# CITRIS: Causal Identifiability from Temporal Intervened Sequences

Phillip Lippe  Sara Magliacane  Sindy Löwe  Yuki M. Asano  Taco Cohen
Efstratios Gavves

###### Abstract

Understanding the latent causal factors of a dynamical system from visual
observations is considered a crucial step towards agents reasoning in complex
environments. In this paper, we propose CITRIS, a variational autoencoder
framework that learns causal representations from temporal sequences of images
in which underlying causal factors have possibly been intervened upon. In
contrast to the recent literature, CITRIS exploits temporality and observing
intervention targets to identify scalar and multidimensional causal factors,
such as 3D rotation angles. Furthermore, by introducing a normalizing flow,
CITRIS can be easily extended to leverage and disentangle representations
obtained by already pretrained autoencoders. Extending previous results on
scalar causal factors, we prove identifiability in a more general setting, in
which only some components of a causal factor are affected by interventions.
In experiments on 3D rendered image sequences, CITRIS outperforms previous
methods on recovering the underlying causal variables. Moreover, using
pretrained autoencoders, CITRIS can even generalize to unseen instantiations
of causal factors, opening future research areas in sim-to-real generalization
for causal representation learning.

Causal Representation Learning; Identifiability; Normalizing Flows;
Variational Autoencoders; Generalization

##  1 Introduction

Figure 1: From a sequence of images with interventions (orange), CITRIS learns
a normalizing flow, which maps a pretrained autoencoder’s entangled latent
representation to a causal representation. In this, multiple latent dimensions
(blue blocks) can represent a single causal factor, where causal relations
exist over time.

Causal representation learning (Schölkopf et al., 2021; Locatello et al.,
2020a; Yang et al., 2021; Khemakhem et al., 2020a; Lachapelle et al., 2022)
focuses on learning representations of causal factors from high-dimensional
observations, such as images. Commonly, in learning causal representations,
causal factors are assumed to be scalars. As settings become more complex and
high-dimensional, however, so do the causal dynamics, where estimating every
scalar causal variable becomes impractical. Consider for instance a set of
objects interacting in a three-dimensional space. For each object, we can
describe its position by three variables $x,y,z$ and its rotation in multiple
angles. However, it is often more natural and sufficient to consider those
factors as two _multidimensional_ variables, _i.e._ position and rotation,
especially when the definition of the default axes is ambiguous. Such
multidimensional causal factors are reminiscent of macrovariables explored
before in causality (Chalupka et al., 2016a, b, 2015; Hoel et al., 2013;
Höltgen, 2021), however, not in causal representation learning yet.

Hence, different from previous causal representation learning approaches, we
consider causal factors as potentially multidimensional vectors in this paper.
This requires a representation model to learn a latent space of disentangled
factors of variation, where latent variables of the same causal factor are
grouped together. To identify these multidimensional causal variables, we use
sequences of observations where interventions with known targets may have been
performed at any time step. This setup resembles a reinforcement learning
environment or an interactive real-world setting, with an agent performing
actions over time representing interventions on multidimensional causal
factors. However, instead of learning a policy to optimize a reward function,
we aim at using this setting to identify the causal factors. We assume we can
observe the intervention targets in our data, which occurs for instance when
we have access to a set of observation and action trajectories. We refer to
this setup as TempoRal Intervened Sequences (TRIS).

Under this setting and common assumptions like an invertible observation
function and a stationary causal process, we can identify the minimal causal
variables, which only model the information of a causal factor that is
strictly affected by a provided intervention. With this, we identify the
factors which can be directly and independently influenced by, _e.g._ , the
action of an agent, and thus of the most practical relevance. Meanwhile, all
information that cannot be assigned to a causal factor with certainty and
hence not directly influenced, is collected in a separate group of latent
variables. As a practical implementation of this, we propose CITRIS for Causal
Identifiability from TempoRal Intervened Sequences. CITRIS is a variational
autoencoder that learns an assignment of latent variables to causal factors,
and promotes disentanglement by conditioning each latent’s prior distribution
only on its respective intervention target. In experiments on realistically
rendered video datasets, CITRIS is able to find and disentangle the causal
factors with high accuracy. Moreover, we extend CITRIS to pretrained
autoencoders. By using a normalizing flow (Rezende & Mohamed, 2015), CITRIS
learns a mapping from the entangled autoencoder representation to a
disentangled causal representation, see Figure 1. We empirically show that the
normalizing flow can even generalize its disentanglement to unseen
instantiations of causal factors, holding promise for future work on
generalization of causal representations.

Our contributions are summarized as follows:

  * •

We show that multidimensional causal factors can be identified from temporal
sequences with interventions up to their minimal causal variables.

  * •

We propose CITRIS, a VAE architecture for disentangling causal factors in
latent space based on this setup.

  * •

Finally, we extend CITRIS to pretrained autoencoders by learning a map from an
entangled to a causally disentangled latent space using normalizing flows.

##  2 Preliminaries and Causal Assumptions

We assume that the underlying latent causal process is a dynamic Bayesian
network (DBN) (Dean & Kanazawa, 1989; Murphy, 2002) $G$ over a set of $K$
causal variables. In the corresponding graph $G=(V,E)$, each node $i\in V$ is
associated with a causal variable $C_{i}$, which can be scalar or vector
valued. Each edge $(i,j)\in E$ represents a causal relation from $C_{i}$ to
$C_{j}$: $C_{i}\to C_{j}$, where $C_{i}$ is a _parent_ of $C_{j}$ and
$\text{pa}_{G}(C_{i})$ are all parents of $C_{i}$ in $G$. Further, we assume
that the DBN is first-order Markov, stationary, and without instantaneous
effects. This means that in $G$ each causal factor $C_{i}$ is instantiated at
each time step $t$, denoted by $C_{i}^{t}$, and its causal parents can only be
causal factors at time $t-1$, denoted as $C_{j}^{t-1}$, including its own
previous value $C_{i}^{t-1}$. In other words, for $t\in\llbracket
1..T\rrbracket$ and for each causal factor $i\in\llbracket 1..K\rrbracket$ we
can model $C_{i}^{t}=f_{i}(\text{pa}_{G}(C_{i}^{t}),\epsilon_{i})$, where
$\text{pa}_{G}(C_{i}^{t})\subseteq\\{C_{1}^{t-1},\dots,C^{t-1}_{K}\\}$. We
denote the set of all causal variables at time $t$ as
$C^{t}=(C^{t}_{1},\dots,C^{t}_{K})$, where $C^{t}$ inherits all edges from its
components $C_{i}$ for $i\in\llbracket 1..K\rrbracket$ without introducing
cycles. In this setting the structure of the graph is time-invariant, _i.e._ ,
$\text{pa}_{G}(C_{i}^{t})=\text{pa}_{G}(C_{i}^{1})$ for any $t\in\llbracket
1..T\rrbracket$. We also assume all $\epsilon_{i}$ for $i\in\llbracket
1..K\rrbracket$ are mutually independent noises.

We use a binary intervention vector $I^{t}\in\\{0,1\\}^{K}$ to indicate that a
variable $C^{t}_{i}$ in $G$ is intervened upon if and only if $I^{t}_{i}=1$.
We consider that the intervention vector components $I^{t}_{i}$ might be
confounded by another $I^{t}_{j},i\neq j$, and represent these dependencies
with an unobserved regime variable $R^{t}$ (Didelez et al., 2006; Mooij et
al., 2020). With this, we construct an augmented DAG
$G^{\prime}=(V^{\prime},E^{\prime})$, where
$V^{\prime}=\\{\\{C^{t}_{i}\\}_{i=1}^{K}\cup\\{I^{t}_{i}\\}_{i=1}^{K}\cup
R^{t}\\}_{t=1}^{T}$ and $E^{\prime}=\\{\\{\text{pa}_{G}(C^{t}_{i})\to
C^{t}_{i}\\}_{i=1}^{K}\cup\\{I^{t}_{i}\to
C^{t}_{i}\\}_{i=1}^{K}\cup\\{R^{t}\to I^{t}_{i}\\}_{i=1}^{K}\\}_{t=1}^{T}$. We
say that a distribution $p$ is _Markov_ w.r.t. the augmented DAG $G^{\prime}$
if it factors as $p(V^{\prime})=\prod_{j\in
V^{\prime}}p(V_{j}\mid\text{pa}_{G^{\prime}}(V_{j}))$, where $V_{j}$ includes
the causal factors $C^{t}_{i}$, the intervention vector components
$I^{t}_{i}$, and the regime $R^{t}$. Moreover, we say that $p$ is _faithful_
to a causal graph $G^{\prime}$, if there are no additional conditional
independences to the d-separations one can read from the graph $G^{\prime}$.
The augmented graph $G^{\prime}$ can model interventions with an arbitrary
number of targets, including observational data. In this paper we will
consider _soft_ interventions (Eberhardt, 2007), in which the conditional
distribution changes, _i.e._ ,
$p(C^{t}_{i}|\text{pa}_{G}(C^{t}_{i}),I^{t}_{i}=1)\not=p(C^{t}_{i}|\text{pa}_{G}(C^{t}_{i}),I^{t}_{i}=0)$,
which include as a special case _perfect_ interventions
$\text{do}(C_{i}=c_{i})$ (Pearl, 2009).

##  3 Identifiability of Minimal Causal Variables

We first describe our setting, TempoRal Intervened Sequences (TRIS). In this
setting, we show that identifying the underlying causal factors is not always
possible, especially when considering multidimensional causal factors.
Therefore, we define the concept of _minimal causal variable_ , which
represent the manipulable part of each causal factor. Finally, we show under
which conditions we can recover the minimal causal variables in TRIS.

###  3.1 TempoRal Intervened Sequences (TRIS)

$C^{t}_{1}$$C^{t}_{2}$$\cdots$$C^{t}_{K}$$C^{t+1}_{1}$$C^{t+1}_{2}$$\cdots$$C^{t+1}_{K}$$X^{t+1}$$E^{t+1}$$X^{t}$$E^{t}$$I^{t+1}_{1}$$I^{t+1}_{2}$$\cdots$$I^{t+1}_{K}$$R^{t+1}$Temporal
causalrelationsInterventionsLatentconfoundingObservations$C^{t+1}$$C^{t}$

Figure 2: An example causal graph in TRIS, with observed variables shown in
gray and latent variables in white. A latent causal factor $C^{t+1}_{i}$ has
as parents a subset of the causal factors at the previous time step
$C_{1}^{t},\dots,C_{K}^{t}$, and its intervention target $I^{t+1}_{i}$. All
causal variables $C^{t+1}$ and noise $E^{t+1}$ cause the observation
$X^{t+1}$. $R^{t+1}$ is a latent confounder between the intervention targets.

In TRIS, we consider data generated by an underlying latent temporal causal
process with $K$ _causal factors_
$(C^{t}_{1},C^{t}_{2},...,C^{t}_{K})_{t=1}^{T}$. At each time step $t$, we
observe a high-dimensional observation $X^{T}$ representing a noisy, entangled
view of all causal factors. The following paragraphs describe the details of
this setup, visualized in Figure 2.

##### Multidimensional Causal Factors:

As opposed to most work on causal representation learning, which considers
causal factors to be one-dimensional (Klindt et al., 2021; Lachapelle et al.,
2022; Khemakhem et al., 2020a), we allow them to be potentially
multidimensional, _i.e._ , $C_{i}\in\mathcal{D}_{i}^{M_{i}}$ with $M_{i}\geq
1$ and in practice we let $\mathcal{D}_{i}$ be $\mathbb{R}$ for continuous
variables (_e.g._ , spatial position), $\mathbb{Z}$ for discrete variables
(_e.g._ , the score of a player) or mixed. This allows modeling different
levels of causal variables (_e.g._ a 2D-position encoded in a single factor
with two dimensions instead of two different causal factors). We define the
causal factor space as
$\mathcal{C}=\mathcal{D}_{1}^{M_{1}}\times\mathcal{D}_{2}^{M_{2}}\times...\times\mathcal{D}_{K}^{M_{K}}$.

##### Observation Function:

We define the observation function
$h(C^{t}_{1},C^{t}_{2},...,C^{t}_{K},E^{t}_{o})=X^{t}$, where $E^{t}_{o}$
represents any noise independent of the causal factors that influence the
observations, and $h:\mathcal{C}\times\mathcal{E}\rightarrow\mathcal{X}$ is a
function from the causal factor space $\mathcal{C}$ and the space of the noise
variables $\mathcal{E}$ to the observation space $\mathcal{X}$. We assume that
$h$ is bijective, implying that the joint dimensionality of the noise and
causal model is limited to the image size. This allows us to identify each
causal factor uniquely from observations by learning an approximation of $f$,
while disregarding irrelevant features in the observation space.

##### Availability of Intervention Targets:

Crucially, we assume that in each time-step some causal factors might (or
might not) have been intervened upon and that we have access to the
corresponding intervention targets, but not the intervention values. We denote
these intervention targets by the binary vector $I^{t}\in\\{0,1\\}^{K}$ where
$I^{t}_{i}=1$ refers to an intervention on the causal variable $C_{i}^{t}$.

###  3.2 Necessary Condition for Disentanglement in TRIS

In TRIS, we generally cannot disentangle two causal factors if they are always
intervened upon jointly, or, on the contrary, if they are never intervened
upon.

######  Proposition 3.1.

In TRIS, if two causal factors $C_{i}$ and $C_{j}$ have only been jointly
intervened on or not at all, then there exists a causal graph in which $C_{i}$
and $C_{j}$ cannot be uniquely identified from observations $X$ and
intervention targets $I$.

We provide an example of such a graph in Figure 3, where a ball can move in
two dimensions, $x$ and $y$. If both $x$ and $y$ follow a Gaussian
distribution with stationary variances over time, then any two orthogonal axes
can describe the distribution equally well (Hyvarinen & Morioka, 2017;
Belouchrani et al., 1997), making it impossible to uniquely identify them
without interventions. Similarly, if we only observe joint interventions on
$x,y$ together, we cannot identify them either due to the same reasoning. We
include the proof for this proposition in Section B.6.

Additionally, in TRIS where the latent causal factors may correspond to
multidimensional vectors, we cannot even completely reconstruct said factors,
when by the nature of the system the provided interventions leave some of the
causal factor’s dimensions unaffected. In the next section, we will instead
introduce the concept of minimal causal variables to characterize what we can
identify instead.

Figure 3: Example causal process of a ball with a two-dimensional position
$x,y$. The ball can only swap between the two boxes (blue and orange) through
an intervention, which does not influence the relative position within a box.

###  3.3 Minimal Causal Variables

To visualize this scenario, consider again the example in Figure 3 with a ball
in one of two boxes. Over time, the ball can move freely within the box it is
currently in, but it can only jump into another box if there is an
intervention. The intervention moves the ball to the other box, but keeps the
relative position of the ball within the box intact. While one could define
this process by a single causal variable $x$ over time, it can also be
described by two causal variables: the relative position within the box
$x^{\prime}$ and the current box $b$. Since only $b$ is affected by the
intervention, and we consider causal factors to potentially be
multidimensional, we could not identify which causal factor $x^{\prime}$
belongs to.

We formalize this intuition as follows. Suppose for each causal factor
$C_{i}\in\mathcal{D}^{M_{i}}$, there exists an invertible map
$s_{i}:\mathcal{D}_{i}^{M_{i}}\rightarrow\mathcal{D}_{i}^{\textup{var}}\times\mathcal{D}_{i}^{\textup{inv}}$
that splits the domain $\mathcal{D}^{M_{i}}$ of $C_{i}$ into a part that is
variant and a part that is invariant under intervention. We denote the two
parts of this map as

$s_{i}(C^{t}_{i})=(s^{\mathrm{var}}_{i}(C^{t}_{i}),s^{\mathrm{inv}}_{i}(C^{t}_{i}))$
(1)

The split $s$ must be invertible, so that we can map back and forth between
$\mathcal{D}_{i}^{M_{i}}$ and
$\mathcal{D}_{i}^{\textup{var}}\times\mathcal{D}_{i}^{\textup{var}}$ without
losing information. Furthermore, to be called a split in this setup, $s$ must
satisfy $s^{\mathrm{inv}}_{i}(C_{i}^{t})\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}I_{i}^{t}\mid\text{pa}(C_{i}^{t})$, _i.e._ ,
$s^{\mathrm{inv}}_{i}(C_{i}^{t})$ is independent of the intervention variable
$I_{i}^{t}$ given the parents of $C_{i}^{t}$. Also, both parts of the split
must be conditionally independent, _i.e._
$s^{\mathrm{inv}}_{i}(C_{i}^{t})\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}s^{\mathrm{var}}_{i}(C_{i}^{t})\mid\text{pa}(C_{i}^{t}),I_{i}^{t}$.
This means that $s^{\mathrm{var}}_{i}(C_{i}^{t})$ will contain the
manipulable, or _variable_ , part of $C^{t}_{i}$. In contrast,
$s^{\mathrm{inv}}_{i}(C_{i}^{t})$ is the _invariable_ part of $C_{i}^{t}$
which is independent of the intervention.

For any causal variable, there may exist multiple possible splits. There is
always at least the trivial split where
$\mathcal{D}_{i}^{\textup{var}}=\mathcal{D}_{i}^{M_{i}}$ is the original
domain of $C_{i}$, and $\mathcal{D}_{i}^{\textup{inv}}=\\{0\\}$ is the one-
element set (no invariant information). But not all splits are trivial: For
the example in Figure 3, we can split the causal factor $x$ in
$s^{\mathrm{var}}(x)$ such that the box identifier $b$ is in
$s^{\mathrm{var}}(x)$ and the relative position in the box $x^{\prime}$ is in
$s^{\mathrm{inv}}(x)$.

Intuitively, we want to identify the split where $s^{\mathrm{var}}_{i}$
contains _only_ the manipulable information:

######  Definition 3.2.

The _minimal causal split_ of a variable $C^{t}_{i}$ with respect to its
intervention variable $I^{t}_{i}$ is the split $s_{i}$ which maximizes the
information content $H(s^{\mathrm{inv}}_{i}(C^{t}_{i})|\text{pa}(C^{t}_{i}))$.
Under this split, $s^{\mathrm{var}}_{i}(C^{t}_{i})$ is defined as the _minimal
causal variable_ and denoted by $s_{i}^{\mathrm{var}^{*}}(C^{t}_{i})$.

Here $H$ denotes the entropy in the discrete case, or the limiting density of
discrete points (LDDP) (Jaynes, 1957, 1968) which measures information content
of a continuous variable and is invariant under the change of variables
induced by the map $s$. Intuitively, this ensures that only the information
which truly depends on the intervention is represented in
$s^{\mathrm{var}}_{i}(C_{i})$. The definition of the minimal causal split
depends on the characteristics of the provided intervention. In our previous
example, when an intervention on $x$ would also change the internal box
position, the minimal causal variable would contain the full causal factor
$x$. Our goal becomes to identify these minimal causal variables.

Depending on the interventions, the causal graph among the minimal causal
graph may differ from the original graph on $C_{1},...,C_{K}$. If the
interventions are soft, the minimal causal variable,
$s^{\mathrm{var}}_{i}(C_{i})$, has as parents the subset of
$\text{pa}(C_{i})$, whose relation to $C_{i}$ is influenced by the
intervention. For instance, consider a three-dimensional causal variable, of
which each dimension has a different set of parents. If an intervention only
affects the first dimension, $s^{\mathrm{var}}_{i}(C_{i})$ has the same
parents as the first dimension, and $s^{\mathrm{inv}}_{i}(C_{i})$ the same
parents as the last two dimensions. For perfect interventions, the
intervention-invariant part, $s^{\mathrm{inv}}_{i}(C_{i})$, has no parents
since all temporal dependencies are influenced by the intervention. Thus, in
this case, the parents of $s^{\mathrm{var}}_{i}(C_{i})$ are the same as of the
true causal variable, $\text{pa}(C_{i})$.

###  3.4 Learning Minimal Causal Variables

As a practical example of TRIS, we consider a dataset $\mathcal{D}$ of tuples
$\\{x^{t},x^{t+1},I^{t+1}\\}$ where $x^{t},x^{t+1}\in\mathbb{R}^{N}$ represent
the observations at time step $t$ and $t+1$ respectively, and $I^{t+1}$
describes the targets of the interventions performed on $C^{t+1}$. To learn a
causal representation, we consider a latent space $\mathcal{Z}$ larger than
the latent causal factor space $\mathcal{C}$, i.e.
$\mathcal{Z}\subseteq\mathbb{R}^{M},M\geq\text{dim}(\mathcal{E})+\sum_{i=1}^{K}M_{i}=\text{dim}(\mathcal{E})+\text{dim}(\mathcal{C})$.
In this latent space, we aim to disentangle the causal factors. Since we may
not know the exact dimensions $M_{1},...,M_{K}$, we overestimate its size in
the latent space $\mathcal{Z}$.

Our goal is to approximate the inverse of the observation function $h$ by
learning two components. First, we learn an invertible mapping from
observations to latent space, $g_{\theta}:\mathcal{X}\to\mathcal{Z}$. Second,
we learn an assignment function $\psi:\llbracket 1..M\rrbracket\to\llbracket
0..K\rrbracket$ that maps each dimension of $\mathcal{Z}$ to a causal factor.
Learning a flexible assignment function $\psi$ allows us to allocate any
dimension size per causal factor without knowing the individual causal factor
dimensions $M_{1},...,M_{K}$ in advance. Further, it also benefits the
optimization process, since some variables like circular angles or categorical
factors with many categories can have simpler distributions when modelled in
more dimensions, which we verify empirically in Section 6.2. In addition to
the $K$ causal factors, we use $\psi(j)=0,j\in\llbracket 1..M\rrbracket$ to
indicate that the latent dimension $z_{j}$ does not belong to any minimal
causal variable. Instead, those dimensions might model
$s^{\mathrm{inv}}_{i}(C_{i})$ for some causal factor $C_{i}$ or the
observation noise $E^{t}_{o}$. Finally, we denote the set of latent variables
that $\psi$ assigns to the causal factor $C_{i}$ with
$z_{\psi_{i}}=\\{z_{j}|j\in\llbracket 1..M\rrbracket,\psi(j)=i\\}$.

To enforce a disentanglement of causal factors, we model a prior distribution
in latent space, $p_{\phi}(z^{t+1}|z^{t},I^{t+1})$, with
$z^{t},z^{t+1}\in\mathcal{Z}$,
$z^{t}=g_{\theta}(x^{t}),z^{t+1}=g_{\theta}(x^{t+1})$. This transition prior
enforces a disentanglement by conditioning each latent variable on exactly one
of the intervention targets:

$p_{\phi}\left(z^{t+1}|z^{t},I^{t+1}\right)=\prod_{i=0}^{K}p_{\phi}\left(z_{\psi_{i}}^{t+1}|z^{t},I_{i}^{t+1}\right)$
(2)

where $I_{0}^{t+1}=0$. Then, the objective of the model is to maximize the
likelihood:

$p_{\phi,\theta}(x^{t+1}|x^{t},I^{t+1})=\left|\frac{\partial
g_{\theta}(x^{t+1})}{\partial x^{t+1}}\right|p_{\phi}(z^{t+1}|z^{t},I^{t+1})$

(3)

Under the assumptions stated in Section 3.1, we can prove the following
identifiability result for this setup:

######  Theorem 3.3.

Suppose that $\phi^{*}$, $\theta^{*}$ and $\psi^{*}$ are the parameters that,
under the constraint of maximizing the likelihood
$p_{\phi,\theta}(x^{t+1}|x^{t},I^{t+1})$, maximize the information content of
$p_{\phi}(z_{\psi_{0}}^{t+1}|z^{t})$. Then, with sufficient latent dimensions,
the model $\phi^{*},\theta^{*},\psi^{*}$ learns a latent structure where
$z_{\psi_{i}}^{t+1}$ models the minimal causal variable of $C_{i}$ if
$C^{t+1}_{i}\not\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}I^{t+1}_{i}|C^{t},I^{t+1}_{j}$ for any $i\neq
j$. All remaining information is modeled in $z_{\psi_{0}}$.

We provide the proof for this statement in Appendix B, which relies on
$I^{t+1}_{i}$ not being a deterministic function of any other intervention
target. A sufficient condition for this is the interventions being independent
of each other, or single-target interventions with observational data. Finding
the minimal variables intuitively means that the latent variables
$z_{\psi_{i}}$ model only the information of $C_{i}$ which strictly depends on
the intervention target $I_{i}^{t+1}$, thus defining causal variables by their
intervention dependency. Going back to the example of the $x$ position of the
ball in Figure 3, we would model the box identifier in $z_{\psi_{1}}$ if
$C_{1}=x$, while the internal box position is modeled in $z_{\psi_{0}}$. We
empirically verify the learned split for this example in Section D.3.
Nonetheless, one can always ensure that for any definition of the causal
process, $z_{\psi_{i}}$ only contains information of causal variable $C_{i}$
of that process, and no other causal variable.

##  4 Causal Identifiability from Temporal Intervened Sequences

To identify causal factors from high-dimensional temporal observations with
interventions, we propose CITRIS (Causal Identifiability from Temporal
Intervened Sequences). Below, we discuss its architecture and variants.

|
---|---
Figure 4: Comparing the VAE and AE+NF setup of CITRIS. Left: In the VAE, the
encoder and decoder provide an approximate invertible mapping. The transition
prior promotes disentanglement in the latent space by conditioning each latent
variable on only one intervention target. Right: The Normalizing Flow setup
uses a pretrained autoencoder, which remains frozen during training. The flow
learns to map the autoencoder latents to a new space, that promotes the
disentanglement similar to the VAE. The decoder is not needed during training.

###  4.1 Variational Autoencoder Setup

Inspired by several previous works (Locatello et al., 2020a; Higgins et al.,
2017; Träuble et al., 2021), we implement the framework of Section 3.4 by
learning a variational autoencoder (VAE) (Kingma & Welling, 2014), visualized
in Figure 4. The encoder $q_{\theta}$ and decoder $p_{\theta}$ approximate the
invertible mapping $g_{\theta}$ from observations to latent space, and
$p_{\phi}\left(z^{t+1}|z^{t},I^{t+1}\right)$ is the transition prior on the
latent variables. In this VAE setup, the objective of the model becomes the
Evidence Lower Bound (ELBO):

$\begin{aligned} &\mathcal{L}_{\text{ELBO}}=-\mathbb{E}_{z^{t+1}}\left[\log
p_{\theta}\left(x^{t+1}|z^{t+1}\right)\right]+\\\
&\mathbb{E}_{z^{t},\psi}\left[\sum_{i=0}^{K}D_{\mathrm{KL}}\left(q_{\theta}(z_{\psi_{i}}^{t+1}|x^{t+1})||p_{\phi}(z_{\psi_{i}}^{t+1}|z^{t},I_{i}^{t+1})\right)\right]\end{aligned}$

(4)

The KL divergence uses the prior definition of Equation 2. This ensures that,
conditioned on the previous time step and the interventions, the different
blocks of latent variables are independent. Thereby, the assignment function
of latent to causal variables, $\psi$, is learned via a Gumbel-Softmax
distribution (Jang et al., 2017) per latent variable. Hence, during training,
we sample a latent-to-causal variable assignment from these distributions,
while for inference, we can use the argmax to obtain a unique assignment. To
encourage information independent of any intervention to be modeled in
$z_{\psi_{0}}$, we weight the KL divergence of $z_{\psi_{0}}$ with
$1-\lambda$, where $\lambda>0$ is a hyperparameter (usually $\lambda=0.01$).

The prior $p_{\phi}$ for each set of latents $z_{\psi_{i}}^{t+1}$ is
implemented by an autoregressive model. For each set of latents
$z_{\psi_{i}}^{t+1}$, the model takes $z^{t}$, $I_{i}^{t+1}$ and $z^{t+1}$ as
input, where we sample from $\psi$ and mask the dimensions of $z^{t+1}$ for
which $\psi(j)\neq i$. From this input, the model predicts one Gaussian per
latent variable. The autoregressive nature of the prior allows complex
distributions over the multiple latent dimensions, while still being
independent across causal variables.

##### Target Classifier

Since a causal variable $C_{i}$ is independent of any other target variable
when conditioned on $I_{i}$, we can use those independence relations to guide
the disentanglement in latent space. For this, we propose a target classifier,
which is an additional small network trained to predict the intervention
targets from the latent variables over time, _i.e._ modeling
$p(I^{t+1}|z^{t},z_{\psi_{i}}^{t+1})$ for $i\in\llbracket 0..K\rrbracket$. To
guide the disentanglement, we take the gradients of the same objective with
respect to $z^{t},z_{\psi_{i}}^{t+1}$ and $\psi$, but only for $I^{t+1}_{i}$.
In other words, we explicitly train $z_{\psi_{i}}^{t+1}$ having a higher
mutual information with the intervention target $I^{t+1}_{i}$ of its intended
causal variable $C_{i}$. For all other targets $I_{j}^{t+1}$, we change the
label to the conditional distribution $p(I_{j}|I_{i})$, since $I^{t+1}_{j}$
should not be fully identifiable by any other set of latents except
$z_{\psi_{j}}^{t+1}$. This way, the target classifier ensures that information
about all causal variables is encoded in the latent space, while also guiding
the disentanglement in latent space. We provide further details in Section
C.3.3, and empirically verify its benefits in Section 6.2.

###  4.2 Using Pretrained Autoencoders

Despite the improved optimization process by the addition of the target
classifiers, VAEs can still struggle to model high-dimensional complex images,
especially when small details in the image are relevant. To overcome this
issue, we propose an adaptation of CITRIS to pretrained autoencoders. In this
setting, the invertible map $g_{\theta}$ is implemented by a deep autoencoder,
which is trained to encode and decode the high-dimensional observations to
low-dimensional feature vectors, independently of any disentanglement. During
training, we add small Gaussian noise to the latents to prevent the latent
distribution from collapsing to single delta peaks. However, we do not enforce
a certain prior like in a VAE, hence allowing complex marginal distributions.

In a second step, after the autoencoder converged, we freeze its parameters
and learn a normalizing flow (Rezende & Mohamed, 2015) that maps the entangled
latent representation to a disentangled version. The invertibility of the
normalizing flow ensures that no information is lost when mapping from the
entangled to the disentangled latent space, and thus we can use the pretrained
decoder to reconstruct the observations without requiring any fine-tuning.
Compared to the VAE setup in Section 4.1, we replace the encoder by a
successive application of the frozen encoder and a normalizing flow on the
encoded latents, shown in Figure 4. Besides that, we deploy the same setup,
using both the transition prior structure and target classifier.

Compared to the VAE setup, this approach has three main benefits. First, the
autoencoder can be trained on observational data alone, potentially reducing
the amount of interventional data required. Second, learning a separate
autoencoder provides an opportunity for generalizing causal factors beyond the
known dataset. For instance, one could train an autoencoder on two datasets,
where only one has interventions, _e.g._ synthetic and real-world data. Then,
since the autoencoder uses a joint latent space for both datasets, training
the normalizing flow on only the dataset with interventions can lead to a
disentanglement function that generalizes to the purely observational dataset.
We verify the viability of this approach in a restricted setting in Section
6.2, which opens up great potential to practical applications. Finally, the
setup is easier to optimize since the autoencoder can compress the information
in an almost unrestricted latent space, while the normalizing flow solely
focuses on disentanglement.

##  5 Related Work

Identifying independent factors of variations from data is a well-studied
field in machine learning (Reed et al., 2014; Higgins et al., 2017; Kumar et
al., 2018; Locatello et al., 2019, 2020b; Klindt et al., 2021; Locatello et
al., 2020a). One of the first lines of work is Independent Component Analysis
(ICA) (Comon, 1994; Hyvärinen et al., 2001). ICA tries to recover independent
latent variables that were transformed by some invertible transformation.
Although not generally possible in the non-linear case (Hyvärinen & Pajunen,
1999), ICA was recently extended to this setting by exploiting auxiliary
variables under which the latents become conditionally mutually independent
(Hyvärinen & Morioka, 2016; Hyvärinen et al., 2019). Several follow-up works
extended this work to deep learning architectures like VAEs (Khemakhem et al.,
2020a; Sorrenson et al., 2020; Khemakhem et al., 2020b; Zimmermann et al.,
2021). Recent works draw a connection between causality and ICA (Gresele et
al., 2021; Monti et al., 2019). In particular, Lachapelle et al. (2022); Yao
et al. (2022) discuss identifiability from temporal sequences and bring it
into context to causality. While Lachapelle et al. (2022) can model
interventions as external actions, Yao et al. (2022) can model soft
interventions through their non-stationary noise. On the other hand, they do
not exploit the knowledge of the intervention targets as we do and therefore
require additional assumptions in terms of sufficient variation. Moreover,
both of these works require scalar causal variables, while CITRIS generalizes
to multidimensional causal factors.

A second, related line of work is causal representation learning (Schölkopf et
al., 2021), which aims at discovering causal structures and variables from
data. For instance, Locatello et al. (2020a) showed that one can identify
independent latent causal factors from pairs of observations that only differ
in $k$ causal factors. Yang et al. (2021) propose a VAE that integrates a
structural causal model in its prior, but requires the true causal variables
as labels during training. von Kügelgen et al. (2021) demonstrated that common
contrastive learning methods can block-identify the causal variables that
remain unchanged under augmentations. CITRIS similarly identifies
multidimensional causal factors as a block, and, furthermore, disentangles
individual causal factors by grouping the latent space.

##  6 Experiments

We evaluate CITRIS on two video datasets, and compare it to common
disentanglement methods. We include further details for reproducibility in
Appendix C, and make our code publicly
available.111https://github.com/phlippe/CITRIS

###  6.1 Experimental Setup

Temporal Causal3DIdent We evaluate CITRIS on an adapted, temporal version of
the Causal3DIdent identifiability benchmark (Zimmermann et al., 2021; von
Kügelgen et al., 2021). The dataset consists of 3D renderings of objects, for
which we consider seven different causal factors: the object position as
multidimensional vector $[x,y,z]\in[-2,2]^{3}$; the object rotation with two
dimensions $[\alpha,\beta]\in[0,2\pi)^{2}$; the hue of the object, background
and spotlight in $[0,2\pi)$; the spotlight’s rotation in $[0,2\pi)$; and the
object shape (categorical with seven values). The relations among those
variables are shown in Figure 5. Each continuous variable follows a Gaussian
distribution over time, where the mean is a (non-linear) function of the
parents. Overall, this causes strong dependencies among the variables. We
perform perfect interventions with $I^{t}_{i}\sim\text{Bernoulli}(0.1)$,
ensuring the minimal causal variables to be the true factors. Experiments with
confounded interventions are included in Section D.1 with similar results.
Further details on this dataset are summarized in Appendix C.1.

pos_orot_orot_shue_shue_bhue_oobj_spos_orot_orot_shue_shue_bhue_oobj_s

Figure 5: The temporal causal graph among the 7 dimensions of variation. The
graph structure covers a wide range of scenarios, such as confounders, chains,
and up to 4 parents per variable.

Interventional Pong As a second benchmark, we adapt the popular Atari game
Pong (Bellemare et al., 2013), in which we define 5 causal factors: the $x,y$
position of the ball, the $y$-positions of the two paddles, and the velocity
direction of the ball (angle in $[0,2\pi)$). Further, we add the score as a
sixth causal factor, which is part of the game’s dynamics. However, we do not
provide any interventions on it. Hence, in CITRIS, this information should be
modeled in $z_{\psi_{0}}$. The temporal dependencies follow the dynamics of
Pong, where the ball can collide with the paddles and walls, and the two
paddles move towards the ball to hit it. We provide single-target
interventions for the variables. For the paddles, the imperfect interventions
override the policy and take a random action (up/down), while keeping a smooth
motion. For the causal factors of the ball, we uniformly sample a new value
when intervened, causing a possible jump. Example frames of the environment
are shown in Figure 6.

|  |  |
---|---|---|---
Figure 6: Examples of Interventional Pong. The score represents a causal
factor on which we do not have any interventions. The velocity of the ball is
encoded in a 4th channel, not visualized here.

##### Baselines

We compare CITRIS to SlowVAE (Klindt et al., 2021), a state-of-the-art
disentanglement method for temporal sequences. Notably, SlowVAE assumes that
the factors of variation are independent. As we show, this assumption, which
often cannot be met in more complex settings like Temporal Causal3DIdent, has
detrimental effects on modelling, underlining the importance of modelling the
correlations between causal variables. The second baseline we consider is iVAE
(Khemakhem et al., 2020a), a VAE model which conditions its components on
additional observed variables $u$. In our setup, $u$ corresponds to the
previous time step observation $x^{t}$ and the intervention targets $I^{t+1}$.
As we aim to find a mapping from image to a causal space, which is independent
of those factors, we must adapt iVAE to only condition its prior on $u$. We
refer to this model variant as iVAE∗. With this change, the main difference
between iVAE∗ and CITRIS becomes the structure of the prior
$p(z_{t+1}|z_{t},I_{t+1})$, showcasing the significance of CITRIS’s
identifiability guarantees in this setup. Hyperparameter details are listed in
Appendix C, and the code is provided in the supplementary material.

Table 1: Results on the Temporal-Causal3DIdent dataset. All triplet distances
have a distance at random of $1$ and optimum of $0$. Oracle represents the
accuracy of a CNN trained supervised to predict the causal factors from
images, hence constituting a lower error limit. In the correlation metrics,
diag refers to the average score of the predicted causal factor to its true
value (optimal 1), and sep for the average of the maximum correlation per
predicted causal variable besides its true factor (optimal 0). Standard
deviations over 3 seeds shown in Section C.1. CITRIS is able to disentangle
the causal factors well, and CITRIS-NF even accurately models all 7 shapes.

|  Triplet evaluation distances $\downarrow$ | Correlation metrics
---|---|---
| pos_x | pos_y | pos_z | rot_$\alpha$ | rot_$\beta$ | rot_s | hue_s | hue_b | hue_o | obj_s | Mean |  $R^{2}$ diag $\uparrow$ |  $R^{2}$ sep $\downarrow$ | Spearman diag $\uparrow$ | Spearman sep $\downarrow$
Temporal-Causal3DIdent Teapot
Oracle | 0.02 | 0.02 | 0.02 | 0.02 | 0.03 | 0.01 | 0.02 | 0.01 | 0.02 | - | 0.02 | - | - | - | -
SlowVAE | 0.13 | 0.10 | 0.12 | 0.50 | 0.59 | 0.22 | 0.64 | 0.21 | 0.17 | - | 0.30 | 0.65 | 0.20 | 0.62 | 0.27
iVAE∗ \- 9dim | 0.11 | 0.09 | 0.12 | 0.70 | 0.76 | 0.06 | 0.67 | 0.02 | 0.12 | - | 0.30 | 0.65 | 0.11 | 0.65 | 0.13
iVAE∗ \- 32dim | 0.04 | 0.03 | 0.04 | 0.25 | 0.31 | 0.03 | 0.58 | 0.02 | 0.05 | - | 0.15 | 0.78 | 0.21 | 0.77 | 0.17
CITRIS-VAE | 0.05 | 0.04 | 0.05 | 0.10 | 0.20 | 0.03 | 0.08 | 0.02 | 0.05 | - | 0.07 | 0.96 | 0.02 | 0.95 | 0.04
\- No target classifier | 0.05 | 0.04 | 0.05 | 0.62 | 0.66 | 0.19 | 0.04 | 0.02 | 0.17 | - | 0.20 | 0.79 | 0.15 | 0.76 | 0.12
CITRIS-NF | 0.04 | 0.03 | 0.04 | 0.06 | 0.10 | 0.03 | 0.04 | 0.01 | 0.04 | - | 0.04 | 0.98 | 0.01 | 0.97 | 0.05
Temporal-Causal3DIdent 7-shapes
Oracle | 0.08 | 0.06 | 0.08 | 0.06 | 0.09 | 0.04 | 0.04 | 0.01 | 0.04 | 0.00 | 0.05 | - | - | - | -
SlowVAE | 0.44 | 0.25 | 0.41 | 0.69 | 0.75 | 0.25 | 0.57 | 0.10 | 0.14 | 0.37 | 0.40 | 0.61 | 0.23 | 0.59 | 0.27
iVAE∗ | 0.26 | 0.23 | 0.34 | 0.58 | 0.65 | 0.10 | 0.31 | 0.02 | 0.09 | 0.14 | 0.27 | 0.80 | 0.29 | 0.77 | 0.28
CITRIS-VAE | 0.15 | 0.13 | 0.23 | 0.54 | 0.71 | 0.07 | 0.05 | 0.02 | 0.06 | 0.18 | 0.21 | 0.89 | 0.10 | 0.88 | 0.12
CITRIS-NF | 0.12 | 0.08 | 0.11 | 0.09 | 0.14 | 0.05 | 0.05 | 0.02 | 0.06 | 0.00 | 0.07 | 0.98 | 0.04 | 0.97 | 0.08

##### Correlation Metrics

Following common practice, we report the correlation of the learned latent
variables to the ground truth causal factors. Since in our setup, multiple
latent variables can jointly describe a single causal variable, we first learn
a mapping between such, _e.g._ , with an MLP. For CITRIS, we apply one MLP per
set of latent variables that are assigned to the same causal factor by $\psi$.
The MLP is then trained to predict _all_ causal factors per set of latents, on
which we measure the correlation. Thereby, no gradients are propagated through
the model. iVAE∗ and SlowVAE do not learn an assignment of latent to causal
factors. As an alternative, we assign each latent dimension to the causal
factor it has the highest correlation with. Although this gives the baselines
a considerable advantage, it shows whether CITRIS can improve upon the
baselines beyond finding a good latent to causal factor assignment. We report
both the $R^{2}$ coefficient of determination (Wright, 1921) and the
Spearman’s rank correlation coefficient (Spearman, 1904) of the predicted
values against the ground truth causal factors. However, in our datasets, the
causal factors are correlated themselves. This makes it difficult to spot
spurious correlations between latents and causal factors. To overcome this
issue, we measure the correlations on a test dataset for which we sample the
causal factors independently.

Triplet Evaluation To also evaluate the decoding part of the model, we propose
another parameter-free evaluation, triplet evaluation, to reveal complex
dependencies between latent variables. For this, we create triplets of images:
the first two are randomly sampled test images, while the third one is created
based on a random combination of causal factors of the first two images. For
example, in Figure 7, we take the spotlight rotation and object shape from
image 1, and all other causal factors from image 2. For evaluation, we then
encode the two test images independently, perform the combination of ground-
truth causal factors as done for the third image in latent space, and use the
decoder to generate a new image, which ideally resembles the ground truth
third image. Since the reconstruction error is not descriptive of the errors
being made, _e.g._ the rotation in Figure 7, we train an additional CNN in a
supervised manner that maps images to the causal factors. With this model, we
can extract the causal factors from the generated image, and report the
average distance of these to the ground truth causal factors.

|  |  |
---|---|---|---
Image 1 | Image 2 | Ground Truth | Prediction
|  |  |
Image 1 | Image 2 | Ground Truth | Prediction
|  |  |
Image 1 | Image 2 | Ground Truth | Prediction
Figure 7: Example triplet evaluations on the Temporal-Causal3DIdent. In the
first row, the ground truth combines the spotlight rotation and object shape
from Image 1 with all other causal factors from Image 2. The prediction is
generated by encoding the two images, and performing the same combination of
causal factors in latent space. The shown prediction by CITRIS-NF only
slightly differs in the object rotation. For details on the other predictions,
see Section D.1.

###  6.2 Temporal Causal3DIdent Experiments

##### Teapot Experiments

The Causal3DIdent benchmark constitutes a challenging dataset due to its
various interactions of causal factors in the high-dimensional observational
space. To slightly simplify the problem, especially for learning a VAE on the
high-dimensional observational space, we first show experiments on the teapot
shape only, Temporal-Causal3DIdent Teapot. All results are summarized in Table
1. Both the VAE and Normalizing Flow version of CITRIS considerably
outperformed the two baselines and were able to achieve an average $R^{2}$ and
Spearman correlation above 0.9, while keeping the correlation between factors
low. Moreover, CITRIS-NF achieves close-to optimal scores on the triplet
evaluation, and especially outperforms the VAE-based approaches in modeling
the rotations. This is because the autoencoder is able to reconstruct images
with negligible error, while the VAE optimization involves balancing the small
difference in output space with its KL divergence regularization in latent
space. Further, an ablation study on training CITRIS-VAE without the target
classifier verifies that the additional guidance can considerably help in
optimization. Finally, in Section D.1.4, we show that the original causal
graph can be found from CITRIS’s latents.

In contrast, the SlowVAE entangles the causal factors due to their strong
correlation over time. The iVAE∗ disentangles the position and the rotation of
the spotlight well. However, the hue of the spotlight was highly entangled
since its appearance differs with different background and object colors. We
further report the performance of an iVAE∗ with equal number of latent
dimensions as causal factors. It is apparent that the smaller latent space is
particularly insufficient for modeling the rotation angles, showing the need
for higher dimensional latent spaces, not only in CITRIS.

Table 2: Results on the Temporal-Causal3DIdent dataset with CITRIS-NF trained
on 5 object shapes. The same metrics as in Table 1 are reported, with “Spr”
being the Spearman correlation score. The full table is in Section D.1. For 2
unseen shapes, CITRIS-NF can yet disentangle their causal factors well.

CITRIS-NF |  Triplets $\downarrow$ |  $R^{2}$ diag $\uparrow$ |  $R^{2}$ sep $\downarrow$ | Spr diag $\uparrow$ | Spr sep $\downarrow$
---|---|---|---|---|---
5 seen shapes | 0.09 | 0.98 | 0.05 | 0.97 | 0.10
2 unseen shapes | 0.23 | 0.94 | 0.15 | 0.93 | 0.19

##### 7-shapes Experiments

We apply all models on the Temporal Causal3DIdent dataset with all seven
shapes (Table 1). The models have to align the rotation axes and central
points of all shapes, which caused higher triplet distances for rotations
across models. While CITRIS-VAE still outperformed the other baselines,
CITRIS-NF significantly improved upon that, still maintaining a
disentanglement similar to the single shape experiments. This underlines the
optimization benefits of using pretrained autoencoders for disentanglement
learning on complex, high-dimensional observations.

##### Generalization of Causal Representations

Since the autoencoder is trained on observational data, we evaluate whether
its causal representation can generalize to new, unseen settings. For this, we
reuse the same autoencoder as before, but train the Normalizing Flow on an
interventional dataset which excludes any observations from two shapes (Head
and Cow). Afterwards, we test its zero-shot generalization to the two unseen
shapes (see Table 2). Note that optimal performance cannot be achieved here,
since the central point and default rotation of an object cannot be
generalized to other objects. Nonetheless, the results in Table 2 indicate a
strong disentanglement among factors, with slight decreases in position and
rotation due to the forementioned limitations. This shows that the learned
disentanglement function can indeed generalize to unseen instantiations of
causal factors, promising potential for future work on generalizing causal
representations to unseen settings with CITRIS.

Table 3: Results on the Interventional Pong dataset. The same metrics as in
Table 1 are reported, with “Spr” being the Spearman correlation score. The
full table is in Section D.2. The results how that CITRIS can handle imperfect
interventions.

|  Triplets $\downarrow$ |  $R^{2}$ diag $\uparrow$ |  $R^{2}$ sep $\downarrow$ | Spr diag $\uparrow$ | Spr sep $\downarrow$
---|---|---|---|---|---
SlowVAE | 0.34 | 0.61 | 0.17 | 0.66 | 0.23
iVAE∗ | 0.09 | 0.91 | 0.04 | 0.92 | 0.06
CITRIS-VAE | 0.03 | 0.99 | 0.01 | 0.99 | 0.05
CITRIS-NF | 0.02 | 1.00 | 0.04 | 1.00 | 0.10

###  6.3 Interventional Pong

Finally, we report the disentanglement results of CITRIS on the Interventional
Pong dataset in Table 3. The challenge of this dataset is its imperfect and
correlated interventions, and the score being an unintervened causal variable.
Still, CITRIS is able to disentangle the factors and also assign the score
variables to its correct set of latents, _i.e._ $z_{\psi_{0}}$, showing that
it can indeed handle imperfect interventions. Although CITRIS-NF obtains a
higher correlation among learned latents, it again did not affect its triplet
generations. SlowVAE entangles the two paddles, since they are strongly
correlated by following the same movement policy and having imperfect
interventions. iVAE∗ showed unstable behavior over seeds, and entangled the
ball position and velocity.

##  7 Conclusion

We propose CITRIS, a VAE framework for learning causal representations. CITRIS
identifies the minimal causal variables of a dynamical system from temporal,
intervened sequences. Furthermore, by using normalizing flows, CITRIS learns
to disentangle the representation of pretrained autoencoders. In experiments,
CITRIS reliably recovered the causal factors of 3D rendered images. Moreover,
we empirically showed that CITRIS can generalize to unseen instantiations of
causal factors. This promises great potential for future work on simulation-
to-real generalization research for causal representation learning. As future
work, CITRIS can be extended to an active learning setup, allowing for more
data-efficient causal identifiability methods in practice. Further, future
work could consider the setup of robotic simulators (Szot et al., 2021;
Makoviychuk et al., 2021), where interventions are available through sequences
of actions.

## Acknowledgements

We thank Johann Brehmer and Pim de Haan for valuable discussions throughout
the project. We also thank SURFsara for the support in using the Lisa Compute
Cluster. This work is financially supported by Qualcomm Technologies Inc., the
University of Amsterdam and the allowance Top consortia for Knowledge and
Innovation (TKIs) from the Netherlands Ministry of Economic Affairs and
Climate Policy.

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Appendix

[sections]  Table of Contents [sections]l1  Appendix A \- Reproducibility
Statement A
Appendix B \- Proofs B
B.1. Preliminaries B.1
B.2. Proof Outline B.2
B.3. Step 1: True Disentanglement $\delta^{*}$ is one of the Global Maxima B.3
B.4. Step 2: Characterizing the Disentanglement Class $\Delta$ B.4
B.5. Step 3: Deriving the Final Theorem B.5
B.6. Non-Identifiability without Interventions B.6
Appendix C \- Experimental Details C
C.1. Temporal Causal3DIdent Dataset C.1
C.2. Interventional Pong C.2
C.3. Experimental Design C.3
C.4. Hyperparameters C.4
Appendix D \- Additional Experiments and Results D
D.1. Temporal Causal3DIdent D.1
D.2. Interventional Pong D.2
D.3. Ball-in-Boxes Example D.3

##  Appendix A Reproducibility Statement

To ensure reproducibility, we publish the code for all models used in this
paper at https://github.com/phlippe/CITRIS. Further, we include the code for
generating the Interventional Pong dataset and the sequences of the causal
factors in the Temporal-Causal3DIdent dataset. The complete datasets of this
paper are released with corresponding licenses, and links to those datasets
are available in the code repository. Moreover, we give a detailed overview of
the datasets and more visual examples in Section C.1 and Section C.2.

Further, for all experiments of Section 6, we have included a detailed
overview of the hyperparameters in C.4 and additional implementation details
of the evaluation metrics and model architecture components in Section C.3.
All experiments have been repeated with 3 seeds to obtain stable, reproducible
results. We provide an overview of the standard deviations, as well as
additional results in Appendix D.

Finally, all experiments in this paper were performed on a single NVIDIA
TitanRTX GPU with a 6-core CPU. The overall computation time of all
experiments together in this paper correspond to approximately $80$ GPU days
(excluding hyperparameter search and trials during the research).

##  Appendix B Proofs

The following section contains the proof for Theorem 3.3. We first give an
overview of the notation and additional preliminary discussions in Section
B.1. Then, we give an outline of the proof. The remaining sections provide the
details of the proof.

Additionally, we provide the proof and more details on Proposition 3.1 in
Section B.6.

###  B.1 Preliminaries

####  B.1.1 Summary of Notation

We summarize the notation, which is the same as used for the main paper, as
follows:

  * •

We assume $K$ causal factors $C_{1},\dots,C_{K}$ such that
$C_{i}\in\mathcal{D}_{i}^{M_{i}}$ with $M_{i}\geq 1$;

  * •

We can group all causal factors in a single variable
$C=(C_{1},\dots,C_{K})\in\mathcal{C}$, where $\mathcal{C}$ is the causal
factor space
$\mathcal{C}=\mathcal{D}_{1}^{M_{1}}\times\mathcal{D}_{2}^{M_{2}}\times...\times\mathcal{D}_{K}^{M_{K}}$;

  * •

The data is generated by a latent Dynamic Bayesian network with variables
$(C^{t}_{1},C^{t}_{2},...,C^{t}_{K})_{t=1}^{T}$;

  * •

We assume to know at each time step the binary intervention vector
$I^{t}\in\\{0,1\\}^{K+1}$ where $I^{t}_{i}=1$ refers to an intervention on the
causal factor $C_{i}^{t+1}$. As a special case $I^{t}_{0}=0$ for all $t$;

  * •

For each causal factor $C_{i}$, there exists a split
$s^{\mathrm{var}}_{i}(C_{i}),s^{\mathrm{inv}}_{i}(C_{i})$ such that
$s^{\mathrm{var}}_{i}(C_{i})$ represents the variable/manipulable part of
$C_{i}$, while $s^{\mathrm{inv}}_{i}(C_{i})$ represents the invariable part of
$C_{i}$;

  * •

The minimal causal split is defined as the one which only contains the
intervention-dependent information in $s^{\mathrm{var}}_{i}(C_{i})$, and
everything else in $s^{\mathrm{inv}}_{i}(C_{i})$. This split is denoted by
$s_{i}^{\mathrm{var}^{*}}(C_{i})$ and $s_{i}^{\mathrm{inv}^{*}}(C_{i})$

  * •

At each timestep we can access observations
$x^{t},x^{t+1}\in\mathcal{X}\subseteq\mathbb{R}^{N}$;

  * •

$\mathcal{E}$ is the space of the noise variables that affect the observation
without changing the encoding of the causal factors. For example, this could
be random color shifts in Pong, or brightness shifts in Causal3D, since no
causal factor is encoded in color and brightness in these setups respectively;

  * •

Observation function $h:\mathcal{C}\times\mathcal{E}\rightarrow\mathcal{X}$,
where $\mathcal{E}$ is the space of the noise variables;

  * •

Latent vector $z^{t}\in\mathcal{Z}\subseteq\mathbb{R}^{M}$, where
$\mathcal{Z}$ is the latent space of dimension
$M\geq\text{dim}(\mathcal{E})+\text{dim}(\mathcal{C})$;

  * •

Inverse of the observation function in the latent space
$g_{\theta}:\mathcal{X}\to\mathcal{Z}$;

  * •

Assignment function from latent dimensions to causal factors $\psi:\llbracket
1..M\rrbracket\to\llbracket 0..K\rrbracket$;

  * •

Disentanglement function
$\delta^{*}:\mathcal{X}\to\mathcal{\tilde{C}}\times\mathcal{\tilde{E}}$ with
$\mathcal{\tilde{C}}=\mathcal{D}^{\tilde{M_{1}}}\times...\times\mathcal{D}^{\tilde{M_{K}}}$
and $\tilde{M_{i}}$ being the number of latent dimensions assigned to causal
factor $C_{i}$ by $\psi^{*}$. We denote the output of $\delta^{*}$ for an
observation $X$ as
$\delta^{*}(X)=(\tilde{C}_{1},\tilde{C}_{2},...,\tilde{E})$. Then,
$\delta^{*}$ is a disentanglement function if there exist a set of
deterministic functions $h_{0},h_{1},...,h_{K}$ for which, for any $X=h(C,E)$,
$h_{i}(\tilde{C}_{i})=C_{i}$ for all $i\in\llbracket 1..K\rrbracket$, and
$h_{0}(\tilde{E})=E$.

  * •

The representation of $\delta^{*}$ in terms of the learnable function is
denoted by $g_{\theta}^{*}$ and $\psi^{*}$;

  * •

Latent variables assigned to each causal factor $C_{i}$ by $\psi$ are denoted
as $z_{\psi_{i}}=\\{z_{j}|j\in\llbracket
1..M\rrbracket,\psi(j)=i\\}=\\{g_{\theta}(x^{t})_{j}|j\in\llbracket
1..M\rrbracket,\psi(j)=i\\}$;

  * •

The remaining latent variables that are not assigned to any causal factor are
denoted as $z_{\psi_{0}}$;

  * •

The goal is to learn for each $C_{i}$:
$p_{\phi}\left(z_{\psi_{i}}^{t+1}|z^{t},I_{i}^{t+1}\right)\approx
p\left(s^{\mathrm{var}}_{i}(C^{t+1}_{i})|C^{t},I^{t+1}_{i}\right)$;

####  B.1.2 Limiting Density of Discrete Points

In this section, we give a short overview on the difference between
differential entropy and the limiting density of discrete points approach,
introduced by Jaynes (1957, 1968). Differential entropy on a continuous random
variable $X$ with a distribution $p(X)$ is defined as:

$H(X)=-\int p(X)\log p(X)dx$ (5)

While for discrete variables, entropy has the intuitive explanation of an
uncertainty measure, or the ’information’ of a variable, one cannot draw the
same relation so easily for continuous variables. This is because differential
entropy lacks properties that would be necessary for that. For one, the
entropy can become negative. Secondly, and most importantly for the use case
in this paper, it is not invariant under invertible transformations, _i.e._ a
change of variables. For the example of the random variable $X$, the entropy
of $H(X)$ does not necessarily equal to $H(aX)$ where $a$ is a constant
factor, _e.g._ $a=2$. Thus, it becomes difficult to use differential entropy
as a measure of information content of a continuous variable, like in the
discrete case.

One approach that was proposed to overcome these issues is the limiting
density of discrete points (LDDP) Jaynes (1957, 1968). It adjusts the
definition of differential entropy by introducing an _invariant measure_
$m(X)$, which can be seen as a reference distribution we measure the entropy
of $p(X)$ to. Intuitively, the LDDP adjustment is derived from arguing that
the continuous entropy should be derived by taking the limit of increasingly
dense discrete distributions. In the limit of infinitely many discrete points,
one arrives at the entropy for continuous functions, which becomes:

$H(X)=-\int p(X)\log\frac{p(X)}{m(X)}dx$ (6)

Note that in some formulations, a constant $\log N$ is added to this equation,
where $N$ is the number of discrete points considered which goes against
infinity in the limit. Since for this paper, we only require to compare two
entropy values with each other and do not require the entropy to take a
specific value, we can neglect this constant.

One crucial property of LDDP, which we use in the following proof, is that the
entropy stays invariant under a change of variable. This is achieved by
transforming the invariant measure $m(X)$ by the exact same invertible
transformation as done for $p(X)$. Therefore, when coming back to the example
of scaling $X$ by a constant factor, both $p(X)$ and $m(X)$ change in the same
way, resulting in $H(X)=H(aX)$.

###  B.2 Proof Outline

The goal of this section is to proof Theorem 3.3: the global optimum of CITRIS
will find the minimal causal variables. We will take the following steps in
the proof:

  1. 1.

(Section B.3) Firstly, we show that the function $\delta^{*}$ that
disentangles the true latent variables $C_{1},...,C_{K}$ and assigns them to
the corresponding sets $z_{\psi_{1}},...,z_{\psi_{K}}$ constitutes a global,
but not necessarily unique, optimum for maximizing the likelihood of Equation
2.

  2. 2.

(Section B.4) Next, we characterize the class of disentanglement functions
$\Delta^{*}$ which all represent a global maximum of the likelihood, _i.e._
get the same score as the true disentanglement. In particular, we show that in
all optimal disentanglement functions, each assignment set $z_{\psi_{i}}$
contains the variable part of the causal factor $s^{\mathrm{var}}_{i}(C_{i})$,
but that it might contain also the invariable parts of any other causal
factor, thus creating multiple optimal solutions. We do this in two sub-steps:

    1. (a)

First, we assume that all intervention targets are independent, _i.e._
$I^{t+1}_{i}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}I^{t+1}_{j}|C^{t}$ for any $i\neq j$.

    2. (b)

Secondly, we extend it to a wider group of intervention settings where
interventions might be confounded, and show that all of them fall in the same
class $\Delta^{*}$.

  3. 3.

(Section B.5) Finally, we derive Theorem 3.3 by showing that the function
$\hat{\delta}\in\Delta^{*}$, which maximizes the entropy of $z_{\psi_{0}}$,
identifies the minimal causal mechanisms, which intuitively represent the
parts of the causal factors that are affected by the available interventions.

Along the way, we will make use of Figure 8 summarizing the temporal causal
graph. For the remainder of the proof, we assume that the prior
$p_{\phi}\left(z^{t+1}|z^{t},I^{t+1}\right)$ and the invertible map
$g_{\theta}$ are sufficiently complex to approximate any possible function and
distribution one might encounter. To simplify the exposition, we also assume
that the latent dimension size is unlimited, _i.e._ $M=\infty$, so there are
no limitations on how many latent variables $z_{\psi_{i}}$ can be used to
represent a causal factor $C_{i}$. In practice, however, this is not a
limiting factor as long as we can overestimate the dimensions of the causal
factors and noise variables.

Throughout the proof, we will use $C^{t}$ to refer to the set of all causal
factors at time step $t$, _i.e._ $C^{t}=\\{C_{1}^{t},...,C_{K}^{t}\\}$.
Similarly, we define $I^{t+1}=\\{I^{t+1}_{1},...,I^{t+1}_{K}\\}$.

$C^{t}_{1}$$C^{t}_{2}$$\cdots$$C^{t}_{K}$$C^{t+1}_{1}$$C^{t+1}_{2}$$\cdots$$C^{t+1}_{K}$$X^{t+1}$$E^{t+1}$$X^{t}$$E^{t}$$I^{t+1}_{1}$$I^{t+1}_{2}$$\cdots$$I^{t+1}_{K}$$R^{t+1}$Temporal
causalrelationsInterventionsLatentconfoundingObservations$C^{t+1}$$C^{t}$

Figure 8: An example temporal causal graph in TRIS, with observed variables
shown in gray and latent variables in white. A latent causal factor
$C^{t+1}_{i}$ has as parents a subset of the causal factors at the previous
time step $C_{1}^{t},\dots,C_{K}^{t}$, and its intervention target
$I^{t+1}_{i}$. All causal variables $C^{t+1}$ and the noise $E^{t+1}$ cause
the observation $X^{t+1}$. $R^{t+1}$ is a latent confounder between the
intervention targets, but is not strictly necessary (_i.e._ $R^{t+1}$ may also
be constant). The goal is to identify the causal factors $C_{1},...,C_{K}$.

###  B.3 Step 1: True Disentanglement $\delta^{*}$ is one of the Global Maxima
of the Conditional Likelihood $p(X^{t+1}|X^{t},I^{t+1})$

We start by proving the following Lemma:

######  Lemma B.1.

The true disentanglement function $\delta^{*}$ that correctly disentangles the
true causal factors $C^{t+1}_{1},...,C^{t+1}_{K}$ from observations
$X^{t},X^{t+1}$ using the true $\psi^{*}$ assignment function on the true
latent variables $Z^{t+1}$ is one of the global maxima of the likelihood of
$p(X^{t+1}|X^{t},I^{t+1})$.

We are interested in optimizing $p(X^{t+1}|X^{t},I^{t+1})$. We can first try
to simplify this equation with the knowledge of the causal graph in Figure 8,
_i.e._ using the true underlying generative model, since we aim to show that
learning the causal factors and aligning them correspondingly in the prior of
Equation 2 represents a global optimum of maximizing
$p(X^{t+1}|X^{t},I^{t+1})$. Using the conditional independence relations of
the graph in Figure 8, we write the joint distribution of all the variables in
the true generative model as:

$p(X^{t},X^{t+1},C^{t},C^{t+1},I^{t+1})=p(X^{t+1}|C^{t+1})\cdot\left[\prod_{i=1}^{K}p(C_{i}^{t+1}|C^{t},I_{i}^{t+1})\right]\cdot
p(X^{t}|C^{t})\cdot p(C^{t})\cdot p(I^{t+1})$ (7)

We can now condition on $X^{t}$ and $I^{t+1}$, marginalize out $C^{t}$ and
$C^{t+1}$ and write the conditional likelihood as:

$p(X^{t+1}|X^{t},I^{t+1})=\int_{C^{t+1}}\int_{C^{t}}p(X^{t+1}|C^{t+1})\cdot\left[\prod_{i=1}^{K}p(C_{i}^{t+1}|C^{t},I_{i}^{t+1})\right]\cdot
p(C^{t}|X^{t})dC^{t}dC^{t+1}$ (8)

In our assumptions of Section 3.1, we have defined the observation function
$h$ to be bijective, meaning that there exists an inverse $f$ that can
identify the causal factors $C^{t}$ and noise variable $E^{t}_{o}$ from
$X^{t}$. Thus, we can write $p(C^{t}|X^{t})=\delta_{f(X^{t})=C^{t}}$, where
$\delta$ is a Dirac delta. Since the noise on the observations, $E^{t}_{o}$,
is said to be independent of $X^{t+1}$ and $C^{t+1}$, we can remove it from
being in the conditioning set. This leads us to:

$p(X^{t+1}|X^{t},I^{t+1})=\int_{C^{t+1}}\left[\prod_{i=1}^{K}p(C_{i}^{t+1}|C^{t},I_{i}^{t+1})\right]\cdot
p(X^{t+1}|C^{t+1})dC^{t+1}$ (9)

Since we have assumed $h$ to be bijective, we know that for each $X^{t+1}$,
there exist only one combination of $C^{t+1}$ and $E^{t+1}_{o}$. Thus, by
using the change of variables formula, we can rewrite the equation above by:

$p(X^{t+1}|X^{t},I^{t+1})=\left|J_{h}\right|^{-1}\cdot\left[\prod_{i=1}^{K}p(C_{i}^{t+1}|C^{t},I_{i}^{t+1})\right]\cdot
p(E^{t+1}_{o})$ (10)

where $J_{h}=\frac{\partial h(C^{t+1},E^{t+1}_{o})}{\partial C^{t+1}\partial
E^{t+1}_{o}}$ denotes the Jacobian of the bijective/invertible observation
function $h$. Equation 10 constitutes a global optimum of the maximum
likelihood, since it represents the true underlying dynamics.

We relate this conditional likelihood to the prior setup of CITRIS. We show
that assigning $C^{t+1}_{i}$ to $z_{\psi_{i}}^{t+1}$, _i.e._ , learning the
true assignment function $\psi^{*}$, provides us with the same maximum
likelihood solution as in Equation 10. We have defined our objective in
Section 3 in Equation 2 as:

$p_{\phi}\left(z^{t+1}|z^{t},I^{t+1}\right)=\prod_{i=0}^{K}p_{\phi}\left(z_{\psi_{i}}^{t+1}|z^{t},I_{i}^{t+1}\right)$
(11)

Since we know that $g_{\theta}^{*}$ is an invertible function between
$\mathcal{X}$ and $\mathcal{Z}$, we know that $z^{t}$ must include all
information of $X^{t}$. Thus, we can also replace it with
$z^{t}=[C^{t},E_{o}^{t}]$, giving us:

$p_{\phi}\left(z^{t+1}|C^{t},E_{o}^{t},I^{t+1}\right)=\prod_{i=0}^{K}p_{\phi}\left(z_{\psi_{i}}^{t+1}|C^{t},E_{o}^{t},I_{i}^{t+1}\right)$
(12)

The optimal assignment function $\psi^{*}$ assigns sufficient dimensions to
each causal factor $C_{1},...,C_{K}$. Since $\mathcal{Z}$ can have a larger
space than $\mathcal{E}\times\mathcal{C}$, but $\mathcal{E}\times\mathcal{C}$
is sufficient to describe $\mathcal{X}$, we know that the remaining dimensions
of $\mathcal{Z}$ do not contain any information. Thus, the assignment function
$\psi^{*}$ can map them to any causal factor without a change in distribution.
Using this assignment function, we now consider
$z_{\psi^{*}_{i}}^{t+1}=C^{t+1}_{i}$ for $i=1,...,K$. Then, Equation 12
becomes:

$p_{\phi}\left(z^{t+1}|C^{t},E_{o}^{t},I^{t+1}\right)=\left[\prod_{i=1}^{K}p_{\phi}\left(z_{\psi^{*}_{i}}^{t+1}=C^{t+1}_{i}|C^{t},I_{i}^{t+1}\right)\right]\cdot
p(z_{\psi^{*}_{0}}^{t+1}|C^{t},E_{o}^{t})$ (13)

where we remove $E^{t}_{o}$ from the conditioning set for the causal factors,
since know that $C^{t+1}$ and $E_{o}^{t+1}$ is independent of $E_{o}^{t}$. We
further simplify by noting that $z_{\psi^{*}_{0}}^{t+1}=E_{o}^{t+1}$ is
independent of any other factor.

$p_{\phi}\left(z^{t+1}|C^{t},E_{o}^{t},I^{t+1}\right)=\left[\prod_{i=1}^{K}p_{\phi}\left(z_{\psi^{*}_{i}}^{t+1}=C^{t+1}_{i}|C^{t},I_{i}^{t+1}\right)\right]\cdot
p(z_{\psi^{*}_{0}}^{t+1}=E_{o}^{t+1})$ (14)

Finally, by using $g_{\theta}^{*}$, we can replace the distribution on
$z^{t+1}$ by a distribution on $X^{t+1}$ by the change of variables formula:

$p_{\phi}\left(X^{t+1}|C^{t},E_{o}^{t},I^{t+1}\right)=\left|\frac{\partial
g_{\theta}^{*}(z^{t+1})}{\partial
z^{t+1}}\right|\cdot\left[\prod_{i=1}^{K}p_{\phi}\left(z_{\psi^{*}_{i}}^{t+1}=C^{t+1}_{i}|C^{t},I_{i}^{t+1}\right)\right]\cdot
p(z_{\psi^{*}_{0}}^{t+1}=E_{o}^{t+1})$ (15)

Thereby, it is apparent that $g_{\theta}^{*}$ is equal to $h^{-1}$, since both
are identical invertible functions between the same spaces ($\mathcal{Z}$
becomes $\mathcal{E}\times\mathcal{C}$ here). Hence, Equation 15 represents
the exact same distribution as Equation 10. Therefore, we have shown that the
function $\delta^{*}$ that disentangles the true latent variables
$C_{1},...,C_{K}$ and assigns them to the corresponding sets
$z_{\psi_{1}},...,z_{\psi_{K}}$ constitutes a global, but not necessarily
unique, optimum for maximizing the likelihood of Equation 2.

Note that while assigning $C^{t+1}_{i}$ to $z_{\psi_{i}}^{t+1}$ provides us
with the same maximum likelihood solution as in Equation 10, this is not the
only possible representation. Additional possible representation will be
discussed in Step 2.

###  B.4 Step 2: Characterizing the Disentanglement Class $\Delta$

Showing that the correct disentanglement constitutes a global optimum is not
sufficient for showing that a model trained on solving the maximum likelihood
solution converges to it, since there might potentially be multiple global
optima. Hence, this section discusses the class of causal representation
functions $\delta\in\Delta^{*}$ which can achieve the same maximum likelihood
optimum as the true causal factor disentanglement discussed in Section B.3.
For this, we first need to distinguish between the _variable_ and _invariable_
information of a causal variable $C_{i}$, which is introduced in Section
B.4.1. Next, we will discuss the causal representation function class $\Delta$
for the setting where interventions are independent, _i.e._
$I^{t+1}_{i}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}I^{t+1}_{j}|C^{t}$ for any $i\neq j$, and
finally extend it to confounded interventions.

####  B.4.1 Intervention-Independent Variables

$C^{t}$ $I^{t+1}_{i}$$C^{t+1}_{i}$ (a) Original causal graph of $C_{i}$

$C^{t}$
$I^{t+1}_{i}$$s^{\mathrm{var}}_{i}\left(C^{t+1}_{i}\right)$$s^{\mathrm{inv}}_{i}\left(C^{t+1}_{i}\right)$
(b) Minimal causal split graph of $C_{i}$

Figure 9: Splitting the causal variable $C_{i}$ in its minimal causal split.
(a) In the original causal graph, $C_{i}^{t+1}$ has $C^{t}$ (or an eventual
subset of it) and $I_{i}^{t+1}$ as its parents. (b) In the minimal causal
split, only the variable part $s^{\mathrm{var}}_{i}(C_{i}^{t+1})$ depends on
the intervention. The invariable part, $s^{\mathrm{inv}}_{i}(C_{i}^{t+1})$, is
independent of $I_{i}^{t+1}$, hence giving us an additional conditional
independence. Note that $s^{\mathrm{var}}_{i}(C_{i}^{t+1})$ and
$s^{\mathrm{inv}}_{i}(C_{i}^{t+1})$ are conditionally independent.

Interventions allow us to identify a causal variable by seeing the caused
change in its conditional distribution. However, especially when talking about
multidimensional causal variables, one might have interventions that only
affect a subset of the actual causal variable dynamics, while the rest remains
independent of the intervention. As we will see later, this can have an
influence on the identifiability result, making the found causal factors
intervention-dependent.

We start by considering a single causal factor
$C_{i}\in\mathcal{D}_{i}^{M_{i}}$ in the setup of Figure 8 under our
previously discussed assumptions. Suppose for each causal factor
$C_{i}\in\mathcal{D}^{M_{i}}$, there exists an invertible map
$s_{i}:\mathcal{D}_{i}^{M_{i}}\rightarrow\mathcal{D}_{i}^{\textup{var}}\times\mathcal{D}_{i}^{\textup{inv}}$
that splits the domain $\mathcal{D}^{M_{i}}$ of $C_{i}$ into a part that is
invariant and a part that is variant under intervention. We denote the two
parts of this map as

$s_{i}(C^{t}_{i})=(s^{\mathrm{var}}_{i}(C^{t}_{i}),s^{\mathrm{inv}}_{i}(C^{t}_{i}))$
(16)

The split $s$ must be invertible, so that we can map back and forth between
$\mathcal{D}_{i}^{M_{i}}$ and
$\mathcal{D}_{i}^{\textup{var}}\times\mathcal{D}_{i}^{\textup{var}}$ without
losing information. Furthermore, to be called a split, $s$ must satisfy
$s^{\mathrm{inv}}_{i}(C_{i}^{t})\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}I_{i}^{t}\mid\text{pa}(C_{i}^{t})$, _i.e._ ,
$s^{\mathrm{inv}}_{i}(C_{i}^{t})$ is independent of the intervention variable
$I_{i}^{t}$ given the parents of $C_{i}^{t}$. Further, both parts of the split
must be conditionally independent, _i.e._
$s^{\mathrm{inv}}_{i}(C_{i}^{t})\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}s^{\mathrm{var}}_{i}(C_{i}^{t})\mid\text{pa}(C_{i}^{t}),I_{i}^{t}$.
Hence, we can write their distributions as:

$p\left(s_{i}(C^{t+1}_{i})|C^{t},I^{t+1}_{i}\right)=p\left(s^{\mathrm{var}}_{i}(C^{t+1}_{i})|C^{t},I^{t+1}_{i}\right)\cdot
p\left(s^{\mathrm{inv}}_{i}(C^{t+1}_{i})|C^{t}\right)$ (17)

This means that $s^{\mathrm{var}}_{i}(C_{i}^{t})$ will contain the
manipulable, or _variable_ , part of $C^{t}_{i}$. In contrast,
$s^{\mathrm{inv}}_{i}(C_{i}^{t})$ is the _invariable_ part of $C_{i}^{t}$
which is independent of the intervention. This relation is visualized in
Figure 9.

For any causal variable, there may exist multiple possible splits, but there
is always at least the trivial split where
$\mathcal{D}_{i}^{\textup{var}}=\mathcal{D}_{i}^{M_{i}}$ is the original
domain of $C_{i}$, and $\mathcal{D}_{i}^{\textup{inv}}=\\{0\\}$ is the one-
element set (no invariant information). However, there might also exist splits
in which $s^{\mathrm{inv}}_{i}(C^{t+1}_{i})\neq\emptyset$. For instance, in a
multidimensional causal variable $\hat{C}\in\mathbb{R}^{3}$, if an
intervention only affects the first two dimensions while the last one remains
unaffected, we obtain the split
$s^{\mathrm{var}}([\hat{C}_{1},\hat{C}_{2}]),s^{\mathrm{inv}}(\hat{C}_{3})$.
Nonetheless, this can even happen for scalar variables, since we do not
constraint the possible distributions of $C_{i}$. We give an example for such
a case below.

Figure 10: Example for splitting a causal variable into an intervention-
dependent and -independent part. The two ground truth causal variables are the
$x$ and $y$ positions of the ball. See Section B.4.1 for details.

##### Example 1

Consider the scenario in Figure 10 where we have a ball with its two
positional dimensions $x$ and $y$ as its causal factors. For now, we only
focus on its $x$ position (in the remainder of the section, $x^{t}$ refers to
the position of the ball on the $x$-axis, not the full observation $X^{t}$
which we denote by a capital letter). Over time, the ball moves within one of
the two boxes, but cannot jump in between boxes. An example of such a
conditional could be:

$p(x^{t+1}|x^{t},I_{x}^{t+1}=0)=\begin{cases}\min(\max(x^{t}+\epsilon,0),1)&\text{if
}x^{t}<1\\\ \min(\max(x^{t}+\epsilon,1),2)&\text{otherwise}\\\ \end{cases}$
(18)

with $\epsilon\sim\mathcal{N}(0,0.1)$. Intuitively, the ball therefore moves
randomly around its previous position, while being bounded by the box it is
in. Due to its modular conditional distribution, we can rewrite the causal
variable $x$ and its distribution in terms of two different variables: its
position within its current box, $u\in[0,1]$, and a binary variable indicating
in which box the ball is, $b\in{0,1}$ (left/blue vs right/orange in Figure
10). Then, its conditional distribution becomes:

$p(x^{t+1}|x^{t},I_{x}^{t})=p\left(b^{t+1}|x^{t},I^{t+1}_{x}\right)\cdot
p\left(u^{t+1}|x^{t},I^{t+1}_{x}\right)$ (19)

Now, suppose that an intervention $I^{t+1}_{x}$ changes the box the ball is
in, while the relative position keeps evolving as it would under no
intervention, _i.e._ still depending on its parents. Then, we can yet write
its conditional distribution as:

$p(x^{t+1}|x^{t},I_{x}^{t})=p\left(b^{t+1}|x^{t},I^{t+1}_{x}\right)\cdot
p\left(u^{t+1}|x^{t}\right)$ (20)

Using the notation above, we therefore can define the split
$s^{\mathrm{var}}(x)=b,s^{\mathrm{inv}}(x)=u$, where $b$ depends on the
intervention, while $u$ does not. Note that
$s^{\mathrm{var}}(x)=x,s^{\mathrm{inv}}(x)=\emptyset$ is yet another valid
split in this case.

##### Example 2

Consider the same example as before, however, now with a different
intervention setup. Suppose that an intervention $I^{t+1}_{x}$ constitutes a
perfect intervention on $x$, under which $x^{t+1}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}x^{t}|I^{t+1}_{x}=1$. Then, the previous split
$s^{\mathrm{var}}(x)=b,s^{\mathrm{inv}}(x)=u$ is not valid anymore, if the
intervention target $I^{t+1}_{x}$ cannot be deterministically deduced from
$x^{t}$, since an intervention changes the distribution of the relative
position $u^{t+1}$. Hence, the only valid split is
$s^{\mathrm{var}}(x)=x,s^{\mathrm{inv}}(x)=\emptyset$. This shows that the
possible space of such splits depends on the available interventions.

##### Minimal causal variables

In the examples above, one can see that for certain situations, a causal
variable can have multiple valid splits
$s^{\mathrm{var}}_{i}(C_{i}),s^{\mathrm{inv}}_{i}(C_{i})$ since intervention-
independent information can be modeled in either $s^{\mathrm{var}}_{i}(C_{i})$
or $s^{\mathrm{inv}}_{i}(C_{i})$. The split that will be the most relevant for
the identifiability discussion here is the one that assigns only the
intervention-dependent information to $s^{\mathrm{var}}_{i}(C_{i})$, and the
rest to $s^{\mathrm{inv}}_{i}(C_{i})$. We define this as follows:

######  Definition B.2.

The _minimal causal split_ of a variable $C^{t}_{i}$ with respect to its
intervention variable $I^{t}_{i}$ is the split $s_{i}$ which maximizes the
entropy of $H(s^{\mathrm{inv}}_{i}(C^{t}_{i})|\text{pa}(C^{t}_{i}))$. Under
this split, $s^{\mathrm{var}}_{i}(C^{t}_{i})$ is defined as the _minimal
causal variable_ and denoted by $s_{i}^{\mathrm{var}^{*}}(C^{t}_{i})$.

Additionally, we also define:

######  Definition B.3.

The minimal causal mechanism of a variable $C_{i}$ with respect to its
intervention $I_{i}$ is defined as the conditional distribution
$p\left(s_{i}^{\mathrm{var}^{*}}(C^{t}_{i})|\text{pa}(C^{t+1}_{i}),I^{t+1}_{i}\right)$.

We refer to
$p\left(s_{i}^{\mathrm{var}^{*}}(C^{t}_{i})|C^{t},I^{t+1}_{i}\right)$ as
minimal causal mechanism, since it is the distribution for which as little as
possible information depends on $I^{t+1}_{i}$. Hence, the definition of this
mechanism depends on the characteristics of the provided intervention. As we
will see later, while we cannot guarantee to find the full causal mechanism,
we can yet identify the minimal causal mechanism.

The existence of a split where $s^{\mathrm{inv}}(C_{i})\neq\emptyset$ for any
causal factor $C_{i}$ creates additional, possible solutions that obtain the
same maximum likelihood as the true split. This is because
$s^{\mathrm{inv}}(C_{i})$ is independent of the intervention target $I_{i}$,
allowing it to be modeled by any set $z_{\psi_{j}}$ without losing
information. The following subsections further characterize the space of new
solutions with the existence of such splits.

####  B.4.2 Independent Interventions

In this section, we show by which class of disentanglement functions $\Delta$
a maximum likelihood solution of the generative model can be found.

However, to simplify the first steps, we assume that all intervention targets
are independent of each other given the causal factors of the previous time
step, _i.e._ $I^{t+1}_{i}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}I^{t+1}_{j}|C^{t}$ for any $i\neq j\in
1,\dots,K$. By construction, we also assume $I^{t+1}_{0}=0$ for all $t$. We
will extend it afterwards in Section B.4.3.

##### Solutions for $s^{\mathrm{inv}}_{i}(C_{i})=\emptyset$ for all
$i\in\llbracket 1..K\rrbracket$

As a first step, we assume that for all causal factors $C_{1},...,C_{K}$,
there does not exist any minimal causal mechanism split besides
$s^{\mathrm{inv}}_{i}(C_{i})$ being the empty set. Therefore, all of $C_{i}$
is dependent on $I^{t+1}_{i}$. For this case, consider an arbitrary partition
of $C_{i}$, $s^{0}(C_{i}),s^{1}(C_{i})$, with the same invertibility
constraints as $s^{\mathrm{var}},s^{\mathrm{inv}}$ and conditionally
independence between $s^{0}(C_{i}),s^{1}(C_{i})$, but with a non-empty
invariable part, _i.e._ , $s^{1}(C_{i})\neq\emptyset$. For this partition, the
conditional entropy of $s^{1}(C_{i})$ given $C^{t}$ must be strictly lower
when conditioning also on $I^{t+1}_{i}$:

$H\left(s^{1}(C^{t+1}_{i})|C^{t},I^{t+1}_{i}\right)<H\left(s^{1}(C^{t+1}_{i})|C^{t}\right)$
(21)

If the conditional entropy was equal, then $s^{1}(C^{t+1}_{i})$ would be
independent of $I^{t+1}_{i}$ given $C^{t}$, which is only true for
$s^{\mathrm{inv}}_{i}(C_{i})$. Since we assume $s^{\mathrm{inv}}_{i}(C_{i})$
is empty, while $s^{1}(C^{t+1}_{i})$ is not, this can never happen. Thus, to
model a causal factor $C_{i}$ where $s^{\mathrm{inv}}_{i}(C_{i})=\emptyset$, a
maximum likelihood solution can only be found if all of
$s^{\mathrm{var}}_{i}(C_{i})$ is conditioned on $I^{t+1}$. Further, since in
this setting $C_{j}^{t+1}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}C_{i}^{t+1}|C^{t},I^{t+1}$ for all
$i,j\in\llbracket 1..K\rrbracket,i\neq j$, there cannot exist any split across
multiple causal factors that violate the entropy inequality above for
$I^{t+1}_{i}$ and $I^{t+1}_{j}$ while still modeling the true conditional
distributions.

Similarly, in this setting under the specified assumptions, the information of
$I^{t+1}_{i}$ cannot be determined by any other target variable
$I^{t+1}_{j},i\neq j$, since otherwise, the targets would not be independent.
Hence, we can write the following entropy inequality for any $i\neq j$:

$H\left(C^{t+1}_{i}|C^{t},I^{t+1}_{i}\right)<H\left(C^{t+1}_{i}|C^{t}\right)=H\left(C^{t+1}_{i}|C^{t},I^{t+1}_{j}\right)$
(22)

Therefore, one can only achieve the maximum likelihood solution (_i.e._ , the
minimum entropy solution) if all information of $C^{t+1}_{i}$ is conditioned
on $I^{t+1}_{i}$ for $i=1,...,K$ in addition to $C^{t}$. This implies that
each factor $p_{\phi}(z_{\psi_{i}}^{t+1}|z^{t},I^{t+1}_{i})$ described in
Equation 2 will have $p(C^{t+1}_{i}|z^{t},I^{t+1}_{i})$, and therefore
$z_{\psi_{i}}^{t+1}=C^{t+1}_{i}$ as its maximum likelihood solution.

Nonetheless, this excludes $z_{\psi_{0}}$, _i.e._ the factors independent of
any intervention, in this case being the noise $E_{o}^{t+1}$. Since it is
independent of any interventions, any distribution of $E_{o}^{t+1}$ across the
different causal factor sets $z_{\psi_{0}},...,z_{\psi_{K}}$ will achieve the
same likelihood score, as long as the part of $E_{o}^{t+1}$ across factors is
independent. Hence, in conclusion for this scenario, we can guarantee that
$z_{\psi_{i}}$ will model all information of $C_{i}$ and no other causal
factor $C_{j},i\neq j$, but can contain additional information from $E^{t+1}$.

##### Solutions with invariable parts

Next, we consider the scenario where there exists a split with
$s^{\mathrm{inv}}_{i}(C_{i})\neq\emptyset$ for some causal variables in
$i\in\llbracket 1..K\rrbracket$. For this case, we can write the maximum
likelihood solution of Equation 10 as:

$\displaystyle p(X^{t+1}|X^{t},I^{t+1})$ $\displaystyle=\left|\frac{\partial
g_{\theta}^{*}(z^{t+1})}{\partial
z^{t+1}}\right|\cdot\left[\prod_{i=1}^{K}p(C_{i}^{t+1}|I_{i}^{t+1},C^{t})\right]\cdot
p(E^{t+1})$ (23) $\displaystyle=\left|\frac{\partial
g_{\theta}^{*}(z^{t+1})}{\partial
z^{t+1}}\right|\cdot\left[\prod_{i=1}^{K}p(s^{\mathrm{var}}_{i}(C_{i}^{t+1})|I_{i}^{t+1},C^{t})\right]\cdot\left[\prod_{i=1}^{K}p(s^{\mathrm{inv}}_{i}(C_{i}^{t+1})|C^{t})\right]\cdot
p(E^{t+1})$ (24)

This equation shows that one can assign
$s^{\mathrm{inv}}_{1}(C_{1}),...,s^{\mathrm{inv}}_{K}(C_{K})$ to any latent
variable set $z_{\psi_{0}},...,z_{\psi_{K}}$ or split it across, while
achieving the same optimal likelihood, since they are independent of any
intervention target. The remaining information in
$s^{\mathrm{inv}}_{1}(C_{1}),...,s^{\mathrm{inv}}_{K}(C_{K})$ thereby acts the
same way as the noise variable $E^{t+1}$. Thus, there exist multiple maximum
likelihood solutions with different splits of information to causal factors.

However, on the other hand, the solution space is yet restricted by the
assignment of $s^{\mathrm{var}}_{i}(C_{i})$. In particular, if
$s^{\mathrm{var}}_{i}(C_{i})$ cannot be split further into an
invariable/intervention-independent part, we can rely on the same results from
the previous setting, when considering $s^{\mathrm{var}}_{i}(C_{i})$ as new
causal variables. In case there exist another split of $C_{i}$ which would add
more information to $s^{\mathrm{inv}}_{i}(C_{i})$, this part could not be
guaranteed to be matched to the causal factor $C_{i}$ due to its independence.
Hence, in conclusion here, we can guarantee that $z_{\psi_{i}}$ will model all
information of $s^{\mathrm{var}}(C_{i})$ and no other causal factor
$s^{\mathrm{var}}(C_{j}),i\neq j$, if there does not exist another split of
$s^{\mathrm{var}}(C_{i})$. The additional information of $E^{t+1}$ as well as
$s^{\mathrm{inv}}(C_{j})$ can be assigned to any causal variable. In the third
step of the proof (Section B.5), we discuss how one can yet obtain a unique
solution.

####  B.4.3 Confounded Interventions

In the previous discussion, we have used the assumption that interventions are
independent of each other: $I^{t+1}_{i}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}I^{t+1}_{j}|C^{t}$. This assumption was
required for showing that conditioning information of $C^{t+1}$ on any other
target will lead to the same entropy as having it without a target, _i.e._
$H\left(C^{t+1}_{i}|C^{t},I^{t+1}_{j}\right)=H\left(C^{t+1}_{i}|C^{t}\right)$.
In this section, however, we consider a wider range of interventions.
Specifically, we assume that the intervention targets
$I^{t+1}_{1},...,I^{t+1}_{K}$ are confounded by some unobserved variable
$R^{t+1}$ besides $C^{t}$. This allows the modeling of, for example, single-
target interventions or groups of interventions, _e.g._
$I^{t+1}\in\\{[0,0,0],[1,1,0],[0,1,1]\\}$ for a three-variable case. Under
such a setup, the entropy equation from before, _i.e._
$H\left(C^{t+1}_{i}|C^{t},I^{t+1}_{j}\right)\neq
H\left(C^{t+1}_{i}|C^{t}\right)$, is not valid anymore since $I^{t+1}_{j}$ and
$I^{t+1}_{i}$ are not necessarily independent anymore and hence
$C^{t+1}_{i}\not\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}I^{t+1}_{j}$ can occur for some
$i,j\in\llbracket 1..K\rrbracket,i\neq j$.

Despite that, a causal factor $C^{t+1}_{i}$ is still independent of any other
target $I^{t+1}_{j},i\neq j$, as long as it is conditioned on its true target
and previous time step: $C^{t+1}_{i}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}I^{t+1}_{j}|C^{t},I^{t+1}_{i}$. This is
because $C^{t}$ and $I^{t+1}_{i}$ are all the parents of $C^{t+1}_{i}$, as
shown in the causal graph of Figure 8. Further, suppose that there exist
information of $C^{t+1}_{i}$ which is statistically independent of the
intervention $I^{t+1}_{i}$, _i.e._ $s^{\mathrm{inv}}_{i}(C_{i})\neq\emptyset$.
Then, this will also be independent of any other intervention target
$I^{t+1}_{j}$, since $s^{\mathrm{inv}}_{i}(C_{i})\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}I^{t+1}_{i}|C^{t}$, and all paths from $C_{i}$
to $I^{t+1}_{j}$ include $I^{t+1}_{i}$. Hence, our discussion of the
intervention-independent parts follow the same logic as in Section B.4.2, and
we are left with showing that $s^{\mathrm{var}}_{i}(C_{i})$ is modeled by
$z_{\psi_{i}}$ in any maximum likelihood solution.

For this, we consider a pair of variables $C_{i},C_{j}$, for which
$I^{t+1}_{i}\not\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}I^{t+1}_{j}|C^{t}$, and show under which
circumstances we can guarantee that no information of
$s^{\mathrm{var}}_{i}(C_{i})$ will be modeled in $z_{\psi_{j}}$. It is
sufficient to limit the discussion to pairs of variables, since one latent
variable can be only assigned to a single causal variable, hence to the actual
one it belongs to, $C_{i}$, or any other variable $C_{j}$ here. A crucial
insight to the discussion will be that the influence of $I^{t+1}_{j}$ to
$C^{t+1}_{i}$ solely relies on $I^{t+1}_{i}$ being correlated to both
variables. Further, one requirement is that no additional conditional
independence relations exist, such as
$s^{\mathrm{var}}_{i}(C_{i})\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}I^{t+1}_{i}|I^{t+1}_{j},C^{t}$, which is
covered by our faithfulness assumption. Now, under this setup, we consider
three cases:

  1. 1.

for every time step $t$, the two variables $C_{i},C_{j}$ have always been
intervened on together, _i.e._ $I^{t+1}_{i}=I^{t+1}_{j}$ for any $t$;

  2. 2.

there exist a time step $t$ at which $I^{t+1}_{i}=0,I^{t+1}_{j}=1$;

  3. 3.

there exist a time step $t$ at which $I^{t+1}_{i}=1,I^{t+1}_{j}=0$.

Note that the only excluded case is when for every time step $t$,
$I^{t+1}_{i}=0,I^{t+1}_{j}=0$. This case refers to not having observed
interventions for any of the two variables, and goes back to Section B.4.2,
where the variable part of $C_{i}$ is empty, _i.e._
$s^{\mathrm{var}}_{i}(C_{i})=\emptyset$. Hence, in that case,
$s^{\mathrm{var}}_{i}(C_{i})$ would have no influence on the modeled solution.

In the first case, since the two factors have always been intervened on
together, we know that $I^{t+1}_{i}=I^{t+1}_{j}$ for any time step $t$. Hence,
one can assign the information of $s^{\mathrm{var}}_{i}(C^{t+1}_{i})$,
$s^{\mathrm{var}}_{j}(C^{t+1}_{j})$, or the union of both to either
intervention target $I^{t+1}_{i}$ or $I^{t+1}_{j}$, without losing any
information. Moreover, if $s^{\mathrm{var}}_{i}(C^{t+1}_{i})$ has multiple
independent dimensions, _i.e._ can be written as a product of multiple,
conditionally independent variables, one can even split information of $C_{i}$
over the two targets. This shows that in the general case, we cannot
disentangle between two variables which have always been intervened on
together. Similarly, if more than 2 variables have always been intervened on
together, we cannot disentangle among all those variables.

For the second case, we can deduce that there must be interventions provided
for at least the observational case, _i.e._ $I^{t+1}_{i}=0,I^{t+1}_{j}=0$, the
case where $C_{j}$ is intervened on but not $C_{i}$, _i.e._
$I^{t+1}_{i}=0,I^{t+1}_{j}=1$, and either the joint intervention on both
$C_{i},C_{j}$ or only interventions on $C_{i}$, not $C_{j}$. The reason why
one of the two latter cases needs to exist is that if it would not be the
case, $I^{t+1}_{i}=0$ would be zero for any $t$. In that case, the minimal
causal mechanism of $C_{i}$ uses $s^{\mathrm{var}}_{i}(C_{i})=\emptyset$,
hence making the modeling of $s^{\mathrm{var}}_{i}(C_{i})$ irrelevant for the
maximum likelihood solution.

Thus, from these different intervention settings, it is apparent that there
cannot exist a deterministic function $f$ with which we can determine
$I^{t+1}_{i}$ from seeing $I^{t+1}_{j}$. If we observe joint interventions on
both variables, then for $I^{t+1}_{j}=1$, both $I^{t+1}_{i}=0$ and
$I^{t+1}_{i}=1$ can occur. Similarly, if we observe interventions on $C_{i}$
when $C_{j}$ is not intervened on, then both $I^{t+1}_{i}=0$ and
$I^{t+1}_{i}=1$ can occur for $I^{t+1}_{j}=0$. If both joint interventions and
single interventions on $C_{i}$ have been observed, we cannot determine
$I^{t+1}_{i}$ from $I^{t+1}_{j}$ at either $I^{t+1}_{j}=0$ or $I^{t+1}_{j}=1$.
Since $s^{\mathrm{var}}_{i}(C^{t+1}_{i})\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}I_{j}^{t+1}|C^{t},I^{t+1}_{i}$ by definition
and $s^{\mathrm{var}}_{i}(C^{t+1}_{i})\not\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}I^{t+1}_{i}|C^{t},I_{j}^{t+1}$ (the latter
because $I_{i}^{t+1}$ is not a deterministic function of $I_{j}^{t+1}$, and
therefore the dependence holds), we can write:

$H\left(s^{\mathrm{var}}_{i}(C^{t+1}_{i})|C^{t},I^{t+1}_{i}\right)=H\left(s^{\mathrm{var}}_{i}(C^{t+1}_{i})|C^{t},I^{t+1}_{i},I^{t+1}_{j}\right)<H\left(s^{\mathrm{var}}_{i}(C^{t+1}_{i})|C^{t},I^{t+1}_{j}\right)$
(25)

In conclusion, we cannot find the maximum likelihood solution if any
information of $C^{t+1}_{i}$, which depends on $I^{t+1}_{i}$, is assigned to
latent variables $z_{\psi_{j}}$. Hence, the maximum likelihood solution will
strictly model $s^{\mathrm{var}}_{i}(C_{i})$ in $z_{\psi_{i}}$.

Finally, in the third case, we can take a similar argument as for the second
case. The only difference is that any of the additional intervention cases
(joint or single on $C_{j}$), we have that from $I^{t+1}_{j}=0$, both
$I^{t+1}_{i}=0$ and $I^{t+1}_{i}=1$ can occur. Hence, the inequality in
Equation 25 is still valid, and we cannot replace $I^{t+1}_{i}$ by
$I^{t+1}_{j}$ for any subset of information of $s^{\mathrm{var}}_{i}(C_{i})$.
In summary, the maximum likelihood solution will strictly model
$s^{\mathrm{var}}_{i}(C_{i})$ in $z_{\psi_{i}}$ also for this case.

Therefore, we can summarize the results in the following statement. We can
disentangle the intervention-dependent part of any two variables
$C_{i},C_{j}$, if there does not exist a deterministic function $f$ for which
$I_{i}^{t}=f(I_{j}^{t})$ holds for every time step $t$.

###  B.5 Step 3: Deriving the Final Theorem

Now that we have discussed the class of disentanglement functions $\Delta^{*}$
with their corresponding solutions, we can take the final step by adding
constraints that ensure a unique solution. In all the settings discussed in
Section B.4.2 and Section B.4.3, the problem is that intervention-independent
information can be represented in any of the latent sets
$z_{\psi_{0}},...,z_{\psi_{K}}$ without affecting the optimal likelihood.
However, our main goal in getting a causal representation is that we
disentangle information from different causal factors, meaning that we want to
guarantee that the latents of $z_{\psi_{i}}$ will only model information of
the causal factor $C_{i}$, and no other causal factor $C_{j},i\neq j$. Thus,
we can do this by collecting all intervention-independent information in
$z_{\psi_{0}}$. In other words, our ideal solution would be to have the
latents of $z_{\psi_{i}}$ model $s^{\mathrm{var}}_{i}(C_{i})$ ($i=1,...,K$),
and $z_{\psi_{0}}$ to model
$\\{s^{\mathrm{inv}}_{1}(C_{1}),...,s^{\mathrm{inv}}_{K}(C_{K})\\}$, where the
split $s^{\mathrm{var}}_{i},s^{\mathrm{inv}}_{i}$ was chosen to maximize the
entropy of $p(s^{\mathrm{inv}}_{i}(C_{i}^{t+1})|C^{t})$. To find both the
right splits and collecting all intervention-independent information in
$z_{\psi_{0}}$, we thus want to find the representation function
$\hat{\delta}\in\Delta^{*}$ which maximizes the entropy of $z_{\psi_{0}}$
while maintaining the optimal likelihood. If any intervention-independent
information would not be modeled in $z_{\psi_{0}}$, it implies that there must
exist another solution with greater entropy in $z_{\psi_{0}}$, since all
$s^{\mathrm{inv}}_{1}(C_{1}),...,s^{\mathrm{inv}}_{K}(C_{K})$ as well as
$s^{\mathrm{var}}_{1}(C_{1}),...,s^{\mathrm{var}}_{K}(C_{K})$ are
conditionally independent of each other (_i.e._ adding parts to $z_{\psi_{0}}$
cannot reduce the entropy). Further, since we try to maximize the entropy of
$z_{\psi_{0}}$, we find the information splits
$s^{\mathrm{var}}_{i},s^{\mathrm{inv}}_{i}$ that maximize the entropy of its
intervention-independent part. This is the same split as we had defined as
minimal causal mechanisms in Section B.4.1. Thus, we can summarize this result
as follows:

######  Theorem B.4.

Suppose that $\phi^{*}$, $\theta^{*}$ and $\psi^{*}$ are the parameters that,
under the constraint of maximizing the likelihood
$p_{\phi}(g_{\theta}(x^{t+1})|g_{\theta}(x^{t}),I^{t+1})$, maximize the
entropy of $p_{\phi}(z_{\psi_{0}}^{t+1}|z^{t},I^{t+1})$. Then, with sufficient
latent dimensions, the model $\phi^{*},\theta^{*},\psi^{*}$ learns a latent
structure where $z_{\psi_{i}}^{t+1}$ models the minimal causal variable of
$C_{i}$ if $C^{t+1}_{i}\not\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}I^{t+1}_{i}|C^{t},I^{t+1}_{j}$ for any $i\neq
j$. All remaining information independent of any interventions is modeled in
$z_{\psi_{0}}$.

The conditional independence $C^{t+1}_{i}\not\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}I^{t+1}_{i}|C^{t},I^{t+1}_{j}$ ensures that
there exists no deterministic function $f$ for which
$I^{t+1}_{i}=f(I^{t+1}_{j})$. This also includes when $I^{t+1}_{i}$ is
constant, _i.e._ , when $C_{i}$ is intervened all the time or not at all,
since then, $C_{i}$ becomes independent of $I^{t+1}_{i}$.

###  B.6 Non-Identifiability without Interventions

This section discusses Proposition 3.1 and its accompanying proof. Proposition
3.1 states that in TRIS, if two causal factors $C_{i}$ and $C_{j}$ have only
been jointly intervened on or not at all, then there exists a causal graph in
which $C_{i}$ and $C_{j}$ cannot be uniquely identified from observations $X$
and intervention targets $I$. In other words, without interventions, it cannot
be guaranteed that for all possible settings, the causal variables can be
uniquely identified. A similar situation occurs in (non-linear) independent
component analysis (Hyvärinen et al., 2001, 2019), where the goal is to
disentangle and identify $N$ independent sources/signals from observations in
which the signals are entangled. In Hyvärinen et al. (2001, Section 7.5), it
is shown that if we have two, independent variables $s_{1},s_{2}$ that follow
a standard Gaussian distribution $\mathcal{N}(0,1)$, then any mixing of the
two variables with an orthogonal matrix $\bm{A}$ results in two variables,
$x_{1},x_{2}$ with $\bm{x}=\bm{A}\bm{s}$, which yet again follow a standard
Gaussian distribution. Thus, from the perspective of the probability density
function, the two representations cannot be distinguished, and hence no unique
solution can be found. This also implies that simply from samples of this
distribution, one cannot uniquely identify the true sources.

In TRIS, we do not have two independent variables, but instead variables that
have dependencies over time. This corresponds to the setting described in
Hyvärinen et al. (2001, Section 18), Belouchrani et al. (1997), or Hyvarinen &
Morioka (2017). Suppose that we have two variables $s^{t}_{1},s^{t}_{2}$ at a
time step $t$, which follow a Gaussian distribution with constant variance
over time:

$s^{t}_{1}\sim\mathcal{N}(\mu_{1}(s^{t-1}_{1},s^{t-1}_{2}),\sigma^{2}),x^{t}_{2}\sim\mathcal{N}(\mu_{2}(s^{t-1}_{1},s^{t-1}_{2}),\sigma^{2})$
(26)

where $\mu_{1},\mu_{2}$ are arbitrary functions on the variables of the
previous time step, $s^{t-1}_{1},s^{t-1}_{2}$. The probability density
function of it is:

$p(s^{t}_{1},s^{t}_{2}|\bm{s}^{t-1})=\frac{1}{\sqrt{\det(2\pi\bm{\Sigma})}}\exp\left(-\frac{1}{2}(\bm{s}^{t}-\bm{\mu}^{t})^{T}\bm{\Sigma}^{-1}(\bm{s}^{t}-\bm{\mu}^{t})\right)$
(27)

where $\bm{\mu}^{t}$ represents the mean based on $\bm{s}^{t-1}$, and
$\bm{\Sigma}=\begin{bmatrix}\sigma^{2}&0\\\ 0&\sigma^{2}\end{bmatrix}$. Now,
consider a mixing of the sources by an orthogonal matrix $\bm{A}$, i.e.
$\bm{x}^{t}=\bm{A}\bm{s}^{t}$. Then, the probability density function of those
are:

$\displaystyle p(x^{t}_{1},x^{t}_{2}|\bm{s}^{t-1})$
$\displaystyle=\frac{1}{\sqrt{\det(2\pi\bm{A}^{T}\bm{\Sigma}\bm{A})}}\exp\left(-\frac{1}{2}(\bm{A}\bm{s}^{t}-\bm{A}\bm{\mu}^{t})^{T}(\bm{A}^{T}\bm{\Sigma}\bm{A})^{-1}(\bm{A}\bm{s}^{t}-\bm{A}\bm{\mu}^{t})\right)$
(28)
$\displaystyle=\frac{1}{\sqrt{2\pi\det(\bm{A}^{T})\det(\bm{\Sigma})\det(\bm{A})}}\exp\left(-\frac{1}{2}(\bm{s}^{t}-\bm{\mu}^{t})^{T}\bm{A}^{T}\bm{A}\bm{\Sigma}^{-1}\bm{A}^{T}\bm{A}(\bm{s}^{t}-\bm{\mu}^{t})\right)$
(29)
$\displaystyle=\frac{1}{\sqrt{2\pi\det(\bm{\Sigma})}}\exp\left(-\frac{1}{2}(\bm{s}^{t}-\bm{\mu}^{t})^{T}\bm{\Sigma}^{-1}(\bm{s}^{t}-\bm{\mu}^{t})\right)$
(30)

where $\bm{A}^{T}\bm{A}=\bm{I}$ since $\bm{A}^{T}=\bm{A}^{-1}$ for rotation
matrices, and $\det(\bm{A})=1$. This shows that the density functions for the
rotated and original signal are identical. Thus, similar to the time-
independent case, we cannot uniquely identify the original sources.

Coming back to the setting of TRIS, the two sources $s^{t-1}_{1},s^{t-1}_{2}$
can represent two causal variables, $C_{1}$ and $C_{2}$, and the orthogonal
matrix $\bm{A}$ represents a valid observation function $h$. Hence, the
previous conclusion also holds here, meaning that there exist a setting in
which $C_{1}$ and $C_{2}$ cannot be uniquely identified from observational
data only. This concludes the proof for Proposition 3.1.

##  Appendix C Experimental Details

In this section, we describe the datasets and hyperparameter used in the
experiments of Section 6 in detail. First, we discuss the two datasets, the
Temporal Causal3DIdent dataset (Section C.1) and the Interventional Pong
dataset (Section C.2), including giving more examples and insights in the
underlying dynamics. The datasets will be publicly released upon publication.
In the second part, we discuss the experimental design (Section C.3), which
includes a closer discussion of the correlation metrics as well as
implementation details of CITRIS. Finally, we give an overview of the used
hyperparameters in Section C.4.

###  C.1 Temporal Causal3DIdent Dataset

|  |  |  |  |  |
---|---|---|---|---|---|---
(a) Teapot | (b) Cow | (c) Head | (d) Horse | (e) Armadillo | (f) Dragon | (g) Hare
Figure 11: An example image for each object shape in the Temporal-
Causal3DIdent dataset.

The creation of the Temporal Causal3DIdent dataset closely followed the setup
of von Kügelgen et al. (2021). We used the code provided by Zimmermann et al.
(2021)222https://github.com/brendel-group/cl-ica to render the images via
Blender (Blender Online Community, 2021). We will publish the adapted code for
this dataset generation as well as the full datasets used here upon
publication with an accompanying license.

####  C.1.1 Causal Factor Description

pos_x | pos_y | pos_z | rot_$\alpha$ | rot_$\beta$ | rot_s | hue_s | hue_b | hue_o | obj_s
---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  |  |
|  |  |  |  |  |  |  |  |
|  |  |  |  |  |  |  |  |
|  |  |  |  |  |  |  |  |
Figure 12: Visualizing the different factors of variation in the Temporal
Causal3DIdent dataset. Each column represents one dimension of a causal
factor, and the different rows show the original image (first row) with only
the corresponding causal factor being changed.

To begin with, we give a more detailed description of the 7 causal factors
here, and provide examples of varying individual factors in Figure 12:

  * •

The object position (pos_o) is modeled in 3 dimensions ($x$ \- depth
dimension, $y$ \- horizontal, $z$ \- vertical). All values are scaled between
-2 as minimum and a maximum of 2, following Zimmermann et al. (2021). For the
$y$ and $z$ dimension, this ensures that the object stays within the frame.
For the $x$ dimensions, it ensures that the object does not cover the whole
camera image, but also does not become too small in resolution for recognizing
rotations and shapes.

  * •

The object rotation (rot_o) is modeled in 2 dimensions ($\alpha$ \- roll
angle, $\beta$ \- pitch angle). All dimensions use circular values of
$[0,2\pi)$, meaning that in distributions, we consider values close to $0$ and
$2\pi$ as close as well. The rotation is restricted to two dimensions to
guarantee that every object rotation has a unique value assignment of the
angles. This can be violated when modeling three angles with a value range of
$[0,2\pi)$.

  * •

The spotlight rotation (rot_s) is the positioning of the spotlight as an
angle. The value range is $[0,2\pi)$, where, similarly as the object rotation,
we consider it to be circular.

  * •

The spotlight hue is the color of the spotlight. The value range is, again,
$[0,2\pi)$, where $0$ corresponds to red. Note that the color appearance of
the spotlight changes with the object and background color, since we only see
the combined reflected color.

  * •

The background hue (hue_b) is the color of the background. The value range is
$[0,2\pi)$ with the same color spectrum as the spotlight hue.

  * •

The object hue (hue_o) is the color of the object, and follows the same setup
as the background hue.

  * •

The object shape (obj_s) is a categorical variable describing the object
shape. For the 7-shape dataset version, we consider the same object shapes as
von Kügelgen et al. (2021): Cow (Crane, 2021), Head (Rusinkiewicz et al.,
2021), Dragon (Curless & Levoy, 1996), Hare (Turk & Levoy, 1994), Armadillo
(Krishnamurthy & Levoy, 1996), Horse (Praun et al., 2000), Teapot (Newell,