markovian SCM (M), semi-Markovian SCM (SM)
1. Symbolic identifiability and point identification with probabilistic expressions. Complete and sound algorithms were proposed for symbolic (non-parametric) identifiability of the counterfactuals based on observational  $(\mathcal{L}_1)$  or interventional  $(\mathcal{L}_2)$  information, and causal diagram of a semi-Markovian SCM. For example, [121] adapted d-separation for the parallel-worlds networks [3] for simple counterfactual queries, and [27] extended it to nested counterfactuals with counterfactual unnesting theorem. These methods aim to provide a probabilistic expression, in the case the query is identifiable, which is then used in downstream point identification methods. Rather rare examples of identifiable queries include the treatment effect of the treated (ETT) [119] and path-specific effects [122, 146] for certain SCMs, e. g., when the treatment is binary. If the query is non-identifiable, like in the case with ECOU [ECOT] as in our paper, no guidance is provided on how to perform partial identification and methods can not be used.
2(i). Point counterfactual identification with SCMs. Another way to perform a point identification when the counterfactual query is symbolically non-identifiable is to restrict a functional class in SCMs. Examples of restricted Markovian SCMs include nonlinear additive noise models (ANMs) [66, 123], post nonlinear models (PNL) [148], location-scale noise models (LSNMs) [59], and bijective generation mechanisms (BGMs) [93, 94]. The latest, BGMs, also work in semi-Markovian settings with additional assumptions. Numerous deep generative models were also proposed for
point identification in both Markovian and semi-Markovian settings. Examples are normalizing flows [100]; diffusion models [21, 113]; variational inference [70, 100, 114, 134, 135]; adversarial learning [29, 84, 100, 117, 118, 140], and neural expectation-maximization [31]. As it was shown in [94], all those deep generative models above need to assume the BGM of the data to yield valid counterfactual inference. $^{4}$  The assumption of monotonicity of BGMs effectively sets the dimensionality of the latent noise to the same as the observed variables, which is rarely a realistic assumption. In our paper, we relax this assumption, and let the latent noise have arbitrary dimensionality.
4In semi-Markovian SCM additional assumptions are needed about the shared latent noise [31, 93].
In the context of the non-SCM-based counterfactuals, the monotonicity assumption assumes a stochastic dominance of one potential outcome over the other [87, 103], i.e.,  $\mathbb{P}(Y_a\geq Y_{a'}) = 1$
A.2 Identifiability of latent variable models and disentanglement
The question of identifying latent noise variables was also studied from the perspective of nonlinear independent component analysis (ICA) [47, 57, 67]. Khemakhem et al. [67] showed that the joint distribution over observed and latent noise variables is in general unidentifiable. Although, nonlinear ICA was applied for interventional inference [133], these works did not consider SCMs or counterfactual queries.
B Background materials
Geometric measure theory. Let  $\delta(x): \mathbb{R} \to \mathbb{R}$  be a Dirac delta function, defined as zero everywhere, except for  $x = 0$ , where it has a point mass of 1. Dirac delta function induces a Dirac delta measure, so that (with the slight abuse of notation)
$$
\int_ {\mathbb {R}} f (x) \delta (\mathrm {d} x) = \int_ {\mathbb {R}} f (x) \delta (x) \mathrm {d} x = f (0), \tag {12}
$$
where  $f$  is a  $C^0$  function with compact support. Dirac delta function satisfies the following important equality
$$
\int_ {\mathbb {R}} f (x) \delta (g (x)) \mathrm {d} x = \sum_ {i} \frac {f \left(x _ {i}\right)}{\left| g ^ {\prime} \left(x _ {i}\right) \right|}, \tag {13}
$$
where  $f$  is a  $C^0$  function with compact support,  $g$  is a  $C^1$  function,  $g'(x_i) \neq 0$ , and  $x_i$  are roots of the equation  $g(x) = 0$ .
In addition, we define the  $s$ -dimensional Hausdorff measure  $\mathcal{H}^s$ , as in [95]. Let  $E \subseteq \mathbb{R}^n$  be a  $s$ -dimensional smooth manifold ( $s \leq n$ ) embedded into  $\mathbb{R}^n$ . Then,  $E$  is a Borel subset in  $\mathbb{R}^n$ , is Hausdorff-measurable, and  $s$ -dimensional Hausdorff measure  $\mathcal{H}^s(E)$  is the  $s$ -dimensional surface volume of  $E$ . For example, if  $s = 1$ , the Hausdorff measure coincides with a line integral, and, if  $s = 2$ , with surface integral:
$$
s = 1: \quad E = \binom {x _ {1} (t)} {\vdots} = \mathbf {x} (t) \quad \Rightarrow \quad \mathrm {d} \mathcal {H} ^ {1} (\mathbf {x}) = \left\| \frac {\mathrm {d} \mathbf {x} (t)}{\mathrm {d} t} \right\| _ {2} \mathrm {d} t, \tag {14}
$$
$$
s = 2: \quad E = \binom {x _ {1} (s, t)} {\vdots} = \mathbf {x} (s, t) \quad \Rightarrow \quad \mathrm {d} \mathcal {H} ^ {2} (\mathbf {x}) = \left\| \frac {\mathrm {d} \mathbf {x} (s , t)}{\mathrm {d} s} \times \frac {\mathrm {d} \mathbf {x} (s , t)}{\mathrm {d} t} \right\| _ {2} \mathrm {d} s \mathrm {d} t, \tag {15}
$$
where  $\times$  is a vector product,  $x_{i}(t)$  is a  $t$ -parametrization of the line, and  $x_{i}(s,t)$  is a  $(s,t)$ -parametrization of the surface.
Also,  $\mathcal{H}^n (E) = \mathrm{vol}^n (E)$  for Lebesgue measurable subsets  $E\subseteq \mathbb{R}^n$  , where  $\mathrm{vol}^n$  is a standard  $n$  -dimensional volume (Lebesgue measure). In the special case of  $s = 0$  , Hausdorff measure is a counting measure:  $\int_{E}f(x)\mathrm{d}\mathcal{H}^{0}(x) = \sum_{x\in E}f(x)$
Importantly, the Hausdorff measure is related to the high-dimensional Dirac delta measure via the coarea formula [53] (Theorem 6.1.5), [95] (Theorem 3.3):
$$
\int_ {\mathbb {R} ^ {n}} f (\mathbf {x}) \delta (g (\mathbf {x})) \mathrm {d} \mathbf {x} = \int_ {E: \{g (\mathbf {x}) = 0 \}} \frac {f (\mathbf {x})}{\| \nabla_ {\mathbf {x}} g (\mathbf {x}) \| _ {2}} \mathrm {d} \mathcal {H} ^ {n - 1} (\mathbf {x}), \tag {16}
$$
where  $f$  is  $C^0$  function with compact support, and  $g$  is a  $C^1$  function with  $\nabla_{\mathbf{x}}g(\mathbf{x}) > 0$  for  $\mathbf{x} \in E$ . Functions for which  $\nabla_{\mathbf{x}}g(\mathbf{x}) > 0$  for  $\mathbf{x} \in \mathbb{R}^d$  holds are called regular.
We define a bundle of level sets of the function  $y = f(x)$  as a set of the level sets, indexed by  $y$ , i.e.,  $\{E(y); y \in \mathcal{Y} \in \mathbb{R}\}$ , where  $E(y) = \{y \in \mathbb{R}: f(x) = y\}$  and  $\mathcal{Y} \subseteq \mathbb{R}$ . A bundle of level sets of regular functions are closely studied in the Morse theory. In particular, we further rely on the fundamental Morse theorem, i.e., the level set bundles of a regular function are diffeomorphic to each other. This means that there exists a continuously differentiable bijection between every pair of the bundles.
Differential geometry of manifolds. In the following, we formally define the notion of the curvature for level sets. As the result of the implicit function theorem, level sets  $E(y)$  of a regular function  $f$  of class  $C^1$  are Riemannian manifolds [78], namely smooth differentiable manifolds. Riemannian manifolds can be locally approximated via Euclidean spaces, i.e., they are equipped with a tangent space  $T_{\mathbf{x}}(E(y))$  at a point  $\mathbf{x}$ , and a dot product defined on the tangent space. The tangent space,  $T_{\mathbf{x}}(E(y))$ , is orthogonal to the normal of the manifold,  $\nabla_{\mathbf{x}}f(\mathbf{x})$ .
For the level sets of regular functions of the class  $C^2$ , we can define the so-called curvature [43, 78]. Informally, curvature defines the extent a Riemannian manifold bends in different directions.
Convex regions of the manifold correspond to a negative curvature (in all directions), and concave regions to positive curvature, respectively. Saddle points have curvatures of different signs in different directions. Formally, the curvature is defined via the rate of change of the unit normal of the manifold, which is parameterized with the orthogonal basis  $\{\tilde{x}_1,\dots ,\tilde{x}_{n - 1}\}$  of the tangent space,  $T_{\mathbf{x}}(E(y))$  This rate of change, namely a differential, forms a shape operator (or second fundamental form) on the tangent space,  $T_{\mathbf{x}}(E(y))$  . Then, eigenvalues of the shape operator are called principal curvatures,  $\kappa_{i}(\mathbf{x})$  for  $i\in \{1,\ldots ,n - 1\}$  . Principal curvatures are a measure of the extrinsic curvature, i. e., the curvature of the manifold with respect to the embedding space,  $\mathbb{R}^n$
Principal curvatures for Riemannian manifolds, defined as the level sets of a regular  $C^2$  function  $f$ , can be also expressed via the gradient and the Hessian of the following function [43]:
$$
\kappa_ {i} (\mathbf {x}) = - \frac {\operatorname {r o o t} _ {i} \left\{\det  \binom {\operatorname {H e s s} _ {\mathbf {x}} f (\mathbf {x}) - \lambda I} {(\nabla_ {\mathbf {x}} f (\mathbf {x})) ^ {T}} \begin{array}{c c} & \nabla_ {\mathbf {x}} f (\mathbf {x}) \\ & 0 \end{array} \right\}}{\| \nabla_ {\mathbf {x}} f (\mathbf {x}) \| _ {2}}, \quad i \in \{1, \dots , n - 1 \}, \tag {17}
$$
where  $\mathrm{root}_i$  are roots of the equation with respect to  $\lambda \in \mathbb{R}$ . For  $\mathbf{x} \in \mathbb{R}^2$ , the level sets  $E(y)$  are curves, and there is only one principal curvature
$$
\kappa_ {1} (\mathbf {x}) = - \frac {1}{2} \nabla_ {\mathbf {x}} \left(\frac {\nabla_ {\mathbf {x}} f (\mathbf {x})}{\| \nabla_ {\mathbf {x}} f (\mathbf {x}) \| _ {2}}\right). \tag {18}
$$
One of the important properties of the principal curvatures is that we can locally parameterize the manifold as the second-order hypersurface, i.e.,
$$
y = \frac {1}{2} \left(\kappa_ {1} \tilde {x} _ {1} ^ {2} + \dots + \kappa_ {n - 1} \tilde {x} _ {n - 1} ^ {2}\right) + O \left(\| \tilde {\mathbf {x}} \| _ {2} ^ {3}\right), \tag {19}
$$
where  $\{\tilde{x}_1,\dots ,\tilde{x}_{n - 1}\}$  is the orthogonal basis of the tangent space  $T_{\mathbf{x}}(E(y))$
Example 1 (Counterfactual non-identifiability in Markovian-SCMs (continued)). Here, we continue the example and provide the inference of observational (interventional) and counterfactual distributions for the SCMs  $\mathcal{M}_1$  and  $\mathcal{M}_2$ .
Let us consider  $\mathcal{M}_1$  first. It is easy to see that a pushforward of uniform distribution in a unit square with  $f_{Y}(A,U_{Y^{1}},U_{Y^{2}}) = A(U_{Y^{1}} - U_{Y^{2}} + 1) + (1 - A)(U_{Y^{1}} + U_{Y^{2}} - 1)$  induces triangular distributions. For example, for  $f_{Y}(0,U_{Y^{1}},U_{Y^{2}})$ , a cumulative distribution function (CDF) of  $\mathbb{P}^{\mathcal{M}_1}(Y\mid A = 0) = \mathbb{P}^{\mathcal{M}_1}(Y_{a = 0})$  will have the form
$$
\begin{array}{l} \mathbb {F} ^ {\mathcal {M} _ {1}} (y \mid A = 0) = \mathbb {P} ^ {\mathcal {M} _ {1}} (Y \leq y \mid A = 0) = \mathbb {P} \left(U _ {Y ^ {1}} + U _ {Y ^ {2}} - 1 \leq y\right) (20) \\ = \left\{ \begin{array}{l l} 0, & (y \leq - 1) \vee (y > 1), \\ \int_ {\left\{u _ {Y ^ {1}} + u _ {Y ^ {2}} - 1 \leq y \right\}} \mathrm {d} u _ {Y ^ {1}} \mathrm {d} u _ {Y ^ {2}}, & \text {o t h e r w i s e} \end{array} \right. (21) \\ = \left\{ \begin{array}{l l} 0, & (y \leq - 1) \vee (y > 1), \\ \frac {(y + 1) ^ {2}}{2}, & y \in (- 1, 0 ], \\ 1 - \frac {(1 - y) ^ {2}}{2}, & y \in (0, 1 ], \end{array} \right. (22) \\ \end{array}
$$
which is the CDF of a triangular distribution. Analogously, a CDF of  $\mathbb{P}^{\mathcal{M}_1}(Y\mid A = 1) = \mathbb{P}^{\mathcal{M}_1}(Y_{a = 1})$  is
$$
\begin{array}{l} \mathbb {F} ^ {\mathcal {M} _ {1}} (y \mid A = 1) = \mathbb {P} ^ {\mathcal {M} _ {1}} (Y \leq y \mid A = 1) = \mathbb {P} \left(U _ {Y ^ {1}} - U _ {Y ^ {2}} + 1 \leq y\right) (23) \\ = \left\{ \begin{array}{l l} 0, & (y \leq 0) \vee (y > 2), \\ \int_ {\left\{u _ {Y ^ {1}} - u _ {Y ^ {2}} + 1 \leq y \right\}} \mathrm {d} u _ {Y ^ {1}} \mathrm {d} u _ {Y ^ {2}}, & \text {o t h e r w i s e} \end{array} \right. (24) \\ = \left\{ \begin{array}{l l} 0, & (y \leq 0) \vee (y > 2), \\ \frac {y ^ {2}}{2}, & y \in (0, 1 ], \\ 1 - \frac {(2 - y) ^ {2}}{2}, & y \in (1, 2 ]. \end{array} \right. (25) \\ \end{array}
$$
To infer the counterfactual outcome distribution of the untreated,  $\mathbb{P}^{\mathcal{M}_1}(Y_{a = 1}\mid A' = 0,Y' = 0)$ , we make use of Lemma 2 and properties of the Dirac delta function (e.g., Eq. (13)). We yield
$$
\mathbb {P} ^ {\mathcal {M} _ {1}} \left(Y _ {a = 1} = y \mid A ^ {\prime} = 0, Y ^ {\prime} = 0\right) = \int_ {\left\{u _ {Y 1} + u _ {Y 2} - 1 = 0 \right\}} \frac {\delta \left(u _ {Y 1} - u _ {Y 2} + 1 - y\right)}{\left\| \nabla_ {u _ {Y}} \left(u _ {Y 1} + u _ {Y 2} - 1\right) \right\| _ {2}} d \mathcal {H} ^ {1} \left(u _ {Y}\right) \tag {26}
$$
$$
\stackrel {(*)} {=} \frac {1}{\sqrt {2}} \int_ {0} ^ {2} \sqrt {\left(\frac {1}{2}\right) ^ {2} + \left(\frac {1}{2}\right) ^ {2}} \delta (t - y) d t = \frac {1}{2} \int_ {0} ^ {2} \delta (t - y) d t = \left\{ \begin{array}{l l} \frac {1}{2}, & y \in [ 0, 2 ], \\ 0, & \text {o t h e r w i s e}, \end{array} \right. \tag {27}
$$
where  $(\ast)$  introduces a parametrization of the line  $\{u_{Y^1} + u_{Y^2} - 1 = 0\}$  with  $t$ , namely
$$
\binom {u _ {Y ^ {1}}} {u _ {Y ^ {2}}} = \binom {\frac {1}{2} t} {1 - \frac {1}{2} t}, \quad t \in [ 0, 2 ] \quad \Rightarrow \quad \mathrm {d} \mathcal {H} ^ {1} (u _ {Y}) = \sqrt {\left(\frac {1}{2}\right) ^ {2} + \left(\frac {1}{2}\right) ^ {2}} \mathrm {d} t. \tag {28}
$$
Therefore, the ECOU for  $\mathcal{M}_1$  is
$$
Q _ {0 \rightarrow 1} ^ {\mathcal {M} _ {1}} (0) = \int_ {0} ^ {2} \frac {1}{2} y \mathrm {d} y = 1. \tag {29}
$$
Now, let us consider  $\mathcal{M}_2$ . Hence,  $f_{Y}(0,U_{Y^{1}},U_{Y^{2}})$  is the same for both  $\mathcal{M}_1$  and  $\mathcal{M}_2$ , so are the observational (interventional) distributions, i.e.,  $\mathbb{P}^{\mathcal{M}_1}(Y\mid A = 0) = \mathbb{P}^{\mathcal{M}_2}(Y\mid A = 0)$ . The same is true for  $f_{Y}(1,U_{Y^{1}},U_{Y^{2}})$  with  $(0\leq U_{Y^1}\leq 1)\wedge (U_{Y^1}\leq U_{Y^2}\leq 1)$  or, equivalently,  $\mathbb{P}^{\mathcal{M}_1}(Y = y\mid A = 1) = \mathbb{P}^{\mathcal{M}_2}(Y = y\mid A = 1)$  with  $y\in (0,1]$ .
Now, it is left to check whether  $\mathbb{P}^{\mathcal{M}_2}(Y = y\mid A = 1)$  is the density of a triangular distribution for  $y\in (1,2]$ . For that, we define level sets of the function  $f_{Y}(1,U_{Y^{1}},U_{Y^{2}})$  in the remaining part of the unit square in a specific way. Formally, they are a family of "bent" lines in the transformed two-dimensional space  $\tilde{u}_{Y^1},\tilde{u}_{Y^2}$  with
$$
\tilde {u} _ {Y ^ {2}} = 8 t ^ {2} | \tilde {u} _ {Y ^ {1}} | + b (t), \quad t \in [ 0, 1 ]; \tilde {u} _ {Y ^ {1}} \in [ - 1, 1 ]; \quad \tilde {u} _ {Y ^ {2}} \in [ 0, 1 - | \tilde {u} _ {Y ^ {1}} | ], \tag {30}
$$
where  $b(t)$  is a bias, depending on  $t$ . The area under the line should change with quadratic speed, as it will further define the CDF of the induced observational distribution,  $\mathbb{F}^{\mathcal{M}_2}(y\mid A = 1), y\in (1,2]$ , i.e.
$$
S (t) = 2 t - t ^ {2}, \tag {31}
$$
so that  $S(0) = 0$  and  $S(1) = 1$ . On the other hand, the area under the line from the family is
$$
S (t) = 2 \int_ {0} ^ {\tilde {u} _ {*}} \left(8 t ^ {2} \tilde {u} _ {Y ^ {1}} + b (t)\right) \mathrm {d} \tilde {u} _ {Y ^ {1}} + \left(1 - \tilde {u} _ {*}\right) ^ {2} = 8 t ^ {2} \tilde {u} _ {*} ^ {2} + 2 \tilde {u} _ {*} b (t) + \left(1 - \tilde {u} _ {*}\right) ^ {2}, \tag {32}
$$
where  $\tilde{u}_{*} = \frac{1 - b(t)}{8t^{2} + 1}$ . Therefore, we can find the dependence of  $b(t)$  on  $t$ , so that the area  $S(t)$  is preserved:
$$
2 t - t ^ {2} = 8 t ^ {2} \tilde {u} _ {*} ^ {2} + 2 \tilde {u} _ {*} b (t) + \left(1 - \tilde {u} _ {*}\right) ^ {2} \Longleftrightarrow b (t) = 1 - \sqrt {(t - 1) ^ {2} \left(8 t ^ {2} + 1\right)}. \tag {33}
$$
Let us reparametrize  $t$  with  $t = y - 1$ , so that the line from a family with  $t = 0$  corresponds to  $y = 1$  and  $t = 1$  to  $y = 2$ , respectively. We then yield
$$
\tilde {u} _ {Y ^ {2}} = 2 (y - 1) | \tilde {u} _ {Y ^ {1}} | + 1 - \sqrt {(y - 2) ^ {2} (8 (y - 1) ^ {2} + 1)}. \tag {34}
$$
In order to obtain the function  $f_{Y}(1, U_{Y^{1}}, U_{Y^{2}})$  in the original coordinates  $u_{Y^{1}}, u_{Y^{2}}$ , we use a linear transformation  $T$  given by
$$
T: \left(\begin{array}{c}\tilde {u} _ {Y ^ {1}}\\\tilde {u} _ {Y ^ {2}}\end{array}\right)\rightarrow \left(\begin{array}{c}- u _ {Y ^ {1}} - u _ {Y ^ {2}} + 1\\u _ {Y ^ {1}} - u _ {Y ^ {2}}\end{array}\right). \tag {35}
$$
Hence, the family of "bent" lines can be represented as the following implicit equation:
$$
F \left(y, u _ {Y ^ {1}}, u _ {Y ^ {2}}\right) = u _ {Y ^ {1}} - u _ {Y ^ {2}} - 2 (y - 1) | - u _ {Y ^ {1}} - u _ {Y ^ {2}} + 1 | - 1 + \sqrt {(y - 2) ^ {2} (8 (y - 1) ^ {2} + 1)} = 0. \tag {36}
$$
Importantly, the determinant of the Jacobian of the transformation  $T$  is equal to 2; therefore, the area under the last line  $S(1)$  shrinks from 1 (in space  $\tilde{u}_{Y^1},\tilde{u}_{Y^2}$ ) to 0.5 (in space  $u_{Y^1},u_{Y^2}$ ). Thus, we can also easily verify with Eq. (31) that the CDF of the induced observational distribution coincides with the CDF of the triangular distribution for  $y\in (1,2]$ :
$$
\mathbb {F} ^ {\mathcal {M} _ {2}} (y \mid A = 1) = \frac {1}{2} + \mathbb {P} ^ {\mathcal {M} _ {2}} (1 \leq Y \leq y \mid A = 1) = \frac {1}{2} + \frac {1}{2} S (y - 1) = 1 - \frac {(2 - y) ^ {2}}{2}. \tag {37}
$$
To infer the counterfactual outcome distribution of the untreated, i.e.,  $\mathbb{P}^{\mathcal{M}_2}(Y_{a = 1}\mid A' = 0,Y' = 0)$  we again use the Lemma 2 and properties of Dirac delta function (e.g., Eq. (13)). We yield
$$
\mathbb {P} ^ {\mathcal {M} _ {2}} \left(Y _ {a = 1} = y \mid A ^ {\prime} = 0, Y ^ {\prime} = 0\right) = \frac {1}{2}, \quad y \in (0, 1 ], \tag {38}
$$
$$
\mathbb {P} ^ {\mathcal {M} _ {2}} \left(Y _ {a = 1} = y \mid A ^ {\prime} = 0, Y ^ {\prime} = 0\right) = \int_ {\left\{u _ {Y ^ {1}} + u _ {Y ^ {2}} - 1 = 0 \right\}} \frac {\delta \left(F \left(y , u _ {Y ^ {1}} , u _ {Y ^ {2}}\right)\right)}{\| \nabla_ {u _ {Y}} \left(u _ {Y ^ {1}} + u _ {Y ^ {2}} - 1\right) \| _ {2}} d \mathcal {H} ^ {1} \left(u _ {Y}\right) \tag {39}
$$
$$
\stackrel {(*)} {=} \frac {1}{\sqrt {2}} \int_ {0} ^ {1} \sqrt {\left(\frac {1}{2}\right) ^ {2} + \left(\frac {1}{2}\right) ^ {2}} \delta \left(t - 1 + \sqrt {(y - 2) ^ {2} (8 (y - 1) ^ {2} + 1)}\right) d t \tag {40}
$$
$$
= \frac {1}{2} \left| - \left(\sqrt {(y - 2) ^ {2} (8 (y - 1) ^ {2} + 1)}\right) ^ {\prime} \right| = \frac {(5 - 4 y) ^ {2} (2 - y)}{2 \sqrt {(y - 2) ^ {2} (8 (y - 1) ^ {2} + 1)}}, \quad y \in (1, 2 ], \tag {41}
$$
$$
\binom {u _ {Y ^ {1}}} {u _ {Y ^ {2}}} = \binom {\frac {1}{2} t + \frac {1}{2}} {\frac {1}{2} - \frac {1}{2} t}, \quad t \in [ 0, 1 ] \quad \Rightarrow \quad \mathrm {d} \mathcal {H} ^ {1} (u _ {Y}) = \sqrt {\left(\frac {1}{2}\right) ^ {2} + \left(\frac {1}{2}\right) ^ {2}} \mathrm {d} t. \tag {42}
$$
Finally, the ECOU for  $\mathcal{M}_2$  can be calculated numerically via
$$
Q _ {0 \rightarrow 1} ^ {\mathcal {M} _ {2}} (0) = \int_ {0} ^ {1} \frac {1}{2} y \mathrm {d} y + \int_ {1} ^ {2} \frac {(5 - 4 y) ^ {2} (2 - y)}{2 \sqrt {(y - 2) ^ {2} (8 (y - 1) ^ {2} + 1)}} \mathrm {d} y \approx \frac {1}{4} + 0. 8 6 4 \approx 1. 1 1 4. \tag {43}
$$
Example 2 (Box-Müller transformation). Here, we demonstrate the application of the Lemma 1 to infer the standard normal distribution with the Box-Müller transformation. The Box-Müller transformation is a well-established approach to sample from the standard normal distribution, which omits the usage of the inverse CDF of the normal distribution. Formally, the Box-Müller transformation is described by an SCM  $\mathcal{M}_{\mathrm{bm}}$  of class  $\mathfrak{B}(C^1, 2)^6$  with the following function for  $Y$ :
$$
\mathcal {M} _ {\mathrm {b m}}: f _ {Y} \left(A, U _ {Y ^ {1}}, U _ {Y ^ {2}}\right) = f _ {Y} \left(U _ {Y ^ {1}}, U _ {Y ^ {2}}\right) = \sqrt {- 2 \log \left(U _ {Y ^ {1}}\right)} \cos (\pi U _ {Y ^ {2}}). \tag {44}
$$
We will now use the Lemma 1 to verify that  $\mathbb{P}^{\mathcal{M}_{\mathrm{bm}}}(Y\mid a) = N(0,1)$ . We yield
$$
\begin{array}{l} \mathbb {P} ^ {\mathcal {M} _ {\mathrm {b m}}} (Y = y \mid a) = \int_ {\left\{\sqrt {- 2 \log \left(u _ {Y 1}\right)} \cos \left(\pi u _ {Y 2}\right) = y \right\}} \frac {1}{\left\| \nabla_ {u _ {Y}} \left(\sqrt {- 2 \log \left(u _ {Y 1}\right)} \cos \left(\pi u _ {Y 2}\right)\right) \right\| _ {2}} \mathrm {d} \mathcal {H} ^ {1} \left(u _ {Y}\right) (45) \\ = \int_ {\left\{\sqrt {- 2 \log \left(u _ {Y 1}\right)} \cos \left(\pi u _ {Y ^ {2}}\right) = y \right\}} \frac {1}{\sqrt {- \frac {\cos^ {2} \left(\pi u _ {Y ^ {2}}\right)}{2 u _ {Y ^ {1}} ^ {2} \log \left(u _ {Y ^ {1}}\right)} - 2 \pi^ {2} \log \left(u _ {Y ^ {1}}\right) \sin^ {2} \left(\pi u _ {Y ^ {2}}\right)}} d \mathcal {H} ^ {1} \left(u _ {Y}\right) (46) \\ \stackrel {(*)} {=} \int_ {0} ^ {1 / 2} \frac {\sqrt {1 + \left(- \frac {\pi y ^ {2} \sin (\pi t)}{\cos^ {3} (\pi t)} \exp \left(- \frac {y ^ {2}}{2 \cos^ {2} (\pi t)}\right)\right) ^ {2}}}{\sqrt {\frac {\cos^ {4} (\pi t)}{y ^ {2}}} \exp \left(\frac {y ^ {2}}{\cos^ {2} (\pi t)}\right) + \frac {\pi^ {2} \sin^ {2} (\pi t) y ^ {2}}{\cos^ {2} (\pi t)}} d t = \int_ {0} ^ {1 / 2} \frac {| y | \exp \left(- \frac {y ^ {2}}{2 \cos^ {2} (\pi t)}\right)}{\cos^ {2} (\pi t)} d t (47) \\ = \frac {1}{\sqrt {2 \pi}} \exp \left(- \frac {y ^ {2}}{2}\right) = N (y; 0, 1), (48) \\ \end{array}
$$
where  $(*)$  introduces a parametrization of the level set  $\{\sqrt{-2\log(u_{Y^1})}\cos (\pi u_{Y^2}) = y\} ,y > 0$  with  $t$ , namely
$$
\begin{array}{l} \binom {u _ {Y ^ {1}}} {u _ {Y ^ {2}}} = \binom {\exp \left(- \frac {y ^ {2}}{2 \cos^ {2} (\pi t)}\right)} {t}, \quad t \in [ 0, \frac {1}{2}) (49) \\ \Rightarrow \mathrm {d} \mathcal {H} ^ {1} \left(u _ {Y}\right) = \sqrt {\left(- \frac {\pi y ^ {2} \sin (\pi t)}{\cos^ {3} (\pi t)} \exp \left(- \frac {y ^ {2}}{2 \cos^ {2} (\pi t)}\right)\right) ^ {2} + 1} \mathrm {d} t. (50) \\ \end{array}
$$
This parametrization is also valid for  $y < 0$  and  $t \in (\frac{1}{2}, 1]$ . Due to the symmetry of the function  $f_{Y}$ , we only consider one of the cases (see Fig. 8 (left) with the level sets).
Figure 8: Box-Muller transformation as an example of the SCM of class  $\mathfrak{B}(C^1,2)$  (see Example 2). Here,  $N(y;0,1)$  is the density of the standard normal distribution.
Formally,  $f_{Y}\in C^{1}$  only for  $u_{Y}\in (0,1)^{2}$