SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Valentyn Melnychuk, Dennis Frauen & Stefan Feuerriegel
LMU Munich & Munich Center for Machine Learning (MCML)
melnychuk@lmu.de
Counterfactual inference aims to answer retrospective "what if" questions and thus belongs to the most fine-grained type of inference in Pearl's causality ladder. Existing methods for counterfactual inference with continuous outcomes aim at point identification and thus make strong and unnatural assumptions about the underlying structural causal model. In this paper, we relax these assumptions and aim at partial counterfactual identification of continuous outcomes, i.e., when the counterfactual query resides in an ignorance interval with informative bounds. We prove that, in general, the ignorance interval of the counterfactual queries has non-informative bounds, already when functions of structural causal models are continuously differentiable. As a remedy, we propose a novel sensitivity model called Curvature Sensitivity Model. This allows us to obtain informative bounds by bounding the curvature of level sets of the functions. We further show that existing point counterfactual identification methods are special cases of our Curvature Sensitivity Model when the bound of the curvature is set to zero. We then propose an implementation of our Curvature Sensitivity Model in the form of a novel deep generative model, which we call Augmented Pseudo-Invertible Decoder. Our implementation employs (i) residual normalizing flows with (ii) variational augmentations. We empirically demonstrate the effectiveness of our Augmented Pseudo-Invertible Decoder. To the best of our knowledge, ours is the first partial identification model for Markovian structural causal models with continuous outcomes.
Counterfactual inference aims to answer retrospective "what if" questions. Examples are: Would a patient's recovery have been faster, had a doctor applied a different treatment? Would my salary be higher, had I studied at a different college? Counterfactual inference is widely used in data-driven decision-making, such as root cause analysis [18, 139], recommender systems [16, 36, 83], responsibility attribution [50, 73, 77], and personalized medicine [84, 138]. Counterfactual inference is also relevant for various machine learning tasks such as safe policy search [108], reinforcement learning [19, 38, 63, 86, 91], algorithmic fairness [76, 104, 146], and explainability [42, 44, 64, 65, 75].
Counterfactual queries are located at the top of Pearl's ladder of causation [8, 51, 101], i.e., at the third layer  $\mathcal{L}_3$  of causation [8] (see Fig. 1, right). Counterfactual queries are challenging as they do reasoning in both the actual world and a hypothetical one where variables are set to different values than they have in reality.
State-of-the-art methods for counterfactual inference typically aim at point identification. These works fall into two streams. (1) The first stream [21, 29, 31, 70, 84, 100, 113, 114, 117, 118,
Figure 1: Pearl's ladder of causation [8, 51, 101] comparing observational, interventional, and counterfactual queries corresponding to the SCM  $\mathcal{M}$  with two observed variables, i.e., binary treatment  $A \in \{0, 1\}$  and continuous outcome  $Y \in \mathbb{R}$ . We also plot three causal diagrams,  $\mathcal{G}(\mathcal{M})$ , corresponding to each layer of causation, namely, Bayesian network, causal Bayesian network, and parallel worlds network. Queries with gray background can be simplified, i.e., be expressed via lower-layer distributions. The estimation of the queries with yellow background requires additional assumptions or distributions from the same layer. In this paper, we focus on partial identification of the expected counterfactual outcome of [un]treated,  $\mathbb{E}(Y_a \mid a', y')$ ,  $a' \neq a$ , shown in orange.
134, 135, 140] makes no explicit assumptions besides assuming a structural causal model (SCM) with Markovianity (i.e., independence of the latent noise) and thus gives estimates that can be invalid. However, additional assumptions are needed in counterfactual inference of layer  $\mathcal{L}_3$  to provide identifiability guarantees [94, 137]. (2) The second stream [4, 32, 59, 66, 93, 96, 123, 148] provides such identifiability guarantees but makes strong assumptions that are unnatural or unrealistic. Formally, the work by Nasr-Esfahany et al. [93] describes bijective generation mechanisms (BGMs), where, in addition to the original Markovianity of SCMs, the functions in the underlying SCMs must be monotonous (strictly increasing or decreasing) with respect to the latent noise. The latter assumption effectively sets the dimensionality of the latent noise to the same dimensionality as the observed (endogenous) variables. However, this is highly unrealistic in real-world settings and is often in violation of domain knowledge. For example, cancer is caused by multiple latent sources of noise (e.g., genes, nutrition, lifestyle, hazardous exposures, and other environmental factors).
In this paper, we depart from point identification for the sake of more general assumptions about both the functions in the SCMs and the latent noise. Instead, we aim at partial counterfactual identification of continuous outcomes. Rather than inferring a point estimation expression for the counterfactual query, we are interested in inferring a whole ignorance interval with informative bounds. Informative bounds mean that the ignorance interval is a strict subset of the support of the distribution. The ignorance interval thus contains all possible values of the counterfactual query for SCMs that are consistent with the assumptions and available data. Partial identification is still very useful for decision-making, e. g., when the ignorance interval for a treatment effect is fully below or above zero.
We focus on a Markovian SCM with two observed variables, namely, a binary treatment and a continuous outcome. We consider a causal diagram as in Fig. 1 (left). We then analyze the expected counterfactual outcome of [un]treated abbreviated by ECOU [ECOT]. This query is non-trivial in the sense that it can not be simplified to a  $\mathcal{L}_1 / \mathcal{L}_2$  layer, as it requires knowledge about the functions in SCM, but can still be inferred by means of a 3-step procedure of abduction-action-prediction [102]. ECOU [ECOT] can be seen as a continuous version of counterfactual probabilities [6] and allows to answer a retrospective question about the necessity of interventions: what would have been the expected counterfactual outcome for some treatment, considering knowledge about both the factual treatment and the factual outcome?
In our paper, we leverage geometric measure theory and differential topology to prove that, in general, the ignorance interval of ECOU [ECOT] has non-informative bounds. We show theoretically that this happens immediately when we relax the assumptions of (i) that the latent noise and the outcome have the same dimensionality and (ii) that the functions in the SCMs are monotonous (and assume they are continuously differentiable). As a remedy, we propose a novel Curvature Sensitivity Model (CSM), in which we bound the curvature of the level sets of the functions and thus yield informative bounds. We further show that we obtain the BGMs from [93] as a special case when setting the curvature to zero. Likewise, we yield non-informative bounds when setting it to infinity. Therefore, our CSM provides a sufficient condition for the partial counterfactual identification of the continuous outcomes with informative bounds.
We develop an instantiation of CSM in the form of a novel deep generative model, which we call Augmented Pseudo-Invertible Decoder (APID). Our APID uses (i) residual normalizing flows with (ii) variational augmentations to perform the task of partial counterfactual inference. Specifically, our APID allows us to (1) fit the observational/interventional data, (2) perform abduction-action-prediction in a differentiable fashion, and (3) bound the curvature of the SCM functions, thus yielding informative bounds for the whole ignorance interval. Finally, we demonstrate its effectiveness across several numerical experiments.
Overall, our main contributions are following:
1. We prove that the expected counterfactual outcome of [un]treated has non-informative bounds in the class of continuously differentiable functions of SCMs.
2. We propose a novel Curvature Sensitivity Model (CSM) to obtain informative bounds. Our CSM is the first sensitivity model for the partial counterfactual identification of continuous outcomes in Markovian SCMs.
3. We introduce a novel deep generative model called Augmented Pseudo-Invertible Decoder (APID) to perform partial counterfactual inference under our CSM. We further validate it numerically.
We briefly summarize prior works on (1) point and (2) partial counterfactual identification below but emphasize that none of them can be straightforwardly extended to our setting. We provide an extended literature overview in Appendix A.
(1) Point counterfactual identification has been recently addressed through neural methods [21, 29, 31, 70, 84, 100, 113, 114, 117, 118, 134, 135, 140] but without identifiability results.
To ensure identifiability, prior works usually make use of (i) symbolic identifiability methods or (ii) put restrictive assumptions on the model class, if (i) led to non-identifiability (see Fig. 2).
(i) Symbolic (non-parametric) identifiability methods [27, 121] aim to provide a symbolic probabilistic expression suitable for point identification if a counterfactual query can be expressed via lower-layer information only. Examples of the latter include the effect of treatment of the treated (ETT) [119] and path-specific effects [122, 146]. However, these are not suited for partial counterfactual identification.
Alternatively, identifiability can be achieved by (ii) making restrictive but unrealistic assumptions about the SCMs or the data generating mechanism [4, 32, 59, 66, 93, 96, 123, 148]. A notable example is the BGMs [93], which assumes a Markovian SCM where the functions in the SCMs must be monotonous (strictly increasing or decreasing) with respect to the latent noise. The latter assumption effectively sets the dimensionality of the latent noise to the same dimensionality as the observed (endogenous) variables, yet this is unrealistic in medicine. As a remedy, we depart from point identification and instead aim at partial counterfactual identification, which allows us to relax the assumptions of BGMs.
Kilbertus et al. [69] build sensitivity models for unobserved confounding in semi-Markovian SCMs, but still assume a restricted functional class of SCMs, namely, an additive noise model [66, 123]. We, on the other hand, build a sensitivity model around the extended class of functions in the SCMs, which is non-trivial even in the Markovian SCMs.
(2) Partial counterfactual identification has been studied previously, but only either for (i) discrete SCMs [97, 103, 129, 137, 141, 142, 143, 147], or for (ii) specific counterfactual queries with informative bounds [2, 4, 37]. Yet, these (i) do not generalize to the continuous setting; and (ii) are specifically tailored for certain queries with informative bounds, such as the variance of the treatment effects, and, thus, are not applicable to our (non-informative) ECOU [ECOT].
Likewise, it is also not possible to extend partial interventional identification of continuous outcomes [35, 49, 60, 61, 126] from the  $\mathcal{L}_2$  to  $\mathcal{L}_3$ , unless it explicitly assumes an underlying SCM.
Research gap. To the best of our knowledge, we are the first to propose a sensitivity model for partial counterfactual identification of continuous outcomes in Markovian SCMs.
Figure 2: Flow chart of identifiability for a counterfactual query.
$^{1}$ Code is available at https://github.com/Valentyn1997/CSM-APID.
3 Partial Counterfactual Identification of Continuous Outcomes
In the following, we derive one of our main results: the ignorance interval of the ECOU [ECOT] has non-informative bounds if we relax the assumptions that (i) both the outcome and the latent noise are of the same dimensionality and that (ii) the functions in the SCMs are monotonous.
Notation. Capital letters  $A, Y, U$  denote random variables and small letters  $a, y, u$  their realizations from corresponding domains  $\mathcal{A}, \mathcal{Y}, \mathcal{U}$ . Bold capital letters such as  $\mathbf{U} = \{U_1, \ldots, U_n\}$  denote finite sets of random variables. Further,  $\mathbb{P}(Y)$  is an (observational) distribution of  $Y$ ;  $\mathbb{P}(Y \mid a) = \mathbb{P}(Y \mid A = a)$  is a conditional (observational) distribution;  $\mathbb{P}(Y_{A=a}) = \mathbb{P}(Y_a)$  an interventional distribution; and  $\mathbb{P}(Y_{A=a} \mid A' = a', Y' = y') = \mathbb{P}(Y_a \mid a', y')$  a counterfactual distribution. We use a superscript such as in  $\mathbb{P}^{\mathcal{M}}$  to indicate distributions that are induced by the SCM  $\mathcal{M}$ . We denote the conditional cumulative distribution function (CDF) of  $\mathbb{P}(Y \mid a)$  by  $\mathbb{F}_a(y)$ . We use  $\mathbb{P}(Y = \cdot)$  to denote a density or probability mass function of  $Y$  and  $\mathbb{E}(Y) = \int y \, \mathrm{d}\mathbb{P}(y)$  to refer to its expected value. Interventional and counterfactual densities or probability mass functions are defined accordingly.
A function is said to be in class  $C^k$ ,  $k \geq 0$ , if its  $k$ -th derivative exists and is continuous. Let  $\| \cdot \|_2$  denote the  $L_2$ -norm,  $\nabla_x f(x)$  a gradient of  $f(x)$ , and  $\mathrm{Hess}_x f(x)$  a Hessian matrix of  $f(x)$ . The pushforward distribution or pushforward measure is defined as a transfer of a (probability) measure  $\mathbb{P}$  with a measurable function  $f$ , which we denote as  $f_{\sharp} \mathbb{P}$ .
SCMs. We follow the standard notation of SCMs as in [8, 13, 101, 137]. An SCM  $\mathcal{M}$  is defined as a tuple  $\langle \mathbf{U},\mathbf{V},\mathbb{P}(\mathbf{U}),\mathcal{F}\rangle$  with latent (exogenous) noise variables  $\mathbf{U}$ , observed (endogenous) variables  $\mathbf{V} = \{V_1,\dots ,V_n\}$ , a distribution of latent noise variables  $\mathbb{P}(\mathbf{U})$ , and a collection of functions  $\mathcal{F} = \{f_{V_1},\ldots ,f_{V_n}\}$ . Each function is a measurable map from the corresponding domains of  $\mathbf{U}_{V_i}\cup \mathbf{Pa}_{V_i}$  to  $V_{i}$ , where  $\mathbf{U}_{V_i}\subseteq \mathbf{U}$  and  $\mathbf{Pa}_{V_i}\subseteq \mathbf{V}\setminus V_i$  are parents of the observed variable  $V_{i}$ . Therefore, the functions  $\mathcal{F}$  induce a pushforward distribution of observed variables, i.e.,  $\mathbb{P}(\mathbf{V}) = \mathcal{F}_{\sharp}\mathbb{P}(\mathbf{U})$ . Each  $V_{i}$  is thus deterministic (non-random), conditionally on its parents, i.e.,  $v_{i}\gets f_{V_{i}}(\mathbf{pa}_{V_{i}},\mathbf{u}_{V_{i}})$ . Each SCM  $\mathcal{M}$  induces an (augmented) causal diagram  $\mathcal{G}(\mathcal{M})$ , which is assumed to be acyclic.
We provide a background on geometric measure theory and differential geometry in Appendix B.
3.2 Counterfactual Non-Identifiability
In the following, we relax the main assumption of bijective generation mechanisms (BGMs) [93], i.e., that all the functions in Markovian SCMs are monotonous (strictly increasing) with respect to the latent noise variable. To this end, we let the latent noise variables have arbitrary dimensionality and further consider functions to be of class  $C^k$ . We then show that counterfactual distributions under this relaxation are non-identifiable from  $\mathcal{L}_1$  or  $\mathcal{L}_2$  data.
Definition 1 (Bivariate Markovian SCMs of class  $C^k$  and  $d$ -dimension latent noise). Let  $\mathfrak{B}(C^k, d)$  denote the class of SCMs  $\mathcal{M} = \langle \mathbf{U}, \mathbf{V}, \mathbb{P}(\mathbf{U}), \mathcal{F} \rangle$  with the following endogenous and latent noise variables:  $\mathbf{U} = \{U_A \in \{0,1\}, U_Y \in [0,1]^d\}$  and  $\mathbf{V} = \{A \in \{0,1\}, Y \in \mathbb{R}\}$ , for  $d \geq 1$ . The latent noise variables have the following distributions  $\mathbb{P}(\mathbf{U})$ :  $U_A \sim \mathrm{Bern}(p_A)$ ,  $0 < p_A < 1$ , and  $U_Y \sim \mathrm{Unif}(0,1)^d$  and are all mutually independent. The functions are  $\mathcal{F} = \{f_A(U_A), f_Y(A, U_Y)\}$  and  $f_Y(a, \cdot) \in C^k$  for  $u_Y \in (0,1)^d \forall a \in \{0,1\}$ .
All SCMs in  $\mathfrak{B}(C^k,d)$  induce similar causal diagrams  $\mathcal{G}(\mathcal{M})$ , where only the number of latent noise variables  $d$  differs. Further, it follows from the above definition that BGMs are a special case of  $\mathfrak{B}(C^1,1)$ , where the functions  $f_{Y}(a,\cdot)$  are monotonous (strictly increasing).
Task: counterfactual inference. Counterfactual queries are at layer  $\mathcal{L}_3$  of the causality ladder and are defined as probabilistic expressions with random variables belonging to several worlds, which are in logical contradiction with each other [8, 27]. For instance,  $\mathbb{P}^{\mathcal{M}}(Y_a = y, Y_{a'} = y), a \neq a'$  for an SCM  $\mathcal{M}$  from  $\mathfrak{B}(C^k, d)$ . In this paper, we focus on the expected counterfactual outcome of [un]treated ECOU [ECOT], which we denote by  $Q_{a' \to a}^{\mathcal{M}}(y') = \mathbb{E}^{\mathcal{M}}(Y_a \mid a', y')$ .
2Uniformity of the latent noise does not restrict the definition, see the discussion in Appendix D.
Figure 3: Inference of observational and interventional distributions (left) and counterfactual distributions (right) for SCMs  $\mathcal{M}_1$  and  $\mathcal{M}_2$  from Example 1. Left: the observational query  $\mathbb{P}^{\mathcal{M}}(Y = y\mid a)$  coincides with the interventional query  $\mathbb{P}^{\mathcal{M}}(Y_a = y)$  for each  $\mathcal{M}_1$  and  $\mathcal{M}_2$ . Right: the counterfactual queries can still differ substantially for  $\mathcal{M}_1$  and  $\mathcal{M}_2$ , thus giving a vastly different counterfactual outcome distribution of the untreated,  $\mathbb{P}^{\mathcal{M}}(Y_1\mid A' = 0,Y' = 0)$ . Thus,  $\mathcal{L}_3$  queries are non-identifiable from  $\mathcal{L}_1$  or  $\mathcal{L}_2$  information.
For an SCM  $\mathcal{M}$  from  $\mathfrak{B}(C^k,d)$ , ECOU [ECOT] can be inferred by the following three steps. (1) The abduction step infers a posterior distribution of the latent noise variables, conditioned on the evidence, i.e.,  $\mathbb{P}(U_Y\mid a',y')$ . This posterior distribution is defined on the level set of the factual function given by  $\{u_{Y}:f_{Y}(a^{\prime},u_{Y}) = y^{\prime}\}$ , i.e., all points in the latent noise space mapped to  $y^\prime$ . (2) The action step alters the function for  $Y$  to  $f_{Y}(a,U_{Y})$ . (3) The prediction step is done by a pushforward of the posterior distribution with the altered function, i.e.  $f_{Y}(a,\cdot)_{\sharp}\mathbb{P}(U_{Y}\mid a^{\prime},y^{\prime})$ . Afterwards, the expectation of it is then evaluated.
The existence of counterfactual queries, which are non-identifiable with  $\mathcal{L}_1$  or  $\mathcal{L}_2$  data in Markovian SCMs, was previously shown in [137] (Ex. 8, App. D3) for discrete outcomes and in [13, 30] (Ex. D.7) for continuous outcomes. Here, we construct an important example to (1) show non-identifiability of counterfactual queries from  $\mathcal{L}_1$  or  $\mathcal{L}_2$  data under our relaxation from Def. 1, and to consequently (2) give some intuition on informativity of the bounds of the ignorance interval, which we will formalize later.
Example 1 (Counterfactual non-identifiability in Markovian SCMs). Let  $\mathcal{M}_1$  and  $\mathcal{M}_2$  be two Markovian SCMs from  $\mathfrak{B}(C^0, 2)$  with the following functions for  $Y$ :
$$
\mathcal {M} _ {1}: f _ {Y} \left(A, U _ {Y ^ {1}}, U _ {Y ^ {2}}\right) = A \left(U _ {Y ^ {1}} - U _ {Y ^ {2}} + 1\right) + (1 - A) \left(U _ {Y ^ {1}} + U _ {Y ^ {2}} - 1\right),
$$
$$
\mathcal {M} _ {2}: f _ {Y} (A, U _ {Y ^ {1}}, U _ {Y ^ {2}}) = \left\{ \begin{array}{l l} U _ {Y ^ {1}} + U _ {Y ^ {2}} - 1, & A = 0, \\ U _ {Y ^ {1}} - U _ {Y ^ {2}} + 1, & A = 1 \wedge (0 \leq U _ {Y ^ {1}} \leq 1) \wedge (U _ {Y ^ {1}} \leq U _ {Y ^ {2}} \leq 1), \\ F ^ {- 1} (0, U _ {Y ^ {1}}, U _ {Y ^ {2}}), & o t h e r w i s e, \end{array} \right.
$$
where  $F^{-1}(0, U_{Y^1}, U_{Y^2})$  is the solution in  $Y$  of the implicitly defined function  $F(Y, U_{Y^1}, U_{Y^2}) = U_{Y^1} - U_{Y^2} - 2(Y - 1) | - U_{Y^1} - U_{Y^2} + 1| - 1 + \sqrt{(Y - 2)^2(8(Y - 1)^2 + 1)} = 0$ .
It turns out that the SCMs  $\mathcal{M}_1$  and  $\mathcal{M}_2$  are observationally and interventionally equivalent (relative to the outcome  $Y$ ). That is, they induce the same set of  $\mathcal{L}_1$  and  $\mathcal{L}_2$  queries. For both SCMs, it can be easily inferred that a pushforward of uniform latent noise variables  $U_{Y^1}$  and  $U_{Y^2}$  with  $f_{Y}(A,U_{Y^{1}},U_{Y^{2}})$  is a symmetric triangular distribution with support  $\mathcal{V}_0 = [-1,1]$  for  $A = 0$  and  $\mathcal{V}_1 = [0,2]$  for  $A = 1$ , respectively. We plot the level sets of  $f_{Y}(A,U_{Y^{1}},U_{Y^{2}})$  for both SCMs in Fig. 3 (left). The pushforward of the latent noise variables preserves the transported mass; i. e., note the equality of (1) the area of each colored band between the two level sets in the latent noise space and (2) the area of the corresponding band under the density graph of  $Y$  Fig. 3 (left).
Despite the equivalence of  $\mathcal{L}_1$  and  $\mathcal{L}_2$ , the SCMs differ in their counterfactuals; see Fig. 3 (right). For example, the counterfactual outcome distribution of untreated,  $\mathbb{P}^{\mathcal{M}}(Y_1 = y \mid A' = 0, Y' = 0)$  has different densities for both SCMs  $\mathcal{M}_1$  and  $\mathcal{M}_2$ . Further, the ECOU,  $Q_{0 \to 1}^{\mathcal{M}}(0) = \mathbb{E}^{\mathcal{M}}(Y_1 \mid A' = 0, Y' = 0)$ , is different for both SCMs, i.e.,  $Q_{0 \to 1}^{\mathcal{M}_1}(0) = 1$  and  $Q_{0 \to 1}^{\mathcal{M}_2} \approx 1.114$ . Further details for the example are in Appendix C.
The example provides an intuition that motivates how we generate informative bounds later. By "bending" the bundle of counterfactual level sets (in blue in Fig. 3 left) around the factual level set
(in orange in Fig. 3, right), we can transform more and more mass to the bound of the support. We later extend this idea to the ignorance interval of the ECOU. Importantly, after "bending" the bundle of level sets, we still must make sure that the original observational/interventional distribution is preserved.
3.3 Partial Counterfactual Identification and Non-Informative Bounds
We now formulate the task of partial counterfactual identification. To do so, we first present two lemmas that show how we can infer the densities of observational and counterfactual distributions from both the latent noise distributions and  $C^1$  functions in SCMs of class  $\mathfrak{B}(C^1,d)$ .
Lemma 1 (Observational distribution as a pushforward with  $f_{Y}$ ). Let  $\mathcal{M} \in \mathfrak{B}(C^1, d)$ . Then, the density of the observational distribution, induced by  $\mathcal{M}$ , is
$$
\mathbb {P} ^ {\mathcal {M}} (Y = y \mid a) = \int_ {E (y, a)} \frac {1}{\| \nabla_ {u _ {Y}} f _ {Y} (a , u _ {Y}) \| _ {2}} d \mathcal {H} ^ {d - 1} \left(u _ {Y}\right), \tag {1}
$$
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).
We provide an example in Appendix C where we show the application of Lemma 1 (therein, we derive the standard normal distribution as a pushforward using the Box-Muller transformation). Lemma 1 is a generalization of the well-known change of variables formula. This is easy to see, when we set  $d = 1$ , so that  $\mathbb{P}^{\mathcal{M}}(Y = y \mid a) = \sum_{u_Y \in E(y,a)} |\nabla_{u_Y} f_Y(a,u_Y)|^{-1}$ . Furthermore, the function  $f_Y(a,u_Y)$  can be restored (up to a sign) from the observational distribution, if it is monotonous in  $u_Y$ , such as in BGMs [93]. In this case, the function coincides (up to a sign) with the inverse CDF of the observed distribution, i.e.,  $f_Y(a,u_Y) = \mathbb{F}_a^{-1}(\pm u_Y \mp 0.5 + 0.5)$  (see Corollary 1 in Appendix D).