# From Identifiable Causal Representations to Controllable Counterfactual
Generation: A Survey on Causal Generative Modeling

Aneesh Komanduri<EMAIL_ADDRESS>
Department of Electrical Engineering and Computer Science
University of Arkansas Xintao Wu<EMAIL_ADDRESS>
Department of Electrical Engineering and Computer Science
University of Arkansas Yongkai Wu<EMAIL_ADDRESS>
Department of Electrical and Computer Engineering
Clemson University Feng Chen<EMAIL_ADDRESS>
Department of Computer Science
University of Texas at Dallas

###### Abstract

Deep generative models have shown tremendous success in data density
estimation and data generation from finite samples. While these models have
shown impressive performance by learning correlations among features in the
data, some fundamental shortcomings are their lack of explainability, the
tendency to induce spurious correlations, and poor out-of-distribution
extrapolation. In an effort to remedy such challenges, one can incorporate the
theory of causality in deep generative modeling. Structural causal models
(SCMs) describe data-generating processes and model complex causal
relationships and mechanisms among variables in a system. Thus, SCMs can
naturally be combined with deep generative models. Causal models offer several
beneficial properties to deep generative models, such as distribution shift
robustness, fairness, and interpretability. We provide a technical survey on
causal generative modeling categorized into causal representation learning and
controllable counterfactual generation methods. We focus on fundamental
theory, formulations, drawbacks, datasets, metrics, and applications of causal
generative models in fairness, privacy, out-of-distribution generalization,
and precision medicine. We also discuss open problems and fruitful research
directions for future work in the field.

##  1 Introduction

With the growing interest in using deep learning for domain-specific tasks in
the natural and medical sciences, deep generative models have become a popular
area of research. Deep generative models are a class of methods that either
explicitly estimate the probability density or implicitly learn to sample from
some data distribution. Such models have demonstrated impressive performance,
especially in generating high-quality image samples. State-of-the-art
generative models such as GPT-4 (OpenAI (2023)) and CLIP (Radford et al.
(2021)) have taken the advances in deep generative modeling to new heights.
Recent research in deep generative modeling focuses on important properties of
the data-generating process that lead to interpretable representations and
controllable generation. Despite such advances, most current methods neglect
the fact that systems in the real world are complex, dynamic, and may have
underlying causal dependencies. The failure to account for this has made deep
learning models prone to shortcut learning and spurious associations.

Many have argued that we must go beyond learning mere statistical correlations
toward causal models that capture the causal-effect relationship between
influential variables in a system. In the 1950s, Alan Turing asked what it
would take for a computer to think like a human. Turing proposed what many
today refer to as the “Turing Test.” However, a more practical version of this
test would be to ask the following question: how can machines represent causal
knowledge of the world in a way that would enable them to access the necessary
information quickly, answer questions accurately, and do so with ease (Pearl &
Mackenzie (2018))? Pearl’s Causal Hierarchy (Pearl (2019); Bareinboim et al.
(2022)) lays out the three levels of causal models that provide a mechanism to
progressively get closer to modeling human-level reasoning: observations
(seeing), interventions (doing), and counterfactuals (imagining). Intervening
on causal variables of a system leads to the ability to reason about
hypothetical scenarios in the world (counterfactuals).

Deep generative models such as variational autoencoders (VAE), generative
adversarial networks (GANs), normalizing flows, and diffusion models have been
at the forefront of machine learning for approximating complex data
distributions. A traditional deep generative model would perform quite well in
generating new data for an application such as generating novel images from
brain MRI data. However, understanding what and how underlying factors truly
influence the generative process can only be captured by modeling the causal
mechanisms between variables. If certain markers or labeled factors causally
influence cancer diagnoses, incorporating them into a causal model will more
robustly encode the system’s information and its complex relationships. If we
want to know the effect of changing certain factors on the structure of the
brain MRI, we can easily perform interventions to infer such relationships.
Further, one could “simulate” patients by generating counterfactual instances
of tumors under different interventions that lie outside the support of the
training data. This would greatly reduce the cost associated with collecting
and annotating patient data with constraints such as privacy. Thus, a shift
towards causal modeling is imperative to learn more robust models that reflect
the complexities in the real world.

Figure 1: Taxonomy of Causal Generative Modeling

In an effort to improve the interpretability and robustness of generative
models, recent research has focused on learning causal generative models using
Pearl’s structural causal model (SCM) formalism (Pearl (2009)). The SCM
inherently describes a data-generating process consisting of a set of
endogenous variables that influence each other through causal mechanisms. The
SCM describes a generative model of the world and, as such, can be naturally
utilized in deep generative models. Causal models offer several beneficial
properties to deep generative models, such as distribution shift robustness,
fairness, and interpretability (Scholkopf et al. (2021)). The interpretability
of models can be viewed in terms of both learning a semantically meaningful
representation of a system and controllably generating realistic
counterfactual data. In the context of representation learning, SCMs can be
used to model the causal dependencies among encoded latent variables. In the
context of controllable generation, SCMs can be used to model the mechanism
that generates high-dimensional observational data as a function of causal
variables to enable counterfactual inference.

We take this opportunity to survey the exciting and ever-growing field of
causal generative models based on the Pearlian theory of structural causal
models, discuss the benefits and drawbacks of different approaches, and
highlight open problems and applications of causal generative models in a
variety of domains for real-world impact. Unlike other surveys (Kaddour et al.
(2022); Zhou et al. (2023)), in addition to covering the methodology of causal
generative models, we also cover foundational identifiability theory in causal
representation learning with a careful emphasis on intuition. We also outline
several challenging open problems from the perspective of both application and
theory. We focus our survey on generative models that utilize fundamental
principles of causality to enable causal representation learning (CRL) and
controllable counterfactual generation. We do not cover causality-inspired
frameworks or causal discovery in this survey.

We break down causal representation learning into the three different classes
from Pearl’s causal hierarchy based on the assumptions of the data-generating
process to achieve identifiability: observational, interventional, and
counterfactual. Causal representation learning is concerned with learning
semantically meaningful causally related latent variables and their causal
structure from high dimensional data. Identifiability is an important property
of representations learned by generative models since it ensures that we learn
independently manipulable unique factors that describe the observational data.
This is important for causal representations since we are often interested in
performing interventions on causal variables to observe causal effects. We
break down controllable counterfactual generation methods by GAN-based,
Flow/VAE-based, and Diffusion-based models. Controllable counterfactual
generation methods focus more on modeling known causal variables (such as
labels) and learning a mapping to the observed data. The main difference
between the two classes of methods is that causal representation learning
assumes a causal structure among latent variables, whereas controllable
generation approaches model the causal process between known/unknown labels
and data as a form of structural mechanism learning. For the most part, our
survey focuses on feature-level causal variables and not on more general
causal modeling schemes. The taxonomy of methods is shown in Figure 1.

Contributions. (1) We taxonomize causal generative modeling work into the
identifiable causal representation learning and controllable counterfactual
generation tasks. (2) We further categorize causal representation learning
methods according to the type of data they assume access to from Pearl’s
Causal Hierarchy and break down controllable generation approaches based on
the type of generative model utilized. (3) We discuss the existing evaluation
metrics, benchmark datasets, and results of causal generative models. (4)
Finally, we lay out current limitations and fruitful research directions in
both causal representation learning and controllable counterfactual
generation.

Structure of survey. The survey is structured as follows. In Section 2, we
discuss the background necessary to understand the rest of the paper,
including formal definitions of SCM, common deep generative models, and
disentangled representation learning. In Section 3, we formulate the causal
representation learning problem and discuss methods categorized according to
the type of data they assume access to from Pearl’s Causal Hierarchy. Then, in
Section 4, we discuss the controllable counterfactual generation task and
methods from GAN-based, Flow/VAE-based, and Diffusion-based frameworks. In
Section 5, we give an overview of the most common metrics used in causal
generative modeling. In Section 6, we discuss synthetic and real-world
datasets that are often used in causal representation learning and
controllable generation tasks. In Section 7, we highlight applications of
causal generative models in trustworthy and robust AI and precision medicine.
In Section 8, we outline several open research directions in causal generative
modeling. Finally, we conclude the survey in Section 9.

[b]  Notation Description $x$ observed data $z$ endogenous latent causal
variables $\epsilon$ exogenous noise variables $y$ labels
$\mathcal{X}\subseteq\mathbb{R}^{d}$ domain of observational data
$\mathcal{Z}\subseteq\mathbb{R}^{n}$ domain of causal variables
$\mathcal{E}\subseteq\mathbb{R}^{n}$ domain of noise variables $\mathcal{Y}$
domain of labels $A$ causal DAG adjacency matrix $z_{\textbf{pa}_{i}}$ causal
parents of variable $z_{i}$ $f_{i}$ causal mechanism deriving variable $z_{i}$
$f_{RF}$ reduced form mapping noise to causal variables $\tilde{f}_{i}$ new
mechanism replacing $f_{i}$ after intervention $F$ set of all causal
mechanisms $\tilde{x}$ counterfactual data $\tilde{z}$ post-intervention
causal representation $\tilde{\epsilon}$ post-intervention noise variable
$\mathcal{C}$ structural causal model $\mathcal{\tilde{C}}$ intervened
structural causal model $\textbf{do}(\cdot)$ perfect hard intervention
$g(\cdot)$ mixing function $p(z)$ (or $p_{z}$)1 joint distribution over set of
variables $z$ $\circ$ composition of functions $\odot$ Hadamard product

Table 1: Main notation used for causal representation learning

  * 1

Note that the notations $p(z)$ and $p_{z}$ are used interchangeably throughout
the survey and are equivalent

##  2 Background

In this section, we introduce notation and key concepts in generative modeling
and representation learning important for understanding the rest of the paper,
including Pearl’s structural causal model, variational autoencoders,
normalizing flows, generative adversarial networks, diffusion probabilistic
models, and disentangled representation learning. A list of the main notation
used in the paper is provided in Table 1.

###  2.1 Structural Causal Model

All the methods discussed in this survey utilize the structural causal model
(SCM) formalism from Pearl (2009), which formally describes the relationship
among a set of variables. The SCM induces a hierarchy of causal reasoning
referred to as Pearl’s Causal Hierarchy.

######  Definition 1 (Structural Causal Model)

A structural causal model (SCM) is defined by a tuple
$\mathcal{C}=\langle\mathcal{Z},\mathcal{E},F,p_{\mathcal{E}}\rangle$, where
$\mathcal{Z}$ is the domain of $n$ endogenous causal variables
$z=\\{z_{1},\dots,z_{n}\\}$, $\mathcal{E}$ is the domain of $n$ exogenous
noise variables $\epsilon=\\{\epsilon_{1},\dots,\epsilon_{n}\\}$, and
$F=\\{f_{1},\dots,f_{n}\\}$ is a collection of $n$ causal mechanisms of the
form

$z_{i}=f_{i}(\epsilon_{i},z_{\textbf{pa}_{i}})$ (1)

where $\forall i$,
$f_{i}:\mathcal{E}_{i}\times\prod_{j\in\textbf{pa}_{i}}\mathcal{Z}_{j}\to\mathcal{Z}_{i}$
are causal mechanisms that determine each causal variable as a function of the
parents and noise, $z_{\textbf{pa}_{i}}$ are the parents of causal variable
$z_{i}$; and $p_{\mathcal{E}}(\epsilon)$ is a probability measure of
$\mathcal{E}$.

An SCM where the exogenous noise variables are jointly independent (no hidden
confounders) is known as a Markovian model. The assumption of no hidden
confounding is also referred to as causal sufficiency. We depict the causal
structure of $z$ by a causal directed acyclic graph (DAG) $\mathcal{G}$ with
adjacency matrix $A\in\\{0,1\\}^{n\times n}$. A $2$-variable SCM is shown in
Figure 2(a). The distribution over endogenous variables $p^{\mathcal{C}}(z)$
is called the entailed distribution. We have that the joint distribution of
endogenous variables can be written as the following causal Markov
factorization

$p^{\mathcal{C}}(z)=\prod_{i=1}^{n}p^{\mathcal{C}}(z_{i}|z_{\textbf{pa}_{i}})$
(2)

Interventions. An SCM enables us to go beyond observational distributions to
construct interventional distributions that reflect changes in a system at the
population level. Interventions enable one to answer the question: “What would
$z_{j}$ be if we change $z_{i}$?” We can construct interventional
distributions by making modifications to an SCM $\mathcal{C}$ and considering
the entailed distribution.

Let $\mathcal{C}=\langle\mathcal{Z},\mathcal{E},F,p_{\mathcal{E}}\rangle$ be
an SCM and $p^{\mathcal{C}}(z)$ be its entailed distribution. We replace any
number of structural assignments to obtain a new SCM $\tilde{\mathcal{C}}$.
Suppose we replace the mechanism that generates causal variable $z_{i}$ by

$z_{i}=\tilde{f}_{i}(z_{\tilde{\textbf{pa}}_{i}},\tilde{\epsilon_{i}})$ (3)

We call the entailed distribution of the new SCM an interventional
distribution, which is denoted by

$p^{\tilde{\mathcal{C}}}(z)=p^{\mathcal{C};do(z_{i}=\tilde{f}(z_{\tilde{\textbf{pa}}_{i}},\tilde{\epsilon}_{i}))}(z)$
(4)

(a) 2-variable SCM

(b) do-intervention (hard)

(c) soft (imperfect)

Figure 2: Types of Interventions

The two main types of interventions are hard interventions, which remove
dependency from parents, and soft interventions, which alter the dependency on
direct parents, as depicted in Figure 2.

  * •

Hard intervention. A hard intervention, or do-intervention, modifies an SCM by
deterministically replacing for a subset of causal variables the variable
$z_{i}$ with a constant $c$, denoted $\textbf{do}(z_{j}=c)$, such that the
causal variable is no longer dependent on its parents. The interventional
distribution given a hard intervention on variable $z_{i}$ can be expressed
as:

$p(z_{1},\dots,z_{n}|\textbf{do}(z_{i}=c))=\prod_{j\neq
i}p(z_{j}|z_{\textbf{pa}_{j}})\delta_{z_{i}=c}$ (5)

  * •

Soft intervention. Unlike hard interventions, which remove all direct causal
edges of the intervened target, a soft intervention, or mechanism change,
simply changes the dependency of the causal variable on its direct parents,
where the variable can still be dependent on a subset of the original parents.

We introduce two additional subclasses of interventions that are of special
interest in the causal generative modeling literature: perfect and imperfect
interventions (Peters et al. (2017)).

  * •

Perfect intervention. A perfect intervention modifies an SCM by replacing for
a subset of causal variables the causal mechanism $f_{i}$ with a new mechanism
$\tilde{f}_{i}$ that is no longer dependent on the original parents. With
perfect interventions, all causal dependencies between the intervened target
$z_{i}$ and its causes are removed, but the variable can still depend on its
corresponding noise term, and its value may still be random. Thus, perfect
interventions can be seen as a generalization of hard interventions where
stochasticity is allowed.

  * •

Imperfect intervention. An imperfect intervention modifies an SCM by replacing
for a subset of causal variables the causal mechanism $f_{i}$ with a different
causal mechanism $\tilde{f}_{i}$ that preserves the dependence on the causal
variable’s original parents. We note that imperfect interventions are a
special case of soft interventions where the variable is still dependent on
all of its original parents.

Counterfactuals. Beyond the ability to compute causal effects from
interventional distributions, SCMs enable reasoning about counterfactuals. A
counterfactual is a query at the unit level that asks the question: “Given an
arbitrary unit, what would $z_{j}$ have been if $z_{i}$ were different?”.
Counterfactuals, unlike interventions, are relative to a factual observation.
That is, given an observed set of variables that reflect reality,
counterfactuals aim to infer what would happen if the original observations
were intervened upon while keeping the original noise terms. Let
$\mathcal{C}=\langle\mathcal{Z},\mathcal{E},F,p_{\mathcal{E}}\rangle$ be an
SCM over variables $z$. Given some observations $z$, Pearl (2009) outlines the
following three-step procedure to perform counterfactual inference:

  * •

Abduction: Infer noise terms $\epsilon$ compatible with observations $z$
(i.e., $p(\epsilon|z)$)

  * •

Action: Perform an intervention $\textbf{do}(z_{i}=c)$ corresponding to the
desired manipulation, which results in a modified interventional SCM
$\tilde{\mathcal{C}}=\mathcal{C}_{\textbf{do}(z_{i}=c)}$

  * •

Prediction: Compute the counterfactual quantity based on the modified SCM
$\tilde{\mathcal{C}}$ and $p(\epsilon|z)$

######  Definition 2 (Pearl’s Causal Hierarchy - Pearl (2019))

The observational $(L_{1})$, interventional $(L_{2})$, and counterfactual
$(L_{3})$ distributions entailed by the SCM form a hierarchy in the sense that
$L_{1}\subset L_{2}\subset L_{3}$, where each level encodes richer information
that the previous level cannot express. The three layers are collectively
referred to as Pearl’s Causal Hierarchy (PCH).

Figure 3: Pearl’s Causal Hierarchy

On Pearl’s Causal Hierarchy. $L_{1}$ (observational) is concerned with
modeling mere statistical correlation and does not in itself provide a causal
perspective. $L_{2}$ (interventions) focuses on manipulating a system of
causal variables to observe downstream effects on other variables.
Interventions are often performed using the do-operator and are population-
level operations. Finally, $L_{3}$ (counterfactuals) is the most complex layer
and deals with retrospective thinking. That is, counterfactuals are unit-level
queries that imagine alternative situations contrary to factual observations
under a specific intervention. For instance, a question such as “Would I have
gotten an A grade on my exam had I studied for one more hour?” would be a
counterfactual query. An illustration of Pearl’s causal hierarchy is shown in
Figure 3.

###  2.2 Variational Autoencoders

The variational autoencoder (VAE) (Kingma & Welling (2013)), shown in Figure
4(a), is a framework for optimizing latent variable models inspired by the
theory of variational inference and energy-based models. Similar to a standard
autoencoder, the VAE is composed of both an encoder and a decoder and is
trained to minimize the reconstruction error. However, instead of encoding an
input as a single point, the VAE encodes it as a distribution over the latent
space. Formally, suppose the observed data $x$ is mapped to some low-
dimensional latent space via an encoder $h:\mathcal{X}\to\mathcal{Z}$ and
reconstructed by a decoder $g:\mathcal{Z}\to\mathcal{X}$. The goal of the VAE
is to approximate the likelihood of the observed data, namely $p_{\theta}(x)$,
parameterized by $\theta$. Given some prior over the latent variable
$p_{\theta}(z)$, we can introduce a variational distribution $q_{\phi}(z|x)$
with parameters $\phi$ that approximates an intractable posterior
$p_{\theta}(z|x)$. Since we cannot tractably estimate the exact likelihood, we
alternatively maximize the evidence lower bound (ELBO), which is a lower bound
to the log-likelihood of the data formulated as follows:

$\log p_{\theta}(x)\geq\mathbb{E}_{q_{\phi}(z|x)}[\log
p_{\theta}(x|z)]-\mathcal{D}_{KL}(q_{\phi}(z|x)||p_{\theta}(z))$ (6)

where the first term corresponds to the reconstruction loss (likelihood) and
the second term encourages the learned posterior to match a suitable prior of
the latent variables.

(a) VAE

(b) Normalizing Flow

(c) GAN

(d) Diffusion

Figure 4: Overview of generative models (a) Variational Autoencoder (VAE), (b)
Normalizing Flow, (c) Generative Adversarial Network (GAN), and (d) Diffusion
models

###  2.3 Normalizing Flows

Normalizing flows (Tabak & Turner (2013); Tabak & Vanden-Eijnden (2010);
Papamakarios et al. (2021)), shown in Figure 4(b), are an expressive and
general way of constructing flexible probability distributions over continuous
random variables. Flow-based models can transform a simple base distribution,
e.g., Gaussian, into a complex distribution that resembles the distribution of
the underlying data. Unlike VAEs and GANs, flow-based models explicitly learn
the data distribution and thus the loss function is simply the negative log-
likelihood. Consider the observed data $x$ and suppose, similar to the VAE,
that we would like to estimate the likelihood of this data. In flow-based
models, we express x as an invertible transformation of some real-valued
variable $z$ as follows:

$x=T(z)$ (7)

where $z$ is sampled from some distribution $p(z)$, known as the base
distribution of the flow-based model. Further, we require that $T$ is a
diffeomorphic function and that $z$ is the same dimension as $x$. The density
of $x$ is well-defined and can be estimated exactly by a change of variables:

$p(x)=p(z)|\det J_{T}(z)|^{-1}$ (8)

or, equivalently,

$p(x)=p(T^{-1}(x))|\det J_{T}(T^{-1}(x))|$ (9)

The Jacobian, which in practice can be efficiently computed by restricting it
to a triangular structure, is a $d\times d$ matrix consisting of all partial
derivatives of the transformation $T$

$J_{T}(z)=\begin{bmatrix}\frac{\partial T_{1}}{\partial
z_{1}}&\dots&\frac{\partial T_{1}}{\partial z_{d}}\\\ \vdots&\ddots&\vdots\\\
\frac{\partial T_{d}}{\partial z_{1}}&\dots&\frac{\partial T_{d}}{\partial
z_{d}}\end{bmatrix}$ (10)

Flows can be thought of as warping space to mold the base density $p(z)$,
which is usually a simple distribution such as a Gaussian, into a complex
distribution $p(x)$. The determinant of the Jacobian is the volumetric change
in $z$ due to the transformation $T$. Flow-based models are capable of both
generating new samples from the data distribution by sampling from the base
distribution and computing the forward transformation $T$ and exact density
estimation by computing the inverse transformation $T^{-1}$. Normalizing flows
are easier to converge but less expressive when compared to GANs and VAEs
because the transformations need to be invertible with a tractable Jacobian
determinant.

###  2.4 Generative Adversarial Networks

Generative adversarial networks (GANs) (Goodfellow et al. (2014)), shown in
Figure 4(c), are an example of implicit generative models, which are capable
of sampling from probability distributions but cannot learn the likelihoods
for data points. Thus, GANs are quite different than VAEs in their
applications. GANs utilize an adversarial training procedure to produce
realistic samples from distributions over a very high dimensional space, such
as images. To learn the generator’s distribution $p_{g}$ over data $x$, a
prior is defined on input noise variables $p(z)$ that is mapped to the data
space via $G(z)$, where $G$ is a differentiable function (implemented as a
multilayer perceptron) known as the generator network. The goal of $G$ is to
generate samples close to the input distribution $p(x)$. Another function
$D(x)$ (implemented as a multilayer perceptron), known as the discriminator
network, is defined to represent the probability that $x$ indeed came from the
data distribution $p_{\text{data}}$ rather than the generator $p_{g}$. The
discriminator $D$ is trained to maximize the probability of assigning the
correct label to both training examples from input distribution and samples
generated from $G$. Simultaneously, $G$ is trained to minimize
$\log(1-D(G(z)))$. Formally, $G$ and $D$ play the following minimax zero-sum
game:

$\min_{G}\max_{D}\;\mathbb{E}_{x\sim p_{\text{data}}(x)}[\log
D(x)]+\mathbb{E}_{z\sim p(z)}[\log(1-D(G(z)))]$ (11)

To optimize with respect to both networks, gradient descent is performed to
update the generator and gradient ascent is performed to update the
discriminator. GAN models are known for potentially unstable training and
relatively less diversity in generation due to the adversarial training
process.

###  2.5 Diffusion Models

Inspired by non-equilibrium statistical physics (Jarzynski (1997)), diffusion
models (Sohl-Dickstein et al. (2015); Ho et al. (2020)), shown in Figure 4(d),
were proposed as a likelihood-based approach to achieve both flexible model
structure and tractable estimation of probability distributions. Unlike VAEs,
diffusion models are learned with a fixed procedure, and the latent variable
has the same dimension as the original data. Diffusion models have shown
impressive results in image generation tasks, even outperforming GANs in many
cases (Dhariwal & Nichol (2021)). The idea of the diffusion model is to define
a Markov chain of diffusion steps to slowly destroy the structure in a data
distribution through a forward diffusion process by adding noise (Ho et al.
(2020)) and learn a reverse diffusion process that restores the structure of
the data. Some proposed methods, such as DDIM (Song et al. (2021a)), break the
Markov assumption111In the context of diffusion models, the Markov assumption
refers to the distribution over $x_{t}$ depending only on $x_{t-1}$ to speed
up the sampling in the diffusion process.

Forward Diffusion. Given some input data sampled from a distribution
$x_{0}\sim q(x)$, the forward diffusion process is defined by adding small
amounts of Gaussian noise to the sample in $T$ steps, thereby producing noisy
samples $x_{1},\dots,x_{T}$. The distribution of noisy sample $x_{t}$ is
defined as a conditional distribution as follows:

$q(x_{t}|x_{t-1})=\mathcal{N}(x_{t};\sqrt{1-\beta_{t}}x_{t-1},\beta_{t}I)$
(12)

where $\beta_{t}\in(0,1)$ is a variance parameter that controls the step size
of noise. As $t\to\infty$, the input sample $x_{0}$ loses its distinguishable
features. In the end, when $t=T$, $x_{T}$ is equivalent to an isotropic
Gaussian. From Eq. (12), we can then define a closed-form tractable posterior
over all time steps factorized as follows:

$q(x_{1:T}|x_{0})=\prod_{t=1}^{T}q(x_{t}|x_{t-1})$ (13)

Now, $x_{t}$ can be sampled at any arbitrary time step $t$ using the well-
known reparameterization trick (Kingma & Welling (2013), Rezende et al.
(2014)). Let $\alpha_{t}=1-\beta_{t}$ and
$\bar{\alpha}_{t}=\prod_{i=1}^{t}\alpha_{i}$:

$q(x_{t}|x_{0})=\mathcal{N}(x_{t};\sqrt{\bar{\alpha}_{t}}x_{0},(1-\bar{\alpha}_{t})I)$
(14)

Reverse Diffusion. In the reverse process to sample from $q(x_{t-1}|x_{t})$,
the goal is to recreate the true sample $x_{0}$ from a Gaussian noise input
$x_{T}\sim\mathcal{N}(0,I)$. Unlike the forward diffusion, $q(x_{t-1}|x_{t})$
is not tractable and thus requires learning a model $p_{\theta}$ to
approximate the conditional distributions as follows:

$\begin{split}p_{\theta}(x_{0:T})&=p(x_{T})\prod_{t=1}^{T}p_{\theta}(x_{t-1}|x_{t})\\\
p_{\theta}(x_{t-1}|x_{t})&=\mathcal{N}(x_{t-1};\mu_{\theta}(x_{t},t),\Sigma_{\theta}(x_{t},t))\end{split}$
(15)

where $\mu_{\theta}$ is learned via neural networks, typically a UNet
(Ronneberger et al. (2015)). Further, it turns out that conditioning on the
input $x_{0}$ yields a tractable reverse conditional probability

$q(x_{t-1}|x_{t},x_{0})=\mathcal{N}(x_{t-1};\tilde{\mu}(x_{t},x_{0}),\tilde{\beta}_{t}I)$
(16)

where $\tilde{\mu}$ and $\tilde{\beta}$ are the analytical mean and variance
parameters derived as a result of conditioning on $x_{0}$. We can learn the
reverse denoising process by optimizing the following variational lower bound:

$\mathbb{E}[\log
p_{\theta}(x_{0})]\geq-\mathbb{E}\bigg{[}\log\frac{q(x_{1:T}|x_{0})}{p_{\theta}(x_{0:T})}\bigg{]}=\mathcal{L}_{\text{VLB}}$
(17)

For a detailed derivation, see Sohl-Dickstein et al. (2015). Diffusion models
often require longer training and sampling times, making them computationally
expensive.

###  2.6 Disentangled Representation Learning

For the longest time, the efficacy of machine learning models relied on human
ingenuity and prior knowledge to perform feature engineering on data. The
realization that such engineering is inefficient and demands a more automated
approach gave rise to the field of representation learning (Bengio et al.
(2013)). Representation learning aims to extract meaningful information (or
features) from data that can be used downstream to build robust classifiers or
predictors. Thus, it is desirable to build representation-learning algorithms
that incorporate certain priors about the world to automatically learn useful
representations. In the probabilistic sense, which is what we focus on in this
survey, a good representation is one that should ideally learn the posterior
distribution of the underlying factors of variation for the observed input
distribution. Further, a representation that is disentangled such that each
factor is independently manipulable enables a robust representation useful for
downstream tasks such as learning fair and robust predictors (Locatello et al.
(2019a)).

The problem of representation learning can be linked to nonlinear ICA
(Hyvärinen & Pajunen (1999)). Suppose that our low-level observations
$x\in\mathbb{R}^{d}$ are explained by a small number of latent variables
$z=(z_{1},\dots,z_{n})$, where $n\ll d$ and $x$ is generated by applying an
injective (or sometimes diffeomorphic) map
$g:\mathbb{R}^{n}\to\mathbb{R}^{d}$, also called the mixing function, to $z$
such that

$x=g(z)$ (18)

In nonlinear ICA or disentangled representation learning, $z_{i}$ are assumed
to be mutually independent such that

$p(z_{1},\dots,z_{n})=\prod_{i=1}^{n}p(z_{i})$ (19)

Although there is no one commonly accepted definition of disentanglement,
Locatello et al. (2020) propose to formalize the notion in the following
general definition:

######  Definition 3 (Locatello et al. (2020))

The goal of disentangled representation learning is to learn a mapping
$g^{-1}(x)$ such that the effect of the different factors of variation is
axis-aligned with different coordinates. That is, each factor of variation is
associated with exactly one group of coordinates of $g^{-1}(x)$ and vice-
versa, where the groups are non-overlapping. Thus, varying one factor of
variation while keeping the others fixed results in a variation of exactly one
group of coordinates.

Several proposed VAE variants, such as $\beta$-VAE (Higgins et al. (2017)) and
FactorVAE (Kim & Mnih (2018)), have proposed objectives to disentangle the
learned latent codes via independence constraints. However, such models fail
to disentangle factors and do not address one major issue in representation
learning: unidentifiability of representations.

####  2.6.1 The Identifiability Problem

A significant barrier to representation learning is the identifiability
problem. Probabilistic generative models suffer from indeterminacy, which
refers to the situation where the underlying factors of variation cannot be
uniquely inferred from empirical data. Characterizing and reducing these
indeterminacies is the endeavor of identifiability. Identifiability is an
important property to achieve in representation learning since it guarantees
the stability and robustness of a model. When different representations can
explain the same observational data equally well and cannot identify the true
factors of variation, we say that the learned representation is
unidentifiable. In general, it is infeasible to always uniquely identify the
underlying factors and remove all indeterminacies since it is often task-
dependent. Thus, identifiable representation learning typically aims to
recover the true factors (and mixing function) up to tolerable ambiguities
(i.e., transformations, reordering, etc.). It is important to note that model
identifiability is an asymptotic property that can be achieved in its
strictest form only in the limit of an infinite amount of data (Xi & Bloem-
Reddy (2023)). However, weaker identifiability results often suffice to
recover ground-truth factors. The theory of identifiability stems from the
literature on nonlinear independent component analysis (Hyvärinen & Pajunen
(1999)). It has been shown that the identifiability of nonlinear ICA is, in
fact, not possible unless certain restrictions are made on the mixing function
or other auxiliary information is provided (Hyvarinen et al. (2018)).
Locatello et al. (2019b) extend this result to representation learning and
show that learning disentangled representations in an unsupervised manner
without any inductive biases is impossible. It turns out that the notion of
disentanglement and identifiability are actually quite related. Formally,
following Khemakhem et al. (2020) and Gresele et al. (2021), identifiability
is defined as follows:

(a) Darmois CDF Transform

(b) Rotated Gaussian

Figure 5: An illustration of (a) the Darmois construction and (b) rotated
Gaussian MPA that are counterexamples to blind-source separation of nonlinear
ICA. The Darmois CDF transform maps the data to independent intervals of a
uniform distribution. The Gaussian is invariant to rotations and thus
represents the same observational distribution after rotation (the left circle
and right circle are equivalent distributions).

######  Definition 4 (Identifiability)

Let $\mathcal{G}$ be the set of all smooth, mixing functions
$g:\mathbb{R}^{n}\to\mathbb{R}^{d}$, and $\mathcal{P}$ be the set of all
smooth, factorized densities $p(z)$. Let
$\Theta\subseteq\mathcal{G}\times\mathcal{P}$ be the domain of parameters and
let $\sim$ be an equivalence relation on $\Theta$. Then, the generative
process in (18) is said to be $\sim$-identifiable on $\Theta$ if

$\forall\theta,\tilde{\theta}\in\Theta:\qquad
p_{\theta}(x)=p_{\tilde{\theta}}(x)\implies\theta\sim\tilde{\theta}$ (20)

If the true model belongs to $\Theta$, then identifiability implies that any
model in $\Theta$ learned from infinite amounts of data will be
$\sim$-equivalent to the true model. In the context of the VAE222Recall that
the VAE model actually learns a full generative model
$p_{\theta}(x,z)=p_{\theta}(x|z)p_{\theta}(z)$ and an inference model
$q_{\phi}(z|x)$ which approximates its posterior $p_{\theta}(z|x)$. However,
we generally have no guarantees about what these learned distributions
actually are and only the marginal distribution over $x$ is meaningful., if
two different choices of model parameters $\theta$ and $\tilde{\theta}$ lead
to the same marginal density $p_{\theta}(x)$, then they are equal and the
joint distributions should match (i.e.,
$p_{\theta}(x,z)=p_{\tilde{\theta}}(x,z)$). If the joint distribution matches,
then we have found the correct priors $p_{\theta}(z)=p_{\tilde{\theta}}(z)$
and correct posteriors $p_{\theta}(z|x)=p_{\tilde{\theta}}(z|x)$. Practically
speaking, identifiability essentially guarantees that regardless of how many
times we train the representation learner, we will always obtain the same
unique solution (up to tolerable ambiguities). Thus, if identifiability didn’t
hold, learning representations would not be as useful for downstream tasks.

Linear ICA is well-known to be identifiable given non-Gaussian noise (Comon
(1994)). That is, there exists a unique unmixing function that maps the given
observations to a unique set of sources. However, in the general case,
nonlinear ICA is ill-defined and not equivalent to blind-source separation
(BSS). That is, in the $n=d$ case, different mixtures of $z_{i}$ and $z_{j}$
can be independent. Some classic counterexamples include the Darmois
construction (Darmois (1953); Hyvärinen & Pajunen (1999)) and rotated Gaussian
measure-preserving automorphisms (Gresele et al. (2021)). The following two
examples show why nonlinear ICA is unidentifiable in the general case and the
need for additional signals and strategies to resolve identifiability.

######  Example 1 (Darmois Construction - Gresele et al. (2021))

The Darmois construction $g^{D}:\mathbb{R}^{n}\to(0,1)^{n}$ is defined as the
recursive application of the following conditional CDF:

$g^{D}_{i}(x_{1:i})=p(X_{i}\leq
x_{i}|x_{1:i-1})=\int_{-\infty}^{x_{i}}p(x_{i}^{\prime}|x_{1:i-1})\;dx_{i}^{\prime}\qquad(i=1,\dots,n)$
(21)

The Darmois solution maps the marginal distribution over observations to a
uniform distribution using the CDF above, as shown in Figure 5(a). The
estimated sources are thus independent of the conditioning variables
$x_{1:i-1}$. Thus, $y^{D}=g^{D}(x)$ are mutually independent uniform random
variables that may not be meaningfully related to the true sources of
variation. Therefore, the Darmois solution is a counterexample to blind source
separation since we can always find another set of independent sources that
explain the data equally as well.

######  Example 2 (Rotated Gaussian MPA - Gresele et al. (2021))

Another class of counterexamples involves measure-preserving automorphisms
(MPA), which are identity functions on the sources that preserve the
distribution of the source variables. Let $R$ be an orthogonal matrix, and
denote by $F_{z}(z)=(F_{z_{1}}(z_{1}),\dots,F_{z_{n}}(z_{n}))$ and
$\Phi(z)=(\phi(z_{1}),\dots,\phi(z_{n}))$ the elementwise CDF of a smooth,
factorized density $p_{z}$ and of a Gaussian, respectively. Then, the rotated
Gaussian MPA $a^{R}(p_{z})$ is

$a^{R}(p_{z})=F_{z}^{-1}\circ\Phi\circ R\circ\Phi^{-1}\circ F_{z}$ (22)

$a^{R}(p_{z})$ maps to a standard Gaussian (rotationally invariant), then
applies a rotation, and finally maps back to the sources. In this case, there
exists a different set of independent sources through $a^{R}(p_{z})$ that
explain the data equally as well. Interestingly, principal component analysis
(PCA) is not identifiable precisely because it is a rotation of a Gaussian
through the orthogonal rotation matrices $U$ and $V$ obtained through singular
value decomposition of the data covariance matrix. Figure 5(b) shows the
rotational indeterminacy of Gaussian i.i.d. models, where each point in the
circle represents an equivalent solution. In other words, the joint
distribution will always be equivalent no matter what rotations are performed
on the Gaussian, so uniquely identifying the sources is hopeless.

Therefore, additional constraints and assumptions are necessary if we hope to
uniquely identify the factors of variation. The goal is to get closer to
recovering the true underlying factors up to some trivial transformation.
Under certain conditions, we can often recover the underlying factors up to
some affine transformation of the learned factors as formalized in Definition
5.

(a) Affine equivalence

(b) Permutation equivalence

Figure 6: A simple $2$-variable example of (a) affine equivalence vs. (b)
permutation equivalence, where $\tilde{z}_{i}$ denotes the learned latent
factor and $z_{i}$ denotes the ground truth underlying factor. Note that the
arrows in the diagram indicate mappings, not causal relations.

######  Definition 5 (Affine equivalence)

We say $\theta$ and $\tilde{\theta}$ are affine equivalent if

$(g,p_{z})\sim_{A}(\tilde{g},p_{\tilde{z}})\Longleftrightarrow\exists A\text{
s.t. }(g,p_{z})=(\tilde{g}\circ A^{-1},p_{\tilde{z}})$ (23)

where $A$ is a linear invertible transformation.

However, to disentangle the sources, we must go beyond affine equivalent
identifiability since it allows for any linear combination of learned factors
to recover underlying factors, which leads to entangled solutions. For most
settings, the best we can do is to recover the factors up to a simple
reordering (permutation) and scaling (elementwise reparameterization) of the
learned factors as formulated in Definition 6.

######  Definition 6 (Permutation equivalence)

We say $\theta$ and $\tilde{\theta}$ are permutation equivalent if

$(g,p_{z})\sim_{P}(\tilde{g},p_{\tilde{z}})\Longleftrightarrow\exists
P,h\text{ s.t. }(g,p_{z})=(\tilde{g}\circ h^{-1}\circ P^{-1},p_{\tilde{z}})$
(24)

where $P$ is a permutation matrix and $h(z)=(h_{1}(z_{1}),\dots,h_{n}(z_{n}))$
are invertible elementwise reparameterizations. A learned model
$\tilde{\theta}$ is said to be disentangled if $\theta\sim_{P}\tilde{\theta}$.

According to the above definition, it turns out that identifying latent
factors up to permutation (and possibly elementwise reparameterization) of the
ground-truth is equivalent to disentangling the factors of variation.
Componentwise equivalence is a special case of permutation equivalence where
$P=I$ (identity permutation). Block identifiability is a form of
identifiability where a group of latents is uniquely identifiable, but latents
within the group may not be. Since disentanglement is impossible to achieve
without some form of inductive bias, we next briefly discuss works that
propose additional weak supervision and other strategies to learn provably
identifiable representations. An illustrative example of affine and
permutation equivalence is shown in Figure 6.

####  2.6.2 Towards Identifiability in Disentangled Representation Learning

The natural question becomes: Can we utilize additional structure in data to
attain identifiability? The following approaches have been the most promising
strategies for dealing with identifiability challenges in representation
learning. Many of the identifiability results in disentangled and causal
representation learning utilize the general structure of the following
proposals.

Latent Conditioning. Based on Hyvarinen et al. (2018), Khemakhem et al. (2020)
propose a unified theory connecting VAEs to nonlinear ICA called identifiable
VAE (iVAE). The authors propose conditioning the latent variable on some
auxiliary information, such as time steps or class labels, with an exponential
family prior such that the underlying factors are rendered conditionally
independent:

$p_{T,\lambda}(z|u)=\prod_{i}p_{\theta}(z_{i}|u)=\prod_{i}h_{i}(z_{i})\exp\Bigg{[}\sum_{j=1}^{k}T_{i,j}(z_{i})\lambda_{i,j}(u)-\psi_{i}(u)\Bigg{]}$
(25)

where $h_{i}(z_{i})$ is the base measure, $T_{i}:\mathcal{Z}\to\mathbb{R}^{k}$
and $T_{i}=(T_{i,1},\dots,T_{i,k})$ are the sufficient statistics,
$\lambda_{i}(u)=(\lambda_{i,1}(u),\dots,\lambda_{i,k}(u))$ are the
corresponding natural parameters, $k$ is the dimension of each sufficient
statistic, and the remaining term $\psi_{i}(u)$ acts as a normalizing
constant. Consider the well-known Gaussian distribution, which is in the
exponential family. We have that the sufficient statistic is
$T_{i}(z)=(z,z^{2})$ and the base measure is $h_{i}(z)=\frac{1}{\sqrt{2\pi}}$.
The function $\lambda$ outputs the natural parameter vector for the
conditional distribution, which is
$(\lambda_{1},\lambda_{2})=(\frac{\mu}{\sigma^{2}},-\frac{1}{\sigma^{2}})$ in
the Gaussian case.

Contrastive Data Pairs. Inspired by multi-view nonlinear ICA (Gresele et al.
(2019)), Locatello et al. (2020) propose considering pairs of contrastive data
$(x_{1},x_{2})$ consisting of before and after changes (interventions) have
occurred to a system as an inductive bias for representation learning. The
generative model is defined as sampling two images from the causal generative
process with an intervention on a random subset of factors of variation. The
auxiliary information in this setting is the positive pair. Let $k$ be the
number of factors in which the two observations differ, $S$ denote the subset
of shared factors of size $n-k$, and $\bar{S}=[n]\setminus S$. Then, we have
the following generative process:

$p(z)=\prod_{i=1}^{n}p(z_{i}),\quad p(\tilde{z})=\prod_{i=1}^{k}p(z_{i}),\quad
S\sim p(S)$ (26) $x_{1}=g^{*}(z),\quad x_{2}=g^{*}(f(z,\tilde{z},S))$ (27)
$f(z,\tilde{z},S)_{S}=z_{S}\quad\text{and}\quad
f(z,\tilde{z},S)_{\bar{S}}=\tilde{z}$ (28)

where $g^{*}$ is a function that maps the latent variable to observations and
$f$ makes the relation between $x_{1}$ and $x_{2}$ explicit. Intuitively, when
generating $x_{2}$, $f$ selects entries from $z$ with index in $S$ and
replaces the remaining factors with $\tilde{z}$. Thus the two observations
$x_{1}$ and $x_{2}$ have the same shared factors indexed by $S$. Contrastive
learning has been recently shown to provably invert the data-generating
process (Zimmermann et al. (2021)).

Restricting Mixing Function. Gresele et al. (2021) and Reizinger et al. (2022)
propose a new theory called independent mechanism analysis (IMA) to learn
identifiable representations based on restricting the function class of the
mixing function and placing orthogonality constraints on the columns of the
Jacobian of the mixing function. Inspired by the Independence of Causal
Mechanisms (ICM) principle from causality (Scholkopf et al. (2021)), the
authors propose a new form of independence between the influences of sources
on the observation rather than statistical independence. Specifically, the
authors restrict the mixing function $f$ to satisfy mutual independence of
$\frac{\partial f}{\partial z_{i}}$ for all $i=\\{1,\dots,n\\}$. Formally, we
have that the mechanisms by which each source $z_{i}$ influences the observed
distribution are independent of each other in the sense that for all $z$:

$\log|J_{f}(z)|=\sum_{i=1}^{n}\log\Bigg{\|}\frac{\partial f}{\partial
z_{i}}(z)\Bigg{\|}$ (29)

This implies that the absolute value of the determinant of the Jacobian should
decompose as the product of the norms of $\frac{\partial f}{\partial z_{i}}$.
Reizinger et al. (2022) show that the evidence lower bound in VAEs converges
to a regularized version of the log-likelihood. The regularization term gives
VAEs the ability to perform independent mechanism analysis and serves as an
inductive bias for decoders with column-orthogonal Jacobians. The authors show
that the regularized objective enables the identifiability of the ground-truth
latent factors. Further, they show that VAEs satisfy the self-consistency
property in near-deterministic regimes (i.e., the VAE encoder inverts the
decoder even if the VAE is not exactly deterministic).

For a more rigorous treatment of identifiability and indeterminacies in
generative models, we refer the reader to Xi & Bloem-Reddy (2023).

Table 2: Summary of Identifiability Results of Causal Representation Learning Methods Approach |  Identifiability Result |  Key Assumptions |  Main Contributions
---|---|---|---
CausalVAE (§3.2.1) |  component-wise correspondence |  obs. labels, linear SCM, injective mixing |  Framework
DEAR (§3.2.2) |  component-wise correspondence |  obs. labels, non-linear SCM, injective mixing |  Framework
SCM-VAE (§3.2.3) |  component-wise correspondence |  obs. labels, non-linear additive noise SCM, injective mixing |  Framework
ICM-VAE (§3.2.4) |  mechanism perm. eq. & componentwise reparam. |  obs. labels, non-linear general SCM, diffeomorphic mixing |  Framework, Identifiability
Multi-domain CRL (§3.2.5) |  perm. eq. via linear ICA |  observational data from $d^{\prime}$ unpaired domains, injective mixing |  Identifiability
Interventional CRL (§3.3.1) |  perm. eq. given do-interventions |  interventional data from do-interventions, polynomial mixing |  Identifiability
Linear CRL (linear mixing) (§3.3.2) |  perm. eq. given 1-node perfect interventions |  interventional data, linear SCM, linear injective mixing |  Identifiability
Linear CRL (general mixing) (§3.3.3) |  perm. eq. given 1-node perfect interventions |  interventional data, linear SCM, general injective mixing |  Identifiability
Nonparametric CRL (§3.3.4) |  perm. eq. given 2-node perfect interventions |  heterogenous multi-environment interventional data, unknown interventions, diffeomorphic mixing |  Identifiability
CITRIS (§3.4.1) |  recoverability of minimal causal variables |  temporal obs. w/ known intervention targets, bijective mixing |  Framework, Identifiability
iCITRIS (§3.4.1) |  recoverability of minimal causal variables and parental sets |  temporal obs. w/ known intervention targets, bijective mixing |  Framework, Identifiability
BISCUIT (§3.4.2) |  perm. eq. given binary interaction pattern |  temporal obs. w/ action variable, unknown intervention targets, bijective mixing |  Framework, Identifiability
Mechanism Sparsity (§3.4.3) |  perm. eq. given sparse temporal/action causal graph |  temporal sequence w/ action variable, diffeomorphic mixing |  Identifiability
SSL (§3.5.1) |  block-ident. of content up to invertible function |  augmented counterfactual data, bijective mixing |  Identifiability
LCM (§3.5.2) |  perm. eq. given atomic perfect interventions |  paired counterfactual data, diffeomorphic mixing |  Framework, Identifiability

##  3 Causal Representation Learning

Until recently, most representation learning methods assumed the underlying
factors of variation to be independent (Khemakhem et al. (2020)) and that the
data is independently and identically distributed, which is often impractical
in real-world scenarios. The world is full of dynamically changing systems
consisting of complex interactions that induce correlations or causal
dependencies between factors. Träuble et al. (2021) show that existing methods
are not sufficient to disentangle the factors when the data is non-iid and
factors are correlated. Thus, a more realistic endeavor would be to design
models that assume the factors underlying a generative process are causally
related. Combining causal modeling and representation learning allows one to
develop machine learning models for high-dimensional inputs whose underlying
factors are governed by an SCM. The goal of causal representation learning is
to learn high-level causally related factors that describe meaningful
semantics of the data and their causal structure (Scholkopf et al. (2021)).
Semantically meaningful can refer to several properties of a representation,
such as robustness, fairness, transferability, interpretability, or
explainability. Causal representation learning (CRL) is an amalgamation of
multiple lines of work, including identifiable representation learning, causal
structure learning, and latent causal DAG learning. In this survey, we do not
cover causal discovery methods and refer the reader to other surveys that
comprehensively cover this landscape (Vowels et al. (2021)). The key feature
of causal variables is that they are variables on which interventions are
defined and whose relations are of interest. It is worth noting that
independent factors of variation are a special case of causal models where the
causal graph is trivial. In the following sections, we build up causal
representation learning from nonlinear ICA and discuss methods according to
their assumptions on the data-generating process: observational,
interventional, or counterfactual. Table 2 summarizes the identifiability
results, key assumptions, and main contributions of the causal representation
learning methods discussed.

###  3.1 Formulating Causal Representation Learning

####  3.1.1 Linking Causal Representations to Nonlinear Independent Component
Analysis

Recall that in traditional ICA, the ground-truth factors are assumed to be
statistically independent. For the setting of causal representation learning,
we diverge from the independent factors of variation assumption and suppose
that $z_{i}$ are causal variables that may be dependent. The joint
distribution for $z$ can thus be expressed as the following Markov
factorization

$p(z_{1},\dots,z_{n})=\prod_{i=1}^{n}p(z_{i}|z_{\textbf{pa}_{i}})$ (30)

induced by an underlying acyclic SCM
$\mathcal{C}=(\mathcal{Z},\mathcal{E},F,p_{\mathcal{E}})$ with jointly
independent noise $\epsilon_{i}$ and a set of causal mechanisms

$F=\\{z_{i}=f_{i}(z_{\textbf{pa}_{i}},\epsilon_{i})\\}$ (31)

The goal of causal representation learning is to (1) learn the causal
representation $z=g^{-1}(x)$, (2) the corresponding causal DAG, and (3) the
causal mechanisms $p(z_{i}|z_{\textbf{pa}_{i}})$ (Parascandolo et al. (2018);
Bengio et al. (2020)). Assuming a reduced form SCM where we recursively
substitute the structural assignments in the topological order of the causal
graph, we can write the latent variables $z$ directly as a function of the
noise

$z=f_{\text{RF}}(\epsilon)$ (32)

The mapping $f_{\text{RF}}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ has a lower
triangular Jacobian. So, we can write the mapping from noise to data space as
the following composition

$x=g\circ f_{\text{RF}}(\epsilon)$ (33)

Learning (32) can be seen as a structured form of nonlinear ICA with an
intermediate representation learned through $f_{\text{RF}}$ (Schölkopf & von
Kügelgen (2022)).

####  3.1.2 Principles of Causal Representations

The following two principles describe the key properties of causal
representations.

######  Principle 1 (Independent Causal Mechanisms (ICM) - (Scholkopf et al.
(2021)))

The causal generative process of a system’s variables is composed of
autonomous modules that do not inform or influence each other. In the
probabilistic case, this means that the conditional distribution of each
variable given its causes (i.e., its mechanism) does not inform or influence
the other mechanisms.

The independence of causal mechanisms is a statement about the algorithmic
independence between mechanisms that generate causal variables. That is, the
ICM principle implies the locality of interventions where changing a variable
only affects the causal mechanism for that variable. In other words, the
components, or mechanisms, are modular, autonomous, and reusable. Statistical
dependencies are a result of introducing unexplained random variables into the
system. However, a deterministic system consists of physical mechanisms. As a
result, an algorithmic model of causal structure can be devised in terms of
Kolmogorov complexity (Janzing & Schölkopf (2010)), which is the length of a
bit string’s shortest compression on a Turing machine and a measure of its
information content. However, the Kolmogorov complexity is known not to be
computable. Thus, we must resort to statistical and other information-
theoretic notions of quantifying independence.

Figure 7: Relationship between altitude and temperature (Peters et al. (2017))

A two-variable example of ICM Principle (Peters et al. (2017)). In the two-
variable case, where one is referred to as the cause and the other as the
effect, Definition 1 boils down to independence between the cause distribution
and the mechanism producing the effect from the cause. The factors are
independent in two senses: (i) intervening on one mechanism
$p(z_{i}|z_{\textbf{pa}_{i}})$ does not change the other mechanisms
$p(z_{j}|z_{\textbf{pa}_{j}})$ for $i\neq j$, and (ii) knowing information
about some mechanism $p(z_{i}|z_{\textbf{pa}_{i}})$ does not give us any
information about a mechanism $p(z_{j}|z_{\textbf{pa}_{j}})$ for $i\neq j$.
Suppose we are interested in analyzing average annual temperature $T$ vs.
altitude $A$ from different locations. Altitude and temperature are correlated
variables. Intuitively, we can infer that this correlation must arise from the
fact that changes in altitude cause changes in temperature. For example, at
higher altitudes, the temperatures tend to be lower. Figure 7 illustrates the
relationship between the altitude and temperature variable at a given
location. Suppose we collect two datasets from location R and location S.
Then, we can say that $p_{R}(A,T)\neq p_{S}(A,T)$ since the altitude marginal
distributions likely differ, i.e., $p_{R}(A)\neq p_{S}(A)$. However, the
conditionals (or mechanisms) that characterize the physical mechanism of
generating temperature from altitude, $p(T|A)$, will likely be almost
invariant to shifts in the marginal, i.e., $p_{S}(T|A)\approx p_{R}(T|A)$.
Thus, if we assume that $A\to T$, in the joint factorization
$p(A,T)=p(A)p(T|A)$, we can say that the mechanisms are independent and do not
influence each other. However, if we assume an alternate factorization, where
$T\to A$, namely $p(A,T)=p(T)p(A|T)$, we do not enjoy the same invariance
property. That is, it would not make sense for the temperature to cause a
physical change in the altitude and we would not observe much change in
altitude when intervening on temperature. Thus, the correct causal
factorization will always yield independent causal mechanisms.

######  Principle 2 (Sparse Mechanism Shift (SMS) - (Scholkopf et al. (2021)))

Small distribution changes tend to manifest themselves in a sparse way in the
Markov/disentangled factorization in Eq. (30), i.e., they should usually not
affect all factors simultaneously.

Given that the ICM principle holds, the SMS hypothesis is a statement about
distribution shifts. That is, distribution shifts can often be attributable to
sparse changes in causal conditionals of a causal disentangled factorization
(or interventions on causal variables). To learn models that are generalizable
to distribution shifts, it is desirable to learn causal representations due to
their invariance to distribution shifts. There have been several works that
use causal models to study distribution shift robustness (Lu et al. (2022))
and how to mitigate spurious correlations (Wang & Jordan (2021)).

Guided by these principles, a causal representation should consist of
independent mechanisms and must be robust to downstream tasks under
distribution shifts.

####  3.1.3 Intuition behind Causal Disentanglement

Until recently, most existing work formulated disentanglement from the
perspective of independent factors of variation. So, the notion of causal
disentanglement can seem counterintuitive or even contradictory. As mentioned
before, disentanglement is not a completely well-defined concept. One approach
could be to think about it in terms of simply enforcing statistical
independence among all the learned factors, as done in BetaVAE (Higgins et al.
(2017)). However, such an approach does not isolate the learned factors and is
effectively semantically meaningless; hence, the impossibility implication for
arbitrary unsupervised disentanglement (Locatello et al. (2019b)). Thus,
Definition 3 aims to formulate a consistent definition through a non-
overlapping condition on the information that each group of latent codes
encodes. Causal representations adhere to the principle of independent causal
mechanisms (Definition 1). Thus, causal disentanglement should be viewed
through the lens of the ICM principle. In a setting where the factors of
variation are causally related, the mechanisms that produce each causal
variable as a function of its parents should be independent. Thus, we must
ensure not only that each group of latent codes captures distinct information
but also the recoverability of each independent causal mechanism (Komanduri et
al. (2023)) and the causal structure among factors (if learned). In a setting
where the causal graph is learned jointly with the representation, the
recoverability of the causal relations can be formulated as a graph
isomorphism between the true causal model and the learned causal model
(Brehmer et al. (2022)).

####  3.1.4 Identifiability Problem in Causal Representation Learning

In causal representation learning, on top of identifying the latent
representation, the causal graph encoding their relations must also be
identifiable. The task of causal structure identification is already difficult
in the purely observational setting since we can only recover a DAG up to
Markov equivalence (Spirtes et al. (2000)). However, it gets significantly
more challenging when jointly learning latent causal variables. In the
following sections, we summarize work done towards achieving identifiable and
disentangled causal representations from various weak-supervision signals and
data-generating assumptions. We primarily focus our attention on VAE-based
models used to learn disentangled causal representations, but some methods
restrict the mixing function to be linear and thus only use matrix
decompositions. For motivation of the challenges we face in causal
representation learning, consider the following simple example that
illustrates unidentifiability.

######  Example 3 (Unidentifiability in CRL Simple Linear Example)

Suppose we have observations $x\in\mathbb{R}^{2}$ and latent variables
$z\in\mathbb{R}^{2}$ related by $x=Gz$, where $G$ is a linear mixing function.
For simplicity, let the causal variables be related by the following linear
additive noise structural causal model

$z=A^{T}z+\epsilon=(I-A^{T})^{-1}\epsilon,\qquad\epsilon\sim\mathcal{N}(0,I)$
(34)

where $A$ is the causal adjacency matrix. Suppose we have the following
solution to the system

$A=\begin{bmatrix}0&0\\\ -1&0\end{bmatrix},\qquad G=\begin{bmatrix}-2&2\\\
2&-1\end{bmatrix}\implies\Sigma=\begin{bmatrix}20&-16\\\ -16&13\end{bmatrix}$
(35)

where $\Sigma$ is the covariance of the observations $x$ derived as
$\Sigma=\mathbb{E}[xx^{T}]=G(I-A^{T})^{-1}(I-A^{T})^{-T}G^{T}$. Now, consider
the solution

$\hat{A}=\begin{bmatrix}0&0\\\
0&0\end{bmatrix},\qquad\hat{G}=\begin{bmatrix}-2&4\\\
2&-3\end{bmatrix}\implies\hat{\Sigma}=\begin{bmatrix}20&-16\\\
-16&13\end{bmatrix}$ (36)

We have that the covariance of the observation is identical for both solutions
(i.e., $\Sigma=\hat{\Sigma}$). In the first solution, we assume the structure
$z_{2}\to z_{1}$; in the second solution, we assume independent factors of
variation. However, we still obtain the same observational distribution. This
implies that the independent factors of variation represent observations
equally as well as the causally related factors, and we cannot identify a
unique solution.

The following sections outline causal representation learning methods that
rely on certain data-generating assumptions (i.e., observational,
interventional, counterfactual) and weak supervision signals to remedy the
above unidentifiability issue and achieve disentanglement of causal factors.
The majority of causal representation learning methods utilize VAE-based
frameworks and maximize a suitable ELBO loss.

###  3.2 Learning from Observational Data

One approach to learning causal representations relies on purely observational
data from $L_{1}$ of Pearl’s Causal Hierarchy. This could be in the form of
supervised labels or potentially from different sources. The following methods
utilize such auxiliary information to provably disentangle the causal factors
in this paradigm.

Figure 8: CausalVAE Architecture (Yang et al. (2021))

####  3.2.1 CausalVAE

Learning Framework. Yang et al. (2021) propose CausalVAE, a framework to learn
causal representations and causal structure simultaneously given auxiliary
information in the form of labels $y$ corresponding to causal factors of
interest $z$. CausalVAE assumes a linear SCM with additive noise describing
the structure between causal variables, parameterized by a causal adjacency
matrix $A$, as follows:

$z=A^{T}z+\epsilon=(I-A^{T})^{-1}\epsilon$ (37)

where $\epsilon\sim\mathcal{N}(0,I)$. CausalVAE proposes to encode high-
dimensional data into low-dimensional noise variables $\epsilon$ and utilize
an analytical mapping, $(I-A^{T})$, to project the noise variables to causal
variables $z$. The weighted adjacency matrix $A$ is learned via an acyclicity
constraint and the augmented Lagrangian method (Yu et al. (2019)). For
identifiability, the labels $y$ are incorporated into a conditional prior,
similar to Khemakhem et al. (2020), defined as follows:

$\begin{split}p(z|y)=\prod_{i=1}^{n}p(z_{i}|y_{i}),\qquad
p(z_{i}|y_{i})=\mathcal{N}(\lambda_{1}(y_{i}),\lambda_{2}^{2}(y_{i}))\end{split}$
(38)

where the latent variables $z$ are assumed to be mutually independent when
conditioned on their corresponding labels. The conditional prior is used to
regularize the posterior over the latent variables using the auxiliary labels
as a supervision signal. During inference, counterfactual samples can be
generated by encoding an image to an exogenous noise term, transforming it to
causal variables, intervening on a dimension of the causal representation,
propagating effects, and pushing it through the trained decoder to generate a
counterfactual instance. Although CausalVAE is capable of learning causal
representations and causal structure simultaneously, the linear SCM assumption
is often unrealistic in practice. Further, continuous optimization through the
acyclicity constraint (Zheng et al. (2018); Yu et al. (2019); Ng et al.
(2022)) does not necessarily guarantee an accurate learned DAG (recoverable
only up to a supergraph). The quality of the representation learned is
evaluated using the mutual information coefficient (MIC), a mutual information
measure, which does not capture the disentanglement of causal factors. The
overall design of the CausalVAE framework is shown in Figure 8.

Identifiability Result. The authors show that the labels enable the
identifiability of causal factors up to component-wise reparameterization.
They extend the identifiability result from Khemakhem et al. (2020) and show
the identifiability of their framework when given the auxiliary label
information.

####  3.2.2 DEAR

Learning Framework. Shen et al. (2022) propose disentangled generative causal
representation learning (DEAR), a more general bidirectional generative model
(BGM) to learn causal representations by adding a regularization term in the
form of a GAN loss (Figure 9). Similar to other approaches, the DEAR framework
requires access to all causal variables. They propose an SCM prior that can
derive causal variables based on known causal orderings and a supergraph of
the true causal graph. The authors propose the following cross-entropy loss-
based supervised regularizer to learn to predict each generative factor,
thereby taking the place of a Gaussian prior, and disentangle the causal
factors

Figure 9: DEAR Architecture (Shen et al. (2022))

$\mathcal{L}_{\text{sup}}=\sum_{i=1}^{n}\mathbb{E}_{x,z}[\text{CrossEntropyLoss}(y_{i},z_{i})]$
(39)

Similar to Yang et al. (2021) and Komanduri et al. (2023), each latent $z_{i}$
is assumed to be supervised by its corresponding annotated label $y_{i}$ of
each ground-truth factor, which they use to show the identifiability of latent
factors up to component-wise correspondence. To learn causal representations,
the authors propose to learn a nonlinear SCM prior, where causal orderings are
given apriori (similar to Komanduri et al. (2023)). The SCM is assumed to
follow a post-nonlinear additive noise model, with learnable parameters
$\beta=(f,h,A)$, as follows:

$z=F_{\beta}(\epsilon)=f((I-A^{T})^{-1}h(\epsilon))$ (40)

where $f$ and $h$ are elementwise nonlinear transformations and $f$ is assumed
to be invertible. The causal ordering of the supergraph is used as an
initialization for causal structure learning using the acyclicity constraint.
Unlike CausalVAE, DEAR does not learn intermediate noise encodings. Rather,
noise is arbitrarily sampled from a standard Gaussian and mapped to generate
causal variables through the learned structural assignments. The authors
propose a generative model $p_{\theta}(x,z)=p_{\theta}(x|z)p_{\beta}(z)$ and a
stochastic encoder $q_{\phi}(z|x)$ to learn the posterior over the latent
causal variables. The objective of the generative model is to minimize the
following KL-divergence

$\mathcal{L}_{gen}=\mathcal{D}_{KL}(q(x,z),p(x,z))$ (41)

with the following variational lower bound

$\mathbb{E}_{x\sim q_{x}}\mathbb{E}_{q(z|x)}[\log
p(x|z)+\mathcal{D}_{KL}(q(z|x)||p_{\beta}(z))]$ (42)

where $p_{\beta}(z)$ is the learned SCM prior over causal variables $z$. DEAR
uses a GAN-based method to adversarially estimate the gradients w.r.t the
encoder, decoder (generator), and SCM prior with respective learnable
parameters $\phi$, $\theta$, and $\beta$, through specified gradient formulas.
The overall loss objective of the generative model and supervised regularizer
can be formulated as follows:

$\mathcal{L}=\mathcal{L}_{gen}+\lambda\mathcal{L}_{sup}$ (43)

In order to tractably estimate the gradients of $\mathcal{L}_{gen}$, the
authors adopt an adversarial approach using GANs. They train a discriminator
$D$ using logistic regression and optimize the following objective

$\min_{D^{\prime}}\frac{1}{N_{d}}\Bigg{[}\sum_{i}\log(1+e^{-D^{\prime}(x_{i},z_{i})})+\sum_{i}\log(1+e^{D^{\prime}(x_{i},z_{i})})\Bigg{]}$
(44)

Identifiability Result. The authors provide theoretical analysis suggesting
that learning from an independent prior will always lead to an encoder
yielding an entangled solution. They also show the identifiability of DEAR up
to component-wise correspondences since they assume access to labels as
auxiliary information. Although representations learned using the DEAR method
are shown to induce high distributional robustness, there is no quantitative
evaluation of the disentanglement of causal factors.

####  3.2.3 SCM-VAE

Learning Framework. Komanduri et al. (2022) propose SCM-VAE, a framework for
learning causal representations assuming access to the causal structure. SCM-
VAE attempts to remedy the issues from CausalVAE by proposing to learn a post-
nonlinear additive noise SCM describing the structure between causal variables

$z_{i}=g(A_{i}\circ z)+\epsilon_{i},\quad\forall i\in\\{1,\dots,n\\}$ (45)

where $g$ is some nonlinear function. Further, $A$ is assumed to be a binary
unweighted adjacency matrix and $A_{i}$ is the $i$th column of $A$. SCM-VAE
incorporates a structural causal prior that induces a causal-like
factorization of labels

$p_{\theta}(z|y)=\prod_{i=1}^{n}p_{\theta}(z_{i}|y_{i},y_{\mathbf{pa}_{i}}),\quad
p_{\theta}(z_{i}|y_{i},y_{\mathbf{pa}_{i}})=\mathcal{N}\big{(}\lambda_{1}((A+I_{n\times
n})_{i}\odot y),\lambda_{2}^{2}((A+I_{n\times n})_{i}\odot y)\big{)}$ (46)

By leveraging a causally factorized structural causal prior based on the known
topological orderings of the causal graph, SCM-VAE addresses a concern of
CausalVAE, where the conditional factorized prior simply assumes mutual
independence among factors. One issue with SCM-VAE is that the learned
mechanisms are not guaranteed to be bijective, which leads to issues in
optimization and formulating the variational lower bound. Further, the
structural causal prior only factorizes the labels, which may be redundant and
can potentially induce entangled representations.

Identifiability Result. The authors show, similar to CausalVAE, that the
supervision signal yields a one-to-one correspondence between learned factors
and ground-truth factors. Thus, the causal factors are identifiable up to
component-wise reparameterization.

####  3.2.4 ICM-VAE

Figure 10: ICM-VAE Architecture (Komanduri et al. (2023))

Learning Framework. Komanduri et al. (2023) extend the work from Khemakhem et
al. (2020) and Yang et al. (2021) and propose ICM-VAE, a framework of causal
representation learning under supervision from labels. The ICM-VAE is based on
learning a nonlinear diffeomorphic mapping $f_{RF}:\mathcal{E}\to\mathcal{Z}$
with causal mechanisms of the form

$z_{i}=f_{i}(\epsilon_{i};z_{\textbf{pa}_{i}})=\alpha(z_{\textbf{pa}_{i}})\epsilon_{i}+\beta(z_{\textbf{pa}_{i}})$
(47)

where $\alpha$ and $\beta$ are the neural network parameterized scale and
shift parameters of a general nonlinear affine-form autoregressive flow,
respectively. The diffeomorphic mechanisms are learned via a conditional
autoregressive flow mapping a base noise distribution to a complex
distribution over causal variables, similar to CAREFL (Khemakhem et al.
(2021)). This choice is motivated by the efficient and exact likelihood
estimation and the expressiveness of flow-based models in low-dimensional
settings. The theory from Khemakhem et al. (2020) is extended to include
causal mechanism equivalence towards a principled definition of causal
disentanglement. That is, in a causal model, the standard notion of
identifiability that yields marginal distribution equivalence is not
sufficient to capture the equivalence of each individual mechanism between two
models. To this end, the authors propose a causal disentanglement prior with a
similar structure to the temporal prior from Lippe et al. (2023a) to causally
factorize the latent space and learn causally disentangled mechanisms. The
prior is formulated as follows:

$p_{\theta}(z|y)=\prod_{i=1}^{n}p_{\theta}(z_{i}|z_{\textbf{pa}_{i}},y_{i})=\prod_{i=1}^{n}p(y_{i})\bigg{|}\frac{\partial\lambda_{i}(y_{i};z_{\textbf{pa}_{i}})}{\partial
y_{i}}\bigg{|}^{-1}$ (48)
$p_{\theta}(z_{i}|z_{\textbf{pa}_{i}},y_{i})=h_{i}(z_{i})\exp(T_{i}(z_{i})\lambda_{i}(A_{i}\odot
z,y_{i})-\psi_{i}(z,y))$ (49)

where $\lambda$ is defined as an autoregressive causal flow that derives the
natural parameters of $z_{i}$ as a function of the label $y_{i}$ and its
parents $z_{\textbf{pa}_{i}}$. Note that in this causally factorized prior,
each factor $z_{i}$ can be identified due to the access to auxiliary
information $y_{i}$. The ICM-VAE framework is shown in Figure 10.

Identifiability Result. The authors define a new notion of causal mechanism
disentanglement and show that the causal factors are identifiable up to
permutation and elementwise reparameterization. The intuition is that
identifiable models only guarantee the equivalence of the marginal
distribution of the data and latent variables up to some tolerable ambiguity.
However, the theory from Khemakhem et al. (2020) ignores the case where latent
variables could be causally related and induces a Markov factorization. The
authors in this work propose causal mechanism identifiability as a
reformulation of permutation-equivalent identifiability to take into account
causal conditionals in the Markov factorization. That is, causal mechanism
identifiability requires the equivalence of causal mechanisms up to some
tolerable ambiguity and is a sufficient condition for disentangling the causal
factors. The mechanism equivalence definition proposed by Komanduri et al.
(2023) is fundamentally different than the IMA theory introduced by Gresele et
al. (2021) since IMA falls into the mixing function restriction class of
methods rather than latent conditioning (Khemakhem et al. (2020)).

Figure 11: Unpaired Multi-domain Causal Representation Learning (Sturma et al.
(2023))

####  3.2.5 Learning from Multi-Domain Observational Data

Problem Setting. Sturma et al. (2023) study the setting where one has access
to observational data from multiple domains that may share an underlying
causal representation. The observations from different domains are assumed to
be unpaired (i.e., only the marginal distribution of each domain is observed
but not the joint distribution). The setup is that the latent variables $z$
are sampled from some distribution $p(z)$, where $p(z)$ is determined by an
unknown SCM among latent variables. In each domain
$e\in\\{1,\dots,d^{\prime}\\}$, some domain-specific causal latents
$z_{S_{e}}$ are mapped to the observed vector $x^{(e)}$ via a domain-specific
(injective) mixing function $g_{e}$ such that

$x^{(e)}=g_{e}(z_{S_{e}})$ (50)

where $S_{e}\subseteq\mathcal{H}$ is a subset of indices. The general idea is
that shared latent variables capture the key causal relations, and different
domains give combined information about the relations. That is, the multi-
domain setup “completes the picture” of the causal model in some sense since
relations can arise from different domains, as shown in Figure 11.

Identifiability Result. The authors provide sufficient conditions for uniquely
identifying the joint distribution of the causal factors and the causal graph,
where the causal variables are described by a linear SCM. Assuming that (1)
the marginal distributions are non-degenerate, non-symmetric, with unit
variance, a genericity assumption that the errors must be pairwise different,
and (2) the (linear) mixing function is full-rank, the authors show
identifiability up to permutation. The first assumption restricts symmetric
distributions and ensures that the distribution of errors is non-Gaussian.
This allows for the application of the linear ICA identifiability theory from
Comon (1994). The second assumption requires that for each shared latent node
there is at least a node in every domain such that the latent node is a parent
of the domain-specific node, as illustrated in Figure 11.

###  3.3 Learning from Static Interventional Data

Instead of assuming access to observed labels or only observational data,
several approaches leverage interventional data from $L_{2}$ of Pearl’s
hierarchy to learn identifiable causal representations. The following methods
assume access to interventional data under perfect or imperfect interventions.
This can often be a realistic assumption in practice since several domains,
such as robotics, medicine, and genomics, have an abundance of interventional
data available.

####  3.3.1 Interventional Causal Representation Learning

Problem Setting. Ahuja et al. (2023) explore to what extent access to
interventional data can facilitate causal representation learning. The main
contribution of this work is identifiability results in the specified
observational and interventional data setting.

Observational Setting. The observational data-generating process they consider
is as follows:

$z\sim p(z),\qquad x=g(z)$ (51)

where $x$ is an observational data point rendered from the underlying latent
$z$ via an injective decoder $g$. The authors show that in the purely
observational setting, we can identify the latent factors up to affine
transformation (affine equivalence) if the mixing function is a finite-degree
polynomial. Further, if we assume that the latents have independent support
(Wang & Jordan (2021)), we can identify the factors up to permutation, shift,
and scaling.

Interventional Setting. The interventional data-generating process is given by

$z\sim\prod_{i\neq j}p(z_{i}|z_{\textbf{pa}_{i}})\delta_{Z_{j}=z_{j}},\qquad
x=g(z)$ (52)

where $z$ is sampled from the distribution under intervention on causal factor
$z_{j}$. Access to arbitrary interventional data is a more realistic scenario
since interventions often act sparsely in the real world, and changes in
causal conditionals lead to distribution shifts. Modeling distribution shifts
through access to data from interventional distributions should intuitively be
a sufficient weak supervision signal to learn accurate causal representations.

Identifiability Result. If we also assume access to interventional data from
perfect do-interventions, the authors show identifiability of the intervened
factors up to permutation, shift, and scaling and potentially up to
componentwise correspondence if the mixing function is assumed to be
diffeomorphic. The result suggests that intervened-upon latents are
identifiable up to permutation and the remaining latents up to affine
transformation. This can be formulated as optimizing the following constrained
autoencoder objective that performs reconstruction under the constraint that
an arbitrary latent has been fixed to a certain unknown value

$\min_{f,h}\;\mathbb{E}\Bigg{[}\|h\circ f(x)-x\|^{2}\Bigg{]},\text{ s.t.
}f_{i}(x)=\tilde{z}$ (53)

where $(f,h)$ is an encoder-decoder pair and $f_{i}(x)$ is the $i$th latent
factor with intervened value $\tilde{z}$. However, if we only assume soft
interventions (i.e., imperfect interventions), the authors claim
identifiability of the factors up to block affine transformation.

####  3.3.2 Linear Causal Representation Learning with Linear Mixing Function

Problem Setting. Squires et al. (2023) study identifiability and causal
disentanglement when one has access to unpaired interventional data with
single node interventions. The authors consider a setting with a linear
injective mixing function $G$ (i.e., $x=Gz$) with a pseudoinverse $H$ and a
linear Gaussian additive noise structural causal model. The data-generating
setup in this work considers $n$ latent variables that are generated according
to a linear SCM. Additionally, following the common assumption in
identifiability that variables are observed across multiple contexts, the
authors define several contexts (i.e., environments) that are a result of an
intervention on a causal variable. These contexts are indexed by a variable
$k\in\\{0,\dots,K\\}$, where $k=0$ is the observational context and $k>0$
refers to an interventional context. In this setting, the mixing function is
assumed to be invariant across contexts. The use-case considered in this work
is modeling the internal state of a cell. There are complex interactions
between the concentration of proteins, location of organelles, etc, and each
context is an exposure to a different small molecule. Each molecule has a
highly influential effect, changing only one cellular mechanism.

Identifiability Result. The main identifiability result in this work shows
that having access to an intervened context for each of the $n$ causal
variables is a sufficient condition for identifiability. Furthermore, in the
worst case, they show that at least one intervention per latent causal
variable is, in fact, a necessary condition for identifiability. The intuition
behind CRL unidentifiability is shown in Example 3. If we extend this example
to multiple contexts by defining a context-specific noise scaling such that
the SCM is $z=A_{k}^{T}z+\Omega_{k}\epsilon$, where $A_{k}$ is the weighted
adjacency matrix in context $k$ and $\Omega_{k}$ is diagonal with positive
entries, it is clear that without interventional contexts for each latent
variable, we collapse to a spurious ICA solution. That is, two different
solutions $(\\{A_{k}\\}_{k=0}^{<n+1},G,\\{\Omega_{k}\\})$ and
$(\\{\hat{A}_{k}\\},\hat{G},\\{\hat{\Omega}_{k}\\}_{k=0}^{<n+1})$ can produce
the same observational distribution (covariance of the observed data $X$) in
all contexts. Thus, the authors show that any setting with access to less than
$n+1$ observed contexts renders the latent factors unidentifiable. In Example
3, this means that we must observe at least $3$ contexts: one for the
observational, one for an intervention on $z_{1}$, and one for an intervention
on $z_{2}$ to identify the latent factors. In the general case, where the
covariance of the data $x$ is rank-deficient (i.e., $n<d$), the authors
propose to decompose the pseudoinverse of the covariance matrix of $x$ in each
context via RQ decomposition to recover the mixing function. With the
aforementioned assumptions, the authors show that the latent factors and
invariant mixing function can be recovered up to permutation and scaling.

####  3.3.3 Learning Linear Causal Representations with General Mixing
Function

Problem Setting. Buchholz et al. (2023) study causal representation learning
where the mixing function is assumed to be general (e.g., deep neural
networks) instead of restricting it to be linear (Squires et al. (2023)) or
polynomial (Ahuja et al. (2023)). Consider a data-generating process with $n$
causal variables, where one has access to an observational distribution and
$n$ unpaired interventional distributions (i.e., observed marginal
distributions), each a result of an intervention (unknown target) on exactly
one causal variable. The latent variables are assumed to be described by a
linear Gaussian SCM and the mixing function is injective. The setting
considered in this work is quite similar to Squires et al. (2023) and
generalizes the results to arbitrary mixing functions.

Learning Framework. To implement interventional causal representation
learning, the authors propose a novel contrastive learning approach for
interventional learning. The goal is to train a deep neural network to
distinguish observational samples from interventional samples. Although the
identifiability results suggest that a model learning from the aforementioned
data-generating process can properly disentangle the factors of variation,
practical models are still lacking. The authors design a variational
autoencoder-based approach that still relies on paired counterfactual data to
tractably estimate the log-likelihood of the data.

Identifiability Result. The main identifiability result in this work shows
that the latent factors are identifiable up to permutation and scaling. Three
main assumptions are required to keep the soundness of the proposed
identifiability theory. First, the number of interventions must be at least
the dimension of the number of causal factors. A central assumption is that
there must be at least one intervention on each causal factor. This assumption
has been made in several other works such as Ahuja et al. (2023); Brehmer et
al. (2022); Squires et al. (2023). The authors show that this is a necessary
condition since a violation of this property renders even the weakest form of
identifiability to break down. Second, the interventions must not be pure
shift interventions. The authors show that if interventions are pure shift
(relation to parents stays the same and only mean is changed), any causal
graph is compatible with the observations and will induce an indistinguishable
model (i.e., spurious solution). Lastly, similar to Yang et al. (2021), the
latent variables are assumed to follow a linear SCM with Gaussian noise.
Although post-nonlinear additive noise models have good approximability, an
interesting direction is to explore identifiability when assuming nonlinear
and non-additive noise models (Brehmer et al. (2022); Komanduri et al.
(2023)). However, it is not clear that the same theory would hold in the
nonlinear setting.

####  3.3.4 Learning from Heterogeneous Multi-Environment Data with Unknown
Interventions

Figure 12: Nonparametric multi-environment setup with single node perfect
interventions and shared mixing function, where $z$ are the learned factors
and $v$ are the underlying ground-truth (von Kügelgen et al. (2023)). The red
color indicates the intervened causal variable. We can see that the learned
model (right) identifies the factors up to a permutation of the ground-truth
factors (on the left).

Problem Setting. von Kügelgen et al. (2023) consider learning from
heterogeneous data from multiple related distributions that arise from
interventions in a shared underlying causal model, where intervention targets
are unknown. They consider a nonparametric setup with no assumptions on the
form of the causal model. Since only a dataset of i.i.d observations cannot
yield identifiability, the authors propose learning from multiple environments
given access to heterogeneous data from multiple distinct distributions. This
multi-environment data satisfies the assumption that certain parts of the
causal generative process are shared across environments so that the
environments do not vary arbitrarily. All environments share the same
invariant mixing function and underlying SCM, and any distribution shift
occurs as a result of interventions on some causal mechanisms. Such an
assumption is also consistent with the SMS hypothesis from Definition 2. The
shifts across environments from this principle can be formulated as follows:

######  Definition 7 (von Kügelgen et al. (2023))

Each environment $e$ shares the same mixing function $g$ and each $p^{e}$
results from the same SCM through an intervention on a subset of mechanisms
$\mathcal{I}^{e}\subseteq[n]$:

$p^{e}(z)=p^{e}(z_{1},\dots,z_{n})=\prod_{i\in\mathcal{I}^{e}}p^{e}(z_{i}|z_{\textbf{pa}_{i}})\prod_{j\notin\mathcal{I}^{e}}p(z_{j}|z_{\textbf{pa}_{j}})$
(54)

where intervention targets $\mathcal{I}^{e}$ are unknown.

Learning Framework. The authors propose potential approaches for learning
causal representations from finite interventional datasets sampled from
$\\{p^{e}_{x}\\}_{e\in E}$, where $E$ is the collection of all environments.
For instance, a VAE could be used to learn an encoder, causal graph, and
intervention targets, similar to Brehmer et al. (2022), with $z_{i}\perp E\mid
z_{\textbf{pa}_{i}}$ for unintervened $i$. Another approach is to use
normalizing flows to learn the mixing function and causal mechanisms.

Identifiability Result. The authors highlight that imperfect interventions,
which change the mechanism but preserve parental dependence, are insufficient
for full identifiability, as shown by Brehmer et al. (2022). This is because
arbitrary imperfect interventions that preserve parents and only change the
noise term produce a spurious ICA solution, which would be indistinguishable
from the ground truth. Thus, they consider only perfect interventions (i.e.,
$\forall e\text{ and
}i\in\mathcal{I}^{e},p^{e}(z_{i}|z_{\textbf{pa}_{i}})=p^{e}(z_{i})$). In the
two-variable case, they show that single-node perfect interventions are enough
to identify the causal factors up to permutation. Figure 12 shows a simple
$2$-variable causal graph $v_{1}\to v_{2}$, where $v_{1}$ and $v_{2}$ are the
ground-truth causal variables. In this case, if one has an environment as a
result of an intervention on $v_{1}$ and an environment as a result of an
intervention on $v_{2}$, we can learn latent factors $z_{1}$ and $z_{2}$ that
are identifiable up to permutation of the ground truth factors $v_{1}$ and
$v_{2}$ and their causal structure. For a general number of latent causal
variables, access to pairs of environments corresponding to two distinct
perfect interventions on each node is enough to guarantee identifiability up
to permutation and elementwise reparameterization.

######  Example 4 (Two-variable environment setup)

In the $n=2$ case, we have 3 possible environments
$\\{p^{e_{i}}_{x}\\}_{i=0}^{2}$, where $e_{0}$ is the observational
environment and for $i\neq 0$, $e_{i}$ is the environment induced by a perfect
intervention on one of the two variables. That is, it could be $v_{1}$ or
$v_{2}$ since the intervention target is unknown.

######  Example 5 ($n$-variable environment setup)

In the general $n>2$ variable case, we need two perfect interventions to fully
identify the causal factors. If $n=4$, we have $2n$ environment pairs from two
perfect interventions. That is, we would need the following $8$ pairs:
$e^{\prime}=\\{(e_{i},e^{\prime}_{i})\\}_{i=1}^{n}$ and
$e^{\prime\prime}=\\{(e_{i},e^{\prime\prime}_{i})\\}_{i=1}^{n}$, where
$e^{\prime}_{i}$ is the result of an intervention on one of the $n$ variables
and $e^{\prime\prime}_{i}$ is the result of another distinct intervention on
the same variable.

###  3.4 Learning from Temporally Intervened Data

Sometimes, causality can have a temporal interpretation. That is, causal
factors in a dynamic system may evolve over time and be influenced by their
values at previous time steps. The following methods consider learning causal
representations in the temporal setting by leveraging interventional data in
the form of intervention targets (i.e., an indicator of the factor being
intervened on) from $L_{2}$ of Pearl’s hierarchy.

####  3.4.1 Causal Identifiability for Temporally Intervened Sequences

(a) CITRIS (Lippe et al. (2022b))

(b) iCITRIS (Lippe et al. (2023a))

Figure 13: The data-generating process of (a) CITRIS and (b) iCITRIS

Problem Setting. Lippe et al. (2022b) present causal identifiability for
temporally intervened sequences (CITRIS), a temporal setting for causal
representation learning. The authors consider a data-generating process with
access to a sequence of images $\\{x_{t}\\}_{t=1}^{T}$ that represent temporal
changes (interventions) to a system. The key assumption is that the causal
variables at time step $z^{t+1}$ depend on some subset of causal variables at
the previous time step $z^{t}$ along with an intervention target $I^{t+1}$
(specifying which latent is intervened on). Assuming $n$ multidimensional
causal variables $(z_{\psi_{1}}^{t},\dots,z_{\psi_{n}}^{t})_{t=1}^{T}$
following a dynamic Bayesian network with structural assignments of the form
$z^{t}_{\psi_{i}}=f(z^{t}_{\psi_{i}^{\textbf{pa}}},\epsilon)$, each latent
factor $z^{t}_{i}$ is assigned to one of the $n$ causal variable sets via an
assignment function $\psi$ learned via Gumbel-Softmax per latent variable
(sampling latent-to-causal variable assignment). A set $z^{t}_{\psi_{0}}$ is
also reserved for unintervened factors that do not affect anything (noise).
The causal factors are assumed to satisfy causal sufficiency and the induced
distribution satisfies faithfulness. The data-generating process of CITRIS is
shown in Figure 13(a).

Figure 14: CITRIS/iCITRIS VAE Framework

Learning Framework. CITRIS is a VAE-based method that utilizes a prior over
latent variables conditioned on the corresponding intervention target to learn
to disentangle the causal factors. This ensures that intervened variables are
no longer dependent on parents while unintervened variables preserve their
parental dependencies. The observed data is of the form
$\\{x^{t},x^{t+1},I^{t+1}\\}$. Notably, we have that the intervention targets
are assumed to be known.

The temporal causal structure is time-invariant, so the parents are the same
between all time steps. The authors also propose the notion of “minimal causal
variables”, which are latents in each causal set that split into invariable
(independent of interventions) and variable (sensitive to interventions)
parts. In other words, we are trying to find the minimum factors that are
sensitive to interventions. Further, the following factorized prior
distribution, called the transition prior, consisting of the current latent
conditioned on the latent at the previous time step and intervention target,
is proposed to disentangle the causal factors

$p_{\phi}(z^{t+1}|z^{t},I^{t+1})=\prod_{i=0}^{n}p_{\phi}(z^{t+1}_{\psi_{i}}|z^{t},I^{t+1}_{i})$
(55)

To include instantaneous effects, the authors extend CITRIS and propose
iCITRIS (Lippe et al. (2023a)), which can handle instantaneous causal
relations within a timestep when given perfect interventions with known
intervention targets (e.g., flicking a light switch and the bulb turning on
can happen instantaneously). In this setting, both temporal and instantaneous
causal relations exist, as shown in Figure 13(b). To disentangle and identify
the causal factors, the authors propose the following causally factorized
transition prior

$p_{\phi,A}(z^{t+1}|z^{t},I^{t+1})=p_{\phi}(z^{t+1}_{\psi_{0}}|z^{t})\prod_{i=1}^{n}p_{\phi}(z^{t+1}_{\psi_{i}}|z^{t},z^{t+1}_{\psi^{\textbf{pa}}_{i}},I^{t+1}_{i})$
(56)

where $A$ denotes the instantaneous causal graph. The VAE framework for CITRIS
and iCITRIS is shown in Figure 14.

Identifiability Result. The identifiability analysis shows that although all
causal factors are not identifiable in this setting, the minimal causal
variables (intervention-sensitive components) are uniquely identifiable if we
condition each latent variable on exactly one intervention target (in addition
to instantaneous parents for iCITRIS). In this setting, the authors show that
for every pair of causal variables, exactly one must be intervened on to
achieve identifiability. If both causal variables are simultaneously
intervened on or neither are intervened on, we lose the ability to identify
the factors.

####  3.4.2 Binary Interactions for Causal Identifiability (BISCUIT)

Problem Setting. Lippe et al. (2023b) propose Binary Interactions for Causal
Identifiability (BISCUIT), a method for causal representation learning from
videos of dynamic interactive systems. The setup for BISCUIT is quite similar
to CITRIS, where we have $n$ causal variables
$(z_{\psi_{1}}^{t},\dots,z_{\psi_{n}}^{t})_{t=1}^{T}$ in a dynamic Bayesian
network, with an assignment function $\psi$ mapping latent factors to causal
groups, as shown in Figure 15(a). The authors assume that an agent’s
interaction with a causal variable can be described by an unknown binary
interaction variable similar to the action variable in Lachapelle et al.
(2022). Under this assumption, the causal variables of an environment are
shown to be identifiable. BISCUIT is an extension of the CITRIS (Lippe et al.
(2022b)) framework to the setting where intervention targets are unknown.

Learning Framework. Given a temporal setting, BISCUIT is a VAE that encodes
$x^{t+1}$ and $x^{t}$ to the latent space where each causal variable is
encoded in a separate set of dimensions. The prior over the latent variables
is conditioned on the previous frame and a known regime variable $R^{t}$
consisting of action information. Given the regime variable at each time step,
a binary interaction target for each latent causal variable is predicted via a
multilayer perceptron (MLP) that takes as input the action information and
representation from the previous frame. This can be viewed as each causal
variable having an observational ($I_{i}=0$) and interventional ($I_{i}=1$)
mechanism. The transition prior for the model is defined as follows:

$p_{\omega}(z^{t+1}|z^{t},R^{t+1})=\prod_{i=1}^{m}p_{\phi}(z^{t+1}_{\psi_{i}}|z^{t},\text{MLP}_{\omega}^{I_{i}}(R^{t+1},z^{t}))$
(57)

where $m$ is the dimension of the latent space such that $m\gg n$ and $\omega$
denotes the parameters of the MLP. The “regime” variable and representation of
the previous time step can be considered the auxiliary information that we
condition on to achieve identifiability, consistent with Khemakhem et al.
(2020). Thus, BISCUIT disentangles the causal factors. Their identifiability
result mainly relies on the binary interaction variable $I_{i}^{t}$, which
gives a distinct interaction pattern for a causal variable and a minimum of
$\lfloor\log_{2}n\rfloor+2$ actions. This means that an agent interacts with
each causal variable in a distinct pattern and does not always interact with
any two causal variables at the same time. If $I_{i}^{t}=0$, they claim the
unidentifiability result of nonlinear ICA (Hyvärinen & Pajunen (1999)). The
overall VAE framework for BISCUIT is shown in Figure 15(b). Similar to other
causal representation learning models, during inference, BISCUIT supports the
manipulation of the latent codes to generate counterfactual instances.

(a) BISCUIT (Lippe et al. (2023b))

(b) BISCUIT VAE Framework

Figure 15: BISCUIT (Lippe et al. (2023b))

Identifiability Result. In short, the authors show that if two models have the
same likelihood, each causal variable has a distinct interaction pattern, and
sufficient dynamics and temporal variability conditions are met (similar to
Lachapelle et al. (2022); Komanduri et al. (2023)), then a learned model
identifies the true model up to permutation and componentwise invertible
transformation.

####  3.4.3 Mechanism-Sparsity Regularization

Problem Setting. Lachapelle et al. (2022) propose a new theory of
identifiability from the perspective of temporal dynamics. They explore a
similar setting to Lippe et al. (2022b), where they consider a sequence of
observations $\\{x^{t}\\}_{t=1}^{T}$ that can be explained by semantically
meaningful latent variables $\\{z^{t}\\}_{t=1}^{T}$ and a sequence of
corresponding auxiliary variables in the form of actions
$\\{a^{t}\\}_{t=1}^{T}$ taken by an agent (i.e., an action that changes the
state of the latents). Following the form of the conditionally factorial prior
from Eq. (25) (Khemakhem et al. (2020)), the authors propose a conditional
prior where latent factors at a given time step are conditionally independent
given their state at the previous time step and the action variable as
follows:

$p(z^{t}|z^{<t},a^{<t})=\prod_{i=1}^{n}p(z_{i}^{t}|z^{<t},a^{<t})$ (58)
$p(z_{i}^{t}|z^{<t},a^{<t})=h_{i}(z_{i}^{t})\exp\\{T_{i}(z_{i}^{t})^{T}\lambda_{i}(A_{i}^{z}\odot
z^{<t},A_{i}^{a}\odot a^{<t})-\psi_{i}(z^{<t},a^{<t})\\}$ (59)

where $A^{a}$ denotes the action graph specifying the action variables that
are the causal parents of $z$, $A^{z}$ specifies the causal parents of $z^{t}$
from the latents $z^{t-1}$ at the previous time step, $\psi_{i}$ is the
normalizing constant, and $h_{i},T_{i},\lambda_{i}$ are defined as in Eq.
(25). The authors propose a VAE-based method with the defined conditional
prior to regularize latent mechanisms to be sparse via $A^{z}$ and $A^{a}$,
which act as masks to select the direct parents of $z^{t}$. The key assumption
behind their approach is that the relations between causal variables across
time and the effect of action variables on the latent variables are both
sparse. One limitation of their exploration is ignoring the case where there
could be instantaneous causal effects within a time step, as considered by
Lippe et al. (2023a).

Identifiability Result. Under sufficiently varying natural parameter functions
with respect to both temporal latent factors and actions, the authors show
permutation-equivalent identifiability (disentanglement) of latent factors
when only the mean is learned, the decoder is diffeomorphic onto its image,
and the learned connectivity graphs are sufficiently sparse, which can be
enforced by a sparsity constraint in causal discovery.

###  3.5 Learning from Paired Counterfactual Data

Another class of methods assumes access to counterfactual data from $L_{3}$ of
Pearl’s hierarchy to aid in learning identifiable causal representations. The
following methods assume counterfactual data augmentations or contrastively
paired counterfactual data as weak supervision to learn causal
representations. Works such as Gresele et al. (2019) and Locatello et al.
(2020) have taken a multi-view approach to nonlinear ICA and have indicated
that learning a representation that represents multiple different views of the
same underlying factors is a good inductive bias to learn disentangled
representations.

####  3.5.1 Self-supervised Learning with Data Augmentations

Figure 16: Problem formulation of self-supervised learning for causal
representation learning

Problem Setting. von Kügelgen et al. (2021) study the role of data
augmentations in self-supervised learning (SSL), specifically in the context
of identifiable representation learning. SSL with data augmentations is
closely related to identifiability in the multi-view setting (Gresele et al.
(2019); Locatello et al. (2020)), where we are given a second view $\tilde{x}$
of some observation $x$, resulting from a modified version $\tilde{z}$ of the
underlying factors $z$. In this work, instead of applying transformations at
the observation level (i.e., on $x$), as typically done in self-supervised
learning, the authors propose a process of applying transformations in the
representation space on $z$, where $z=g^{-1}(x)$, to obtain a modified
$\tilde{z}$ which is decoded to obtain the augmented image $\tilde{x}$.

The authors propose a VAE-based objective. First, we train an encoder $g^{-1}$
using neural networks to obtain the representations $(z,\tilde{z})$ for the
pair $(x,\tilde{x})$ and use a negative L2 loss as a similarity measure. The
authors propose splitting the latent variable into a content part $c$, which
would be invariant to augmentations and thus preserve semantic information,
and a style part $s$, which captures the stochastic variation from the
augmentations. Diverging from the typical assumption of independent factors of
variation, the authors assume that there can exist any nontrivial statistical
or causal dependencies in the latent space. Thus, a causal view of the data
augmentation process is counterfactuals generated under soft style
interventions, as shown in Figure 16. Based on this assumption, we can define
an underlying SCM on the content and style latents, with causal graph $c\to
s$, as follows:

$c=f_{c}(u_{c}),\qquad s=f_{s}(c,u_{s}),\qquad(u_{c},u_{s})\sim
p_{u_{c}}\times p_{u_{s}}$ (60)

where $u_{c},u_{s}$ are independent exogenous noise variables, and
$f_{c},f_{s}$ are deterministic functions. Then, The causal latents can be
decoded into observations $x=g(c,s)$. Given the factual observation
$x^{F}=g(c^{F},s^{F})$, which resulted from $(u_{c}^{F},u_{s}^{F})$, we may
ask the counterfactual question: “What would have happened if the style
variables had been randomly perturbed without changing anything else?” Thus,
we can consider a soft (imperfect) intervention on the style variable $s$,
which changes the mechanism $f_{s}$ as follows:

$\textbf{do}(s=\tilde{f}_{s}(c,u_{s},u_{A}))$ (61)

where $u_{A}$ is an additional source to account for the randomness of the
augmentation. After decoding this intervened latent variable, we obtain a
distribution over counterfactual observations $x^{CF}=g(c^{F},s^{CF})$ that
can be computed from the modified SCM by fixing the exogenous noise variables
to their factual values and performing the soft intervention as follows:

$c^{CF}=c^{F},\qquad s^{CF}=\tilde{f}_{s}(c^{F},u_{s}^{F},u_{A}),\qquad
u_{A}\sim p_{u_{A}}$ (62)

Identifiability Result. In this setting, the authors prove that we can
identify the invariant content partition $c$ under a weaker set of assumptions
compared to existing identifiability results from nonlinear ICA. In this work,
rather than disentangling each factor of variation, the focus is to
disentangle the content from style (i.e., block identifiability). This
theoretical guarantee has seen empirical successes in downstream tasks such as
domain generalization and out-of-distribution applications (Chen et al.
(2021); Mahajan et al. (2021)).

####  3.5.2 Latent Causal Models

Problem Setting. Brehmer et al. (2022) consider contrastively paired
counterfactual data and extend the work by Locatello et al. (2020) to the
causal setting where the data pairs $(x,\tilde{x})$ represent a system before
and after a random unknown, perfect, and sparse intervention on a causal
variable, respectively. They propose Latent Causal Models (LCMs) and consider
the following weakly supervised data-generating process.

######  Definition 8 (Weakly-supervised counterfactually paired generative
process)

The weakly supervised generative process of counterfactual paired data
$(x,\tilde{x})$ is defined as follows:

$\begin{split}\epsilon\sim p_{\mathcal{E}},\qquad z=s(\epsilon),\qquad
x\leftarrow g(z)\\\ I\sim p_{\mathcal{I}},\qquad\forall i\in
I,\tilde{\epsilon}_{i}\sim p_{\tilde{\mathcal{E}}_{i}},\qquad\forall i\notin
I,\tilde{\epsilon}_{i}=\epsilon_{i},\qquad\tilde{z}=\tilde{s}_{I}(\tilde{\epsilon}),\qquad\tilde{z}=s(\tilde{\epsilon}),\qquad\tilde{x}\leftarrow
g(\tilde{z})\end{split}$ (63)

where $\epsilon$ is the pre-intervention noise, $\tilde{\epsilon}$ is the
post-intervention noise, $z$ is the pre-intervention causal variables,
$\tilde{z}$ is the post-intervention causal variables, $p_{I}$ is the
distribution over intervention targets, $g$ is the mixing function, and $s$ is
a diffeomorphic solution function that maps the noise variables to causal
variables.

Figure 17: Implicit Latent Causal Model (ILCM) (Brehmer et al. (2022))

Learning Framework. Implicit LCM (ILCM). For the implicit, noise-based
formulation, the pair $(x,\tilde{x})$ is mapped to a pair of noise encodings
$(\epsilon,\tilde{\epsilon})$ using a projective noise encoder to learn a
posterior $p_{\theta}(\epsilon,\tilde{\epsilon}|x,\tilde{x})$. In this
formulation, $\tilde{\epsilon}$ is equivalent to the SCM noise term that would
have generated the post-intervention causal variable $\tilde{z}$ under the
unintervened causal mechanisms. They propose learning an intervention encoder
$p_{\theta}(I|\epsilon,\tilde{\epsilon})$ to predict the (single) intervention
target based on the dimension-pair with the minimum KL-divergence. The pair of
noise encodings are then restricted to differ only in the dimensions of the
intervention target. The remaining dimensions are forced to be the same using
different averaging schemes (Locatello et al. (2020); Hosoya (2019);
Bouchacourt et al. (2018)). Then, the pre-intervention noise is mapped to the
causal variable $z$ via a conditional flow $s$. The causal structure is
inferred based on conditional independence testing to simulate causal parents.
The causal structure is extracted from the learned causal variables after
training the LCM. The noise-centric formulation aims to infer the noise
distribution of the post-intervention data by performing a random intervention
on the causal variable $z$ (based on the predicted intervention target) to
yield $\tilde{z}$ and learning an inverse flow prior to then infer the noise
variable $\tilde{\epsilon}$ for intervened dimensions. The post-intervention
noise prior is defined as follows:

$p(\tilde{\epsilon}|\epsilon,I)=\prod_{i\notin
I}\delta(\tilde{\epsilon}_{i}-\epsilon_{i})\prod_{i\in
I}\tilde{p}(\bar{z}_{i})\bigg{|}\frac{\partial\bar{z}_{i}}{\partial\tilde{\epsilon}_{i}}\bigg{|}$
(64)

where $\bar{z}_{i}=s(\tilde{\epsilon}_{i};\epsilon_{\textbf{pa}_{i}})$ with
base density $\tilde{p}(\bar{z}_{i})=\mathcal{N}(0,1)$ and $\delta$ is the
Dirac delta function with a delta peak where the pre and post-intervention
noise variables are equivalent. The overall architecture of the ILCM is shown
in Figure 17.

Explicit LCM (ELCM). The authors also propose an explicit latent causal model
that does not encode an intermediate noise variable and parameterizes all
components of the SCM. In this formulation, the authors consider only the
causal variables as the latents. The idea is to map the pair $(x,\tilde{x})$
directly to causal latents $(z,\tilde{z})$ via a stochastic encoder $q(z|x)$
and decoder $p(x|z)$. Further, the authors propose a prior $p(z,\tilde{z})$
that consists of a parameterized causal graph and causal mechanisms. The model
is called as an explicit LCM as it directly parameterizes all components of an
LCM and contains an explicit representation of the causal graph and causal
mechanisms.

For both formulations, the authors utilize interventional causal discovery
algorithms, such as ENCO (Lippe et al. (2022a)), as a post-processing step to
extract the latent causal structure from the learned causal variables. The
authors note although optimally trained ELCMs correctly identify the causal
structure and disentangle causal variables on simple dataset, jointly learning
explicit graph and representations presents a challenging optimization
problem.

Identifiability Result. The main identifiability result shows that LCMs are
identifiable up to relabeling and elementwise reparameterizations of the
causal variables under weak supervision from contrastively paired data
samples. The authors prove this by leveraging graph isomorphisms between the
true and learned causal models and showing that the entailed weakly supervised
distributions must be equivalent. The main assumption behind the
identifiability result is that paired data counterfactual data is available
and interventions are perfect.

###  3.6 Other Works

Related to the methods discussed in this section, there have been alternate
formulations of causal disentanglement in representation learning. Suter et
al. (2019) consider a causal perspective of disentangled representation
learning and define the notion of a disentangled causal process as shown in
Figure 18. In this setting, the generative factors of variation
$z_{1},\dots,z_{n}$ are independent when conditioned on a set of confounders
$C$.

Later, Reddy et al. (2022b) extend the disentangled causal process from Suter
et al. (2019) to encoders (to study interventions) and generators (to study
counterfactuals) of latent variable models. Further, the authors propose
concrete metrics to account for confounding in the generative process. Induced
spurious correlations among generative factors in observational data are often
due to the existence of confounders. The authors propose two metrics to
evaluate causal disentanglement in their setting: unconfoundedness and
counterfactual generativeness. The unconfoundedness metric evaluates if the
model is able to map the generative factor to a unique set of latent codes.
The counterfactual generativeness metric is formulated as an average causal
effect between counterfactuals generated by intervened factors and those
generated by unintervened factors. That is, this metric ensures that only
interventions on the desired group of latent variables influence the generated
image.

Figure 18: Disentangled Causal Process (Suter et al. (2019))

Based on the disentangled causal process definition, Wang & Jordan (2021) lay
out desiderata for unsupervised disentanglement in representation learning.
They propose an independence of support score (IOSS) as a metric to
disentangle representations learned from observational data. The IOSS is
defined as the Hausdorff distance, which measures the distance between two
subsets of a metric space, between the joint support of the representation
$\\{z_{1},\dots,z_{n}\\}$ and the product of each individual latent’s
support333Note that support refers to the set of possible values a
representation can take. (i.e.,
$d_{H}(\text{supp}(z_{1},\dots,z_{n}),\text{supp}(z_{1})\times\text{supp}(z_{2})\times\dots\times\text{supp}(z_{n}))$).
This condition essentially enforces that each factor must take on a distinct
set of values (i.e., the set of values that a factor can take does not depend
on the values that a different factor can take). Enforcing IOSS can be seen as
ensuring orthogonality between groups of latent factors and thus satisfying
the disentanglement definition in Definition 3. The authors utilize the IOSS
to prove the permutation-equivalent identifiability of representations under
certain conditions.

Table 3: Summary of Controllable Counterfactual Generation Methods Approach |  Structure |  Key Features
---|---|---
CausalGAN (§4.1.1) |  GAN |  causal generators, interventional and counterfactual sampling
CGN (§4.1.2) |  GAN |  learn invariant shape mechanism, counterfactual generation
CONIC (§4.1.3) |  GAN |  counterfactual generation method to break confounding
DSCM (§4.2.1) |  Flow/Variational Inference |  tractable counterfactual inference
CausalHVAE (§4.2.2) |  VAE |  counterfactual inference, latent mediation, and support for direct, indirect, and total effect
Diff-SCM (§4.3.1) |  Diffusion |  counterfactual inference via anti-causal classifier guidance in bivariate setting

##  4 Controllable Counterfactual Generation

The objective of controllable generation is to produce novel samples based on
specified attributes rather than simply mimicking the characteristics of the
training data in non-controllable generation. However, their ability to
extrapolate beyond the training distribution is constrained because of their
reliance on sampling from observational distribution. A different but related
objective in causal generative modeling is controllable counterfactual
generation. Controllable counterfactual generation incorporates encoded causal
knowledge that can further generate samples in previously unobserved contexts
and enable users to edit specific samples by intervening on targeted
attributes, with downstream causal effects automatically incorporated.
Although causal representation learning methods are also capable of
controllable generation through latent space manipulations and decoding, the
focus is primarily on learning robust and interpretable representations for
downstream tasks. Counterfactual generation methods usually assume access to
ground-truth causal information and thus have the advantage of not being
plagued by unidentifiability issues. In contrast to representation learning,
controllable counterfactual generation models often utilize GANs or diffusion
models to generate realistic counterfactual images. Thus, such models are
often more suited for real-world applications, where domain knowledge can be
exploited to describe the causal dependencies of a system. We cover the main
fundamental controllable counterfactual generation approaches proposed in
recent years. We do not cover application-specific methods in this survey.

###  4.1 GAN-based

GANs have shown good performance in tasks involving sampling from data
distributions. Although the standard GAN does not encode any causal
information, it turns out we can use generators to learn mechanisms between
causal variables and formulate mechanism learning as an adversarial problem.
The following methods focus on utilizing GAN components to learn causal
mechanisms from label space to image space and enable counterfactual
generation.

####  4.1.1 CausalGAN

Problem Setting. As a modification of the ConditionalGAN (Mirza & Osindero
(2014)) to support sampling from interventional distributions, Kocaoglu et al.
(2018) propose CausalGAN. Traditional GANs have shown impressive performance
in generating samples from a given data distribution. GANs are an example of
implicit generative models, which can sample from data distributions but
cannot estimate likelihoods for data points. However, GANs cannot sample from
interventional distributions. Assuming that we have a dataset with labels,
mutually independent labels can be thought of as a trivial causal graph. In
this case, the ability to intervene on a variable and get novel samples is
desirable. We assume the labels cause the image, as shown in Figure 19(b). For
instance, if color and species are independent labels, then color=blue should
give us blue eagles and blue jays. However, if labels are causally related,
such as gender $\to$ mustache, conditioning on the mustache will only give us
images of males. However, intervening on mustache will give us males and
females with mustaches. Interventions preserve ancestral distributions,
whereas conditioning doesn’t.

The main idea of CausalGAN is to model each label using a generator neural
network that uses the causal parents of the label as its inputs. In this
setting, the authors assume access to the complete causal graph among the
labels. Since jointly training the labels and image generators is infeasible,
the authors propose a two-stage framework: (1) train a generative model over
only labels $y$ and (2) use the node labels in the causal graph to train a
conditional generative model for images.

Learning Framework.
Stage 1. First, the authors propose to learn the Causal Controller, a causal
implicit generative model between labels $y$ using a feedforward neural
network for variables such that each variable is conditioned on the outputs of
its parents’ neural network satisfying structural equations of the form
$v=f(v_{\text{pa}},\epsilon)$. The arrangement of the causal variable neural
networks for the graph $y_{1}\rightarrow y_{3}\leftarrow y_{2}$ is shown in
Figure 19(a). This process can be split into two subtasks. First, we learn a
causal implicit generative model over a small set of variables and then learn
the remaining set of variables conditioned on the first set using a
conditional generative network. Each node in the first set should come before
any node in the second set. The Causal Controller module decides what
distribution the image will be sampled from (interventional or observational)
and generates labels according to the causal graph. For example, if we want to
sample from the observational distribution, we can specify the values of each
label. If we want to sample from the interventional distribution, we can fix
the value of a specific label and propagate causal effects via the feedforward
neural networks. This module is pre-trained using a Wasserstein GAN (Arjovsky
et al. (2017)) to produce binary class labels with a specific probability.

(a) CausalGAN generators for causal graph

(b) CausalGAN data-generating process

Figure 19: CausalGAN (Kocaoglu et al. (2018))

Stage 2. Let $p_{data}^{l}$ be the real data distribution between the image
$x$ and the binary label $l\in\\{0,1\\}$. Similarly, let $p_{g}^{l}$ denote
the joint distribution between the labels given to the generator and the
generated images. We also assume a perfect Causal Controller such that
$p_{data}^{1}=p_{g}^{1}=\rho$, where $\rho$ is the probability of class label
$l$. Conditioned on the label distribution from the Causal Controller, the
CausalGAN learns an implicit class conditional generative model for images.
There are three main components in the CausalGAN.

  1. 1.

Labeler. The Labeler discriminator $D_{LR}$ is trained to estimate the labels
of real images in the training dataset. The Labeler discriminator module
solves the following optimization objective

$\max_{D_{LR}}\;\rho\mathbb{E}_{x\sim
p^{1}_{data}(x)}[\log(D_{LR}(x))]+(1-\rho)\mathbb{E}_{x\sim
p^{0}_{data}(x)}[\log(1-D_{LR}(x))]$ (65)

  2. 2.

Anti-Labeler. The Anti-Labeler discriminator $D_{LG}$ is trained to estimate
the labels of the images sampled from the generator (labels produced by Causal
Controller). The Anti-Labeler is especially used to prevent label-conditioned
mode collapse. The Anti-Labeler module solves the following optimization
objective

$\max_{D_{LG}}\;\rho\mathbb{E}_{x\sim
p^{1}_{g}(x)}[\log(D_{LG}(x))]+(1-\rho)\mathbb{E}_{x\sim
p^{0}_{g}(x)}[\log(1-D_{LG}(x))]$ (66)

  3. 3.

Generator. The generator produces realistic images conditioned on labels from
the Causal Controller by minimizing the Labeler loss and avoids unrealistic
image distributions that are easily labeled by maximizing the Anti-Labeler
loss, thereby distinguishing between classes and images conditioned on those
classes.

For a fixed generator, the discriminator $D$ solves the following:

$\max_{D}\mathbb{E}_{x\sim p_{data}(x)}[\log(D(x))]+\mathbb{E}_{x\sim
p_{g}(x)}\Big{[}\log\Big{(}\frac{1-D(x)}{D(x)}\Big{)}\Big{]}$ (67)

For a fixed discriminator, Labeler, and Anti-Labeler, the generator solves the
following:

$\begin{split}\min_{G}\mathbb{E}_{x\sim
p_{data}(x)}[\log(D(x))]+\mathbb{E}_{x\sim
p_{g}(x)}\Big{[}\log\Big{(}\frac{1-D(x)}{D(x)}\Big{)}\Big{]}\\\
-\rho\mathbb{E}_{x\sim
p^{1}_{g}(x)}[\log(D_{LR}(X))]-(1-\rho)\mathbb{E}_{x\sim
p^{0}_{g}(x)}[\log(1-D_{LR}(X))]\\\ +\rho\mathbb{E}_{x\sim
p^{1}_{g}(x)}[\log(D_{LG}(X))]-(1-\rho)\mathbb{E}_{x\sim
p^{0}_{g}(x)}[\log(1-D_{LG}(X))]\end{split}$ (68)

During inference time, one can sample from the interventional label
distribution produced by the Causal Controller and generate novel
counterfactual images unseen during training.

####  4.1.2 Counterfactual Generative Networks (CGN)

Problem Setting. Sauer & Geiger (2021) propose counterfactual generative
networks (CGN), a generative model aimed at modeling images as a realization
of shape, texture, and background variables, where spurious signals are strong
predictors of the class label. In contrast to CausalGAN, CGN aims to obtain
classifiers that are interpretable and robust to spurious correlations that
arise as a result of some confounding variable $C$. The authors design a model
that learns independent causal mechanisms $m=f_{shape}(\cdot)$,
$f=f_{texture}(\cdot)$, $b=f_{background}(\cdot)$, and an analytical
composition mechanism $g$ that combines the three mechanisms as follows:

$\hat{x}=g(m,f,b)=m\odot f+(1-m)\odot b$ (69)

Given some Gaussian noise and labels $y$ sampled uniformly as weak supervision
(labels for high-level concepts like shape and background are hard to obtain),
the architecture consists of BigGAN (Brock et al. (2019)) backbone to model
each mechanism as a generator. The overall data-generating process of CGN is
illustrated in Figure 20.

Figure 20: Counterfactual Generative Networks data generating process

Learning Framework. A loss is specified for each of the mechanisms, which are
then combined together by a composite function (analytical, not learned) to
generate the image as a combination of shape, texture, and background. To
incorporate some form of a supervisory signal, the architecture proposes
simultaneously learning a conditionalGAN to generate ground-truth images. A
corresponding reconstruction loss is also defined between generated images
from mechanisms and those from the conditionalGAN. After training the CGN, the
authors propose learning an invariant classifier that is invariant of any
spurious features such as texture and background and only relies on shape
changes for prediction tasks by using counterfactually generated data to
predict mechanism-specific labels. Take the canonical example of images of
cows standing on a field. Through counterfactually generated data from the
trained CGN model, the invariant classifier is effectively trained to use only
shape attributes to predict labels. This is consistent with preventing the
occurrence of spurious associations in distribution shifts, such as predicting
‘cow’ based on background spurious features. The authors empirically evaluate
their framework on variants of MNIST and ImageNet, which are benchmarks, and
their results seem to show that they are able to randomly sample noise and
labels to controllably generate realistic counterfactuals. Since CGN utilizes
a BigGAN (Brock et al. (2019)) architecture to train each of the generative
mechanisms, it can be computationally demanding, inefficient, and not
generalizable. Further, it does not take into account confounding effects.

####  4.1.3 Counterfactual Generation under Confounding (CONIC)

Figure 21: CONIC data generating process

Problem Setting. Reddy et al. (2022a) study the adversarial effect of
confounding in observational data on a classifier’s performance and how data
augmentation using counterfactual data can be used to mitigate these
confounding effects. The authors assume a causal generative process, based on
Suter et al. (2019) and Reddy et al. (2022b), consisting of generative factors
$z_{1},\dots,z_{n}$ (e.g., background, shape, texture, etc.) and some label
$y$, that determine an observation $x$ through some unknown mechanism $f$.
These variables are considered to be confounded by a set of confounders
$C_{1},\dots,C_{m}$. The generative factors here are arbitrary and can be
viewed as a generalization of the shape, background, and texture features from
Counterfactual Generative Networks (Sauer & Geiger (2021)), and perhaps have
more potential to robustly capture important information about the generative
process. The authors propose Counterfactual Generation under Confounding
(CONIC) to remove the effect of confounding, which can lead to spurious
correlations, by performing counterfactual data augmentation.

Learning Framework. To quantify uncertainty, a connection can be made between
the amount of confounding and the correlation between any pair of generative
factors in the causal process from Figure 21. The authors propose a KL-
divergence metric between the conditional and post-interventional distribution