SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Lemma 2. Let  $\mathcal{M} \in \mathfrak{B}(C^1, d)$ . Then, the density of the counterfactual outcome distribution of the [un]treated is
$$
\mathbb {P} ^ {\mathcal {M}} \left(Y _ {a} = y \mid a ^ {\prime}, y ^ {\prime}\right) = \frac {1}{\mathbb {P} ^ {\mathcal {M}} \left(Y = y ^ {\prime} \mid a ^ {\prime}\right)} \int_ {E \left(y ^ {\prime}, a ^ {\prime}\right)} \frac {\delta \left(f _ {Y} \left(a , u _ {Y}\right) - y\right)}{\left\| \nabla_ {u _ {Y}} f _ {Y} \left(a ^ {\prime} , u _ {Y}\right) \right\| _ {2}} d \mathcal {H} ^ {d - 1} \left(u _ {Y}\right), \tag {2}
$$
where  $\delta (\cdot)$  is a Dirac delta function, and the expected counterfactual outcome of the [un]treated, i.e., ECOU [ECOT], is
$$
Q _ {a ^ {\prime} \rightarrow a} ^ {\mathcal {M}} \left(y ^ {\prime}\right) = \mathbb {E} ^ {\mathcal {M}} \left(Y _ {a} \mid a ^ {\prime}, y ^ {\prime}\right) = \frac {1}{\mathbb {P} ^ {\mathcal {M}} \left(Y = y ^ {\prime} \mid a ^ {\prime}\right)} \int_ {E \left(y ^ {\prime}, a ^ {\prime}\right)} \frac {f _ {Y} \left(a , u _ {Y}\right)}{\left\| \nabla_ {u _ {Y}} f _ {Y} \left(a ^ {\prime} , u _ {Y}\right)\right\| _ {2}} d \mathcal {H} ^ {d - 1} \left(u _ {Y}\right), \tag {3}
$$
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  $a' \neq a$ .
Equations (2) and (3) implicitly combine all three steps of the abduction-action-prediction procedure: (1) abduction infers the level sets  $E(y', a')$  with the corresponding Hausdorff measure  $\mathcal{H}^{d-1}(u_Y)$ ; (2) action uses the counterfactual function  $f_Y(a, u_Y)$ ; and (3) prediction evaluates the overall integral. In the specific case of  $d = 1$  and a monotonous function  $f_Y(a, u_Y)$  with respect to  $u_Y$ , we obtain two deterministic counterfactuals, which are identifiable from observational distribution, i.e.,  $Q_{a' \to a}^{\mathcal{M}}(y') = \mathbb{F}_a^{-1}(\pm \mathbb{F}_{a'}(y') \mp 0.5 + 0.5)$ . For details, see Corollary 3 in Appendix D. For larger  $d$ , as already shown in Example 1, both the density and ECOT [ECOT] can take arbitrary values for the same observational (or interventional) distribution.
Figure 4: "Bending" the bundle of counterfactual level sets  $\{E(y,a):y\in [1,2]\}$  in blue around the factual level set  $E(y^{\prime},a^{\prime})$  in orange.
Definition 2 (Partial identification of ECOU (ECOT) in class  $\mathfrak{B}(C^k,d), k\geq 1$ ). Given the continuous observational distribution  $\mathbb{P}(Y\mid a)$  for some SCM of class  $\mathfrak{B}(C^k,d)$ . Then, partial counterfactual identification aims to find bounds of the ignorance interval  $[Q_{a'\to a}(y'),\overline{Q_{a'\to a}(y')} ]$  given by
$$
\underline {{Q _ {a ^ {\prime} \rightarrow a} (y ^ {\prime})}} = \inf  _ {\mathcal {M} \in \mathfrak {B} \left(C ^ {k}, d\right)} Q _ {a ^ {\prime} \rightarrow a} ^ {\mathcal {M}} \left(y ^ {\prime}\right) \quad s. t. \forall a \in \{0, 1 \}: \mathbb {P} (Y \mid a) = \mathbb {P} ^ {\mathcal {M}} (Y \mid a), \tag {4}
$$
$$
\overline {{Q _ {a ^ {\prime} \rightarrow a} (y ^ {\prime})}} = \sup  _ {\mathcal {M} \in \mathfrak {B} \left(C ^ {k}, d\right)} Q _ {a ^ {\prime} \rightarrow a} ^ {\mathcal {M}} (y ^ {\prime}) \quad s. t. \forall a \in \{0, 1 \}: \mathbb {P} (Y \mid a) = \mathbb {P} ^ {\mathcal {M}} (Y \mid a), \tag {5}
$$
where  $\mathbb{P}^{\mathcal{M}}(Y\mid a)$  is given by Eq. (1) and  $Q_{a^{\prime}\to a}^{\mathcal{M}}(y^{\prime})$  by Eq. (3).
Figure 5: Venn diagram of different SCM classes  $\mathfrak{B}(C^k,d)$  arranged by partial vs. point point identification. See our CSM with different  $\kappa$ . We further show the relationships between different classes of SCMs  $\mathfrak{B}(C^k,d)$  and statements about them, e.g., Lemmas 1 and 2. The referenced examples are in the Appendix C.
Figure 6: Overview of our Augmented Pseudo-Invertible Decoder (APID). Our APID uses (i) two residual normalizing flows for each treatment  $a \in \{0,1\}$ , respectively; and (ii) variational augmentations,  $\hat{Y}_{\mathrm{aug}}$ . Together, both components enable the implementation of CSM under Assumption  $\kappa$ .
Hence, the task of partial counterfactual identification in class  $\mathfrak{B}(C^k,d)$  can be reduced to a constrained variational problem, namely, a constrained optimization of ECOU [ECOT] with respect to  $f_{Y}(a,\cdot)\in C^{k}$ . Using Lemma 2, we see that, e. g., ECOU [ECOT] can be made arbitrarily large (or small) for the same observational distribution in two ways. First, this can be done by changing the factual function,  $f_{Y}(a^{\prime},\cdot)$ , i. e., if we increase the proportion of the volume of the factual level set  $E(y^{\prime},a^{\prime})$  that intersects only a certain bundle of counterfactual level sets. Second, this can be done by modifying the counterfactual function,  $f_{Y}(a,\cdot)$ , by "bending" the bundle of counterfactual level sets around the factual level set. The latter is schematically shown in Fig. 4. We formalize this important observation in the following theorem.
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  $\overline{Q_{a^{\prime}\to a}(y^{\prime})} = u_{a}$ .
Theorem 1 implies that, no matter how smooth the class of functions is, the partial identification of ECOU [ECOT] will have non-informative bounds. Hence, with the current set of assumptions, there is no utility in considering more general classes. This includes various functions, such as  $C^0$  and the class of all measurable functions  $f_{Y}(a,\cdot):\mathbb{R}^{d}\to \mathbb{R}$ , as the latter includes  $C^\infty$  functions. In the following, we introduce a sensitivity model which nevertheless allows us to obtain informative bounds.
4 Curvature Sensitivity Model
In the following, we develop our Curvature Sensitivity Model (CSM) to restrict the class  $\mathfrak{B}(C^k,d)$  so that the ECOU [ECOT] obtains informative bounds. Our CSM uses the intuition from the proof of Theorem 1 in that, to construct the non-informative SCM  $\mathcal{M}_{\mathrm{non - inf}}$ , we have to "bend" the bundle of the counterfactual level sets. As a result, our CSM provides sufficient conditions for informative bounds in the class  $\mathfrak{B}(C^2,d)$  by bounding the principal curvatures of level sets globally.
Assumption  $\kappa$ . Let  $\mathcal{M}$  be of class  $\mathfrak{B}(C^2,d),d\geq 2$ . Let  $E(y,a)$  be the level sets of functions  $f_{Y}(a,u_{Y})$  for  $a\in \{0,1\}$ , which are thus  $d - 1$ -dimensional smooth manifolds. Let us assume that principal curvatures  $i\in \{1,\ldots ,d - 1\}$  exist at every point  $u_{Y}\in E(y,a)$  for every  $a\in \{0,1\}$  and  $y\in (l_a,u_a)\subset \mathcal{V}_a$ , and let us denote them as  $\kappa_i(u_Y)$ . Then, we assume that  $\kappa \geq 0$  is the upper bound of the maximal absolute principal curvature for every  $y$ ,  $a$ , and  $u_{Y}\in E(y,a)$ :
$$
\kappa = \max  _ {a \in \{0, 1 \}, y \in \left(l _ {a}, u _ {a}\right), u _ {Y} \in E (y, a)} \quad \max  _ {i \in \{1, \dots , d - 1 \}} | \kappa_ {i} \left(u _ {Y}\right) |. \tag {6}
$$
Principal curvatures can be thought of as a measure of the non-linearity of the level sets, and, when they are all close to zero at some point, the level set manifold can be locally approximated by a flat hyperplane. In brevity, principal curvatures can be defined via the first- and second-order partial derivatives of  $f_{Y}(a,\cdot)$ , so that they describe the degrees of curvature of a manifold in different directions. We refer to Appendix B for a formal definition of the principal curvatures  $\kappa_{i}$ . An example is in Appendix C.
Now, we state the main result of our paper that our CSM allows us to obtain informative bounds for ECOU [ECOT].
Theorem 2 (Informative bounds with our CSM). Let the continuous observational distribution  $\mathbb{P}(Y\mid a)$  be induced by some SCM of class  $\mathfrak{B}(C^2,d),d\geq 2$  , which satisfies Assumption  $\kappa$  .Let  $\mathbb{P}(Y\mid a)$  have a compact support  $\mathcal{V}_a = [l_a,u_a]$  . Then, the ignorance interval for the partial identification of ECOU [ECOT] of class  $\mathfrak{B}(C^2,d)$  has informative bounds, dependent on  $\kappa$  and  $d,$  which are given by  $Q_{a^{\prime}\to a}(y^{\prime}) = l(\kappa ,d) > l_{a}$  and  $\overline{Q_{a^{\prime}\to a}(y^{\prime})} = u(\kappa ,d) <   u_{a}$
Theorem 2 has several important implications. (1) Our CSM is applicable to a wide class of functions  $\mathfrak{B}(C^2,d)$ , for which the principal curvature is well defined. (2) We show the relationships between different classes  $\mathfrak{B}(C^k,d)$  and our CSM  $(\kappa)$  in terms of identifiability in Fig. 5. For example, by increasing  $\kappa$ , we cover a larger class of functions, and the bounds on ECOU [ECOT] expand (see Corollary 4 in Appendix D). For infinite curvature, our CSM almost coincides with the entire  $\mathfrak{B}(C^2,d)$ . (3) Our CSM  $(\kappa)$  with  $\kappa \geq 0$  always contains both (i) identifiable SCMs and (ii) (informative) non-identifiable SCMs. Examples of (i) include SCMs for which the bundles of the level sets coincide for both treatments, i.e.,  $\{E(y,a):y\in \mathcal{Y}_a\} = \{E(y,a'):y\in \mathcal{Y}_{a'}\}$ . In this case, it is always possible to find an equivalent BGM when the level sets are flat and thus  $\kappa = 0$  (see  $\mathcal{M}_{\mathrm{flat}}$  from Example 6 in the Appendix C) or curved and thus  $\kappa = 1$  (see  $\mathcal{M}_{\mathrm{curv}}$  from Example 7 in the Appendix C). For (ii), we see that, even when we set  $\kappa = 0$ , we can obtain non-identifiability with informative bounds. An example is when we align the bundle of level sets perpendicularly to each other for both treatments, as in  $\mathcal{M}_{\mathrm{perp}}$  from Example 8 in the Appendix C. We formalize this observation with the following Lemma.
Lemma 3 (BGMs-EQTDs identification gap of  $\mathrm{CSM}(\kappa = 0)$ ). Let the assumptions of Theorem 2 hold and let  $\kappa = 0$ . Then the ignorance intervals for the partial identification of ECOU [ECOT] of class  $\mathfrak{B}(C^2,d)$  are defined by min/max between BGMs bounds and expectations of quantile-truncated distributions (EQTDs):
$$
\underline {{Q _ {a ^ {\prime} \rightarrow a} (y ^ {\prime}) / \overline {{Q _ {a ^ {\prime} \rightarrow a} (y ^ {\prime})}}} = \min  / \max  \left\{B G M _ {+} (y ^ {\prime}), B G M _ {-} (y ^ {\prime}), E Q T D _ {l} (y ^ {\prime}), E Q T D _ {u} (y ^ {\prime}) \right\} \tag {7}
$$
$$
B G M _ {+} \left(y ^ {\prime}\right) = \mathbb {F} _ {a} ^ {- 1} \left(\mathbb {F} _ {a ^ {\prime}} \left(y ^ {\prime}\right)\right) \quad B G M _ {-} \left(y ^ {\prime}\right) = \mathbb {F} _ {a} ^ {- 1} \left(1 - \mathbb {F} _ {a ^ {\prime}} \left(y ^ {\prime}\right)\right), \tag {8}
$$
$$
E Q T D _ {l} \left(y ^ {\prime}\right) = \mathbb {E} \left(Y \mid a, Y <   \mathbb {F} _ {a} ^ {- 1} (1 - 2 | 0. 5 - \mathbb {F} _ {a ^ {\prime}} \left(y ^ {\prime}\right) |\right), \tag {9}
$$
$$
E Q T D _ {u} \left(y ^ {\prime}\right) = \mathbb {E} \left(Y \mid a, Y > \mathbb {F} _ {a} ^ {- 1} \left(2 \left| \mathbb {F} _ {a ^ {\prime}} \left(y ^ {\prime}\right) - 0. 5 \right|\right)\right), \tag {10}
$$
where  $\mathbb{E}(Y\mid Y <   \cdot),\mathbb{E}(Y\mid Y > \cdot)$  are expectations of truncated distributions.
Lemma 3 shows how close we can get to the point identification after setting  $\kappa = 0$  in our CSM. In particular, with CSM  $(\kappa = 0)$  the ignorance interval still contains BGMs bounds and EQTDs, i.e., this is the identification gap of CSM. Hence, to obtain full point identification with our CSM, additional assumptions or constraints are required. Notably, we can not assume monotonicity, as there is no conventional notion of monotonicity for functions from  $\mathbb{R}^d$  to  $\mathbb{R}$ .
We make another observation regarding the choice of the latent noise dimensionality  $d$ . In practice, it is sufficient to choose  $d = 2$  (without further assumptions on the latent noise space). This choice is practical as we only have to enforce a single principal curvature, which reduces the computational burden, i.e.,
$$
\kappa_ {1} \left(u _ {Y}\right) = - \frac {1}{2} \nabla_ {u _ {Y}} \left(\frac {\nabla_ {u _ {Y}} f _ {Y} (a , u _ {Y})}{\| \nabla_ {u _ {Y}} f _ {Y} (a , u _ {Y}) \| _ {2}}\right), \quad u _ {Y} \in E (y, a). \tag {11}
$$
Importantly, we do not lose generality with  $d = 2$ , as we still cover the entire identifiability spectrum by varying  $\kappa$  (see Corollary 5 in Appendix D). We discuss potential extensions of our CSM in Appendix E.
Interpretation of  $\kappa$ . The sensitivity parameter  $\kappa$  lends to a natural interpretation. Specifically, it can be interpreted as a level of non-linearity between the outcome and its latent noises that interact with the treatment. For example, when (i) the treatment does not interact with any of the latent noise variables, then we can assume w.l.o.g. a BGM, which, in turn, yields deterministic counterfactual outcomes. There is no loss of generality, as all the other SCMs with  $d > 1$  are equivalent to this
<sup>3</sup>In general, BGMs bounds and EQTDs have different values, especially for skewed distributions. Nevertheless, it can be shown, that they approximately coincide as  $y'$  goes to  $l_{a'}$  or  $u_{a'}$ .
BGM (see Examples 6 and 7 in Appendix C). On the other hand, (ii) when the treatment interacts with some latent noise variables, then the counterfactual outcomes become random, and we cannot assume a BGM but we have to use our CSM. In this case,  $d$  corresponds to the number of latent noise variables which interact with the treatment. Hence,  $\kappa$  bounds the level of non-linearity between the outcome and the noise variables. More formally,  $\kappa$ , as a principal curvature, can be seen as the largest coefficient of the second-order term in the Taylor expansion of the level set (see Eq. (19) in Appendix B). This interpretation of the  $\kappa$  goes along with human intuition [20]: when we try to imagine counterfactual outcomes, we tend to "traverse" all the possible scenarios which could lead to a certain value. If we allow for highly non-linear scenarios, which interact with treatment, we also allow for more extreme counterfactuals, e. g., interactions between treatment and rare genetic conditions.
5 Augmented Pseudo-Invertible Decoder
We now introduce an instantiation of our CSM: a novel deep generative model called Augmented Pseudo-Invertible Decoder (APID) to perform partial counterfactual identification under our CSM  $(\kappa)$  of class  $\mathfrak{B}(C^2, 2)$ .
Architecture: The two main components of our APID are (1) residual normalizing flows with (2) variational augmentations (see Fig. 6). The first component are two-dimensional normalizing flows [107, 125] to estimate the function  $f_{Y}(a,u_{Y}),u_{Y}\in [0,1]^{2}$ , separately for each treatment  $a\in \{0,1\}$ . Specifically, we use residual normalizing flows [24] due to their ability to model free-form Jacobians (see the extended discussion about the choice of the normalizing flow in Appendix F). However, two-dimensional normalizing flows can only model invertible transformations, while the function  $\hat{f}_Y(a,\cdot):[0,1]^2\to \mathcal{Y}_a\subset \mathbb{R}$  is non-invertible. To address this, we employ an approach of pseudo-invertible flows [10, 54], namely, the variational augmentations [22], as described in the following. The second component in our APID are variational augmentations [22]. Here, we augment the estimated outcome variable  $\hat{Y}$  with the variationally sampled  $\hat{Y}_{\mathrm{aug}}\sim N(g^a (\hat{Y}),\varepsilon^2)$  where  $g^{a}(\cdot)$  is a fully-connected neural network, and  $\varepsilon^2 >0$  is a hyperparameter. Using the variational augmentations, our APID then models  $\hat{f}_Y(a,\cdot)$  through a two-dimensional transformation  $\hat{F}_a = (\hat{f}_{Y_{\mathrm{aug}}}(a,\cdot),\hat{f}_Y(a,\cdot)):[0,1]^2\to \mathbb{R}\times \mathcal{Y}_a$ . We refer to the Appendix F for further details.
Inference: Our APID proceeds in first steps: (P1) it first fits the observational/interventional distribution  $\mathbb{P}(Y\mid a)$ , given the observed samples; (P2) it then performs counterfactual inference of ECOU [ECOT] in a differentiable fashion, which, therefore, can be maximized/minimized jointly with other objectives; and (P3) it finally penalize functions with large principal curvatures of the level sets, by using automatic differentiation of the estimated functions of the SCMs.
We achieve (P1)-(P3) through the help of variational augmentations:  $\bullet$  In (P1), we fit two two-dimensional normalizing flows with a negative log-likelihood (for the task of maximizing or minimizing ECOU [ECOT], respectively).  $\bullet$  In (P2), for each normalizing flow, we sample points from the level sets of  $\hat{f}_Y(a,\cdot)$ . The latter is crucial as it follows an abduction-action-prediction procedure and thus generates estimates of the ECOU [ECOT] in a differential fashion. To evaluate ECOU [ECOT], we first perform the abduction step with the inverse transformation of the factual normalizing flow, i.e.,  $\hat{F}_{a'}^{-1}$ . This transformation maps the variationally augmented evidence, i.e.,  $(y',\{Y_{\mathrm{aug}}'\}_{j = 1}^{b}\sim N(g^{a}(y'),\varepsilon^{2}))$ , where  $b$  is the number of augmentations, to the latent noise space. Then, the action step selects the counterfactual normalizing flow via the transformation  $\hat{F}_a$ . Finally, the prediction step performs a pushforward of the latent noise space, which was inferred during the abduction step. Technical details are in Appendix F.  $\bullet$  In (P3), we enforce the curvature constraint of our CSM. Here, we use automatic differentiation as provided by deep learning libraries, which allows us to directly evaluate  $\kappa_1(u_Y)$  according to Eq. (11).
Training: Our APID is trained based on observational data  $\mathcal{D} = \{A_i, Y_i\}_{i=1}^n$  drawn i.i.d. from some SCM of class  $\mathfrak{B}(C^2, 2)$ . To fit our APID, we combine several losses in one optimization objective: (1) a negative log-likelihood loss  $\mathcal{L}_{\mathrm{NLL}}$  with noise regularization [110], which aims to fit the data distribution; (2) a counterfactual query loss  $\mathcal{L}_Q$  with a coefficient  $\lambda_Q$ , which aims to maximize/minimize the ECOU [ECOT]; and (3) a curvature loss  $\mathcal{L}_\kappa$  with coefficient  $\lambda_\kappa$ , which penalizes the curvature of the level sets. Both coefficients are hyperparameters. We provide details about the losses, the training algorithm, and hyperparameters in Appendix G. Importantly, we
incorporated several improvements to stabilize and speed up the training. For example, in addition to the negative log-likelihood, we added the Wasserstein loss  $\mathcal{L}_{\mathbb{W}}$  to prevent posterior collapse [26, 132]. To speed up the training, we enforce the curvature constraint only on the counterfactual function,  $\hat{f}_Y(a,\cdot)$ , and only for the level set, which corresponds to the evaluated ECOU [ECOT],  $u_{Y}\in E(\hat{Q}_{a^{\prime}\to a}(y^{\prime}),a)$ .
Datasets. To show the effectiveness of our APID at partial counterfactual identification, we use two synthetic datasets. This enables us to access the ground truth CDFs, quantile functions, and sampling. Then, with the help of Lemma 3, we can then compare our APID with the BGMs-EQTDs identification gap (= a special case of CSM with  $\kappa = 0$ ). Both synthetic datasets comprise samples from observational distributions  $\mathbb{P}(Y \mid a)$ , which we assume to be induced by some (unknown) SCM of class  $\mathfrak{B}(C^2, 2)$ . In the first dataset,  $\mathbb{P}(Y \mid 0) = \mathbb{P}(Y \mid 1)$  is the standard normal distribution, and in second,  $\mathbb{P}(Y \mid 0)$  and  $\mathbb{P}(Y \mid 1)$  are different mixtures of normal distributions. We draw  $n_a = 1,000$  observations from  $\mathbb{P}(Y \mid a)$  for each treatment  $a \in \{0,1\}$ , so that  $n = n_0 + n_1 = 2,000$ . Although both distributions have infinite support, we consider the finite sample minimum and maximum as estimates of the support bounds  $[\hat{l}_1, \hat{u}_1]$ . Further details on our synthetic datasets are in Appendix H. In sum, the estimated bounds from our APID are consistent with the theoretical values, thus showing
the effectiveness of our method.
Results. Fig. 7 shows the results of point/partial counterfactual identification of the ECOU, i.e.,  $\hat{Q}_{0\rightarrow 1}(y')$ , for different values of  $y'$ . Point identification with BGMs yields two curves, corresponding to strictly increasing and strictly decreasing functions. For partial identification with our APID, we set  $\lambda_{Q} = 2.0$  and vary  $\lambda_{\kappa}\in \{0.5,1.0,5.0,10.0\}$  (higher values correspond to a higher curvature penalization). We observe that, as we increase  $\lambda_{\kappa}$ , the bounds are moving closer to the BGMs, and, as we decrease it, the bounds are becoming non-informative, namely, getting closer to  $[\hat{l}_1,\hat{u}_1]$ . We report additional results in Appendix H (e.g., for APID with  $\lambda_{Q} = 1.0$ ).
Case study with real-world data. In Appendix I, we provide a real-world case study. Therein, we adopt our CSM to answer "what if" questions related to whether the lockdown was effective during the COVID-19 pandemic.
Limitations. Our CSM and its deep-learning implementation with APID have several limitations, which could be addressed in future work. First, given the value of  $\kappa$ , it is computationally infeasible to derive ground-truth bounds, even when the ground-truth observation density is known. The inference of these bounds would require solving a constrained optimization Hausdorff integration. This is intractable, even for such sin Second, the exact relationship between  $\kappa$  and corresponding known to be inversely related. Third, our APID sometimes s e.g., the bounds for the multi-modal distribution (see Fig. 7) happens as the gradual "bending" of the level sets is sometime and some runs omit "bending" at all as it would require passi
Conclusion. Our work is the first to present a sensitivity model for partial counterfactual identification of continuous outcomes in Markovian SCMs. Our work rests on the assumption of the bounded curvature of the level sets, yet which should be sufficiently broad and realistic to cover many models from physics and medicine. As a broader impact, we expect our bounds to be highly relevant for decision-making in safety-critical settings.
Figure 7: Results for partial counterfactual identification of the ECOU across two datasets. Reported: BGMs-EQTDs identification gap and mean bounds of APID over five runs.
SF acknowledges funding from the Swiss National Science Foundation (SNSF) via Grant 186932.
A.1 Counterfactual inference
In the following, we explain why existing work does not straightforwardly generalize to the partial counterfactual identification of continuous outcomes. In particular, we do the following:
1. We pinpoint that symbolic (non-parametric) counterfactual identifiability methods only provide the probabilistic expression suitable for point identification, if a certain query can be expressed via lower layer information.
2. We note that existing point identification methods do not have any guidance towards partial identification, and often do not even provide identifiability guarantees. In this case, valid point identification can only be achieved via additional assumptions either (i) about the SCM or (ii) about the data generating mechanism.
3. We discuss why methods for partial identification of (i) discrete counterfactual queries, (ii) continuous counterfactual queries with informative bounds, and (iii) continuous interventional queries can not be extended to our setting.