x}(\boldsymbol{\mathbf{z}}),\sigma_{x}\big{)},&\text{if $x$ is continuous}\\\
\mathcal{B}\textit{er}\big{(}\sigma(f_{z\to
x}(\boldsymbol{\mathbf{z}}))\big{)},&\text{if $x$ is binary}\\\
\mathcal{C}\textit{at}\big{(}\zeta(f_{z\to
x}(\boldsymbol{\mathbf{z}}))\big{)},&\text{if $x$ is categorical}\end{cases},$
$\displaystyle p(x|\boldsymbol{\mathbf{u}},\boldsymbol{\mathbf{z}})$
$\displaystyle\text{ is defined analogously but with $f_{uz\to x}(f_{u\to
uz}(\boldsymbol{\mathbf{u}}),f_{z\to uz}(\boldsymbol{\mathbf{z}}))$},$
$\displaystyle p(t|\boldsymbol{\mathbf{z}})$
$\displaystyle=\mathcal{B}\textit{er}\big{(}\sigma(f_{z\to
t}(\boldsymbol{\mathbf{z}}))\big{)},$ $\displaystyle
p(y|\boldsymbol{\mathbf{z}})$ $\displaystyle=\begin{cases}\mathcal{N}(f_{z\to
y}(\boldsymbol{\mathbf{z}}),\sigma_{y})&\text{if continuous}\\\ \text{see
\lx@cref{creftype~refnum}{app:survival}}&\text{if time to event}\end{cases},$
$\displaystyle q(\boldsymbol{\mathbf{u}}|\boldsymbol{\mathbf{x}})$
$\displaystyle=\mathcal{N}(\mu^{u}(\boldsymbol{\mathbf{x}}),\sigma^{u}(\boldsymbol{\mathbf{x}})),$
$\displaystyle q(\boldsymbol{\mathbf{u}}^{t}|\boldsymbol{\mathbf{x}})$
$\displaystyle=\mathcal{N}(\mu^{u}_{t}(\boldsymbol{\mathbf{x}}),\sigma^{u}_{t}(\boldsymbol{\mathbf{x}})),$
$\displaystyle q(\boldsymbol{\mathbf{z}}|\boldsymbol{\mathbf{x}})$
$\displaystyle=\mathcal{N}(\mu^{z}(\boldsymbol{\mathbf{x}}),\sigma^{z}(\boldsymbol{\mathbf{x}})),$
$\displaystyle q(\boldsymbol{\mathbf{z}}^{m}|\boldsymbol{\mathbf{m}})$
$\displaystyle=\mathcal{N}(\mu^{z_{m}}(\boldsymbol{\mathbf{x}}),\sigma^{z_{m}}(\boldsymbol{\mathbf{x}})),$
$\displaystyle\hskip
20.00003pt\text{$q(\boldsymbol{\mathbf{z}}|\boldsymbol{\mathbf{c}},\boldsymbol{\mathbf{x}})$
is defined analogously with an additional input $\boldsymbol{\mathbf{c}}$},$
$\displaystyle\hskip 20.00003pt\text{the split $\boldsymbol{\mathbf{z}}$
cases, i.e., \emph{+snet}, \emph{+tedvae} split the output of
$\mu^{z},\sigma^{z}$ accordingly},$

where $\sigma(\cdot)$ is the logistic sigmoid, $\zeta(\cdot)$ the softmax, and
all remaining functions are parameterized via neural networks. Standard
deviations $\sigma_{x}$ and $\sigma_{y}$ are treated as free parameters that
are optimized via gradient descent as part of the training routine.

###  C.3 Modern

_TEDVAE_ (Zhang et al., 2021) and _SNet_ (Curth and van der Schaar, 2021a)
serve as two examples of more modern generative and deterministic models and
how they can easily be adapted to our setup. _SNet_ stands in as an example of
a deterministic neural net with a more advanced architectural structure and
regularization method. Simply interpreting its mapping from $\mathcal{X}$ to
the representation space $\mathcal{Z}$ as the parameterization of an amortized
variational posterior and adding a generator that mirrors it by learning a
mapping from $\mathcal{Z}$ to $\mathcal{X}$, as well as adding a separate
representation $\mathcal{U}$ are all that is needed. _TEDVAE_ provides the
same illustration for generative models, whose adaptation is even simpler.
Here an encoder-decoder structure is already provided lacking simply a
separate $\mathcal{U}$ representation. Other novel approaches can be adapted
to our setup in the same manner. For comparison, we replicate the
corresponding rows from Table 1 below. Our adaptations improve upon these
baselines in all but one setting (highlighted in the table via underlining).

(a) full outcome observation (RMSE of CATE) (b) partial outcome observation
(AE of ATT) all+high subset+low all+high subset+low Method within sample out-
of-sample within sample out-of-sample within sample out-of-sample within
sample out-of-sample SNet
$0.168\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.003$
$0.264\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008$
$0.211\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008$
$\underline{{0.243\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.016}}$
$0.043\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.003$
$0.137\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$0.221\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.015$
$0.366\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.026$ Ours+snet
$0.141\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002$
$0.256\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008$
${0.185\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007}$
${0.253\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.017}$
${0.038\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002}$
${0.117\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007}$
$0.194\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.014$
$0.322\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.020$ TEDVAE
$0.176\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002$
$0.283\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$0.293\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.009$
$0.336\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.018$
$0.052\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.003$
$0.179\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.009$
$0.144\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.009$
$0.272\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.016$ Ours+tedvae
$0.143\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002$
$0.257\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008$
${0.187\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007}$
$0.258\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.019$
${0.039\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.003}$
${0.119\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007}$
${0.103\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008}$
${0.189\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.015}$

###  C.4 Architectures and Hyperparameters

We rely on the same set of architectures and hyperparameters throughout the
various experiments and methods, adapting them only to the specific
dimensionalities. Please refer to Section C.2 for a description of where the
respective nets appear.

##### Neural architectures.

We use the notation $A,B,C$, to indicate fully connected layers of $A,B,C$
neurons respectively. All models use exponentiated linear units (Clevert et
al., 2016) as activation functions between layers. They follow the layer width
and depth of Shalit et al. (2017) as closely as possible.

Function | Mapping | Comments
---|---|---
$\mu(\cdot,\cdot)$ | $\dim(\boldsymbol{\mathbf{x}})+1,200,200,\dim(z),100,100,1$ |
$\mu_{i}(\cdot),i\in\\{0,1\\}$ | $\dim(\boldsymbol{\mathbf{x}}),200,200,\dim(z),100,100,1$ | _TNet_
$\mu_{i}(\cdot),i\in\\{0,1\\}$ | $\dim(z),100,100,1$ | otherwise
$h(\cdot)$ | $\dim(\boldsymbol{\mathbf{x}}),200,200,\dim(z)$ |
$h_{i}(\cdot),i\in\\{1,\ldots,5\\}$ | $\dim(\boldsymbol{\mathbf{x}}),200,200,s_{i}$ |  $s_{i}$ defined in the hyperparameters
$g(\cdot)$, $f_{z\to t}$ | $\dim(\boldsymbol{\mathbf{z}}),1$ |
$\lambda(\cdot)$ | $\dim(\boldsymbol{\mathbf{c}}),10,10,\dim(\boldsymbol{\mathbf{z}})$ |
$f_{z\to uz}(\cdot)$ | $\dim(\boldsymbol{\mathbf{z}}),50$ |
$f_{u\to uz}(\cdot)$ | $\dim(\boldsymbol{\mathbf{z}}),150$ |
$f_{uz\to x}(\cdot)$ | $200,200,\dim(\boldsymbol{\mathbf{x}})$ |
$f_{z\to x}(\cdot)$ | $\dim(\boldsymbol{\mathbf{z}}),200,200,\dim(\boldsymbol{\mathbf{x}})$ |
$f_{z\to y}(\cdot)$ | $\dim(\boldsymbol{\mathbf{z}}),100,100,1$ |
$\mu_{t}^{u}(\cdot),\sigma_{t}^{u}(\cdot)$ | $\dim(\boldsymbol{\mathbf{x}}),200,200,\dim(\boldsymbol{\mathbf{u}})$ |
$\mu_{t}^{z}(\cdot),\sigma_{t}^{z}(\cdot)$ | $\dim(\boldsymbol{\mathbf{x}}),200,200,\dim(\boldsymbol{\mathbf{z}})$ |
$\mu_{t}^{z_{m}}(\cdot),\sigma_{t}^{z_{m}}(\cdot)$ | $\dim(\boldsymbol{\mathbf{m}}),200,200,\dim(\boldsymbol{\mathbf{z}}^{m})$ |

##### Hyperparameters and further settings.

In the fully observed setup, the representation space $\mathcal{Z}$ is five-
dimensional, with $s=(1,1,2,1)$ to specify the split in the SNet
representation space. The dimensionality of $\mathcal{U}$ is fixed at 50
throughout all experiments.

Discrete covariates are treated as Bernoulli or categorical variables turned
into one-hot encodings for all models in the case of categorical covariates.
Continuous covariates are treated as normal variables with a homoscedastic
noise model, with a variance parameter per covariate inferred via gradient
descent. These are modeled as normal variables with a global precision
parameter inferred via gradient descent.

We perform gradient descent via the Adam optimizer (Kingma and Ba, 2015) with
a learning rate of $0.001$ and weight decay of $0.0001$. A random subset of
10% of the training data in each of the replications is used as a validation
set for early stopping. The upper limit for the maximal number of epochs was
set to 500, which was never reached.

For baselines that do not model missing values, we impute the empirical mean
for missing continuous covariates and the median for missing binary ones.
(Categorical covariates were assumed to always be observed for simplicity.)

###  C.5 Semisynthetic Experimental Data

Due to the fundamental problem of causality (Pearl, 2009), the true treatment
effect is not known for real data as the counterfactual outcome can never be
observed. Any experimental evaluation is therefore constrained to rely on
either completely synthetic or semi-synthetic data sets. In the main paper, we
focus on the latter case and rely on real covariates with simulated treatment
outcomes. This allows us to stay realistic with respect to covariate
distributions, while still being able to generate factual and counterfactual
outcomes to simulate various treatment effect scenarios.

We rely on two sets of covariates for this task, which we describe below in
greater detail. The first is based on covariates from the popular IHDP
benchmark (Hill, 2011). The second is a newly curated real-world data set
combining RCT study data with electronic health records. We describe each of
them in turn.

####  C.5.1 IHDP

##### Data.

The Infant Health and Development Program (IHDP) data set as used by Hill
(2011) consists of 25 covariates. 19 of them are binary, five of them
continuous and one categorical.555The categorical covariate is modelled as
continuous in most prior work. This observational data set contains 139
treated children and 608 untreated ones after preprocessing, where Hill (2011)
removes all children with nonwhite mothers from the treated group to create an
artificial reduction in overlap between the two groups.

##### Synthetic outcome creation.

As is common, we follow Hill (2011). Given the covariates
$\boldsymbol{\mathbf{x}}$, an offset vector $\boldsymbol{\mathbf{w}}$, whose
entries all equal $0.5$, and a regression vector $\beta$, whose coefficients
are randomly sampled from $(0,0.1,0.2,0.3,0.4)$ with probabilities
$(0.6,0.1,0.1,0.1,0.1)$, the potential outcomes are generated as

$Y(0)\sim\mathcal{N}(\exp((\boldsymbol{\mathbf{x}}+\boldsymbol{\mathbf{w}})^{\top}\beta,1)\quad\text{and}\quad
Y(1)\sim\mathcal{N}(\boldsymbol{\mathbf{x}}^{\top}\beta-\omega,1),$

where $\omega$ is an offset, chosen so that the true treatment effect on the
treated is given as $4$. For the different scenarios described below,
$\boldsymbol{\mathbf{x}}$ is further modified before creating the synthetic
outcomes $Y(0)$ and $Y(1)$. We rely on the precomputed train/val/test splits
provided by Shalit et al. (2017). The number of random data sets used in each
experiment is marked in the respective result tables or figure captions.

We generate $\boldsymbol{\mathbf{c}}$ by picking three random covariates from
the 19 binary covariates for each seed. Note that given the randomness in
which covariates are predictive, this does not guarantee that the chosen
$\boldsymbol{\mathbf{c}}$ provides good guidance to $\boldsymbol{\mathbf{z}}$.
Similar to how the researcher cannot know a priori which covariates are
useful.

##### All vs Subset.

The _all_ scenario relies on all 25 covariates $\boldsymbol{\mathbf{x}}$ to
create the synthetic outcomes. As the $\beta$ vector is randomly sampled and
can contain zeros, this still leaves us with potentially fewer than 25
predictive covariates.

Compared to this, _subset_ explicitly constraints to fewer than 25 covariates
by choosing a random subset of $n_{\text{pred}}$ covariates for each seed
before following the procedure described above. As before, $n_{\text{pred}}$
only provides us with an upper bound of predictive covariates. Throughout our
experiments, we use $n_{\text{pred}}=15$.

##### High vs Low.

The _high_ setup refers to using the original $\boldsymbol{\mathbf{x}}$ as
defined by Hill (2011). We keep the overlap as is, which has already been
perturbed to a certain degree as discussed above.

For _low_ we try to mimic real-world scenarios where the overlap tends to be
even lower. To increase the divergence between the two data sources we modify
the control group as follows. We shift the five continuous covariates as

$X_{c}\leftarrow
X_{c}+s+\varepsilon,\quad\text{where}\;\;\;s\sim\mathcal{B}\textit{er}(0.5)\cdot
6-3,\quad\varepsilon\sim\mathcal{N}(0,9),$

for the $c$-th continuous covariate. That is, $s\in\\{-3,3\\}$ with equal
probability, and $\varepsilon$ adds a large amount of variation to still keep
some overlap. Of the $19$ binary covariates, we pick a random subset of five
increasing the probability of three of them to be one by $80\%$ and the
probability of the other two to be $0$ by $80\%$.

##### Missingness.

We compare three settings of increasing missing not at random structure. In
each of the three settings, the covariates for each observation are missing
based on the following:

  * •

_Strong_. For discrete covariates: If $x_{i}=1$ it has a probability of
$p=0.2$ of being observed, if $x_{1}=0$ it has a probability of $p=0.3$ of
being observed. A continuous covariate with $x_{i}>0$ has a probability of
$p=0.1$ of being observed.

  * •

_Medium_. For discrete covariates: If $x_{i}=1$ it has a probability of
$p=0.5$ of being observed, if $x_{1}=0$ it has a probability of $p=0.7$ of
being observed. A continuous covariate with $x_{i}>0$ has a probability of
$p=0.6$ of being observed.

  * •

_Weak_. For discrete covariates: If $x_{i}=1$ it has a probability of $p=0.9$
of being observed, if $x_{1}=0$ it has a probability of $p=0.9$ of being
observed. A continuous covariate with $x_{i}>0$ has a probability of $p=0.9$
of being observed.

Missing discrete covariates are imputed by the median of the observed data
set, missing continuous by zeros.

##### Survival analysis.

Data generation details for the survival analysis experiments are given in
Section C.8.

####  C.5.2 Real-world data

##### Data.

_Note. To ensure that the identity of the authors is not leaked during the
submission, the following description is partially obfuscated and anonymized.
Further details and references to related publications will be provided upon
acceptance in the camera-ready version of the paper._

This data set consists of two parts. The single-arm trial data is taken from
Anonymous1 (X), a published RCT study, evaluating the treatment of patients
suffering from chronic kidney disease and type 2 diabetes. For our purpose,
only a subset of the treated group of patients is selected for the model to
train on (N=833). The control group is constructed from electronic health
records provided by Anonymous2, whose ethics committee gave ethical approval
and a study permit. A set of 2646 patients was selected who roughly fulfilled
the inclusion criteria of the RCT study. Although both data sets are
originally longitudinal, the measurements closest to the index date are
collected into a single set of covariates, leaving proper longitudinal
modeling for future work. Note that while the index date is properly defined
in the RCT it remains a noisy choice for the EHR data. For each patient, we
pick the time a subset of the original inclusion criteria is fulfilled. A
subset of 100 binary covariates of the most prevalent diagnosis and medical
history is selected (from approximately 4000), as well as $40$ continuous
covariates (lab measurements age, weight, height, …) giving us a total of 140
covariates. Due to their wide range of potential values, we standardize them
by subtracting the median and scaling them by the interquartile range.666See
sklearn.preprocessing.RobustScaler for an implementation. As the EHR data
contains a lot of missing covariates, we impute missing binary covariates via
their median, while missing continuous ones are imputed with zeros (justified
due to the prior standardization of the data). To create synthetic treatment
outcomes, we require fully observed covariates, i.e., cannot model the true
missingness pattern inherent in the model. We provide a longer discussion on
the differences and similarities in the respective covariate distributions in
Anonymous3 (X).

##### Synthetic outcome creation.

We generate synthetic outcomes as follows. First, a random subset of $30$
covariates is selected from the original 140 and mapped with a single hidden
layer neural network (30,10,$\dim(\boldsymbol{\mathbf{z}})$)777See Section C.4
for details on this notation. into a five-dimensional representation space.
The outcomes are then given as

$Y(0)\sim\mathcal{N}\left(\boldsymbol{\mathbf{a}}^{\top}\boldsymbol{\mathbf{v}},0.3^{2}\right)\qquad\text{and}\quad
Y(1)\sim\mathcal{N}\left((\boldsymbol{\mathbf{a}}+0.5)^{\top}\boldsymbol{\mathbf{w}}-\omega,0.3^{2}\right),$

where
$\boldsymbol{\mathbf{v}},\boldsymbol{\mathbf{w}}\sim\mathcal{N}(0,\mathds{1})$,
$\boldsymbol{\mathbf{a}}$ the five dimensional representation, and $\omega$
such that the ATT is four.

###  C.6 Alternative Matching Approaches

##### Matching without replacement.

Throughout all of our experiments, we conduct matching with replacement, i.e.,
we match control patients to treated patients independently of whether they
have been matched before. While this ensures that every treated patient gets a
control that is closest to them, it effectively reduces the number of used
patients and might induce statistical problems due to multiple samples being
identical. While this is the most common approach, some alternatives can be
considered. The first one would be to pick patients greedily for a random
permutation of the treated patients. While this is simple, it will depend on
the order and thus won’t be optimal. Another approach is to consider the
matching as a task that aims to find the optimal minimal weight solution for a
bipartite graph. Considering each group as one set of vertices with the edges
between the groups given by the respective distances, the optimal matching is
given by solving a straightforward linear program.888Our implementation relies
on a SciPy (Virtanen et al., 2020) routine.

It should be noted that these two approaches solve the task of reusing
patients, but suffer from the problem that the chosen subset is now heavily
interdependent, i.e., the choice of one patient is no longer independent of
the other choices. We evaluate this latter approach on a subset of methods in
Table 3.

Table 3: _Alternative matching._ We compare matching with replacement (_with_)
to matching without replacement (_w/out_). Matching without replacement
performs similar, or even improves upon the default matching with replacement
approach.

_AE of ATT_ all+high subset+low Method within sample out-of-sample within
sample out-of-sample with w/out with w/out with w/out with w/out VAE
${0.085\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004}$
$\underline{0.083\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004}$
${0.266\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012}$
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$0.282\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.014$
$0.427\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.022$
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${0.174\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.009}$
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$\underline{0.150\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.009}$
$0.192\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.011$
$\underline{0.259\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.015}$
$0.263\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.014$ TEDVAE
${0.052\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.003}$
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$\underline{0.179\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.009}$
$0.180\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.009$
$\underline{0.144\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.009}$
$0.198\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.010$
$0.272\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.016$
$\underline{0.271\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.014}$ Ours
$\underline{0.037\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002}$
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$0.113\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
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$0.114\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008$
$\underline{0.107\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008}$
${0.190\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.016}$
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${0.038\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002}$
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${0.122\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007}$
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$0.110\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$\underline{0.108\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008}$
${0.176\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.014}$
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${0.037\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002}$
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${0.111\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007}$
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${0.144\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.009}$
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$0.272\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.016$
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_mean $\pm$ standard error over 300 random replications; the lower of the two
means between each variation is underlined_

##### Varying metrics.

Within the main paper, we always rely on the Euclidian distance

$d(\boldsymbol{\mathbf{a}},\boldsymbol{\mathbf{b}})=\sqrt{\sum_{i=1}^{d}(a_{i}-b_{i})^{2}},$

for $\boldsymbol{\mathbf{a}},\boldsymbol{\mathbf{b}}\in\mathds{R}^{d}$.
However, given that our variational posteriors consist of distributions
instead of point estimates, we can also rely on distributional distance
metrics that take posterior variances into account.

The _Wasserstein distance_ between two univariate Normal distributions is
analytically tractable and given as

$W(\mathcal{N}(\mu_{0},\sigma_{0}),\mathcal{N}(\mu_{1},\sigma_{1}))^{2}=(\mu_{0}-\mu_{1})^{2}+(\sigma_{0}-\sigma_{1})^{2}.$

The squared _Hellinger distance_ between two univariate Normal distributions
is given as

$H(\mathcal{N}(\mu_{0},\sigma_{0}),\mathcal{N}(\mu_{1},\sigma_{1}))^{2}=1-\sqrt{\frac{2\sigma_{0}\sigma_{1}}{\sigma_{0}^{2}+\sigma_{1}^{2}}}\exp\left(-\frac{(\mu_{0}-\mu_{1})^{2}}{4(\sigma_{0}^{2}+\sigma_{1}^{2})}\right).$

Both Wasserstein and Hellinger factorize for mean-field normal distributions
which is why we only state their univariate formulae.

For
$q(\boldsymbol{\mathbf{z}}_{i})=\mathcal{N}(\boldsymbol{\mathbf{z}}_{i},\mu_{i},\sigma_{i}^{2})$,
and we
$q(\boldsymbol{\mathbf{z}}_{j})=\mathcal{N}(\boldsymbol{\mathbf{z}}_{j}|\mu_{j},\sigma_{j}^{2})$,
we compare the ATT performance for matching in the reduced setting via _(i)_
the Euclidean distance $d(\mu_{i},\mu_{j})$, _(ii)_ the Wasserstein distance
$W(q(\boldsymbol{\mathbf{z}}_{i}),q(\boldsymbol{\mathbf{z}}_{j}))$, _(iii)_
the Hellinger distance
$H(q(\boldsymbol{\mathbf{z}}_{i}),q(\boldsymbol{\mathbf{z}}_{j}))$, and _(iv)_
via the Euclidean distance on a propensity estimator $\hat{\pi}$, i.e.,
$d(\hat{\pi}(\boldsymbol{\mathbf{z}}_{i}),\hat{\pi}(\boldsymbol{\mathbf{z}}_{j}))$.

As in the remainder of the paper, we stick to mean-field normal posteriors.
However, the low-dimensional structure of $\boldsymbol{\mathbf{z}}$ allows us
to infer more complex posteriors without too many constraints due to
computational or memory costs. E.g., by switching to a multivariate Normal,
for which the two distances remain analytically tractable, while at the same
time allowing for more structured variance estimation. We leave such
extensions to future work.

The results are summarized in Section D.6.

###  C.7 Predictive Latent Space $\mathcal{Z}$ Regularization

As discussed in our main paper, one crucial assumption for causal inference
from observational data is overlap, i.e., that $0<\pi(x)<1$, $\forall x$ where
$\pi(x)=\mathds{P}(T=1|X=x)$ is the propensity score. It is therefore common
practice (Shalit et al., 2017; Johansson et al., 2016; Lu et al., 2020) to
further regularize the latent space, in our case $\mathcal{Z}$, on top of the
existing regularization through the loss. In our case it is already implicitly
regularized by the KL terms within our ELBO (4), i.e., in the identifiable
setup
$\textrm{KL}\left(q(\boldsymbol{\mathbf{z}})\;\|\;p(\boldsymbol{\mathbf{z}}|\boldsymbol{\mathbf{c}})\right)$
and
$\textrm{KL}\left(q(\boldsymbol{\mathbf{z}})\;\|\;p(\boldsymbol{\mathbf{z}})\right)$
in the unidentifiable one. To further encourage overlap through additional
regularization we can rely on several approaches. One could rely on
distributional approximation methods, such as the maximum mean discrepancy
method (MMD) (Gretton et al., 2012), as was proposed, e.g., by Johansson et
al. (2016); Shalit et al. (2017). MMD aims to minimize the distance between
two aggregate variational posteriors

$q_{t}(\boldsymbol{\mathbf{z}})\triangleq\frac{1}{N_{t}}\sum_{n:t_{n}=t}q(\boldsymbol{\mathbf{z}}_{n}|\boldsymbol{\mathbf{x}}_{n}),\qquad
t\in\\{0,1\\},$

by minimizing $\text{MMD}^{2}(q_{0},q_{1})$ defined as

$\text{MMD}^{2}(q_{0},q_{1})=\mathds{E}_{q_{0}(z)q_{0}(z^{\prime})}\left[k(z,z^{\prime})\right]-2\mathds{E}_{q_{0}(z)q_{1}(z^{\prime})}\left[k(z,z^{\prime})\right]+\mathds{E}_{q_{1}(z)q_{1}(z^{\prime})}\left[k(z,z^{\prime})\right],$

where we use the exponentiated quadratic kernel as $k(\cdot,\cdot)$ and
approximate the expectations via sampling. The loss to be optimized is then
given as

$\mathcal{L}_{\text{elbo}}+\kappa\text{MMD}^{2}(q_{0},q_{1}),$

where $\kappa$ serves as a balancing factor. Throughout our experiments, we
use a fixed $\kappa=1$

Another approach is to use critic-based approaches (e.g., Lu et al., 2020),
i.e., learning to fool a critic whose task is to differentiate between treated
and control observations, as is commonly done in generative adversarial
networks (GANs) (Goodfellow et al., 2014). Within this work, we rely primarily
on the gradient reversal layer formulation introduced by Ganin and Lempitsky
(2015) for GAN-based regularization. An additional discriminator tries to
classify instances $\boldsymbol{\mathbf{z}}_{n}$ correctly as treated/control,
i.e., to maximize the log-likelihood $\log
p(t_{n}|\boldsymbol{\mathbf{z}}_{n})$. The main model tries to maximize its
increasing overlap in the process. These two diverging objectives can be
incorporated in our existing objective (4) by relying on a pseudo-function

$R_{\kappa}(\boldsymbol{\mathbf{z}})\triangleq\boldsymbol{\mathbf{z}}\quad\text{with}\quad\frac{dR_{\kappa}}{d\boldsymbol{\mathbf{z}}}\triangleq-\kappa\mathds{1},$

and using ${\tilde{f}(\boldsymbol{\mathbf{z}})\triangleq(f\circ
R_{\kappa})(\boldsymbol{\mathbf{z}})}$ for the likelihood
${p(t|\boldsymbol{\mathbf{z}})=\mathcal{B}\textit{er}(t|\sigma(\tilde{f}(\boldsymbol{\mathbf{z}}))}$,
and $f$ a neural network. The inherent objective of the discriminator $f$ is
then to maximize $\log p(t|\boldsymbol{\mathbf{z}})$, while the objective of
our generative model becomes a modification of the ELBO, i.e., its objective
is to

$\text{maximize}\quad\mathcal{L}_{\text{elbo}}-2\log
p(t|\boldsymbol{\mathbf{z}}),$

where $\mathcal{L}_{\text{elbo}}$ is given by (4). We use $\kappa=1$ in the
experiments.

These regularization approaches apply to the case where we want full overlap
in the latent space $\mathcal{Z}$. This is the case for having access to
outcome data from both groups, i.e., the single-arm patients as well as the
control. For our estimators $\mu_{t}$ to be predictive of the counterfactual
of the other group, we want the overlap between these groups to be maximal.
However, if we only have access to outcome data $y$ for the control group, we
only require the encodings of the treated group to overlap with the encodings
of the control group, but not vice versa. I.e., as long as we can select a
suitable subset of control patients, it is irrelevant how the remainder of the
controls are encoded. It is therefore sometimes already sufficient to rely on
the indirect regularization provided by the KL terms and the generative
constraints to ensure a sufficient overlap.

We do this in the one-arm experiments for IHDP. Note, that this is also
necessary to fulfill the assumptions of Khemakhem et al. (2020)’s Theorem 4
(see above).

###  C.8 Survival

Our survival analysis experiment uses the IHDP covariates as its starting
point after transforming them following the _subset+low_ strategy. We only
experiment with right-censored observations.

Synthetic outcomes are then generated via the following steps, where we
roughly follow Pölsterl (2019); Manduchi et al. (2022).

  1. 1.

Sample $\beta$ as in the default IHDP setting from $(0,0.1,0.2,0.3,0.4)$ with
probabilities $(0.6,0.1,0.1,0.1,0.1)$

  2. 2.

For every patient $n$ compute a risk score assuming they are treated
$r_{n}^{t}$ or control $r_{n}^{c}$ as

$\displaystyle r_{n}^{c}$
$\displaystyle=\text{clip}\big{(}\text{soft}\big{(}\boldsymbol{\mathbf{x}}_{n}^{\top}\beta+((\boldsymbol{\mathbf{x}}-0.5)^{\top}\beta)^{2}\big{)}\big{)}$
$\displaystyle r_{n}^{t}$
$\displaystyle=\text{clip}\big{(}\text{soft}\big{(}\boldsymbol{\mathbf{x}}_{n}^{\top}\beta+|\boldsymbol{\mathbf{x}}^{\top}\beta|\big{)}\big{)}$

where $\text{soft}(x)=\log(1+\exp(x))$ is the softplus function, and we clip
them to be within $[0.5,15]$.

  3. 3.

For each $\boldsymbol{\mathbf{r}}^{c}=(r_{1}^{c},\ldots,r_{N}^{c})$ and
$\boldsymbol{\mathbf{r}}^{t}=(r_{1}^{t},\ldots,r_{N}^{t})$, we, relying on a
generic $\boldsymbol{\mathbf{r}}$ for notational simplicity in the following
list, create survival times $\boldsymbol{\mathbf{y}}$ and censoring indicators
$\delta$ as follows

    1. (a)

Define mean survival time $T_{0}$ ($=365$ in our experiments) and
$p_{\text{cens}}$ the probability of being censored ($=0.3$ in our
experiments).

    2. (b)

Let $\lambda_{n}=\exp(r_{n})/T_{0}$, for $n=1,\ldots,N$.

    3. (c)

Let $u_{n}=-\frac{\log a_{n}}{\lambda_{n}}$, for $a_{n}\sim\mathcal{U}(0,1)$
for $n=1,\ldots,N$.

    4. (d)

Let $q_{\text{cens}}=q_{1-p_{\text{cens}}}s(\boldsymbol{\mathbf{u}})$, where
$\boldsymbol{\mathbf{u}}=(u_{1},\ldots,u_{N})$ and $q_{\alpha}(\cdot)$ is the
$\alpha$-th quantile.

    5. (e)

Sample $t_{\text{cens}}\sim\mathcal{U}(\min_{n}u_{n},q_{\text{cens}})$.

    6. (f)

Let $\delta_{n}=\begin{cases}1,&\text{if $u_{n}\leq t_{\text{cens}}$}\\\
0,&\text{else}\end{cases}$, for $n\in\\{1,\ldots,N\\}$.

    7. (g)

Let $y_{n}=\begin{cases}u_{n},&\text{if $\delta=1$}\\\
t_{\text{cens}},&\text{else}\end{cases}$, for $n\in\\{1,\ldots,N\\}$.

  4. 4.

Normalize survival times $y$ to the interval $[0.001,1.001]$.

During inference, each model has access to pre-treatment covariates for the
treated group, i.e, $\boldsymbol{\mathbf{x}}$, with post-treatment outcome
data being hidden, and to pre-treatment covariates $\boldsymbol{\mathbf{x}}$,
as well as, (censored) survival times $y$ and event indicators $\delta$ for
the control group.

We model the likelihood $p(y|t,\boldsymbol{\mathbf{z}})$ for each model as a
Weibull distribution, $\mathcal{W}(\lambda,k)$, whose density is given as

$f(y;\lambda,k)=\frac{k}{\lambda}\left(\frac{y}{\lambda}\right)^{k-1}\exp\left(-\left(\frac{y}{\lambda}\right)^{k}\right)I(y\geq
0),$

with an indicator function $I$. For a survival function
$S(y|t,\boldsymbol{\mathbf{z}})=\int_{y}^{\infty}f(\tau;\lambda,k)\operatorname{d}\\!{\tau}$,
the likelihood is given as

$\displaystyle p(y|t,\boldsymbol{\mathbf{z}})$
$\displaystyle=f(y;\lambda(\boldsymbol{\mathbf{z}}),k)^{\delta}S(y|t,\boldsymbol{\mathbf{z}})^{1-\delta}$
$\displaystyle=\left(\frac{k}{\lambda_{t}(\boldsymbol{\mathbf{z}})}\left(\frac{y}{\lambda_{t}(\boldsymbol{\mathbf{z}})}\right)^{k-1}\exp\left(-\left(\frac{y}{\lambda_{t}(\boldsymbol{\mathbf{z}})}\right)^{k}\right)\right)^{\delta}\left(\exp\left(-\left(\frac{y}{\lambda_{t}(\boldsymbol{\mathbf{z}})}\right)^{k}\right)\right)^{1-\delta},$

where the shape parameter $k$ is a fixed hyperparameter ($k=1$ throughout our
experiments), and
${\lambda_{t}(\boldsymbol{\mathbf{z}})=\log(1+\exp(\boldsymbol{\mathbf{z}}^{\top}\beta_{t}))}$.
$\beta_{0}$ and $\beta_{1}$ are optimized via gradient descent together with
the remaining parameters.

See Section D.2 for an extended table corresponding to Figure 3 in the main
paper.

###  C.9 Computation of Performance Metrics

##### RMSE of CATE.

As specified in the main paper, the conditional average treatment effect is
defined as

$\mathds{E}\left[Y(1)-Y(0)|X=x\right]=\mu_{1}(x)-\mu_{0}(x).$

Given outcome information for treated ($t=1$) and untreated ($t=0$)
observations, all our methods and baselines can infer estimators
$\hat{\mu}_{0}(x)$ and $\hat{\mu}_{1}(x)$ to get CATE estimates at test time.
Given that the outcomes are synthetically generated, we know the true CATE for
each observation and can directly compute and report the RMSE.

We report two RMSEs. First the _within sample_ RMSE. During training time, for
observation $(\boldsymbol{\mathbf{x}}_{n},t_{n},y_{n})$, we only observe the
factual outcome $y_{n}=Y(t_{n})$. To estimate the CATE at test time, a model
needs to correctly predict the unseen counterfactual $Y(1-t_{n})$ as well. The
_out-of-sample_ RMSE refers to new covariates $\boldsymbol{\mathbf{x}}_{m}$,
i.e., the model has no never seen them, nor their factual outcome.

##### AE of ATT.

The average treatment effect for the treated is defined as

$\mathds{E}\left[Y(1)-Y(0)|T=1\right]=\mathds{E}\left[Y(1)|T=1\right]-\mathds{E}\left[Y(0)|T=1\right].$

In our specific setting, we assume that outcomes $y$ are only observed for the
control group, i.e., for $t=0$.

Once we are unblinded, we can estimate $\mathds{E}\left[Y(1)|T=1\right]$ by
the sample average $\frac{1}{N_{1}}\sum_{n=1:t_{n}=1}^{N}y_{n}$.

Building an estimator for $\mathds{E}\left[Y(0)|T=1\right]$ requires the
selection of a suitable subset of control observations that match the
characteristics of the treated observations. To avoid any statistical bias, we
require this matching to be based only on pre-treatment information, i.e.,
only on observed covariates for the single-arm trial data. After matching, we
estimate $\mathds{E}\left[Y(0)|T=1\right]$ via the empirical average
$\frac{1}{N_{J}}\sum_{j\in J}y_{j}$, where $J$ are the indices of the matched
patients, and $N_{J}$ the number of matched samples. After matching the
treatment outcome is unblinded and the ATT can be estimated and compared with
the synthetic true ATT.

##### Squared error for Time-to-event data.

For our semi-synthetic survival experiment, the goal is to compare estimated
hazard ratios. The evaluation proceeds in four steps: _(i)_ The true hazard
ratio is estimated by fitting a Cox proportional hazards regression model on
the factual and counterfactual survival curves of the single-arm group; _(ii)_
each model is fit using pre-treatment covariates for both groups and post-
treatment survival outcomes for the control group; _(iii)_ a subset of control
patients is selected and a second cox proportional hazards regression model is
fit on the survival curve for the selected subset as well as the, now
unblinded, observed factual outcomes of the treated group; _(iv)_ the squared
difference between the two hazard ratio estimates is computed and reported.

Cox models are fitted in R using the survival package by Therneau (2023).

##  Appendix D FURTHER EVALUATION

In this section we provide extended results on the experiments provided in the
main paper.

###  D.1 Runtime

##### IHDP.

Given the small size of the models and the small size of the IHDP data set,
models can be trained efficiently and fast on a modern CPU. Training a
deterministic model takes about 30 seconds and about twice as much for a
generative one. Differences between the individual deterministic/generative
approaches are too minuscule to be relevant. These numbers apply to all three
experimental setups, CATE estimation, ATT estimation, and survival analysis.
Depending on the experimental setup we ran 100–300 replications.

##### Real-world data.

Training on the real-world set takes about five minutes for a deterministic
model and about twice as much for a generative one.

###  D.2 Survival

We report results for the survival experiment in Table 4.

Table 4: _Survival._ Our approach improves both upon its deterministic as well
as its generative baselines in most variants.

Method Median Error Naive $0.394$ TARNet $0.162$ CFRNet $0.513$ SNet $0.486$
VAE $0.288$ CEVAE $0.173$ Ours $\mathbf{0.135}$ +I $0.180$ +sep $0.155$ +sep+I
$0.156$ +snet $0.331$ +snet+I $0.201$
_median over 100 random replications;
lowest median marked bold_

###  D.3 Full Results For Both Arms

We report results for all four settings on the IHDP data set in Table 5.

Table 5: _Outcome for both groups._ Extended results related to Table 1 in the
main paper.

_RMSE of CATE_ all+high all+reduced subset+high subset+reduced Method within
sample out-of-sample within sample out-of-sample within sample out-of-sample
within sample out-of-sample CFor
$0.556\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.000$
$0.5967\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.001$
$4.571\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.043$
$4.074\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.047$
$0.508\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.001$
$0.507\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.001$
$3.449\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.052$
$2.597\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.031$ PScov – – – – – – – – PSpca – – – – – – – – PSlat – – – – – – – –
SingleNet
$0.241\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005$
$0.328\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.009$
$0.863\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.013$
$0.839\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.030$
$0.250\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005$
$0.274\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.006$
$0.924\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.016$
$0.878\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.027$ TNet
$0.175\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.003$
$0.275\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008$
$0.210\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008$
$0.346\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.020$
$0.217\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004$
$0.246\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005$
$0.278\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.009$
$0.328\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.017$ TARNet
$0.177\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004$
$0.280\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008$
$0.220\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.009$
$0.343\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.019$
$0.221\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004$
$0.251\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005$
$0.272\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.009$
$0.333\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.019$ CFRNet
$0.171\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.003$
$0.279\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008$
$0.297\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.009$
$0.397\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.020$
$0.179\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.003$
$0.210\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005$
$0.279\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008$
$0.338\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.017$ SNet
$0.168\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.003$
$0.264\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008$
$0.228\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.009$
$\mathbf{0.299\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.018}$
$0.159\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.003$
$0.194\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005$
$0.211\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008$
$\underline{\mathbf{0.243\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.016}}$ VAE – – – – – – – – CEVAE
$0.182\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002$
$0.287\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$0.224\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008$
$0.348\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.019$
$0.223\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.003$
$0.256\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004$
$0.288\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.009$
$0.350\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.017$ TEDVAE
$0.176\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002$
$0.283\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$0.246\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008$
$0.344\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.019$
$0.202\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002$
$0.238\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004$
$0.293\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.009$
$0.336\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.018$ Ours
$0.152\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002$
$0.267\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008$
$0.174\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.006$
$0.304\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.019$
$0.161\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.003$
$0.202\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005$
$0.201\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$0.272\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.016$ +I
$0.141\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002$
${0.251\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008}$
$\underline{\mathbf{0.159\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005}}$
$\mathbf{0.293\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.017}$
$0.157\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.003$
$0.195\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005$
$0.194\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$0.262\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.016$ +sep
$0.151\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002$
$0.267\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008$
$0.184\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$0.310\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.019$
$0.162\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002$
$0.204\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005$
$0.209\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$0.278\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.016$ +sep+I
$0.143\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002$
$0.255\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$\mathbf{0.167\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.006}$
$\mathbf{0.294\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.017}$
$0.162\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.003$
${0.202\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005}$
$0.197\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$0.266\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.017$ +snet
$0.141\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002$
$0.256\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008$
$0.183\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$0.308\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.018$
$0.145\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002$
$\mathbf{0.184\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005}$
$\mathbf{0.185\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007}$
$\mathbf{0.253\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.017}$ +snet+I
$\underline{\mathbf{0.130\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002}}$
$\underline{\mathbf{0.242\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008}}$
$\mathbf{0.161\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005}$
$\underline{\mathbf{0.288\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.017}}$
$\underline{\mathbf{0.139\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002}}$
$\mathbf{0.182\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005}$
$\mathbf{0.181\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.006}$
$\mathbf{0.250\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.017}$ +snet+sep
$0.143\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002$
$0.254\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008$
$0.168\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.006$
$0.297\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.018$
$\mathbf{0.143\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002}$
$\mathbf{0.186\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005}$
$0.191\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$0.258\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.017$ +snet+sep+I
${\mathbf{0.131\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002}}$
${\mathbf{0.247\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008}}$
$\mathbf{0.160\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005}$
$\mathbf{0.290\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.017}$
$\mathbf{0.142\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002}$
$\underline{\mathbf{0.182\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005}}$
$\underline{\mathbf{0.180\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007}}$
${\mathbf{0.251\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.017}}$ +tedvae
$0.143\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002$
$0.257\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008$
$0.186\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$0.310\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.018$
$0.146\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002$
$\mathbf{0.184\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005}$
$\mathbf{0.187\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007}$
$0.258\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.019$ +tedvae+I
$0.142\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002$
$0.255\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008$
$0.174\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$0.305\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.019$
$\mathbf{0.142\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002}$
$\mathbf{0.182\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005}$
$0.191\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.006$
$\mathbf{0.253\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.016}$ +tedvae+sep
$0.143\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002$
$0.257\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008$
$0.192\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$0.317\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.019$
$0.146\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002$
$0.188\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005$
$0.200\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$0.263\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.017$ +tedvae+sep+I
$0.138\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002$
$0.250\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008$
$0.181\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.006$
$0.308\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.018$
$\mathbf{0.142\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002}$
$\mathbf{0.185\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005}$
$0.190\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$0.261\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.016$

_mean $\pm$ standard error over 300 random replications; statistically
significant best models marked bold; lowest mean underlined_

###  D.4 Full Results For Single-arms

We report results for all four settings on the IHDP data set in Table 6.

Table 6: _Outcome only for the control group._ Extended results related to
Table 1 in the main paper.

_AE of ATT_ all+high all+reduced subset+high subset+reduced Method within
sample out-of-sample within sample out-of-sample within sample out-of-sample
within sample out-of-sample CFor – – – – – – – – PScov
$0.115\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005$
$0.336\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.017$
$0.648\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.082$
$0.804\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.265$
$0.111\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005$
$0.373\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.019$
$0.691\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.059$
$0.702\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.063$ PSpca
$0.096\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005$
$0.300\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.014$
$0.556\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.056$
$0.811\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.265$
$0.115\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005$
$0.310\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.014$
$0.496\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.036$
$0.613\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.051$ PSlat
$0.084\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004$
$0.187\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.013$
$0.501\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.039$
$0.290\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.024$
$0.096\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.006$
$0.260\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.016$
$0.276\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.025$
$0.272\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.023$ SingleNet – – – – – – – – TNet – – – – – – – – TARNet
$0.043\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.003$
$0.130\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008$
$0.087\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008$
$0.108\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$\mathbf{0.075\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004}$
$0.223\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012$
$0.140\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.011$
$0.239\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.016$ CFRNet
$0.042\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002$
$0.131\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$0.321\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.015$
$0.327\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.016$
$0.080\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004$
$0.224\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012$
$0.345\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.016$
$0.370\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.017$ SNet
$0.043\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.003$
$0.137\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$0.111\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.010$
$0.178\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.013$
$\mathbf{0.075\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004}$
$0.224\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012$
$0.221\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.015$
$0.366\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.026$ VAE
$0.085\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004$
$0.266\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012$
$0.280\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.015$
$0.360\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.028$
$0.094\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004$
$0.271\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.014$
$0.264\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.014$
$0.427\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.022$ CEVAE
$0.052\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002$
$0.174\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.009$
$0.096\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.006$
$0.178\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.011$
$0.086\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004$
$0.234\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012$
$0.150\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.009$
$0.259\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.015$ TEDVAE
$0.052\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.003$
$0.179\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.009$
$0.108\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$0.188\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012$
$0.082\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004$
$0.233\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012$
$0.144\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.009$
$0.272\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.016$ Ours
$\mathbf{0.037\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002}$
$\mathbf{0.113\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007}$
$0.074\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.006$
$\mathbf{0.091\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008}$
$\mathbf{0.074\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004}$
$\mathbf{0.212\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.011}$
$\mathbf{0.114\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008}$
$\mathbf{0.190\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.016}$ +I
$\mathbf{0.038\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002}$
$\mathbf{0.122\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007}$
$\mathbf{0.055\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005}$
$\mathbf{0.080\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.006}$
$0.079\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004$
$0.229\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.011$
$\mathbf{0.110\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007}$
$\underline{\mathbf{0.176\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.014}}$ +sep
$\mathbf{0.037\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002}$
$\mathbf{0.111\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007}$
$\mathbf{0.063\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005}$
$\mathbf{0.083\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.006}$
$\mathbf{0.074\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004}$
$0.222\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012$
${0.117\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.009}$
$\mathbf{0.193\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.016}$ +sep+I
$\underline{\mathbf{0.035\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002}}$
$\mathbf{0.117\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008}$
$\mathbf{0.055\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005}$
$\underline{\mathbf{0.078\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.006}}$
$\mathbf{0.074\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004}$
$0.226\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012$
${\mathbf{0.104\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007}}$
${\mathbf{0.185\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.015}}$ +snet
$\mathbf{0.038\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002}$
$\mathbf{0.117\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007}$
$0.091\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008$
$0.169\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012$
$\mathbf{0.076\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004}$
$\mathbf{0.217\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.013}$
$0.194\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.014$
$0.322\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.020$ +snet+I
$\mathbf{0.038\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002}$
$\mathbf{0.120\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008}$
$0.094\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$0.159\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012$
$\mathbf{0.077\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004}$
$\mathbf{0.212\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.011}$
$0.190\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012$
$0.322\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.023$ +snet+sep
$0.040\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.003$
$\underline{\mathbf{0.109\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007}}$
$0.095\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$0.173\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012$
$\underline{\mathbf{0.071\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004}}$
$\mathbf{0.211\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.011}$
$0.179\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.013$
$0.316\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.020$ +snet+sep+I
$0.039\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.003$
$\mathbf{0.115\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007}$
$0.085\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.006$
$0.156\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.011$
$\mathbf{0.076\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004}$
$\mathbf{0.211\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.011}$
$0.196\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.015$
$0.292\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.020$ +tedvae
$\mathbf{0.039\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.003}$
$\mathbf{0.119\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007}$
$0.076\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.006$
$0.094\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$\mathbf{0.074\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004}$
$\underline{\mathbf{0.196\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.011}}$
$\mathbf{0.103\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008}$
$\mathbf{0.189\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.015}$ +tedvae+I
$\mathbf{0.036\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002}$
$0.124\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$\underline{\mathbf{0.054\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004}}$
$\mathbf{0.085\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.006}$
$\mathbf{0.072\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004}$
$0.234\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.013$
$\underline{\mathbf{0.102\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007}}$
$\mathbf{0.191\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.017}$ +tedvae+sep
$0.039\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002$
$0.125\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$0.067\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005$
$\mathbf{0.089\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007}$
$\mathbf{0.075\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004}$
$\mathbf{0.209\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.011}$
$\mathbf{0.112\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.009}$
$0.197\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.014$ +tedvae+sep+I
$0.039\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002$
$0.124\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$\mathbf{0.061\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004}$
$\mathbf{0.091\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007}$
$\mathbf{0.074\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004}$
$0.218\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012$
$\mathbf{0.104\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.008}$
$\mathbf{0.173\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012}$

_mean $\pm$ standard error over 300 random replications; statistically
significant best models marked bold; lowest mean underlined_

###  D.5 Missingness

We report results for various degrees of missingness in Table 7.

Table 7: _Missingness._ Increasing the amount of missing covariates decreases
performance as expected. Modeling this in our proposed approach recovers this
to a certain degree.

(a) outcome for both groups (all + high)
_RMSE of CATE_ none weak medium strong Method within sample out-of-sample
within sample out-of-sample within sample out-of-sample within sample out-of-
sample TARNet
$0.176\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.006$
$0.263\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012$
$0.304\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.006$
$0.314\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.010$
$0.462\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.011$
$0.463\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.019$
$0.531\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.014$
$0.607\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.025$ CFRNet
$0.170\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005$
$0.258\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012$
$0.302\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$0.307\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.011$
$0.457\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.011$
$0.452\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.019$
$0.527\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.014$
$0.600\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.025$ SNet
$0.172\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005$
$0.254\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.013$
$0.301\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$0.293\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012$
$0.465\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012$
$0.444\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.018$
$0.536\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.015$
$0.644\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.027$ CEVAE
$0.185\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005$
$0.277\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012$
$0.295\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005$
$0.322\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.010$
$0.438\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.009$
$0.482\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.017$
$0.511\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012$
$0.635\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.022$ TEDVAE
$0.176\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004$
$0.270\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012$
$0.297\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005$
$0.318\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.010$
$0.448\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.011$
$0.464\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.018$
$0.514\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.013$
$0.629\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.023$ Our
$0.152\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004$
$0.249\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012$
$0.297\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005$
$0.318\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.010$
$0.412\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.010$
$0.464\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.019$
$0.492\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.013$
$0.632\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.026$ +I
$0.144\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004$
$0.236\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.013$
$0.268\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.006$
$0.301\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.010$
$0.412\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.011$
$0.453\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.017$
$0.489\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.013$
$0.621\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.026$ +sep
$0.149\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004$
$0.245\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.013$
$0.267\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005$
$0.298\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.011$
$0.412\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.011$
$0.459\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.019$
$0.490\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.013$
$0.622\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.025$ +I+sep
$0.144\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004$
$0.243\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.013$
$0.268\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.006$
$0.300\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.011$
$0.411\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.011$
$0.452\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.018$
$0.488\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.013$
$0.629\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.026$ Our+mask – –
$0.224\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004$
$0.275\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012$
$0.331\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.006$
$0.348\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.013$
$0.430\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.010$
$0.518\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.019$ +I – –
$0.221\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004$
$0.264\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.011$
$0.327\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$0.356\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.014$
$0.415\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.010$
$0.506\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.021$ +sep – –
$0.223\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005$
$0.267\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.011$
$0.329\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.006$
$0.348\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.014$
$0.428\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.010$
$0.495\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.017$ +I+sep – –
$0.217\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004$
$0.257\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.011$
$0.326\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.007$
$0.345\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.013$
$0.426\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.010$
$0.492\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.018$
_mean $\pm$ standard error over 100 random replications_
(b) outcome only for the control group (subset + low)
_AE of ATT_ none weak medium strong Method within sample out-of-sample within
sample out-of-sample within sample out-of-sample within sample out-of-sample
TARNet
$0.138\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.016$
$0.172\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.018$
$0.216\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.016$
$0.238\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.024$
$0.289\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.019$
$0.196\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.020$
$0.448\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.030$
$0.266\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.027$ CFRNet
$0.415\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.031$
$0.414\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.036$
$0.405\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.029$
$0.428\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.031$
$0.318\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.023$
$0.335\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.029$
$0.413\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.027$
$0.313\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.030$ SNet
$0.177\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.020$
$0.240\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.024$
$0.248\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.021$
$0.294\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.027$
$0.305\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.021$
$0.291\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.026$
$0.434\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.026$
$0.390\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.035$ CEVAE
$0.241\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.021$
$0.274\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.024$
$0.268\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.021$
$0.353\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.028$
$0.289\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.021$
$0.302\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.025$
$0.397\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.028$
$0.312\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.033$ TEDVAE
$0.238\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.020$
$0.286\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.025$
$0.267\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.020$
$0.331\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.027$
$0.289\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.022$
$0.274\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.023$
$0.409\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.027$
$0.303\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.034$ Our
$0.106\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.015$
$0.160\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.018$
$0.192\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.014$
$0.215\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.024$
$0.271\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.021$
$0.246\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.020$
$0.442\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.028$
$0.291\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.027$ +I
$0.109\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.016$
$0.185\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.020$
$0.174\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.013$
$0.214\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.020$
$0.280\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.021$
$0.236\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.021$
$0.418\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.027$
$0.365\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.059$ +sep
$0.104\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.016$
$0.181\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.019$
$0.193\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.015$
$0.229\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.023$
$0.288\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.021$
$0.245\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.021$
$0.445\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.029$
$0.268\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.026$ +I+sep
$0.115\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.017$
$0.162\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.019$
$0.191\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.015$
$0.331\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.027$
$0.260\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.020$
$0.215\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.021$
$0.433\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.030$
$0.332\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.056$ Our+mask – –
$0.150\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.013$
$0.200\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.021$
$0.197\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.014$
$0.171\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.019$
$0.378\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.023$
$0.218\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.022$ +I – –
$0.137\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.014$
$0.171\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.020$
$0.176\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.014$
$0.178\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.018$
$0.337\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.023$
$0.239\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.025$ +sep – –
$0.143\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012$
$0.176\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.019$
$0.203\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.015$
$0.165\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.018$
$0.375\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.024$
$0.230\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.022$ +I+sep – –
$0.138\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.013$
$0.175\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.019$
$0.179\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.014$
$0.186\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.019$
$0.337\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.021$
$0.228\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.027$
_mean $\pm$ standard error over 100 random replications_

###  D.6 Matching Metrics

We evaluate two different scenarios on a subset of the methods discussed. ATT
estimation in _(i)_ the _all+high_ scenario, and _(ii)_ in the _subset+low_
scenario. Adding distributional information within the mean-field assumption
of $q(\boldsymbol{\mathbf{Z}})$ provides little benefit and tends to even hurt
performance. Matching via a propensity score estimator in the latent space is
never competitive in our setting. The results following the current mean-field
assumption are summarized in Table 8.

Table 8: _Varying metrics for matching._ We compare relying on the Euclidean
metric using the posterior mean, to using distributional metrics or propensity
score (PS) matching in the latent space.

_(i) ATT estimation for _all + high__
_AE of ATT_ within sample out-of-sample Method Euclidian Wasserstein Hellinger
PS Euclidian Wasserstein Hellinger PS VAE
$\underline{0.085\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004}$
$\underline{0.085\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.004}$
$0.112\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.006$
$0.116\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.005$
$\underline{0.266\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012}$
$\underline{0.266\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.012}$
$0.388\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.021$
$0.315\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.015$ CEVAE
$\underline{0.052\color[rgb]{0.2,0.2,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.2,0.2}\pgfsys@color@gray@stroke{0.2}\pgfsys@color<EMAIL_ADDRESS>0.002}$