#  Causal Deep Learning
Encouraging impact on real-world problems through causality

Jeroen Berrevoets<EMAIL_ADDRESS>
DAMTP, University of Cambridge Krzysztof Kacprzyk<EMAIL_ADDRESS>
DAMTP, University of Cambridge Zhaozhi Qian<EMAIL_ADDRESS>
DAMTP, University of Cambridge Mihaela van der Schaar<EMAIL_ADDRESS>
DAMTP, University of Cambridge
The Alan Turing Institute

###### Abstract

Causality has the potential to truly transform the way we solve a large number
of real-world problems. Yet, so far, its potential largely remains to be
unlocked as causality often requires crucial assumptions which cannot be
tested in practice. To address this challenge, we propose a new way of
thinking about causality-– we call this causal deep learning. Our causal deep
learning framework spans three dimensions: (1) a structural dimension, which
incorporates partial yet testable causal knowledge rather than assuming either
complete or no causal knowledge among the variables of interest; (2) a
parametric dimension, which encompasses parametric forms that capture the type
of relationships among the variables of interest; and (3) a temporal
dimension, which captures exposure times or how the variables of interest
interact (possibly causally) over time. Causal deep learning enables us to
make progress on a variety of real-world problems by leveraging partial causal
knowledge (including independencies among variables) and quantitatively
characterising causal relationships among variables of interest (possibly over
time). Our framework clearly identifies which assumptions are testable and
which ones are not, such that the resulting solutions can be judiciously
adopted in practice. Using our formulation we can combine or chain together
causal representations to solve specific problems without losing track of
which assumptions are required to build these solutions, pushing real-world
impact in healthcare, economics and business, environmental sciences and
education, through causal deep learning.

###### Contents

  1. 1 Introduction
  2. 2 CDL in 3 dimensions: Structural, Parametric, and Temporal
    1. 2.1 Preamble: distribution factorization
    2. 2.2 The structural scale and rung 1.5
    3. 2.3 The parametric scale
    4. 2.4 The temporal scale
    5. 2.5 Characterizing CDL models using the scale
  3. 3 Constructing and testing model pipelines
    1. 3.1 Transition in the knowledge
    2. 3.2 Building CDL model pipelines
    3. 3.3 Non-matching assumptions
  4. 4 Treatment effects: An illustrative example using CDL
    1. 4.1 Treatment effects: the CDL description
  5. 5 Conclusion and guidelines for CDL papers
  6. A Real-world applications
  7. B Categorising CDL methods
    1. B.1 Comparing methods
    2. B.2 Developing methods
    3. B.3 Using methods
    4. B.4 Beyond supervised learning
      1. B.4.1 Forecasting and temporal supervised learning
      2. B.4.2 Bandit algorithms and reinforcement learning
      3. B.4.3 Generative modelling

##  1 Introduction

Causality holds the promise to transform the way we solve a large number of
real-world problems [1, 2]. Unlocking this promise amounts to adapting
causality to the real world with its highly complex, unstructured, and
abundance of data. Beyond rich datasets, causal inference methods typically
rely on diverse and extensive prior knowledge about the causal nature of the
underlying system, many of which are not available or even testable in the
application domain [3]. This has led to two consequences. Firstly,
practitioners may shy away from adopting causal inference in general because
the available causal knowledge is not sufficient, let alone complete.
Secondly, the emphasis on causal knowledge may overshadow other important
considerations such as the statistical, functional, or temporal properties.
These consequences may lead the practitioner to under-utilize these methods
which ultimately results in sub-optimal solutions.

We want to encourage researchers and practitioners to take causality to its
next step: real-world impact. For this, we recognise the need to integrate
causal inference with the sophisticated modelling capabilities of deep
learning. This integration is particularly vital in real-world domains where
understanding causal relationships can lead to better decision-making and
predictions.

In this article, we introduce a new framework which enables pragmatically
adopting causality ideas to solve real-world problems: Causal Deep Learning
(CDL). Informally, CDL leverages partial causal knowledge (among some and not
necessarily all variables of interest), and quantitatively characterises the
functional form (among variables of interest) and the time-evolution of the
variables to provide significant insights to researchers and decision-makers.
The reason why the above properties are important is two-fold: (i) We need a
good way to match a model with prior knowledge in a complex system, this
should allow information of any type (be it no information, partial
information, or full information); and (ii) We need a good way to evaluate a
solution. This latter point is important in any real-world setting, especially
settings where we rely on modelling solutions to support impactful decisions.
Inherently being built on a strong set of assumptions, model validation is a
particularly tricky endeavour in causality and one which CDL can help us solve
[4].

Illustrative examples. Let us explore four diverse domains where causal deep
learning, characterised by its focus on causal structure, functional
relationships among variables of interest, and time, to understand the need
for this new way of thinking.

First, consider the medical and healthcare domain, where determining the
effectiveness of medications is paramount. Traditional models from
pharmacology or physiology often struggle to capture the complexity of drug
interactions and patient responses. To make progress in studying the effects
of medications on health outcomes, CDL might focus on the causal relationship
between the drug dosage and patient recovery rate while controlling for other
variables like age, gender, and pre-existing conditions. In such a case, CDL
would not assert complete causality of overall health factors but rather
isolate the drug’s impact. By employing only a partial causal structure, CDL
can model the nuanced relationship between drug dosage and patient recovery,
considering both linear and non-linear effects. Moreover, by incorporating
temporal dynamics, CDL may unravel how medication effectiveness evolves over
time, providing critical insights for personalized medicine.

In economics and business, the relationship between interest rate changes and
consumer spending is a classic example of a complex causal interaction.
Traditional econometric models might fail to capture the intricacy of this
relationship. CDL, with its ability to handle non-linear and high-dimensional
data, can provide a more robust understanding. By adopting a pragmatic
approach, CDL could enable the investigation of partial causality between
interest rate changes and consumer spending. While acknowledging that other
factors like education, employment rates, inflation, and economic policies
also impact spending, the focus here is on understanding how variations in
interest rates specifically influence consumer behaviour. By considering the
temporal dimension, CDL may also uncover lag effects, where changes in
interest rates take time to manifest in consumer behaviour, a crucial insight
for policymakers.

Environmental science, especially the study of climate change, presents
another compelling case for causal deep learning. An interesting research
aspect might be to explore the causal relationship between carbon dioxide
emissions and global temperature increase, acknowledging that other factors
like deforestation and solar radiation also play roles in climate change.
Using such a pragmatic CDL approach which requires only partial causal
structure would allow researchers to isolate and understand the specific
impact of CO2 emissions. In addition, the relationship between CO2 emissions
and global temperature increase is not straightforward and likely non-linear.
Here, causal deep learning can capture the complexity of this relationship
beyond traditional linear models. Additionally, understanding how this
relationship evolves over time is critical for predicting future climate
patterns and for formulating effective environmental policies.

Lastly, in the education sector, the impact of classroom size on student
performance is a topic of ongoing debate. While simple linear models might
suggest a straightforward inverse relationship, the reality is likely more
nuanced. Causal deep learning can help in identifying not just whether, but
how class size impacts student performance, including potential threshold
effects or non-linear dynamics. Furthermore, by examining how this
relationship changes over an academic year or across different educational
stages, deeper insights can be gained into effective educational planning and
resource allocation.

In these and other real-world domains, causal deep learning can be used as a
powerful framework to make progress. Unlike current causal models in machine
learning, CDL allows for a nuanced understanding of causal relationships
without requiring complete causal knowledge, learns in a data-driven manner
the parametric form of the relationships among the variables of interest using
powerful deep-learning models, and importantly, considers the dimension of
time. This approach can uncover insights that traditional models might miss,
leading to more effective interventions and policies across various sectors.
The exploration of causal deep learning in these diverse real-world domains
not only underscores its versatility but also highlights its potential to
revolutionize how we understand and interact with the world around us.

With our detailed definitions, we can better align a problem with the (causal)
solution. While research in causality has a long history of listing out the
various assumptions necessary to identify causal effects, these assumptions
often do not map easily into practice. Furthermore, deep learning has no such
history, typically because most deep learning architectures are assumed to be
non-parametric. However, not all problems require the enormous flexibility of
deep neural networks, but may still be non-linear. Our proposal encompasses
all the above in hopes of aiding the correct and useful adoption of causal
deep learning methods.

##  2 CDL in 3 dimensions: Structural, Parametric, and Temporal

CDL aims to provide practical guidelines that yield more informed and robust
models to solve real-world problems. We introduce these guidelines under two
main axes: (i) the structural scale concerning the links between variables,
described in section 2.2; and (ii) the parametric scale concerning the shape
of the functions that model links between variables, described in section 2.3.
These two axes describe the first two (of three) directions in which CDL is
best described. The third is time, which due to its special meaning in
causality (causes precede effects), is worthy of a completely separate
discussion. As we will show, CDL’s guidelines help us to combine models, build
pipelines, and test settings of the data-generating process to avoid
unexpected bias and errors.

###  2.1 Preamble: distribution factorization

Before explaining the structural and parametric scale, we revisit the
fundamental concept of distribution factorization, which states how a
distribution can be factorised as a series of functions.

######  Definition 1 (factor).

Let $\mathbf{X}$ be a set of random variables. We define a factor $f$ to be a
function from $\text{Val}(\mathbf{X})\to\mathbb{R}$. A factor is non-negative
if all its entries are non-negative. The set of variables $\mathbf{X}$ is
called the scope of the factor and denoted $\text{Scope}[f]$ [5, Definition
4.1].

In its most general form, a joint distribution admits a factorisation as,

$p(\mathbf{X})=\frac{1}{Z}\prod_{i\in[I]}f_{i}(\mathbf{X}^{(i)}),$ (1)

where $Z$ is a normalising constant, and one factor $f_{i}$ takes a set of
variables, $\mathbf{X}^{(i)}\subseteq\mathbf{X}$, as arguments [5].

Notice that in eq. 1 we iterate over $I$ factors. However, the number of
factors and each $\mathbf{X}^{(i)}$ is, at this stage, not yet fully
determined. A trivial example of such a factorisation is the decomposition of
a joint distribution into conditional distributions:
$p(A,B)=f_{1}(A,B)f_{2}(B)$ with $f_{1}(A,B)=p(A|B)$ and $f_{2}=p(B)$. As
another trivial example, one could have a factorisation which is comprised of
the least amount of factors: $p(\mathbf{X})=\frac{1}{Z}f(\mathbf{X})$.

Factorization plays a central role in CDL because it captures both the
structural and parametric properties of the underlying system. The structural
properties manifest in the scope of each factor $\mathbf{X}^{(i)}$ whereas the
parametric properties manifest in the functional form of $f$. Although
distribution factorization is originally a statistical concept, it also has a
close connection with various causality frameworks, such as structural causal
models. Therefore, we use it as a starting point to introduce the scales.

###  2.2 The structural scale and rung 1.5

Crucial to causal deep learning is the ability to leverage knowledge about the
relationships between variables, i.e., which variables are placed in the same
factor $f$. Conveniently depicted as graphs, we can order these relations on a
scale from less detailed to more detailed – this gives the structural scale,
which is central to our map of causal deep learning.

fig. 1 depicts the structural scale, which ranges from less structure, to more
structure. The structural scale governs information about the statistical or
causal relationships in a system. Essentially, it quantifies how much we know,
expressed in graphical structures. Below, we discuss each level using an
illustrative example of three variables: smoking, denoted as $S$; cancer,
denoted as $C$; and death, denoted as $D$.

Level 1 – no structure. At the lowest end of our scale we have structures that
exhibit zero knowledge, i.e., $p(\mathbf{X})=\frac{1}{Z}f(\mathbf{X})$. These
structures connect all variables because all variables appear in the same
factor. An example of such a structure is denoted as,

$S$$C$$D$(unknown). (2)

While technically a structure, we have no idea about the potential
relationships these variables have concerning each other. Having a fully
connected structure such as in eq. 2 means that we assume potential dependence
between each variable. Assuming a fully connected structure is not a strict
assumption. In our example above, the structure in eq. 2 states that “smoking,
cancer and death” are possibly related, which is a very general statement
indeed.

unknownNo tests needed$S$$C$$D$plausibleCan be
tested!$S$$C$$D$causalUntestable$S$$C$$D$less structurelots of structure

Figure 1: The structural scale. The first axis on the map of causal deep
learning is the structural scale which categorises different structure types.
The two extremes are non-informative structures or a complete causal graph.
Note that the structural scale makes no parametric assumptions on these
structures (parametric assumptions are the focus of the parametric scale in
fig. 2).

That is because relatedness is a flexible term. While having an edge may
appear to be a strict statement, the edge does not express the level at which
two variables are related. In fact, the absence of an edge is a much stronger
assumption which we discuss next.

Level 2 – plausibly causal structures. Let us assume now that we know a little
more: cancer is the reason why smoking is related to death. Statistically, we
can express this assumption as $S\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}D|C$. In Level 2, we assume statistical
independence. Independence is not directly the same as assuming some
structural information, however, it can be expressed as such.

The easiest thing we can do to reflect $S\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}D|C$ is to explicitly remove the edge $S$$D$
from eq. 2 as we have below in eq. 3,

$\leavevmode\hbox to55.16pt{\vbox
to8.83pt{\pgfpicture\makeatletter\hbox{\hskip 27.11644pt\lower-4.41666pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$S$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$C$}}
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}\pgfsys<EMAIL_ADDRESS>(3)

Crucially, however, there exist more structures which reflect
$S\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}D|C$. Consider the three structures below,

$S$$C$$D$

,

$S$$C$$D$

,

$S$$C$$D$

,

which all model the same independencies.

For the sake of our discussion, let us assume $S\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}D|C$ is correct111If cancer renders smoking
and death independent, we imply that cancer is the only path from smoking to
death, which is known to be false [6]. For example, smoking may result in
chronic obstructive pulmonary disease (COPD) or ischemic heart disease, which
can also lead to premature death. However, for illustration, we assume it is
true.. As (in)dependence is a statement of statistical association, we cannot
assume that the structures based on it are causal. For example, $S$$C$$D$,
implies that dying impacts whether or not the deceased is a smoker, which
cannot be true.

The type of structure of interest in Level 2, are probabilistic graphical
models (PGMs) [5]. These include structures such as directed acyclic graphs
(DAGs), undirected graphs, or even factor graphs. Depending on the set of
independence statements one wishes to assume, one may express them as a
certain type of PGM.

For example, if one wishes to assume $S\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}D$ while also assuming
$S\not\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}D|C$, we best use a DAG:

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}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$S$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$C$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$D$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$ (4)

which is often called a “collider” structure. This collider structure (and
more importantly the (in)dependence statements it implies) can only be
modelled with a DAG.

Being able to model independence is a clear step up from Level 1, where we
assumed nothing at all. While independence is a statement of statistical
association (i.e. it is bidirectional and not causal), they do pose
constraints for a later causal model. As we can check (in)dependence in data,
a recovered causal model should respect (or even explain) these (in)dependence
statements. Hence, the above structures are plausibly causal.

In level 2 we have no consensus on one graph in particular, only about
dependence and independence (predominately statistical concepts). To
illustrate Level 2 on our structural scale in fig. 1 above, we denote these
plausible structures as,

$S$$C$$D$(plausible). (5)

where we combined all possible DAGs that respect, $S\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}D|C$ [7]. Applying the rules of d-separation
[8].

While Level 2 includes all types of PGMs, we chose to illustrate Level 2 using
DAGs only, as DAGs are the structure of choice when modelling causality [9].
In eq. 5 all of the structures are DAGs, but only one of them is causal
(likely to be the first one).

Level 3 – causal structure. That leaves assuming the exact causal structure as
opposed to the many plausible causal structures in Level 2. With eq. 6 we say
that smoking causes cancer which causes death. Such knowledge translates into
the directed structure,

$S$$C$$D$(causal). (6)

Which was indeed one of the plausible DAGs in Level 2 (eq. 5).

Comparing eq. 3 with eq. 6 may lead to believe there is little difference
between either end of the structural scale– nothing could be further from the
truth [1, 10]! Rather, the step from association to causation is very large
and is a central argument in causality. So large in fact that we recognise
several intermediary structures.

An important realisation is that each level of information further specifies
the environment, i.e. with information, we restrict possible interpretations
and data distributions. For example, from eqs. 2 and 3 we expect $S$ to change
if $C$ is changed as both variables are associated with each other. However,
eq. 6 tells us this is not necessarily the case: patients do not automatically
start smoking when diagnosed with cancer. However, the reverse is true; when a
cancer-patient stops smoking they have a higher chance of remission [11].

“Rung 1.5” Pearl’s ladder of causation [1, 10] ranks structures in a similar
way as we do, i.e., increasing a model’s causal knowledge will yield a higher
place upon his ladder. Like Pearl, we have three different levels in our
scale. However, they do not correspond one-to-one.

In particular, we find that our Level 2 is not well represented in Pearl’s
ladder of causation222Similarly, his counterfactual rung is not represented in
our structural scale.. We reason that our Level 2 does encode some prior
knowledge into a model, which is more than encoding no prior knowledge at all.
Yet, both would be categorised under Pearl’s first rung. Of course, we
recognise that the models categorised under Level 2 have no embedded causal
knowledge. While Level 2 does not reach the level of a rung 2 model (but Level
3 does), we consider a Level 2 model to correspond with a hypothetical “rung
1.5”.

At the structural level, we find that Pearl’s rung 3 (counterfactuals) is the
same as rung 2 (interventional). Of course, rung 3 goes beyond structure, in
that it also requires functional knowledge. Instead, we dedicate an entirely
new scale to functional knowledge as (like structure) we find there are
different levels of functional knowledge one can assume (or learn). We
introduce this additional scale in the following subsection.

###  2.3 The parametric scale

non-param.$f(\mathbf{X},\mathbf{U}_{\mathbf{X}})$fully
known$f^{*}(\mathbf{X},\mathbf{U}_{\mathbf{X}})$Noise
models$g(\mathbf{X})+\mathbf{U}_{\mathbf{X}}$fully param.$g(\mathbf{X})$No
tests neededCan be tested!Untestableno assumptionsmany assumptions

Figure 2: The parametric scale. A second axis in the map of causal deep
learning. The parametric scale logs the type of assumptions made on the
factors of the assumed distribution or model. The extremes are no assumptions
at all (leaving completely non-parametric factors), or a fully known model.
The parametric scale further discerns assumptions on the way noise interacts
in the system (${\epsilon}$) and the functional shape of the system’s
factorisation.

Our goal in this section is to organise the types of assumptions one can make
at the parametric level of these factors, contrasting the structural
assumptions one makes about dependence (as discussed previously in section
section 2.2). An important differentiating property of these assumptions is:
the strictness expressed in the size of the accompanying hypothesis class
[12], and whether or not one needs to test their validity outside what is
possible given the available observational data. We summarise the assumption
types in fig. 2 and introduce each separate component below.

Level 1 – non-parametric. Level 1 of the parametric scale encompasses the
least strict assumption: namely, there exists a factorisation such as eq. 1.
Beyond that, level 1 assumes no specific functional form of its
factorisations, $f_{i}$.

Note here that assumptions such as iid data or value assumptions (such as
numerical or ordinal variables) fall outside the scope of what is understood
as a parametric scale above.

Level 2 – noise models. The next level in our parametric scale is to first
make assumptions on the composition of the set of random variables,
$\mathbf{X}$. In particular, we assume that $\mathbf{X}$ can be separated in
noise terms or exogenous variables $\mathbf{U}$ and variables
$\mathbf{X}\setminus\mathbf{U}$ where each element in $\mathbf{U}$ corresponds
with exactly one element in $\mathbf{X}\setminus\mathbf{U}$ and
$|\mathbf{X}|=2|\mathbf{U}|$.

The above assumption has implications for the factors. Specifically, if a
factor takes $X\in\mathbf{X}\setminus\mathbf{U}$ as argument, it will always
take $U_{X}\in\mathbf{U}$ (which is the corresponding noise variable of $X$)
as argument also. Note that this separation alone is a structural assumption
[9].

As a parametric scale, the Level 2 further states how the noise variables
$\mathbf{U}_{\mathbf{X}^{(i)}}$ are incorporated in $f_{i}$ [13, 14, 15]. In
the case of causal structural equation models (where a random variable results
from a function of that variable’s causal parents), we have a common example
in additive noise:

$\mathbf{X}=g(\textrm{Pa}(\mathbf{X}))+\mathbf{U}_{\mathbf{X}}.$ (7)

In short, eq. 7 assumes we can decompose a factor into the noise,
$\mathbf{U}_{\mathbf{X}}$, and some (non-parametric) function, $g$. While how
$\mathbf{U}_{\mathbf{X}}$ interacts in $f$ is fixed, $g$ is still non-
parametric and therefore only level 2 of this scale.

Many prior works in causal discovery make assumptions of this level[16]. For
example, in Hoyer et al. [15] $g$ remains non-linear (through the use of
neural networks) but $\mathbf{U}_{\mathbf{X}}$ is assumed additive independent
(as above) resulting in a clear level 2 assumption.

Assumptions on the noise distribution (Gaussian, Uniform), which fall under
the umbrella term of parametric assumptions, can be tested using statistical
tests. These types of distributional assumptions can, for example, be tested
using the classical (non-parametric) Kolmogorov-Smirnov (K-S) test [17], or a
Jarque-Bera test for normality [18].

Level 3 – fully parametric. Naturally, the next level is to make a parametric
assumption on $g$ in eq. 7. These assumptions vary from making linear
assumptions, to $n$ times differentiable, to many more types of assumptions on
the functional form of $g$ and consequentially $f$. Note that because we make
an assumption on $g$ (and not $f$ directly), we automatically assume that $f$
can be decomposed in $g$ and noise, i.e. Level 3 subsumes assumptions from
Level 2.

Less parametric assumptions imply a larger model hypothesis class. For
example, having linear assumptions allows much fewer possible factors than a
non-linear class of assumptions. It has been shown that neural networks333In
particular with a ReLU activation. have a connection to non-parametric
regression, leading to high-performance models [19]. As such, making a
parametric assumption is typically done in favour of easier regression, but at
the cost of less general model classes.

The testability of this type of parametric assumption largely depends on which
assumption we are making. Testing based on, for example, a Savitzky-Golay
filter [20] can be effective in testing a certain type of parametric
assumption (such as a linear assumption). In the specific case of linearity,
the largest CUSUM test [21] in combination with a K-S test can be used for
(in)validation.

Level 4 – fully known factors. Rather than making assumptions on the
functional class of the factors, some methods assume full access to a known
set of factors which we term $f^{*}(\mathbf{X},\mathbf{U}_{\mathbf{X}})$. This
is of course a very strict assumption, resulting in the final level of the
parametric scale.

To validate this type of function, one cannot rely on a standardised
statistical test using only observational data [4]. As is the case in fields
such as theoretical physics, validation of a specific function needs to happen
experimentally (in a clinical trial, or by an experimental physicist).

With level 4 we conclude our parametric axis. This parametric axis has been
the main focus of investigation of deep machine learning and statistics. The
parametric scale is also important in causality (specifically econometric
approaches to causality [3]).

Significance of the parametric scale. We illustrate the benefit of the
parametric scale with a concrete example—causal structure discovery models,
which attempt to uncover causal graphs from observational data. Previously,
all causal structure discovery methods are categorised the same in the ladder
of causation [10]. By introducing a parametric scale, CDL allows for better
differentiation between methods. For example, comparing Zheng et al. [22] with
the later Zheng et al. [23] using the latter of causation, implies that Zheng
et al. [23] wrongly claimed an improvement on Zheng et al. [22] as they both
transition from having no structural knowledge to plausible structural
knowledge. Of course, only when we introduce the parametric scale it becomes
clear that the difference does not lie in structure, but in their parametric
assumptions.

###  2.4 The temporal scale

We explicitly differentiate between temporal and static models because the
presence of temporal structure may lead to alternative considerations in the
causal setting. For example, a typical assumption in the temporal domain is
that causes precede effects [24]. While we do not express any opinion on
whether or not this makes things harder or easier, we do believe it is
important to differentiate to properly compare methods. Furthermore, when
using our guideline to search for an appropriate method from a practical point
of view, it makes sense to search across static methods for a static problem,
and temporal methods for a temporal problem.

While, in our writing, we consider a model temporal or not to be binary, we
wish to emphasise the complexity associated with time. Especially in settings
of causality, we find time to be an incredibly complex and interesting topic.
Consider, for example, fig. 3 which illustrates how a simple time indicator
may make a simple four-variable problem a situation with a complex associated
DAG. Interestingly, time can also help in causality: having $X_{t=1}\to
X_{t=2}$ makes much more sense than $X_{t=2}\to X_{t=1}$ because a cause
cannot manifest later than an effect. The latter brings up an interesting
thought. If there are causes in a static data setting, they too should
manifest before and effect. Hence, having static data with an associated
causal DAG, always implies there is a temporal character to the data. As such,
we define time here as a setting in which we observe the same variable (such
as $X,U,A,Y$ in fig. 3) more than once.

While fig. 3 is illustrating a discrete-time setting, the above can also be
generalised to a continuous time setting. Peters et al. [25] connect the
dynamical systems literature to structural causal models. Modelled as a
dynamical system (such as ODEs), we allow the same variable to be observed
multiple times while also being dictated by a structural causal model.
Searching for ODEs then connects dynamical systems to a causal structural
equation, which translates to an FK model on our parametric scale in section
2.3.

Now that we have described the three-dimensional scale for characterizing
knowledge and assumptions, we will use this scale to characterize, construct,
and test CDL model pipelines.

$X_{t=1}$$X_{t=2}$$Y_{t=1}$$Y_{t=2}$$A_{t=1}$$A_{t=2}$$U_{t=1}$$U_{t=2}$

Figure 3: Confounding over time. We borrow the above DAG from Bica et al. [26]
in two time steps Robins et al. [27]. This figure shows how time complicates
matters. Specifically, because of time, the same variable ($X$) has different
causal edges associated as a function of $t$. See for example how $X_{t=1}$
causes $U_{t=1}$, but not $U_{t=2}$.

###  2.5 Characterizing CDL models using the scale

Using our scales in figs. 1 and 2, we provide an organisational structure
which allows comparing and documenting CDL methods. Each CDL method can be
characterized by answering the following three questions:

  * •

What is the a priori knowledge (or assumptions) according to the structural
and parametric scale?

  * •

Is the model static or temporal?

  * •

What is the a posteriori knowledge after applying the model to data?

This framework is illustrated in Figure 4.

Unknown (U)Plausible (P)Causal (C)(NP)Non-param.(NM)Noise
Models(Pa.)Param.(FK)Fully knownA Priori KnowledgeTimeStaticUPCNPNMPa.FKA
Posteriori Knowledge properties of the data given to a model properties of the
model’s representation or output structural scale (fig. 1) parametric scale
(fig. 2) Example model

Figure 4: Combining axes. We can combine our axes to evaluate a model’s input
and output (or representation). As an example, we show a fictitious method
which assumes no structure but does make an additive noise assumption;
operates in the temporal domain; and provides a truly causal representation
under the same additive noise assumption. Doing so allows easy examination of
a method on many properties at once.

A priori knowledge. The a priori knowledge field (leftmost grid in fig. 4)
reflects the structural and parametric knowledge that the practitioner
possesses or is willing to assume before applying the CDL method to data.
Here, the structural scale measures how much we already know about which
variables are dependent; whereas the parametric scale measures how these
variables are dependent.

Time. This is a binary indicator of whether the CDL method (and the dataset)
is static or temporal.

A posteriori knowledge. The a posteriori knowledge field reflects the
structural and parametric knowledge that one would obtain after applying the
CDL method to the data. It is evaluated using the same scales as the a priori
knowledge, which helps to document how much additional knowledge the CDL
modelling process adds.

With our framework, we can fingerprint each CDL method based on its location
on the map. As we will show in the next sections, this framework not only
allows us to categorize model classes but also to construct and test new CDL
model pipelines.

##  3 Constructing and testing model pipelines

###  3.1 Transition in the knowledge

We recognize that a CDL method’s a priori and a posteriori knowledge may be
different because the method may discover additional knowledge from the data.
We use the term Transition to describe this situation.

######  Definition 2 (Transition).

When a method’s a priori and a posteriori knowledge is different, either
structural (section 2.2) or parametric (section 2.3), we say that a model
invokes a transition in the knowledge.

A transition may occur either in the structural scale or parametric scale (or
both).

Structural Transition. Causal learning is a prime example of a structural
transition, where the input structure (typically unknown) is at a lower level
than the output (a causal graph). This applies to various causal structural
learning methods that learn (or discover) a causal graph from data.

Parametric Transition. Transition on the parametric scale often involves
testing functional form (e.g. linearity) or the distribution of the noise
terms (e.g. Gaussian noise). Symbolic regression methods, which try to
discover closed-form equations from data, would also invoke a transition in
the parametric scale.

In our appendix B we categorise existing models in supervised learning,
forecasting, bandit algorithms, reinforcement learning, and generative
modelling. In none of these methods we find transitions that end up in a fully
known (FK) structural equation model. Clearly this type of transition is an
interesting avenue to explore further in research.

In our treatment effects example in section 4 we perform a similar exercise
and find one model [28] (to our knowledge) that does transition in the
parametric scale with a FK equation model as posteriori knowledge.

No Transition. In general, Causal reasoning, such as counterfactual prediction
and treatment effect estimation, constitutes no transition as the algorithms
aim to estimate the causal effect rather than discovering structural or
parametric properties. Examples of this include bandit algorithms such as
Lattimore et al. [29], which we discuss further using our map of CDL in
appendix B.

Unknown (U)Plausible (P)Causal (C)(NP)Non-param.(NM)Noise
Models(Pa.)Param.(FK)Fully known1st CascadeUPCNPNMPa.FK2nd
CascadeUPCNPNMPa.FK3rd CascadeCausal learning [30]Causal debiasing
[31]Generate fair data

Figure 5: Building pipelines. As the input and representation in our map (fig.
4) is defined on the same domain– the structural scale $\times$ the parametric
scale –we can use the representation of a model as input for a subsequent
model. Documenting these model combinations (i.e. pipelines) is easy with our
scales as we can simply cascade multiple diagrams after one another as we have
done above. Doing so shows us that, to truly generate fair data, one has to be
willing to make either strong assumptions (such as having access to causal
DAGs, or the necessary assumptions to discover them), or use alternative
models in their pipeline that make testable assumptions.

###  3.2 Building CDL model pipelines

Very often a standalone CDL model cannot solve the practical problem because
there is a gap between a priori knowledge of the practitioner and what is
required by the model. With CDL, we can identify the right type of transition
and construct model pipelines to bridge the knowledge gap. We consider the
example below (visualized in Fig 5).

van Breugel et al. [31] proposed a GAN which generates fair synthetic data by
exploiting a known causal graph. While useful, it is hard to apply the
algorithm in practice as a fully known causal graph is seldom available in
many application domains.

From the map, it is clear that the practitioner needs to incorporate a
transition model as van Breugel et al. [31] (illustrated in green) cannot
function with the practitioner’s insufficient knowledge of the causal
structure. As such, we first learn a causal graph, using for example Peters et
al. [30]. With the two-stage model pipeline, we can match the a priori
knowledge of the practitioner.

Naturally, our illustration is quite straightforward and one could likely
arrive at the same solution without our map. However, often real-world
scenarios can be much more complicated and require more than two cascading
models to solve a particular problem [32, 33]. Imagine problems such as data
imputation, transformation, or life-long learning. All of these intermediary
steps have to take into account the assumptions made on each model present in
the pipeline. Building on a set of non-cascading assumptions can have
detrimental consequences as there may be unwanted bias creeping into your
solution.

For example, in the above scenario, the observational data used to learn a
causal graph may have missing values. Depending on which imputation strategy
we use, we may introduce additional parametric assumptions into our pipeline.
Furthermore, depending on which mechanism governs the missingness patterns, we
may have to make additional structural assumptions before imputation can even
begin [34, 35, 36]. Making such structural assumptions, before learning a
graph using Peters et al. [30] would make our solution for the practitioner’s
problem invalid once again.

###  3.3 Non-matching assumptions

In our example in fig. 5, we have not addressed the fact that the used
structure learner yields a graph on a different parametric level than what is
assumed by the reasoning task. Namely, RESIT leans a causal structure assuming
noise model structural equations, while Lattimore et al. [29] assumes a non-
parametric causal graph.

Luckily, because of the composition of both the structural and parametric
scales, each stricter assumption is subsumed by a lower-level assumption. As
such, from left to right, we can always relax the assumptions (such as
allowing non-parametric reasoning based on a noise-model causal structure),
but not the other way around. If, for example, our structure learner yielded a
plausible but non-parametric causal structure, we are not guaranteed optimal
regret from Lattimore et al. [37]. On our map, that would become clear as we
would move from a less strict assumption (plausible causality) to a strict
assumption (full causality).

##  4 Treatment effects: An illustrative example using CDL

We have seen that the CDL framework allows practitioners to verify the
suitability of a model and build model pipelines. Let us discuss how we can
arrive at such a conclusion using a practical example of treatment effects. In
particular, we will focus our CDL efforts on solving the common case of
inferring a treatment effect from the infant health and development program
(IHDP) dataset [38, 39].

Applying treatment effects in practice, the CDL way amounts to the following
steps: (1) A priori knowledge: Identify the structural and parametric
properties of the data. (2) Temporal or static: Is the problem we wish to
solve one that requires inference over time, or not? (3) Posteriori knowledge:
What type of structural and parametric should our resulting model have?

A brief overview of CATE. Before applying our framework as laid out above, let
us first briefly recapitulate the general treatment effects setting, in
particular conditional average treatment effect (CATE) inference, which
focuses on the following object:

$\tau(X)\coloneqq\mathbf{E}[Y_{1}-Y_{0}|X],$ (8)

where $Y_{A}$ (with $A\in\\{0,1\\}$) is the potential outcome given a
treatment $A$ (which in eq. 8 is binary). Equation 8 is a conditional ATE due
to $X$. If $X$ is dropped from eq. 8 we are left with an ATE which covers the
complete population $\mathcal{X}$ with $X\in\mathcal{X}$. Interestingly, our
objective, $\tau(X)$, is impossible to observe! We can only observe parts of
this: $\mathbb{E}[Y_{A}|X]$, again with $A\in\\{0,1\\}$. The tricky thing
about only observing one term in the expectation in eq. 8, is that the per-
term expectations are covering possible different populations. In particular,
we have:

$p(X|A=1)\neq p(X|A=0),$

which does not allow to split eq. 8 into a difference between
$\mathbb{E}[Y_{1}|X]$ and $\mathbb{E}[Y_{0}|X]$– that right is reserved only
for a randomised trial (with $p(X|A=1)=p(X|A=0)$ for all $X\in\mathcal{X}$)
[1].

There are several strategies to infer $\tau(X)$ regardless, either directly or
in parts by inferring each potential outcome separately. For an overview of
different inference tactics, we refer to Curth and van der Schaar [40], Künzel
et al. [41].

###  4.1 Treatment effects: the CDL description

Let us use our CDL perspective to describe what is required to solve the
treatment effects problem given the IHDP dataset [38].

Treatment effects papers make three starting assumptions: Overlap (each
individual in the data has a non-zero probability of receiving treatment),
consistency (the observed outcome corresponds with a potential outcome), and
ignorability (all variables that confound treatment and the potential outcomes
are observed). The last assumption is often expressed as:
$A\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\\{Y_{a}:a\in\mathcal{A}\\}|X$, where $X$ are
the individual’s covariates, $A\in\mathcal{A}$ is the treatment, and
$\\{Y_{a}:a\in\mathcal{A}\\}$ is the set of potential outcomes [42, 43]. The
complication from the above setup is that one can only observe one potential
outcome [44]. Hence, the above conditional independence statement can never be
observed (as it concerns the complete set of potential outcomes).

[ frametitle=CDL Element: A Priori Knowledge ]

The structural scale. Translating this conditional independence statement to a
causal structure, would look like $A\leftarrow
X\rightarrow\\{Y_{a}:a\in\mathcal{A}\\}$ [45]. This is a structural causal
assumption [46]. However, because of the special nature of the potential
outcomes framework (where only one of the potential outcomes can be observed),
one cannot validate this causal statement, even with an elaborate experimental
design [1].

The parametric scale. From IHDP’s description we know that the potential
outcome surfaces follow the following functions:

$\mu_{0}=\exp((X+M)\beta)\quad\text{and}\quad\mu_{1}=X\beta+\omega,$

with $\mu_{a}$ the expectation of $Y_{a}$ given $X$. Given the above outcome
surfaces, it would make sense to use a linear estimator for $\mu_{1}$, and an
exponential estimator for $\mu_{0}$ (or even a linear one for $\ln\mu_{0}$).
Interestingly, most work that currently uses IHDP as a benchmark dataset
mostly focuses on neural networks, which is situated two places left on the
structural scale!

Given the above description, we argue that treatment effects papers which
assume the standard set of the stable unit treatment value assumptions (SUTVA)
from above [47], make causal structural assumptions as is illustrated in fig.
6.

Recall from above that temporal datasets are those that observe an instance of
a variable multiple times. That means that datasets which simply include a
timestamp, are not classified as a temporal dataset.

[ frametitle=CDL Element: Time ] Hence, from this definition, we know that
IHDP is not a temporal dataset, and thus does not require a model that
explicitly models for time. In our fig. 6, that means we should not consider
the models that are illustrated by a dashed line.

We find that CATE models (and TE models in general [48]) are not concerned
with transitioning across structural models. That is, a CATE model does
typically not learn additional causal structure on top of the structure it
assumes a priori.

[ frametitle=CDL Element: Posteriori Knowledge ] A CATE model’s structure
present in its representation remains the same as those assumed at the input,
i.e. the structural assumptions made a priori. As such, CATE models typically
do not add causal knowledge on top of what is assumed at the start.

From a practical perspective, the reason for this is that a CATE-model’s goal
is not to determine causality, but instead to infer a causal effect [3].

UPCNPNMPa.FKA Priori KnowledgeTimeStaticNPNMPa.FKA Posteriori Knowledge⚫[26,
49, 50, 51]⚫[52]⚫[27]⚫[53, 40, 54, 55, 56]⚫[57, 58]⚫[34]⚫[28]

Figure 6: Treatment effects and CDL. We document some of our own, and other
well-known, treatment effects papers in the diagram presented in section 3.
The basic takeaways from doing this are: (1) Learning treatment effects models
requires making causal assumptions about the data-generating process, (2) The
focus of the treatment effects community has been relaxing parametric
assumptions, (3) learning fully known functions (FK) has not been a focus in
this community. In section 2 we document many more papers across many more
fields. Here too: Non-parametric (NP), Noise models (NM), Parametric (Pa.),
and Fully known (FK); and Unknown (U), Plausible (P), and Causal (C).

We hope that the above provides an example of what we advocate any CDL paper
to include (a clear description of the problem and solution spaces discussed
in the paper using our CDL axes), in some form or another. We have performed
this exercise for a few well-known treatment effects models in fig. 6.
Interestingly, only in temporal models did we find a treatment effects models
that makes a transition on the parametric scale, ending up in a fully known
(FK) parametric description. That is because, in Holt et al. [28], the
objective is to explicitly learn underlying equations which are later used for
treatment effects inference.

For more practical ML examples, we refer to our appendix A which lists a few
where we observe CDL to already have made some impact.

##  5 Conclusion and guidelines for CDL papers

In conclusion, our framework for causal deep learning (CDL) represents a major
step forward in the field of machine learning research. By incorporating a
structural dimension to measure absent, present, or partially known structural
causal knowledge in methods, a parametric dimension that maps the strengths of
deep learning methods to causality, and a temporal dimension that accounts for
the impact of time on a causal system, we have provided a comprehensive guide
for using CDL to solve real-world problems.

Our framework extends the levels of structural knowledge in causality by
introducing a new level, "plausible causal knowledge", that accommodates many
existing papers and provides a clear pathway to powerful tools and methods.
With our clear definition, we can now identify real problems and select the
method that best fits the setting of interest, empowering us to better
understand and address the shortcomings of current methods. The addition of
such a scale helps us differentiate between the many works on causal
representation learning [59, 60, 61, 62, 63, 64, 65], and the classical
independent components analysis [66, 67, 68, 69]. While the latter helps
advance the causal representation learning agenda, it recovers components
based on a plausible causal structure (i.e. the second level on our scale in
fig. 1), whereas the former recovers components based on a causal structure.

Essentially, through CDL we propose new research to take into account the
following practical guidelines:

  1. 1.

Be clear about causal assumptions. It is important to leverage the causal
knowledge we have [70] but also to take into account knowledge that we don’t
have; in particular, we need to be careful with using models that ask for
total causal knowledge but only give it partially confirmed structures.

  2. 2.

Avoid model misspecification by carefully documenting and analysing the
parametric setting. Non-parametric models are powerful allies but are
unnecessarily complicated when we already have some parametric knowledge about
our setting. Before designing new methods, we encourage a researcher to first
analyse the setting in which the new method is to be deployed. Their analysis
can vastly reduce model complexity and as a result, allow more powerful
inference.

  3. 3.

Time deserves special attention from a causal perspective. As shown in our
paper, time is more than simply taking into account an observation’s history.
Careful analysis is required when time presents new complexities related to
confounding.

Furthermore, our framework identifies gaps in current research and proposes
avenues for future work, including the need for research on recovering fully
known structural equations and reflection on the relevance of current gaps to
practical problems. We are confident that our framework will make a
significant contribution to the advancement of CDL research and its practical
applications, paving the way for a more efficient and effective approach to
solving complex real-world problems.

#### Acknowledgements

We truly value the input of the CDL community. We wish to explicitly thank the
following community members for their comments (alphabetised by last name):

Alicia Curth (University of Cambridge)
Isabelle Guyon (Google Brain)
Sam Holt (University of Cambridge)
Alihan Hüyük (University of Cambridge)
Trent Kyono (Meta)
Andrew Rashbass (University of Cambridge)
Paulius Rauba (University of Cambridge)
Guillermo Sapiro (Duke University & Apple)
Ricardo Silva (UCL)
Ruibo Tu (KTH)
Boris van Breugel (University of Cambridge)
David Watson (King’s College London & Meta)
Simon Woodhead (Eedi)
Cheng Zhang (Microsoft Research)
Kun Zhang (CMU & MBZUAI)

## References

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##  Appendix A Real-world applications

We discuss the impact of CDL on five different real-world applications:
Imputation, Fairness, Generalisation, Treatment effects, and Recommendation.
We will examine how CDL can help define interaction effects of missingness and
covariates, identify sources of unfairness, constrain methods using a general
multi-domain causal structure, learn models that respect identification
assumptions, and infer the causal effect of recommendation.

Imputation Imputation is a common problem in many datasets, and missing data
can lead to biased solutions. CDL can help define interaction effects of
missingness and covariates to improve the accuracy of imputation methods. For
example, CDL-based approaches allow deep learning to accurately impute missing
values, respecting the causal interaction between missingness indicators and
treatment selection [34, 36, 71]. CDL-based methods outperform existing
imputation techniques in terms of both imputation accuracy and unbiased
estimation from data with missing values. Another approach is to leverage
causal knowledge to perform the imputation task directly [72, 73].

Fairness. CDL can also be used to identify sources of unfairness in machine
learning models. In particular, CDL can help identify which features in the
data are responsible for observed disparities in outcomes. CDL-based
approaches can help mitigate unfairness in machine learning models by learning
a representation of the data that is fair concerning a given set of protected
attributes and their causal neighbours [74, 75, 31, 76]. The authors showed
that their method outperforms existing fairness methods in terms of both
fairness and accuracy. Other CDL techniques evaluate models post-training
using causal evaluation metrics [77].

Generalisation. CDL can help constrain methods using a general multi-domain
causal structure, which can improve generalisation performance in new domains.
Specifically, causal structures can be used to select models in a new domain
as (partial) causal knowledge is assumed invariant [78, 79, 80], yielding more
robust model selections with increased performance as a result. While it would
be beneficial to gather complete causal knowledge, we can already advance in
this application using partial knowledge.

Treatment effects. Perhaps one of the most natural application domains for CDL
is estimating causal treatment effects [58, 45, 81, 82]. With a long history
in statistics, it seems that treatment effect literature is truly benefiting
from taking a causal deep learning approach. With applications in donor-organ
transplantation [53, 57], cancer [83, 84, 85], and COVID-19 [86] and its
impact [87].

Drug repurposing. At the intersection of generalisation and treatment effects
is drug repurposing, an incredibly exciting area for CDL to have an impact
[88, 89]. With applications in COVID-19 [90] and dementia [91, 92], CDL is
already proving to be a useful framework. However, as pointed out in table 1,
these high-stakes application domains are a clear example of why we require a
careful analysis of the made assumptions.

Recommendation. CDL can also be used to infer the causal effect of online
store recommendations [93, 94, 95], which is essential for accurate evaluation
of recommendation algorithms. Furthermore, Basilico and Raimond [96]
illustrate the importance of causality and time to provide meaningful
recommendations, again highlighting the importance of our temporal dimension!
Beyond this, Ghazimatin et al. [97] show how causality can be used to evaluate
a recommendation system by estimating counterfactual explanations.

Table 1: Real-world impact. Some methods in CDL have already impacted the real world. We list some of them in the table above, spanning a few important application areas. In our table, we use the following abbreviations: Unknown (U), Plausible (P), Causal (C), Non-parametric (NP), Noise models (NM), Parametric (Pa.), and Static (St.). These abbreviations correspond with our framework presented in section 2. Application |  How they use causality |  Ref. | Range of assumptions
---|---|---|---
|  |  | Struct. | Param. | Time
Imputation |  Define missingness and covariates interaction |  [98, 72, 34, 71, 36, 99] | U $\to$ C | NP $\to$ Pa. | St.
Fairness |  Identify of sources of unfairness |  [31, 74, 75, 77, 76] | U $\to$ C | NP $\to$ Pa. | St.
Generalisation |  Respect multi-domain causal structure |  [79, 78, 80] | U $\to$ C | NP $\to$ NM | St.
Treatment effects |  Respect causal identification assumptions |  [58, 45, 81, 82, 100, 53, 57, 83, 84, 85, 86, 87, 26, 101] | U $\to$ P | NP $\to$ Pa. | Both
Drug repurposing |  Relate multi-modalities using causal structure |  [90, 91, 92, 88, 89] | U $\to$ C | NP $\to$ Pa. |  Both.
Recommendation |  Infer causal effect of recommendation |  [94, 96, 93, 95, 97] | U $\to$ P | NP $\to$ NM | Both

We summarise the above in table 1, where we also indicate the range of
structural, parametric and temporal assumptions, as per our topographic map.
Clearly, from table 1 as well as the examples presented in appendix B, it
should be clear that we require our framework to truly have detailed
discussions and clear adoption of methods going forward.

##  Appendix B Categorising CDL methods

Unknown (U)Plausible (P)Causal (C)(NP)Non-param.(NM)Noise
Models(Pa.)Param.(FK)Fully
knownInputTimeStaticUPCNPNMPa.FKRepresentationGoalGoal

Figure 7: Goals and direction of the field. We can use the map of causal deep
learning to specify the future of our field. Darker shades of red indicate a
harder, but more desirable goal. In the case of input, we wish to minimise the
required graphical input and made assumptions; with which we hope to achieve a
more knowledgeable representation with minimal assumptions.

Using the map of causal deep learning we can: (i) categorise and compare
literature, as well as (ii) identify some areas that are not well explored.
Using these two functionalities, we can perhaps attribute (i) to
practitioners, and (ii) to researchers. A practitioner is faced with a problem
they wish to solve. Using the map of CDL, the practitioner can map their
available data onto the input field and search for a method in the
representation field that would solve their needs. Perhaps guided by the
required assumptions the practitioner is willing to make, the set of solutions
is more digestible than scanning all the literature in CDL. The researcher may
use the map differently. Rather than scanning the potential solutions, a
researcher can use the map to scan which solutions are still lacking (or more
likely, underrepresented).

Problems and goals. Learning a structured representation space, while
retaining flexible assumptions, is difficult. Consider fig. 7 where we
annotated learning a causally structured representation without any
assumptions from arbitrary data with “Goal”. This learning setup is the most
ambitious setup included in our map. It is also long thought to be impossible
[102, 103, 104, 105].

However, fig. 7 also shows the relative complexity of other setups. In
principle, the less strict assumptions we make, the harder it becomes to
narrow down the structure in our representation space. For some problem setups
it may be sufficient to learn a flexible structure, which in turn allows for
more flexible assumptions on the input. Similarly, if a problem requires a
completely identified causal structure, our map shows that one may have make
to some strict assumptions on the input. As such, the map of CDL exposes a
certain balance between input assumptions and achievable structure.

###  B.1 Comparing methods

To show how one can use the map to compare methodologies, we take supervised
learning as an example, presented in fig. 8. Of course, CDL spans more than
only supervised learning and so does our map.

To remind ourselves, supervised learning is a problem where we wish to map an
input, to a label [106]:

$m:\mathcal{D}\to\mathcal{Y},$

where $\mathcal{D}$ is once again a dataset, but now including labels:
$\\{(X_{i},Y_{i}):i\in[N]\\}$ with $X_{i}\in\mathcal{X}$ and
$Y_{i}\in\mathcal{Y}$. The labels $Y_{i}\in\mathcal{Y}$ can be anything from a
real variable ($\mathcal{Y}=\mathbb{R}$), to a binary label
($\mathcal{Y}=\\{0,1\\}$).

As we discussed in section 2.5, deep learning methods first map the data to a
representation before it is mapped to the outcome-label [107]:
$m:\mathcal{D}\to\mathcal{H}\to\mathcal{Y}$. Here, $\mathcal{H}$ corresponds
to the representation as in LABEL:eq:cdl_model. As such, our discussion here
does not concern $\mathcal{Y}$, but is instead focused on $\mathcal{H}$ and
the structure it may respect.

Why use CDL for supervised learning? A fair question indeed. Typically,
supervised learning models are evaluated only on the accuracy a model achieves
on a hold-out test set. Yet, there is no immediate reason why a causal
representation would help in this regard. Rather than maximising accuracy on a
hold-out test set, one may resort to CDL to also achieve high accuracy on a
subset that is not representative of the training data. Examples such as these
include: domain adaptation [108, 78], transfer learning [109, 110],
interpretability [111, 112, 113, 114], or general robustness [115, 73, 116,
117].

Unknown (U)Plausible (P)Causal
(C)NPNMPa.FKInputTimeStaticNPNMPa.FKRepresentation⚫[118, 119]⚫[120, 99, 121,
122, 123, 124]⚫[125, 126]⚫[127]⚫[128, 129, 130, 131]⚫[132]⚫[133, 134]

Figure 8: Navigating (static) supervised learning. Here we focus only on
methods that share the task of mapping data to a label in a supervised way. We
discern between methods based on what input they expect (or assume) and what
type of representation they learn before mapping to a label. Our map presents
a straightforward way to categorise methods in causal deep learning. Doing
this is useful for practitioners (to identify a suitable method), and to
researchers (to identify a potential research gap). Here too: Non-parametric
(NP), Noise models (NM), Parametric (Pa.), and Fully known (FK).

###  B.2 Developing methods

Let us discuss our map using a non-exhaustive list of some of the key tasks
the map of CDL may help a researcher.

Identification. A big part in a researcher’s workflow is to identify gaps in
the literature. For researchers active in supervised learning, our map of CDL
can be a useful tool to do exactly this. For example, from fig. 8 we learn
that the models that take a fully known function as a parametric assumption
are not well represented– they are not represented at all! Of course, this
makes sense. If one assumes a fully known model, before commencing the
learning task, there is no point learning it anymore.

Looking further, we find that mapping additive noise models with unknown
structure, to additive noise model with structure actually are well
represented. The reason for this is that recent contributions in
differentiable structure learning are a great candidate to regularise models
used for other tasks than structure learning. As such, a researcher would
learn that there is indeed much competition in this field.

Navigation and related works. This brings us to a second task we can employ
our map for: building a body of related works. The map will make it easier for
researchers to find and learn about related works. Essentially enabling a
platform for researchers to share their work with other interested researches.

Furthermore, the map allows researchers to think critically about the work
they present. For example, in fig. 7 we clearly state that the “ultimate goal”
to map data to a non-parametric causal structure without making any
assumptions is thought to be impossible. Not only will this help researchers
to discuss their proposal with a sense of realism, the same can be said about
practitioners (and reviewers) alike.

Data. Beyond the above, the map also provides guidance for model evaluation.
In particular, when evaluating a model, it is important to use data that
actually matches the models input assumptions. Similarly for the benchmarks
the proposal is compared against.

For example, if a method assumes a causal graph as input, there should be some
assurance that the presented graph is indeed causal. One way to do this is by
also provided some empirical evidence that this is the case, such as running
clinical trials or assuming interventional data.

Table 2: Comparing different fields. Section B.4 compares methods in fields beyond supervised learning (cfr. section B.1). Our table below provides a quick overview of the methods and fields we discuss. Beyond that, we also list the target aimed to learn, as well as the minimum data required to learn that target. We have excluded any additional information (beyond the data) such as parametric or structural assumptions, as this is exactly what the map of CDL can be used for.  Field | Data | Target | References | Map
---|---|---|---|---
Supervised learning (SL) | $\\{(X_{i},Y_{i}):i\in[N]\\}$ | $\mathcal{X}\to\mathcal{Y}$ | [118, 119, 120, 99, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134] | fig. 8
Forecasting/temporal SL | $\\{(X_{i},Y_{i})_{1:T_{i}}:i\in[N]\\}$ | $\mathcal{T}\times\mathcal{X}\to\mathcal{Y}$ | [135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151] | fig. 9
Bandits & RL | $\\{(X_{i},A_{i},Y_{i})_{\pi^{b}}:i\in[N]\\}$ | $\mathcal{X}\times\mathcal{A}\times\mathcal{Y}\to\pi^{*}$ | [152, 29, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185] | fig. 10
Generative modelling | $\\{(X_{i}):i\in[N]\\}$ | $\mathcal{X}\to\mathcal{X}$ | [186, 187, 188, 189, 190, 31, 191, 192, 193, 194, 195, 196] | fig. 11

###  B.3 Using methods

Navigate. While the researcher’s perspective is mostly based around
identifying methods that don’t exist, the practitioner’s is the opposite. When
faced with a certain problem, the practitioner may use the map to identify
which methods assume the input the practitioner has available, while also
solving the problem they wish to solve.

Finding a suitable method can be extremely challenging for practitioners as
there are new methods proposed almost daily. This latter point is in fact one
of the prime reasons why we propose the map in the first place.

Validation and education. On the other hand, it may be that the practitioner
wishes to solve a problem that is not yet solved before (stumbling upon a gap
in the literature). Or, more strikingly, they are faced with a problem that is
simply impossible to solve.

Especially the latter could be an interesting use of our map: education. We
strongly believe that the map of CDL can form a bridge between researches and
practitioners. Educating which problems can be solved (which is communicated
from research to practice), and which problems have to be solved (which is in
turn communicated from practice to research).

As we have observed in fig. 8, we find that there is a heavy focus on learning
structure from nothing in the additive noise setting. But, perhaps this is not
interesting from a practical point of view as it could be that practitioners
are well aware of some conditional independence, or some parametric shape of
their environment. Having clear communication between both parties may avoid
researchers to waste valuable resources and time to solve problems that need
not be solved.

Unknown (U)Plausible (P)Causal
(C)NPNMPa.FKInputTimeStaticNPNMPa.FKRepresentation⚫[136]⚫[137, 138]⚫[139, 140,
141, 142]⚫[143]⚫[144, 145]⚫[147]⚫[148, 146]⚫[149]⚫[135]⚫[150]⚫[151]

Figure 9: Navigating (temporal) supervised learning and forecasting. Here we
focus on methods that solve a prediction task in the temporal domain. As in
fig. 8 we discern between methods based on what input they expect (or assume)
and what type of representation they learn before mapping to a label. Here
too: Non-parametric (NP), Noise models (NM), Parametric (Pa.), and Fully known
(FK).

###  B.4 Beyond supervised learning

Having explained how our map can categorise methods within the supervised
learning problem (and what benefits this yields in sections B.1, B.2 and B.3),
let us now consider other machine learning problems. We have listed these in
table 2 where we– like we have for supervised learning in section B.1
–present: the minimal data required to solve the problem, the problem’s
target, and provide a link to the map specific to the problem as well as the
references they categorise. Naturally, more references and problems can be
added, but we believe table 2 spans a broad range of the most researched ML
problems. Thereby showing the versatility of the map of CDL.

####  B.4.1 Forecasting and temporal supervised learning

Perhaps a logical next step beyond supervised learning is supervised learning
in a dynamic setting. In particular, we will now categorise methods that