# Products, Abstractions and Inclusions of Causal Spaces

Simon Buchholz∗ Empirical Inference Department
Max Planck Institute for Intelligent Systems
Tübingen, Germany  Junhyung Park∗ Empirical Inference Department
Max Planck Institute for Intelligent Systems
Tübingen, Germany  Bernhard Schölkopf  Empirical Inference Department
Max Planck Institute for Intelligent Systems
Tübingen, Germany

###### Abstract

Causal spaces have recently been introduced as a measure-theoretic framework
to encode the notion of causality. While it has some advantages over
established frameworks, such as structural causal models, the theory is so far
only developed for single causal spaces. In many mathematical theories, not
least the theory of probability spaces of which causal spaces are a direct
extension, combinations of objects and maps between objects form a central
part. In this paper, taking inspiration from such objects in probability
theory, we propose the definitions of products of causal spaces, as well as
(stochastic) transformations between causal spaces. In the context of
causality, these quantities can be given direct semantic interpretations as
causally independent components, abstractions and extensions.

††*Equal Contribution.

##  1 Introduction

Mathematical modelling of the world allows us to represent and analyse real-
life situations in a quantitative manner. Depending on the aspects that one is
interested in, different mathematical tools are chosen for the modelling; for
example, to model how a system evolves over time, a system of differential
equations can be used, and to model the randomness of events, one can use
probability theory. Another aspect of the world which researchers are
increasingly more interested in modelling is causality [Woodward, 2005, Russo,
2010, Illari et al., 2011, Pearl and Mackenzie, 2018], and this is the focus
of our work.

For the mathematical frameworks used in modelling, there are always ways to
analyse multiple structures in a coherent manner. For example, in vector
spaces, we have the notions of subspaces, product spaces and maps between
vector spaces. In probability spaces, we have the notions of subspaces and
restrictions, product spaces and measurable maps and transition probability
kernels between spaces.

In this work, we consider a modelling of the world through a recently proposed
framework called causal spaces [Park et al., 2023]. Causal spaces are a direct
extension of probability spaces to encode causal information, and as such, are
rigorously grounded in measure-theory. While they have some advantages over
existing frameworks (e.g. structural causal models, or SCMs), such as the fact
that they can easily encode cycles and continuous-time stochastic processes
that are notoriously problematic in SCMs [Halpern, 2000, Bongers et al.,
2021], the theory of causal spaces is still in its infancy. In particular,
Park et al. [2023] only consider the development of single causal spaces, and
omit the discussion of construction of new causal spaces from existing ones or
maps between causal spaces. The latter is of particular interest to
researchers in causality for the purpose of abstraction (see related works in
Section 1.1). When systems, humans or animals perceive the world, they
consider different levels of detail depending on their ability to perceive and
retain information and their level of interest. It is therefore crucial to
connect the mathematical representations at varying levels of granularity in a
coherent way.

In probability spaces, such notions are well-established. Product measures
give rise to independent random variables, and measurable maps and probability
kernels between probability spaces give rise to pushforward measures, which
can be interpreted as abstractions or inclusions. Based on these concepts, and
using the fact that causal spaces are a direct extension of probability
spaces, we develop the notions of product causal spaces and causal
transformations.

###  1.1 Related Works

The theory of causality has two dominant strands [Imbens, 2019], one based on
SCMs [Pearl, 2009, Peters et al., 2017] and another based on potential
outcomes [Hernàn and Robins, 2020, Imbens and Rubin, 2015]. Since concepts
such as abstractions and connected components attract much more attention in
the SCM community than in the potential outcomes community, we focus on
comparisons with the SCM framework.

Seminal works on causal abstraction with SCMs are Rubenstein et al. [2017] and
Beckers and Halpern [2019], where the notions of exact transformations (to be
further discussed in Section 5), uniform transformations, abstractions, strong
abstractions and constructive abstractions are proposed. Beckers et al. [2020]
then relax these to an approximate notion. Massidda et al. [2023] extended the
notions to soft interventions, and Zečević et al. [2023] to continually
updated abstractions. Causal feature learning is a closely related approach,
that also aims to learn higher level features [Chalupka et al., 2015, 2016,
2017] There are also approaches based on category theory [Rischel and
Weichwald, 2021, Otsuka and Saigo, 2022, 2023] and probabilistic logic
[Ibeling and Icard, 2023], all grounded in SCMs; see [Zennaro, 2022] for a
review.

The notion of causal abstraction in the SCM framework has found applications
in interpretations of neural networks [Geiger et al., 2021, 2023] as well as
solving causal inference tasks (identification, estimation and sampling) at
different levels of granularity with neural networks [Xia and Bareinboim,
2024]. Moreover, Zennaro et al. [2023] proposed a way of learning an
abstraction from partial information about the abstraction, and demonstrates
an application of causal abstraction in the SCM framework in the context of
electric vehicle battery manufacturing and Kekić et al. [2023] learn an
abstraction that explains a specific target.

###  1.2 Paper Organisation

The rest of this paper is structured as follows. We first discuss the key
notions from the theory of causal spaces in Section 2. Then we introduce the
extension of product causal spaces in Section 3 followed by our definitions of
transformations of causal spaces in Section 4. We put our definitions into
context by comparing carefully to related works in Section 5. In Section 6, we
then show various properties of our transformations, in particular for the
subclass of abstractions.

##  2 Preliminaries & Notations

We take a probability space as a starting point, namely, a triple
$(\Omega,\mathscr{H},\mathbb{P})$; for a comprehensive introduction, see, for
example, [Cinlar, 2011, Durrett, 2019]. Following [Park et al., 2023], we
additionally insist that $\mathbb{P}$ is defined over the product measurable
space $(\Omega,\mathscr{H})=\otimes_{t\in T}(E_{t},\mathscr{E}_{t})$ with
$(E_{t},\mathscr{E}_{t})$ being the same standard measurable space if $T$ is
uncountable. Denote by $\mathscr{P}(T)$ the power set of $T$, and for
$S\in\mathscr{P}(T)$, we denote by $\mathscr{H}_{S}$ the sub-$\sigma$-algebra
of $\mathscr{H}=\otimes_{t\in T}\mathscr{E}_{t}$ generated by measurable
rectangles $\times_{t\in T}A_{t}$, where $A_{t}\in\mathscr{E}_{t}$ differs
from $E_{t}$ only for $t\in S$ and finitely many $t$. In particular,
$\mathscr{H}_{\emptyset}=\\{\emptyset,\Omega\\}$ is the trivial
sub-$\sigma$-algebra of $\Omega=\times_{t\in T}E_{t}$. Also, we denote by
$\Omega_{S}$ the subspace $\times_{s\in S}E_{s}$ of $\Omega=\times_{t\in
T}E_{t}$, and for $T\supseteq S\supseteq U$, we let $\pi_{SU}$ denote the
natural projection from $\Omega_{S}$ onto $\Omega_{U}$ and we use the
shorthand $\pi_{S}=\pi_{TS}$. We also write
$\omega_{S}=\pi_{S}\omega=(\omega_{s})_{s\in S}$. We write
$[d]=\\{1,\ldots,d\\}$ and $f_{\ast}\mathbb{P}$ denotes the pushforward
measure along the map $f$, namely, $f^{*}\mathbb{P}(A)=\mathbb{P}(f^{-1}(A))$.

We first recall the definition of causal spaces, as given in Park et al.
[2023].

######  Definition 2.1 (Causal Spaces, [Park et al., 2023, Definition 2.2]).

A causal space is defined as the quadruple
$(\Omega,\mathscr{H},\mathbb{P},\mathbb{K})$, where
$(\Omega,\mathscr{H},\mathbb{P})=(\times_{t\in T}E_{t},\otimes_{t\in
T}\mathscr{E}_{t},\mathbb{P})$ is a probability space and
$\mathbb{K}=\\{K_{S}\mathrel{\mathop{\mathchar
58\relax}}S\in\mathscr{P}(T)\\}$, called the causal mechanism, is a collection
of transition probability kernels $K_{S}$ from $(\Omega,\mathscr{H}_{S})$ into
$(\Omega,\mathscr{H})$, called the causal kernel on $\mathscr{H}_{S}$, that
satisfy the following axioms:

  1. (i)

for all $A\in\mathscr{H}$ and $\omega\in\Omega$, we have
$K_{\emptyset}(\omega,A)=\mathbb{P}(A)$;

  2. (ii)

for all $\omega\in\Omega$, and events $A\in\mathscr{H}_{S}$ and
$B\in\mathscr{H}$, we have $K_{S}(\omega,A\cap
B)=\boldsymbol{1}_{A}(\omega)K_{S}(\omega,B)=\delta_{\omega}(A)K_{S}(\omega,B)$.

The causal kernels $K_{S}$ can be defined equivalently as maps from
$(\Omega_{S},\mathscr{H}_{S})$ to $(\Omega,\mathscr{H})$ and it will be
convenient to use this viewpoint occasionally in the following (this was also
used implicitly in Park et al. [2023]). This observation is explained in
Appendix C.

Next, we recall the definition of interventions, which is the central concept
in any theory of causality.

######  Definition 2.2 (Interventions, [Park et al., 2023, Definition 2.3]).

Let $(\Omega,\mathscr{H},\mathbb{P},\mathbb{K})=(\times_{t\in
T}E_{t},\otimes_{t\in T}\mathscr{E}_{t},\mathbb{P},\mathbb{K})$ be a causal
space, $U\subseteq T$ a subset, $\mathbb{Q}$ a probability measure on
$(\Omega,\mathscr{H}_{U})$ and $\mathbb{L}=\\{L_{V}\mathrel{\mathop{\mathchar
58\relax}}V\in\mathscr{P}(U)\\}$ a causal mechanism on
$(\Omega,\mathscr{H}_{U},\mathbb{Q})$. An intervention on $\mathscr{H}_{U}$
via $(\mathbb{Q},\mathbb{L})$ is a new causal space
$(\Omega,\mathscr{H},\mathbb{P}^{\text{do}(U,\mathbb{Q})},\mathbb{K}^{\text{do}(U,\mathbb{Q},\mathbb{L})})$,
where the intervention measure $\mathbb{P}^{\text{do}(U,\mathbb{Q})}$ is a
probability measure on $(\Omega,\mathscr{H})$ defined, for $A\in\mathscr{H}$,
by

$\mathbb{P}^{\text{do}(U,\mathbb{Q})}(A)=\int\mathbb{Q}(d\omega_{U})K_{U}(\omega_{U},A)$

and
$\mathbb{K}^{\text{do}(U,\mathbb{Q},\mathbb{L})}=\\{K^{\text{do}(U,\mathbb{Q},\mathbb{L})}_{S}\mathrel{\mathop{\mathchar
58\relax}}S\subseteq T\\}$ is the intervention causal mechanism whose
intervention causal kernels are

$\displaystyle K^{\text{do}(U,\mathbb{Q},\mathbb{L})}_{S}(\omega_{S},A)$
$\displaystyle\qquad=\int L_{S\cap U}(\omega_{S\cap
U},d\omega^{\prime}_{U})K_{S\cup U}((\omega_{S\setminus
U},\omega^{\prime}_{U}),A).$

We also recall the definition of causal effect.

######  Definition 2.3 (Causal Effects, [Park et al., 2023, Definition B.1]).

Let $(\Omega,\mathscr{H},\mathbb{P},\mathbb{K})=(\times_{t\in
T}E_{t},\otimes_{t\in T}\mathscr{E}_{t},\mathbb{P},\mathbb{K})$ be a causal
space, $U\subseteq T$ a subset, $A\in\mathscr{H}$ an event and $\mathscr{F}$ a
sub-$\sigma$-algebra of $\mathscr{H}$ (not necessarily of the form
$\mathscr{H}_{S}$ for some $S\in\mathscr{P}(T)$).

  1. (i)

If $K_{S}(\omega,A)=K_{S\setminus U}(\omega,A)$ for all $S\in\mathscr{P}(T)$
and all $\omega\in\Omega$, then we say that $\mathscr{H}_{U}$ has no causal
effect on $A$, or that $\mathscr{H}_{U}$ is non-causal to $A$.

We say that $\mathscr{H}_{U}$ has no causal effect on $\mathscr{F}$, or that
$\mathscr{H}_{U}$ is non-causal to $\mathscr{F}$, if, for all
$A\in\mathscr{F}$, the $\sigma$-algebra $\mathscr{H}_{U}$ has no causal effect
on $A$.

  2. (ii)

If there exists $\omega\in\Omega$ such that
$K_{U}(\omega,A)\neq\mathbb{P}(A)$, then we say that $\mathscr{H}_{U}$ has an
active causal effect on $A$, or that $\mathscr{H}_{U}$ is actively causal to
$A$.

We say that $\mathscr{H}_{U}$ has an active causal effect on $\mathscr{F}$, or
that $\mathscr{H}_{U}$ is actively causal to $\mathscr{F}$, if
$\mathscr{H}_{U}$ has an active causal effect on some $A\in\mathscr{F}$.

  3. (iii)

Otherwise, we say that $\mathscr{H}_{U}$ has a dormant causal effect on $A$,
or that $\mathscr{H}_{U}$ is dormantly causal to $A$.

We say that $\mathscr{H}_{U}$ has a dormant causal effect on $\mathscr{F}$, or
that $\mathscr{H}_{U}$ is dormantly causal to $\mathscr{F}$, if
$\mathscr{H}_{U}$ does not have an active causal effect on any event in
$\mathscr{F}$ and there exists $A\in\mathscr{F}$ on which $\mathscr{H}_{U}$
has a dormant causal effect.

Finally, we recall the definition of sources, which allows us to connect the
causal kernels to the probability measure $\mathbb{P}$. For a
sub-$\sigma$-algebra $\mathscr{F}$ of $\mathscr{H}$, we denote the conditional
probability of an event $A\in\mathscr{H}$ given $\mathscr{F}$ by
$\mathbb{P}_{\mathscr{F}}$.

######  Definition 2.4 (Sources, [Park et al., 2023, Definition D.1]).

Let $(\Omega,\mathscr{H},\mathbb{P},\mathbb{K})=(\times_{t\in
T}E_{t},\otimes_{t\in T}\mathscr{E}_{t},\mathbb{P},\mathbb{K})$ be a causal
space, $U\subseteq T$ a subset, $A\in\mathscr{H}$ an event and $\mathscr{F}$ a
sub-$\sigma$-algebra of $\mathscr{H}$. We say that $\mathscr{H}_{U}$ is a
(local) source of $A$ if $K_{U}(\cdot,A)$ is a version of the conditional
probability $\mathbb{P}_{\mathscr{H}_{U}}(A)$. We say that $\mathscr{H}_{U}$
is a (local) source of $\mathscr{F}$ if $\mathscr{H}_{U}$ is a source of all
$A\in\mathscr{F}$. We say that $\mathscr{H}_{U}$ is a global source of the
causal space if $\mathscr{H}_{U}$ is a source of all $A\in\mathscr{H}$.

##  3 Product Causal Spaces and Causal Independence

We first give the definition of the product of causal kernels, and the product
of causal spaces. This constitutes the simplest way of constructing new causal
spaces from existing ones.

######  Definition 3.1 (Product Causal Spaces).

Suppose
$\mathscr{C}^{1}=(\Omega^{1},\mathscr{H}^{1},\mathbb{P}^{1},\mathbb{K}^{1})$
and
$\mathscr{C}^{2}=(\Omega^{2},\mathscr{H}^{2},\mathbb{P}^{2},\mathbb{K}^{2})$
with $\Omega^{1}=\times_{t\in T^{1}}E_{t}$ and $\Omega^{2}=\times_{t\in
T^{2}}E_{t}$ are two causal spaces. For all $S^{1}\subseteq T^{1}$ and
$S^{2}\subseteq T^{2}$, and for a pair of causal kernels
$K^{1}_{S^{1}}\in\mathbb{K}^{1}$ and $K^{2}_{S^{2}}\in\mathbb{K}^{2}$, we
define the product causal kernel $K^{1}_{S^{1}}\otimes K^{2}_{S^{2}}$, for
$\omega=(\omega_{1},\omega_{2})\in\Omega^{1}_{S^{1}}\times\Omega^{2}_{S^{2}}$
and events $A_{1}\in\mathscr{H}^{1}$ and $A_{2}\in\mathscr{H}^{2}$, by

$K^{1}_{S^{1}}\otimes K^{2}_{S^{2}}(\omega,A_{1}\times
A_{2})=K^{1}_{S^{1}}(\omega_{1},A_{1})K^{2}_{S^{2}}(\omega_{2},A_{2}).$

This can then be extended to all of $\mathscr{H}^{1}\otimes\mathscr{H}^{2}$
since the rectangles $A_{1}\times A_{2}$ with $A_{1}\in\mathscr{H}^{1}$ and
$A_{2}\in\mathscr{H}^{2}$ generate $\mathscr{H}^{1}\otimes\mathscr{H}^{2}$.
Then we define the product causal space

$\mathscr{C}^{1}\otimes\mathscr{C}^{2}=(\Omega^{1}\times\Omega^{2},\mathscr{H}^{1}\otimes\mathscr{H}^{2},\mathbb{P}^{1}\otimes\mathbb{P}^{2},\mathbb{K}^{1}\otimes\mathbb{K}^{2})$

where the product causal mechanism $\mathbb{K}^{1}\otimes\mathbb{K}^{2}$ is
the unique family of kernels of the form $(K^{1}\otimes K^{2})_{S^{1}\cup
S^{2}}=K^{1}_{S^{1}}\otimes K^{2}_{S^{2}}$ for $S^{1}\subseteq T^{1}$ and
$S^{2}\subseteq T^{2}$.

We first check that this procedure indeed produces a valid causal space.

######  Lemma 3.2 (Products of Causal Spaces are Causal Spaces).

The product causal space $\mathscr{C}^{1}\otimes\mathscr{C}^{2}$ as defined in
Definition 3.1 is a causal space.

The proof of this lemma can be found in Appendix B.1.

Note that it is only for the sake of simplicity of presentation that we
presented the notion of products only for two probability spaces. Indeed, we
can easily extend the definition to arbitrary products of causal kernels and
causal spaces, just like it is possible for products of probability spaces.

When we take a product of causal spaces, the corresponding components in the
resulting causal space do not have a causal effect on each other, as the
following result shows.

######  Lemma 3.3 (Causal Effects in Product Spaces).

Suppose
$\mathscr{C}^{1}=(\Omega^{1},\mathscr{H}^{1},\mathbb{P}^{1},\mathbb{K}^{1})$
and
$\mathscr{C}^{2}=(\Omega^{2},\mathscr{H}^{2},\mathbb{P}^{2},\mathbb{K}^{2})$
with $\Omega^{1}=\times_{t\in T^{1}}E_{t}$ and $\Omega^{2}=\times_{t\in
T^{2}}E_{t}$ are two causal spaces. Then in
$\mathscr{C}^{1}\otimes\mathscr{C}^{2}$,

  1. (i)

$\mathscr{H}_{T^{1}}$ has no causal effect on $\mathscr{H}_{T^{2}}$, and
$\mathscr{H}_{T^{2}}$ has no causal effect on $\mathscr{H}_{T^{1}}$;

  2. (ii)

$\mathscr{H}_{T^{1}}$ and $\mathscr{H}_{T^{2}}$ are (local) sources of each
other.

The proof of this Lemma is in Appendix B.1.

Product causal spaces are analogous to connected components in graphical
models – see, for example, [Sadeghi and Soo, 2023].

###  3.1 Causal Independence

Recall that, in probability spaces, two events $A$ and $B$ are independent
with respect to the measure $\mathbb{P}$ if $\mathbb{P}(A\cap
B)=\mathbb{P}(A)\mathbb{P}(B)$, i.e. the probability measure is the product
measure. Moreover, two $\sigma$-algebras are independent if each pair of
events from the two $\sigma$-algebras are independent111Many authors take the
view that the notion of independence is truly where probability theory starts,
as a distinct theory from measure theory [Cinlar, 2011, p.82, Section II.5]..
Similarly, for a sub-$\sigma$-algebra $\mathscr{F}$ of $\mathscr{H}$, two
events $A$ and $B$ are conditionally independent given $\mathscr{F}$ if
$\mathbb{P}_{\mathscr{F}}(A\cap
B)=\mathbb{P}_{\mathscr{F}}(A)\mathbb{P}_{\mathscr{F}}(B)$ almost surely, and
two $\sigma$-algebras are conditionally independent given $\mathscr{F}$ if
each pair of events from the two $\sigma$-algebras are conditionally
independent given $\mathscr{F}$.

Now we give a definition of causal independence that is analogous to
conditional independence.

######  Definition 3.4 (Causal Independence).

Consider a causal space $\mathscr{C}=(\Omega=\times_{t\in
T}E_{t},\mathscr{H}=\otimes_{t\in T}\mathscr{E}_{t},\mathbb{P},\mathbb{K})$.
Then for $U\subseteq T$, two events $A,B\in\mathscr{H}$ are causally
independent on $\mathscr{H}_{U}$ if, for all $\omega\in\Omega$,

$K_{U}(\omega,A\cap B)=K_{U}(\omega,A)K_{U}(\omega,B).$

We say that two sub-$\sigma$-algebras $\mathscr{F}_{1}$ and $\mathscr{F}_{2}$
are causally independent on $\mathscr{H}_{U}$ if each pair of events from
$\mathscr{F}_{1}$ and $\mathscr{F}_{2}$ are causally independent on
$\mathscr{H}_{U}$.

Semantically, causal independence should be interpreted as follows: if $A$ and
$B$ are causally independent on $\mathscr{H}_{U}$, then they are independent
once an intervention has been carried out on $\mathscr{H}_{U}$. Note also that
causal independence is really about the causal kernels, and has nothing to do
with the probability measure $\mathbb{P}$ of the causal space. Indeed, it is
possible for $A$ and $B$ to be causally independent but not probabilistically
independent, or causally independent but not conditionally independent, or
vice versa. Let us illustrate with the following simple examples.

$Y_{1}$$Y_{2}$$X$$Y$$X_{1}$$X_{2}$ Figure 1: Graphs of SCMs in Example 3.5.

######  Example 3.5 (Causal Independence).

  1. (i)

Consider three variables $X$, $Y_{1}$ and $Y_{2}$ related through the
equations

$X=N,\quad Y_{1}=X+U_{1},\quad Y_{2}=X+U_{2},$

where $N$, $U_{1}$ and $U_{2}$ are standard normal variables (see Figure 1
left). We denote by $\mathbb{P}$ their joint distribution on $\mathbb{R}^{3}$,
and we identify this SCM with the causal space
$(\mathbb{R}^{3},\mathscr{B}(\mathbb{R}^{3}),\mathbb{P},\mathbb{K})$222Here,
$\mathscr{B}$ represents the Borel $\sigma$-algebra., where $\mathbb{K}$ is
obtained via the above structural equations. Then it is clear to see that
$Y_{1}$ and $Y_{2}$ are causally independent on $\mathscr{H}_{X}$, since, for
every $x$, and $A,B\in\mathscr{B}(\mathbb{R})$, $K_{X}(x,\\{Y_{1}\in
A,Y_{2}\in B\\})$ is bivariate-normally distributed with mean $(x,x)$ and
identity covariance matrix, and so

$\displaystyle K_{X}$ $\displaystyle(x,\\{Y_{1}\in A,Y_{2}\in B\\})$
$\displaystyle=K_{X}(x,\\{Y_{1}\in A\\})K_{X}(x,\\{Y_{2}\in B\\}).$

By the same reasoning, $Y_{1}$ and $Y_{2}$ are conditionally independent given
$\mathscr{H}_{X}$. However, it is clear that they are unconditionally
dependent, because they both depend on the value of $X$.

  2. (ii)

Now consider three variables $X_{1}$, $X_{2}$ and $Y$ related through the
equations

$X_{1}=N_{1},\quad X_{2}=N_{2},\quad Y=X_{1}+X_{2}+U$

where $N_{1}$, $N_{2}$ and $U$ are standard normal variables (see Figure 1
right). We denote by $\mathbb{P}$ their joint distribution on
$\mathbb{R}^{3}$, and we identify this SCM with the causal space
$(\mathbb{R}^{3},\mathscr{B}(\mathbb{R}^{3}),\mathbb{P},\mathbb{K})$, where
$\mathbb{K}$ is obtained via the above structural equations. Then it is clear
that $X_{1}$ and $X_{2}$ are probabilistically independent. They are also
causally independent on $\mathscr{H}_{Y}$, since, for any
$A,B\in\mathscr{B}(\mathbb{R})$,

$\displaystyle K_{Y}(y,\\{X_{1}\in A,X_{2}\in B$
$\displaystyle\\})=\mathbb{P}(X_{1}\in A,X_{2}\in B)$
$\displaystyle=\mathbb{P}(X_{1}\in A)\mathbb{P}(X_{2}\in B).$

However, it is clear that they are conditionally dependent given
$\mathscr{H}_{Y}$.

Again, causal independence is only defined for two $\sigma$-algebras for the
sake of notational convenience; it can easily be extended to arbitrary
collections of $\sigma$-algebra.

##  4 Transformations of Causal Spaces

Consider causal spaces
$\mathscr{C}^{1}=(\Omega^{1},\mathscr{H}^{1},\mathbb{P}^{1},\mathbb{K}^{1})$
and
$\mathscr{C}^{2}=(\Omega^{2},\mathscr{H}^{2},\mathbb{P}^{2},\mathbb{K}^{2})$
with $\Omega^{1}=\times_{t\in T^{1}}E_{t}$ and $\Omega^{2}=\times_{t\in
T^{2}}E_{t}$. We want to define transformations between causal spaces
$\mathscr{C}^{1}$ and $\mathscr{C}^{2}$. These transformations shall, on the
one hand, preserve aspects of the causal structure, i.e., the spaces
$\mathscr{C}^{1}$ and $\mathscr{C}^{2}$ shall still describe essentially the
same system. On the other hand, they shall be flexible so that different types
of mappings between causal spaces can be captured.

We focus on transformations that preserve individual variables or combine them
in a meaningful way. This relation will be encoded by a map
$\rho\mathrel{\mathop{\mathchar 58\relax}}T^{1}\to T^{2}$, which can be
interpreted as encoding the fact that $S\subseteq T^{2}$ depends only on the
variables indexed by $\rho^{-1}(S)$. Deterministic maps are not sufficiently
expressive for our purposes and we therefore focus on stochastic maps, i.e.,
on probability kernels from measurable spaces $(\Omega^{1},\mathscr{H}^{1})$
to $(\Omega^{2},\mathscr{H}^{2})$.

######  Definition 4.1 (Admissible Maps).

Suppose that $\kappa\mathrel{\mathop{\mathchar
58\relax}}\Omega^{1}\times\mathscr{H}^{2}\to[0,1]$ is a probability kernel and
$\rho\mathrel{\mathop{\mathchar 58\relax}}T^{1}\to T^{2}$ is a map. Then we
call the pair $(\kappa,\rho)$ admissible if $\kappa(\cdot,A)$ is
$\mathscr{H}^{1}_{\rho^{-1}(S)}$ measurable for all $S\subset\rho(T^{1})$ and
$A\in\mathscr{H}^{2}_{S}$.

One difference between probability theory and causality seems to be that the
latter requires the notion of variables (equivalently a product structure of
the underlying space) that define entities that can be intervened upon. For a
meaningful relation between two causal spaces, their interventions should be
related, which requires some preservation of variables. The definition of
admissible maps captures the fact that variables from $\rho^{-1}(S)$ are
combined to form a new summary collection of variables indexed by $S$.

We now require maps between causal spaces to respect the distributional and
interventional structure in the following sense.

$\mathscr{C}^{1}$$\mathscr{C}^{2}$$\mathscr{C}^{1,\text{do}(\rho^{-1}(S))}$$\mathscr{C}^{2,\text{do}(S)}$$\kappa$$\kappa$$K^{1}_{\rho^{-1}(S)}$$K^{2}_{S}$
Figure 2: Interventional Consistency Definition 4.2 Equation (2) –
intervention and transformation commute.

######  Definition 4.2 (Causal Transformations).

A transformation of causal spaces, or a causal transformation,
$\varphi\mathrel{\mathop{\mathchar
58\relax}}\mathscr{C}^{1}\to\mathscr{C}^{2}$ is an admissible pair
$\varphi=(\kappa,\rho)$ satisfying the following two properties.

  1. (i)

The map satisfies distributional consistency, i.e., for $A\in\mathscr{H}^{2}$

$\int\mathbb{P}^{1}(\mathrm{d}\omega)\,\kappa(\omega,A)=\mathbb{P}^{2}(A).$
(1)

  2. (ii)

The map satisfies interventional consistency, i.e., for all
$A\in\mathscr{H}^{2}_{\rho(T^{1})}$, $S\subset\rho(T^{1})$, and
$\omega\in\Omega^{1}$ the following holds

$\begin{split}\int
K^{1}_{\rho^{-1}(S)}(\omega,\mathrm{d}\omega^{\prime})&\kappa(\omega^{\prime},A)\\\
&=\int\kappa(\omega,\mathrm{d}\omega^{\prime})K^{2}_{S}(\omega^{\prime},A).\end{split}$
(2)

Interventional consistency requires that interventions and causal
transformations commute, i.e., the result of first intervening and then
applying the transformation is the same as intervening on the target after the
transformation – see Figure 2. We emphasise that in Definition 4.1 and 4.2 we
do not prescribe conditions for added components indexed by
$T^{2}\setminus\rho(T^{1})$. Further, we remark that as a special case, we can
accommodate deterministic maps $f\mathrel{\mathop{\mathchar
58\relax}}\Omega_{1}\to\Omega_{2}$ by considering the associated probability
kernel $\kappa_{f}(\omega,A)=\boldsymbol{1}_{A}(f(\omega))$. In this case, the
admissibility condition reduces to the statement that $\pi_{S}\circ f$ is
measurable with respect to $\mathscr{H}^{1}_{\rho^{-1}(S)}$ for all
$S\subset\rho(T^{1})$ and distributional consistency becomes, for
$A\in\mathscr{H}^{2}$,

$\displaystyle\mathbb{P}^{2}(A)$
$\displaystyle=\int\mathbb{P}^{1}(\mathrm{d}\omega)\kappa(\omega,A)$
$\displaystyle=\int\mathbb{P}^{1}(\mathrm{d}\omega)\boldsymbol{1}_{A}(f(\omega))$
$\displaystyle=\mathbb{P}^{1}(f^{-1}(A))$

so $f_{\ast}\mathbb{P}^{1}=\mathbb{P}^{2}$ is the pushforward measure of
$\mathbb{P}^{1}$ along $f$. Interventional consistency then reads

$K^{1}_{\rho^{-1}(S)}(\omega,f^{-1}(A))=K^{2}_{S}(f(\omega),A)$ (3)

for all $A\in\mathscr{H}^{2}_{\rho(T^{1})}$, $S\subset\rho(T^{1})$, and
$\omega\in\Omega^{1}$. Alternatively this can be expressed as

$f_{\ast}K^{1}_{\rho^{-1}(S)}(\omega,A)=K^{2}_{S}(f(\omega),A).$

where the push-forward acts on the measure defined by the probability kernel
for some fixed $\omega$. Henceforth, with a slight abuse of notation, we
denote deterministic maps by $(f,\rho)$ without resorting to the associated
probability kernel.

###  4.1 Examples

Let us provide four prototypical examples of maps between causal spaces that
are covered by this definition. Here we resort to the language of SCMs because
they are a convenient framework that fits into causal spaces.

$Y_{1}$$Y_{2}$$X_{1}$$X_{2}$$X$$Y$$X=X_{1}+X_{2}$$Y=Y_{1}+2Y_{2}$ Figure 3:
Abstraction of SCMs in Example 4.3.

######  Example 4.3 (Abstraction).

We consider four variables $X_{1}$, $X_{2}$, $Y_{1}$, and $Y_{2}$ which are
related through the equations

$\displaystyle X_{1}$ $\displaystyle=N_{1},$ $\displaystyle X_{2}$
$\displaystyle=N_{2},$ $\displaystyle Y_{1}$
$\displaystyle=3X_{1}+X_{2}+U_{1},\qquad$ $\displaystyle Y_{2}$
$\displaystyle=X_{2}+U_{2}$

where $U_{1}$, $U_{2}$, $N_{1}$, $N_{2}$ are independent standard normal
variables. We denote by $\mathbb{P}$ their joint distribution on
$\mathbb{R}^{4}$. Consider

$X=N,\qquad Y=3X+U$

where $N\sim N(0,2)$ and $U\sim N(0,5)$. Denote their joint distribution on
$\mathbb{R}^{2}$ by $\mathbb{Q}$. We identify the two SCMs with causal spaces
$(\mathbb{R}^{4},\mathscr{B}(\mathbb{R}^{4}),\mathbb{P},\mathbb{K})$ and
$(\mathbb{R}^{2},\mathscr{B}(\mathbb{R}^{2}),\mathbb{Q},\mathbb{L})$ as
explained in [Park et al., 2023, Section 3.1].

Consider the deterministic map $f\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{4}\to\mathbb{R}^{2}$ given by
$f(x_{1},x_{2},y_{1},y_{2})=(x_{1}+x_{2},y_{1}+2y_{2})$ and the map
$\rho\mathrel{\mathop{\mathchar 58\relax}}[4]\to[2]$ given by
$\rho(1)=\rho(2)=1$, $\rho(3)=\rho(4)=2$. Clearly, the pair $(f,\rho)$ is
admissible as defined in Definition 4.1. It can be checked that

$\displaystyle\mathbb{E}[(X_{1}+X_{2})^{2}]$
$\displaystyle=2=\mathbb{E}[X^{2}]$
$\displaystyle\mathbb{E}[(Y_{1}+2Y_{2})^{2}]$
$\displaystyle=23=\mathbb{E}[Y^{2}]$
$\displaystyle\mathbb{E}[(X_{1}+X_{2})(Y_{1}+2Y_{2})]$
$\displaystyle=6=\mathbb{E}[XY]$

which implies that $f_{\ast}\mathbb{P}=\mathbb{Q}$ because both distributions
are centered Gaussian and their covariance matrices agree.

The non-trivial causal consistency relation (2) concerns interventions on
$\\{X_{1},X_{2}\\}$ and $X$ and on $\\{Y_{1},Y_{2}\\}$ and $Y$. Note that

$\displaystyle K_{\\{X_{1},X_{2}\\}}$
$\displaystyle((x_{1},x_{2},y_{1},y_{2}),\cdot)$
$\displaystyle=\delta_{(x_{1},x_{2})}\otimes
N\left(\begin{pmatrix}3x_{1}+x_{2}\\\
x_{2}\end{pmatrix},\mathrm{Id}_{2}\right).$

Then we obtain

$\displaystyle f_{\ast}K_{\\{X_{1},X_{2}\\}}$
$\displaystyle((x_{1},x_{2},y_{1},y_{2}),\cdot)$
$\displaystyle=\delta_{x_{1}+x_{2}}\otimes N(3x_{1}+3x_{2},5).$

On the other hand, we find

$\displaystyle L_{X}((x,y),\cdot)$ $\displaystyle=\delta_{x}\otimes
N\left(3x,5\right)$ $\displaystyle\Rightarrow
L_{X}(f(x_{1},x_{2},y_{1},y_{2}),\cdot)$
$\displaystyle=\delta_{x_{1}+x_{2}}\otimes N(3x_{1}+3x_{2},5)$

so that we see that (3) holds in this case. Similarly, we obtain

$\displaystyle K_{\\{Y_{1},Y_{2}\\}}((x_{1},x_{2},y_{1},y_{2}),\cdot)$
$\displaystyle=N(0,\mathrm{Id}_{1})\otimes\delta_{(y_{1},y_{2})},$
$\displaystyle L_{Y}((x,y),\cdot)$ $\displaystyle=N(0,2)\otimes\delta_{y}.$

We again find

$\displaystyle f_{\ast}K_{\\{Y_{1},Y_{2}\\}}$
$\displaystyle((x_{1},x_{2},y_{1},y_{2}),\cdot)$
$\displaystyle=L_{Y}((x_{1}+x_{2},y_{1}+2y_{2}),\cdot).$

This example shows abstraction, i.e., we obtain a transformation to a more
coarse-grained view of the system. Note that interventional consistency is
quite restrictive to satisfy, e.g., here it is crucial that all distributions
are Gaussian so that all conditional distributions are also Gaussian.

Next, we consider an example that allows us to embed a causal space in a
larger space that adds an independent disjoint system. For this, we make use
of the definition of product causal spaces (Definition 3.1). In this case, the
transformation is stochastic.

$\mathscr{C}^{1}$$\mathscr{C}^{1}\otimes\mathscr{C}^{2}$$\rho(t)=t$$\kappa(\omega,\cdot)=\delta_{\omega}\otimes\mathbb{P}^{2}$
Figure 4: Inclusions of component causal spaces into the product (Example
4.4).

######  Example 4.4 (Inclusion).

Let
$\mathscr{C}^{1}=(\Omega^{1},\mathscr{H}^{1},\mathbb{P}^{1},\mathbb{K}^{1})$
and
$\mathscr{C}^{2}=(\Omega^{2},\mathscr{H}^{2},\mathbb{P}^{2},\mathbb{K}^{2})$
be two causal spaces, with $\Omega^{1}=\times_{t\in T^{1}}E_{t}$ and
$\Omega^{2}=\times_{t\in T^{2}}E_{t}$. We define an inclusion map
$(\kappa,\rho)\mathrel{\mathop{\mathchar
58\relax}}\mathscr{C}^{1}\to\mathscr{C}^{1}\otimes\mathscr{C}^{2}$ by
considering $\rho(t)=t$ for $t\in T^{1}$ and
$\kappa(\omega,\cdot)=\delta_{\omega}\otimes\mathbb{P}^{2}$ (see Figure 4).
This pair is clearly admissible and satisfies distributional consistency:

$\displaystyle\int\mathbb{P}^{1}(\mathrm{d}\omega)\kappa(\omega,A_{1}\times
A_{2})$
$\displaystyle=\int\mathbb{P}^{1}(\mathrm{d}\omega)\boldsymbol{1}_{A_{1}}(\omega)\mathbb{P}^{2}(A_{2})$
$\displaystyle=\mathbb{P}^{1}(A_{1})\mathbb{P}^{2}(A_{2}).$

Moreover, for any $S\subset T^{1}$, $\omega\in\Omega^{1}$,
$A_{1}\in\mathscr{H}^{1}$ and $A_{2}\in\mathscr{H}^{2}$, we have

$\int
K_{S}^{1}(\omega,\mathrm{d}\omega^{\prime})\kappa(\omega^{\prime},A_{1}\times
A_{2})=\mathbb{P}^{2}(A_{2})K_{S}^{1}(\omega,A_{1})$

and also,

$\displaystyle\int$
$\displaystyle\kappa(\omega,d\omega^{\prime}_{1}d\omega^{\prime}_{2})K^{1}_{S}\otimes
K^{2}_{\emptyset}((\omega^{\prime}_{1},\omega^{\prime}_{2}),A_{1}\times
A_{2})$
$\displaystyle=\int\kappa(\omega,d\omega^{\prime}_{1}d\omega^{\prime}_{2})K^{1}_{S}(\omega^{\prime}_{1},A_{1})K^{2}_{\emptyset}(\omega^{\prime}_{2},A_{2})$
$\displaystyle=K^{1}_{S}(\omega,A_{1})\int\mathbb{P}^{2}(d\omega^{\prime}_{2})\mathbb{P}^{2}(A_{2})$
$\displaystyle=\mathbb{P}^{2}(A_{2})K^{1}_{S}(\omega,A_{1}).$

where we used the condition on $K_{\emptyset}$ in Definition 2.1. By the usual
monotone convergence theorem arguments, we have that, for any
$A\in\mathscr{H}^{1}\otimes\mathscr{H}^{2}$,

$\displaystyle\int$ $\displaystyle
K^{1}_{S}(\omega,d\omega^{\prime})\kappa(\omega^{\prime},A)$
$\displaystyle=\int\kappa(\omega,d\omega^{\prime}_{1}d\omega^{\prime}_{2})K^{1}_{S}\otimes
K^{2}_{\emptyset}((\omega^{\prime}_{1},\omega^{\prime}_{2}),A_{1}\times
A_{2}).$

Thus, interventional consistency holds, in this case even for all sets $A$,
not just for those measurable with respect to $\mathscr{H}^{2}_{\rho(T^{1})}$.

This shows that we can consider causal maps including our system into a larger
system containing additional independent components.

Finally, we consider a more involved embedding example.

$Y$$X$$X$$M$$Y$$H$$\rho(\\{X,Y\\})\hookrightarrow\\{X,Y,M,H\\}$$\kappa=\mathbb{P}_{\mathscr{H}^{1}}$
Figure 5: Inclusions of SCMs (Example 4.5).

######  Example 4.5 (Inclusion of SCMs).

Consider the following SCM

$\displaystyle H$ $\displaystyle=N_{H},$ $\displaystyle X$
$\displaystyle=H+N_{X},$ $\displaystyle M$ $\displaystyle=X+N_{M},\qquad$
$\displaystyle Y$ $\displaystyle=M+H+N_{Y}.$

We denote the joint distribution of $(X,Y,M,H)$ by $\mathbb{P}$, and the
marginal distribution on $(X,Y)$ by $\mathbb{P}^{XY}$.

We consider a causal space
$\mathscr{C}^{1}=(\Omega^{1},\mathscr{H}^{1},\mathbb{P}^{XY},\mathbb{K})$ that
represents the pair $(X,Y)$, where $\Omega^{1}=\mathbb{R}^{2}$ and
$\mathscr{H}^{1}=\mathscr{B}(\mathbb{R}^{2})$, and a causal space
$\mathscr{C}^{2}=(\Omega^{2},\mathscr{H}^{2},\mathbb{P},\mathbb{L})$
representing the full SCM, where $\Omega^{2}=\mathbb{R}^{4}$ and
$\mathscr{H}^{2}=\mathscr{B}(\mathbb{R}^{4})$, i.e., it contains in addition a
mediator and a confounder. The causal mechanisms $\mathbb{K}$ and $\mathbb{L}$
are derived from the SCM. Then we consider the obvious $\rho$ that embeds
$\\{X,Y\\}$ into $\\{X,Y,M,H\\}$ and, for $A\in\mathscr{H}^{2}$,

$\kappa(\cdot,A)=\mathbb{P}_{\mathscr{H}^{1}}(A).$

Clearly, this pair is admissible because on the variables $X$ and $Y$ we use
the identity transformation. Distributional consistency follows by

$\displaystyle\int\kappa((x,y),A)\mathbb{P}^{XY}(d(x,y))$
$\displaystyle=\int\mathbb{P}_{\mathscr{H}^{1}}(A)d\mathbb{P}^{XY}$
$\displaystyle=\mathbb{P}(A).$

Interventional consistency also holds so that $(\kappa,\rho)$ is indeed a
causal transformation. For a proof of this fact we refer to the more general
result in Lemma 6.3.

This example therefore shows that we can embed a system in a larger system
that captures a more accurate description.

###  4.2 Abstractions

Note that Example 4.3 is different from Examples 4.4 and 4.5 in that it
compresses the representation while the other two consider an extension of the
system. As these are different objectives, we consider the following
definition.

######  Definition 4.6 (Abstractions).

The pair of maps $(\kappa,\rho)$ between measurable spaces
$(\Omega^{1},\mathscr{H}^{1})=\otimes_{t\in T^{1}}(E_{t},\mathscr{E}_{t})$ and
$(\Omega^{2},\mathscr{H}^{2})=\otimes_{t\in T^{2}}(E_{t},\mathscr{E}_{t})$ is
called an abstraction if $\rho\mathrel{\mathop{\mathchar 58\relax}}T^{1}\to
T^{2}$ is surjective.

In the case of abstractions it is often sufficient to consider deterministic
maps, motivating the following definition.

######  Definition 4.7 (Perfect Abstractions).

An abstraction $(\kappa,\rho)$ is called a perfect abstraction if $\kappa$ is
deterministic, i.e., $\kappa=\kappa_{f}$ for some measurable
$f\mathrel{\mathop{\mathchar 58\relax}}\Omega^{1}\to\Omega^{2}$, and moreover
$f$ is surjective.

We finally remark that one further setting of potential interest would be to
consider the inverse of an abstraction, i.e., a setting where a summary
variable $X$ is mapped to a more detailed description $(X_{1},X_{2})$.
However, to accommodate such transformations we need a slightly different
framework than the one presented here. Roughly, we need to consider
$\rho\mathrel{\mathop{\mathchar 58\relax}}T^{1}\to\mathscr{P}(T^{2})$ with
$\rho(t_{1})\cap\rho(t^{\prime}_{1})=\emptyset$ for $t_{1},t^{\prime}_{1}\in
T_{1}$, and interventions on all sets $S\subset T^{1}$ can be expressed as
interventions on the target $\mathscr{C}^{2}$ (i.e., the more fine-grained
representations), while this is reversed in our case so that those two
settings are dual to each other. We do not pursue this here any further, as
those transformations are of more limited interest and applicability. Let us
emphasise nevertheless that it seems ambitious to handle all cases in one
framework. Indeed, combining variables in a summary variable or splitting
variables in a more fine-grained description are meaningful operations, but it
is less clear to interpret in a causal manner a definition of a transformation
$(X_{1},X_{2})\to(Y_{1},Y_{2})$ that allows both at the same time. For
example, intervening on $X_{1}$, in general, then does not correspond to a
meaningful causal operation on the variables $(Y_{1},Y_{2})$. We also remark
that this attempt has not been made in the SCM literature, where the focus is
almost exclusively on abstractions.

##  5 Comparison with Abstraction in the SCM framework

Rubenstein et al. [2017] gives the definition of exact transformations between
SCMs. While being the seminal work on the theory of causal abstractions, it is
probably also the most relevant to compare to our proposals. We first recall
some essential aspects of their definition of SCMs (or SEMs, for structural
equation models, by their nomenclature)333In this section, some imported
notations might clash with ours; the clashes are restricted to this section
and should not cause any confusion..

######  Definition 5.1 ([Rubenstein et al., 2017, Definition 1]).

Let $\mathbb{I}_{X}$ be an index set. An SEM $\mathcal{M}_{X}$ over variables
$X=(X_{i}\mathrel{\mathop{\mathchar 58\relax}}i\in\mathbb{I}_{X}$ taking
values in $\mathcal{X}$ is a tuple $(\mathcal{S}_{X},\mathbb{P}_{E})$, where

  * •

$\mathcal{S}_{X}$ is a set of structural equations, i.e. the set of equations
$X_{i}=f_{i}(X,E_{i})$ for $i\in\mathbb{I}_{X}$;

  * •

$\mathbb{P}_{E}$ is a distribution over the exogenous variables
$E=(E_{i}\mathrel{\mathop{\mathchar 58\relax}}i\in\mathbb{I}_{X})$.

Note that their definition of SCMs is a bit more general than standard ones in
the literature (e.g. [Peters et al., 2017, p.83, Definition 6.2]), in that
they allow, for example, cycles and latent confounders, but they simply insist
that there must be a unique solution to any interventions. They also consider
a specific set of allowed interventions, rather than considering all possible
interventions. We also recall some essential aspects of the notion of exact
transformations.

######  Definition 5.2 ([Rubenstein et al., 2017, Definition 3]).

Let $\mathcal{M}_{X}$ and $\mathcal{M}_{Y}$ be SCMs, and
$\tau\mathrel{\mathop{\mathchar 58\relax}}\mathcal{X}\rightarrow\mathcal{Y}$ a
function. We say that $\mathcal{M}_{Y}$ is an exact $\tau$-transformation of
$\mathcal{M}_{X}$ if, there exists a surjective mapping $\omega$ of the
interventions such that for any intervention $i$,
$\mathbb{P}^{i}_{\tau(X)}=\mathbb{P}^{\omega(i)}_{Y}$.

Note that this definition is trying to capture the same concept as our notion
of interventional consistency given in (2): that interventions and
transformations commute. However, there are several aspects in which our
proposal is more appealing.

  * •

They only consider deterministic maps $\tau\mathrel{\mathop{\mathchar
58\relax}}\mathcal{X}\rightarrow\mathcal{Y}$, whereas we allow the map $\rho$
to be stochastic.

  * •

They have to find a separate map $\omega$ between the interventions
themselves, whereas our map $\rho$ also determines the transformation of the
causal kernels.

  * •

By insisting on surjectivity of $\omega$, they only allow the consideration of
abstraction, whereas we can consider more general transformations of causal
spaces, such as inclusions considered in Example 4.4.

Nevertheless, restricted to considerations amenable to both approaches, the
notions coincide. For example, we return to Example 4.3, where we already
showed that $f_{*}\mathbb{P}=\mathbb{Q}$, $f_{*}K_{\\{X_{1},X_{2}\\}}=L_{X}$
and $f_{*}K_{\\{Y_{1},Y_{2}\\}}=L_{Y}$, which implies that two-variable SCM is
an exact transformation of the four-variable SCM according to Definition 5.2.

Finally, we mention that Beckers and Halpern [2019] criticise exact
transformations of Rubenstein et al. [2017] on the basis that probabilities
and allowed interventions can mask significant differences between SCMs, and
then proceed to propose definitions of abstractions that depend only on the
structural equations, independently of probabilities. We remark that this
criticism is not valid in our framework, in that the interventional
consistency of our transformations is imposed independently of probabilities,
making it impossible to mask them with the choice of probability measures.
That this is possible with SCMs is an artifact of the fact that in SCMs, the
observational and interventional measures are coupled through the exogenous
distribution, whereas in causal spaces they are completely decoupled.
Moreover, we consider all possible interventions rather than a reduced set of
allowed interventions. We also remark that, since probabilities and causal
kernels are the primitive objects in our framework, rather than being derived
by other primitive objects (namely the structural equations), it does not make
sense for the transformation to be defined independently of probabilities, as
done by Beckers and Halpern [2019].

##  6 Further Properties of Causal Transformations

In this section we investigate various properties of causal transformations
and connect them to the notions introduced in Section 2.

###  6.1 Existence and Uniqueness of Causal Transformations

First, we have the following lemma on the composition of causal
transformations. Recall that for two probability kernels
$\kappa_{1}\mathrel{\mathop{\mathchar
58\relax}}\Omega^{1}\times\mathscr{H}^{2}\to[0,1]$ mapping
$(\Omega^{1},\mathscr{H}^{1})$ to $(\Omega^{2},\mathscr{H}^{2})$ and
$\kappa_{2}\mathrel{\mathop{\mathchar
58\relax}}\Omega^{2}\times\mathscr{H}^{3}\to[0,1]$ mapping
$(\Omega^{2},\mathscr{H}^{2})$ to $(\Omega^{3},\mathscr{H}^{3})$ the
concatenation defined by [Cinlar, 2011, p.39]

$\kappa_{1}\circ\kappa_{2}(\omega_{1},A)=\int\kappa_{1}(\omega_{1},\mathrm{d}\omega_{2})\kappa_{2}(\omega_{2},A)$

defines a probability kernel from $(\Omega^{1},\mathscr{H}^{1})$ to
$(\Omega^{3},\mathscr{H}^{3})$.

######  Lemma 6.1 (Compositions of Causal Transformations).

Let $(\kappa_{1},\rho_{1})\mathrel{\mathop{\mathchar
58\relax}}\mathscr{C}^{1}\to\mathscr{C}^{2}$ and
$(\kappa_{2},\rho_{2})\mathrel{\mathop{\mathchar
58\relax}}\mathscr{C}^{2}\to\mathscr{C}^{3}$ be causal transformations. If
$(\kappa_{1},\rho_{1})$ is an abstraction then
$(\kappa_{3},\rho_{3})=(\kappa_{1}\circ\kappa_{2},\rho_{1}\circ\rho_{2})\mathrel{\mathop{\mathchar
58\relax}}\mathscr{C}^{1}\to\mathscr{C}^{3}$ is a causal transformation.

The proof can be found in Appendix B.2. We remark that, unfortunately, we
cannot remove the assumption that the first transformation is an abstraction.
Let us clarify this through an example.

######  Example 6.2.

Consider an SCM with equations

$\displaystyle X_{1}$ $\displaystyle=N_{1},$ $\displaystyle X_{2}$
$\displaystyle=N_{2},$ $\displaystyle Y$ $\displaystyle=X_{1}+X_{2}+N_{Y}$

where $N_{1}$, $N_{2}$, and $N_{Y}$ follow independent standard normal
distributions. Then we can consider the causal space $\mathscr{C}^{1}$
containing $(X_{1},Y)$, the causal space $\mathscr{C}^{2}$ containing
$(X_{1},X_{2},Y)$ and an abstraction $\mathscr{C}^{3}$ containing
$(X_{1}+X_{2},Y)$. Then we can embed $\mathscr{C}^{1}\to\mathscr{C}^{2}$ and
there is an abstraction $\mathscr{C}^{2}\to\mathscr{C}^{3}$ which are both
transformations of causal spaces.

However, their concatenation is not a causal transformation because it is not
even admissible (and also interventional consistency does not hold for the
intervention $K^{3}_{X}$ as this cannot be expressed by $K^{1}_{X_{1}}$). Note
that $P_{X_{2}|X_{1}=x_{1},Y=y}=N((y-x_{1})/2,1/2)$ and therefore we have
$\kappa_{1}((x_{1},y),\cdot)=\delta_{x_{1}}\otimes
N((y-x_{1})/2,1/2)\otimes\delta_{y}$.

We also have $\kappa_{2}((x_{1},x_{2},y),\cdot)=\delta_{(x_{1}+x_{2},y)}$.
Thus, their concatenation is given by

$\kappa_{3}((x_{1},y),\cdot)=N((y+x_{1})/2,1/2)\otimes\delta_{y}.$

So the first coordinate is not measurable with respect to
$\mathscr{H}^{1}_{X_{1}}$.

This shows that we lose measurability along the concatenation because the
variables added in the more complete description $\mathscr{C}^{2}$ may depend
on all other variables.

Let us now generalise Example 4.5 to general SCMs.

######  Lemma 6.3 (Inclusion of SCMs).

Consider an acyclic SCM on endogenous variables
$(X_{1},\ldots,X_{d})\in\mathbb{R}^{d}$ with observational distribution
$\mathbb{P}$. Let $S\subset[d]$, $R=S^{c}=[d]\setminus S$ and consider causal
spaces
$\mathscr{C}^{1}=(\Omega^{1},\mathscr{H}^{1},\mathbb{P}^{S},\mathbb{K})$ and
$\mathscr{C}^{2}=(\Omega^{2},\mathscr{H}^{2},\mathbb{P},\mathbb{L})$, where we
have
$(\Omega^{1},\mathscr{H}^{1})=(\mathbb{R}^{|S|},\mathscr{B}(\mathbb{R}^{|S|}))$
and
$(\Omega^{2},\mathscr{H}^{2})=(\mathbb{R}^{d},\mathscr{B}(\mathbb{R}^{d}))$.
Moreover, $\mathbb{P}^{S}$ is the marginal distribution on the variables in
$S$, and the causal mechanisms $\mathbb{K}$ and $\mathbb{L}$ are derived from
the SCM. In particular, $\mathbb{K}$ is a marginalisation of $\mathbb{L}$,
namely, for any $\omega\in\Omega^{2}$, any event $A\in\mathscr{H}^{1}$ and any
$S^{\prime}\subseteq S$, we have that
$K_{S^{\prime}}(\omega,A)=L_{S^{\prime}}(\omega,A)$.

Consider the map $\rho\mathrel{\mathop{\mathchar
58\relax}}S\hookrightarrow[d]$ and
$\kappa(\cdot,A)=\mathbb{P}_{\mathscr{H}^{1}}(A)$. Then $(\rho,\kappa)$ is a
causal transformation from $\mathscr{C}^{1}$ to $\mathscr{C}^{2}$.

The proof can be found in Appendix B.2.

We now investigate to what degree distributional and interventional
consistency determines the causal structure on the target space. We show that
generally the causal structure on $\mathscr{H}_{\rho(T)}^{2}$ is quite rigid.

######  Lemma 6.4 (Rigidity of target causal structure).

Let
${\mathscr{C}}^{2}=(\Omega^{2},\mathscr{H}^{2},\mathbb{P}^{2},{\mathbb{K}}^{2})$
and
$\tilde{\mathscr{C}}^{2}=(\Omega^{2},\mathscr{H}^{2},\tilde{\mathbb{P}}^{2},\tilde{\mathbb{K}}^{2})$
be two causal spaces with the same underlying measurable space. Let
$(\kappa,\rho)$ be an admissible pair for the measurable spaces
$(\Omega^{1},\mathscr{H}^{1})$ and $(\Omega^{2},\mathscr{H}^{2})$. Assume that
the pair $(\kappa,\rho)$ defines causal transformations
$\varphi\mathrel{\mathop{\mathchar
58\relax}}\mathscr{C}^{1}\to\mathscr{C}^{2}$ and
$\tilde{\varphi}\mathrel{\mathop{\mathchar
58\relax}}\mathscr{C}^{1}\to\tilde{\mathscr{C}}^{2}$ be a causal
transformations. Then $\mathbb{P}^{2}=\tilde{\mathbb{P}}^{2}$, and for all
$A\in\mathscr{H}^{2}_{\rho(T^{1})}$ and any $S\subset T^{2}$

$\displaystyle K^{2}_{S}(\omega,A)=\tilde{K}^{2}_{S}(\omega,A)\quad\text{for
$\mathbb{P}^{2}=\tilde{\mathbb{P}}^{2}$ a.\ e.\ $\omega\in\Omega^{2}$}.$

The proof is in Appendix B.2. We cannot expect to derive much stronger results
for general causal transformation because interventional consistency does not
restrict $K^{2}(\omega,A)$ for $\omega$ not in the support of $\mathbb{P}^{2}$
or $A\notin\mathscr{H}_{\rho(T)}^{2}$. E.g., in the setting of Example 4.4 the
causal structure on the second factor is arbitrary.

However, when we consider deterministic maps $(f,\rho)$ such that
$f\mathrel{\mathop{\mathchar
58\relax}}(\Omega^{1},\mathscr{H}^{1})\to(\Omega^{2},\mathscr{H}^{2})$ and
$\rho$ are surjective, then there is at most one causal structure on the
target space $(\Omega^{2},\mathscr{H}^{2})$ such that the pair $(f,\rho)$ is a
causal transformation (and thus a perfect abstraction).

######  Lemma 6.5 (Surjective Deterministic Maps).

Suppose $(f,\rho)$ is an admissible pair for the causal space
$\mathscr{C}^{1}=(\Omega^{1},\mathscr{H}^{1},\mathbb{P}^{1},\mathbb{K}^{1})$
to the measurable space $X^{2}=(\Omega^{2},\mathscr{H}^{2})$ and assume that
$\rho$ is surjective and $f\mathrel{\mathop{\mathchar
58\relax}}\Omega_{1}\to\Omega_{2}$ measurable. If $f$ is surjective, there
exists at most one causal space
$\mathscr{C}^{2}=(\Omega^{2},\mathscr{H}^{2},\mathbb{P}^{2},\mathbb{K}^{2})$
such that $(f,\rho)\mathrel{\mathop{\mathchar
58\relax}}\mathscr{C}^{1}\to\mathscr{C}^{2}$ is a causal transformation.

If, in addition, $K^{1}_{\rho^{-1}(S^{2})}(\cdot,A)$ is measurable with
respect to $f^{-1}(\mathscr{H}^{2}_{S^{2}})$ for all $A\in
f^{-1}(\mathscr{H}^{2})$ and all $S^{2}\subset T^{2}$ then a unique causal
space $\mathscr{C}^{2}$ exists such that $(f,\rho)\mathrel{\mathop{\mathchar
58\relax}}\mathscr{C}^{1}\to\mathscr{C}^{2}$ is a causal transformation.

The proof is in Appendix B.2. To motivate the measurability condition for
$K^{1}(\cdot,A)$ we remark that interventional consistency requires
$K^{1}(\omega,A)=K^{1}(\omega^{\prime},A)$ for $\omega$, $\omega^{\prime}$
with $f(\omega)=f(\omega^{\prime})$ and the measurability condition in the
result is a slightly stronger condition than this.

Next, we show that interventions on a space can be pushed forward along a
perfect abstraction.

######  Lemma 6.6 (Perfect Abstraction on Intervened Spaces).

Let
$\mathscr{C}^{1}=(\Omega^{1},\mathscr{H}^{1},\mathbb{P}^{1},\mathbb{K}^{1})$
with $(\Omega^{1},\mathscr{H}^{1})$ a product with index set $T^{1}$ and
$\mathscr{C}^{2}=(\Omega^{2},\mathscr{H}^{2},\mathbb{P}^{2},\mathbb{K}^{2})$
with $(\Omega^{2},\mathscr{H}^{2})$ a product with index set $T^{2}$ be causal
spaces, and let $(f,\rho)\mathrel{\mathop{\mathchar
58\relax}}\mathscr{C}^{1}\to\mathscr{C}^{2}$ be a perfect abstraction.

Let $U^{1}=\rho^{-1}(U^{2})\subseteq T^{1}$ for some $U^{2}\subseteq T^{2}$.
Let $\mathbb{Q}^{1}$ be a probability measure on
$(\Omega^{1},\mathscr{H}^{1}_{U^{1}})$ and $\mathbb{L}^{1}$ a causal mechanism
on $(\Omega^{1},\mathscr{H}^{1}_{U^{1}},\mathbb{Q}^{1})$. Suppose that, for
all $S\subseteq U^{2}$ and $A\in\mathscr{H}^{1}$, the map
$L_{\rho^{-1}(S)}^{1}(\cdot,A)$ is measurable with respect to
$f^{-1}(\mathscr{H}^{2}_{S})$, and consider the intervened causal spaces

$\displaystyle\mathscr{C}^{1}_{I}$
$\displaystyle=(\Omega^{1},\mathscr{H}^{1},(\mathbb{P}^{1})^{\textnormal{do}(U^{1},\mathbb{Q}^{1})},(\mathbb{K}^{1})^{\textnormal{do}(U^{1},\mathbb{Q}^{1},\mathbb{L}^{1})}),$
$\displaystyle\mathscr{C}^{2}_{I}$
$\displaystyle=(\Omega^{2},\mathscr{H}^{2},(\mathbb{P}^{2})^{\textnormal{do}(U^{2},\mathbb{Q}^{2})},(\mathbb{K}^{2})^{\textnormal{do}(U^{2},\mathbb{Q}^{2},\mathbb{L}^{2})}),$

where $\mathbb{Q}^{2}=f_{\ast}\mathbb{Q}^{1}$ and $\mathbb{L}^{2}$ is the
unique family of kernels satisfying
$L^{2}_{S}(f(\omega),A)=L^{1}_{\rho^{-1}(S)}(\omega,f^{-1}(A))$ for all
$\omega\in\Omega^{1}$, $A\in\mathscr{H}^{2}$, and $S\subseteq U^{2}$. Then
$(f,\rho)\mathrel{\mathop{\mathchar
58\relax}}\mathscr{C}^{1}_{I}\to\mathscr{C}^{2}_{I}$ is a perfect abstraction.

The proof of this result is in Appendix B.2.

###  6.2 Sources and causal effects under causal transformations

We now study whether causal effects in target and domain of a causal
transformation can be related. Our first results shows that for perfect
abstractions having no causal effect in the domain implies that there is also
no causal effect in the target.

######  Lemma 6.7 (Perfect Abstraction and No Causal Effect).

Let $(f,\rho)\mathrel{\mathop{\mathchar
58\relax}}\mathscr{C}^{1}\to\mathscr{C}^{2}$ be a perfect abstraction.
Consider two sets $U^{2},V^{2}\subset T^{2}$ and denote
$U^{1}=\rho^{-1}(U^{2})$ and $V^{1}=\rho^{-1}(V^{2})$. If
$\mathscr{H}^{1}_{U^{1}}$ has no causal effect on $\mathscr{H}^{1}_{V^{1}}$ in
$\mathscr{C}^{1}$, then $\mathscr{H}^{2}_{U^{2}}$ has no causal effect on
$\mathscr{H}^{2}_{V^{2}}$ in $\mathscr{C}^{2}$.

The proof can be found in Appendix B.3.

On the other hand, we can show that when there is an active causal effect in
the target space, there is also an active causal effect in the domain.

######  Lemma 6.8 (Perfect Abstraction and Active Causal Effects).

Let $(f,\rho)\mathrel{\mathop{\mathchar
58\relax}}\mathscr{C}^{1}\to\mathscr{C}^{2}$ be a perfect abstraction.
Consider two sets $U^{2},V^{2}\subset T^{2}$ and denote
$U^{1}=\rho^{-1}(U^{2})$ and $V^{1}=\rho^{-1}(V^{2})$. Assume that
$\mathscr{H}^{2}_{U^{2}}$ has an active causal effect on
$\mathscr{H}^{2}_{V^{2}}$ in $\mathscr{C}^{2}$. Then $\mathscr{H}^{1}_{U^{1}}$
has an active causal effect on $\mathscr{H}^{1}_{V^{1}}$ in $\mathscr{C}^{1}$.

The proof is in Appendix B.3.

The reverse statements are not true, i.e., if there is no causal effect in the
target there might be a causal effect in the domain and if there is an active
causal effect in the domain this does not imply that ther is a causal effect
in the target, which can be seen by considering a target space with only a
single point. We can also study causal effects in the context of embedding
transformations, as in Lemma 6.3. Then we see directly that active causal
effects are preserved. On the other hand, it is straightforward to construct
examples where there is no causal effect in a subsystem, but there is a causal
effect in a larger system. This can be achieved by a violation of
faithfulness.

######  Example 6.9.

Consider the SCM

$\displaystyle X$ $\displaystyle=N_{X},$ $\displaystyle M$
$\displaystyle=N_{X}+N_{M},$ $\displaystyle Y$ $\displaystyle=M-X+N_{Y}.$

Then there is no causal effect from $\sigma(X)$ to $\sigma(Y)$ in the system
$(X,Y)$ but there is a causal effect in the complete system.

Finally, we show that similar results can be established for sources. Indeed,
perfect abstraction preserve sources in the following sense.

######  Lemma 6.10 (Perfect Abstraction and Sources).

Let $(f,\rho)\mathrel{\mathop{\mathchar
58\relax}}\mathscr{C}^{1}\to\mathscr{C}^{2}$ be a perfect abstraction.
Consider two sets $U^{2},V^{2}\subset T^{2}$ and denote
$U^{1}=\rho^{-1}(U^{2})$ and $V^{1}=\rho^{-1}(V^{2})$. Assume that
$\mathscr{H}^{1}_{U^{1}}$ is a local source of $\mathscr{H}^{1}_{V^{1}}$ in
$\mathscr{C}^{1}$. Then $\mathscr{H}^{2}_{U^{2}}$ is a local source of
$\mathscr{H}^{2}_{V^{2}}$ in $\mathscr{C}^{2}$.

In particular, this implies that if $\mathscr{H}^{1}_{U^{1}}$ is a global
source then $\mathscr{H}^{2}_{U^{2}}$ also is a global source.

The proof is in Appendix B.3.

Similar to our results for causal effects, the existence of sources in the
abstracted space does not guarantee the existence of sources in the domain
space. Note that local sources are preserved in the setting of Lemma 6.3. On
the other hand, global sources are clearly not preserved, as we can add a
global source to the system.

##  7 Conclusion

In this paper, we developed the theory of causal spaces initiated by Park et
al. [2023] by proposing the notions of products of causal spaces and
transformations of causal spaces. They are defined via natural extensions of
the notions of products and probability kernels in probability theory. Not
only are they mathematically elegant objects, but they have natural and
important semantic interpretations as causal independence and abstraction or
inclusion of causal spaces. Moreover, we explore the connections of these
notions to those of causal effects and sources introduced in [Park et al.,
2023].

Despite the beauty and practical usefulness of the structural causal model and
potential outcomes frameworks, we believe that the theory of causal spaces has
the potential to overcome some of the longstanding limitations of them in
terms of rigour and expressiveness, and the contribution of this paper is to
develop the theory further in terms of treating multiple causal spaces through
products and transformations rather than focusing the investigation to single
causal spaces.

Although probability theory does not seem to be so amenable to a category-
theoretic treatment as other mathematical objects, there have been some
efforts to do so [Lynn, 2010, Adachi and Ryu, 2016, Cho and Jacobs, 2019,
Fritz, 2020]. As future work, it would be interesting to explore extensions of
the transformations proposed here to formal category-theoretic morphisms
between causal spaces.

###### Acknowledgements.

This work was supported by the Tübingen AI Center.

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Products, Abstractions, and Inclusions of Causal Spaces
(Supplementary Material) **footnotetext: Equal Contribution

In the supplementary material, we provide the missing proofs for the results
in our paper, and we recall some definitions from probability theory.

##  Appendix A Background on Probability Theory

Let us collect some definitions and notations from probability theory. Recall
that a probability space $(\Omega,\mathscr{H},\mathbb{P})$ is defined through
the following axioms:

  1. (i)

$\Omega$ is a set;

  2. (ii)

$\mathscr{H}$ is a collection of subsets of $\Omega$, called events, such that

     * •

$\Omega\in\mathscr{H}$;

     * •

if $A\in\mathscr{H}$, then $\Omega\setminus A\in\mathscr{H}$;

     * •

if $A_{1},A_{2},...\in\mathscr{H}$, then $\cup_{n}A_{n}\in\mathscr{H}$;

  3. (iii)

$\mathbb{P}$ is a probability measure on $(\Omega,\mathscr{H})$, i.e. a
function $\mathbb{P}\mathrel{\mathop{\mathchar
58\relax}}\mathscr{H}\rightarrow[0,1]$ satisfying

     * •

$\mathbb{P}(\emptyset)=0$;

     * •

$\mathbb{P}(\cup_{n}A_{n})=\sum_{n}\mathbb{P}(A_{n})$ for any disjoint
sequence $A_{n}$ in $\mathscr{H}$;

     * •

$\mathbb{P}(\Omega)=1$.

Probability kernels (sometimes called Markov kernels) from
$(\Omega^{1},\mathscr{H}^{1})$ to $(\Omega^{2},\mathscr{H}^{2})$ are maps
$\kappa\mathrel{\mathop{\mathchar
58\relax}}\Omega^{1}\times\mathscr{H}^{2}\to[0,1]$ such that

  1. (i)

For every $A\in\mathscr{H}^{2}$ the function

$\omega\to\kappa(\omega,A)$

is measurable with respect to $\mathscr{H}^{1}$.

  2. (ii)

For every $\omega\in\Omega^{1}$ the function

$A\to\kappa(\omega,A)$

defines a measure on $(\Omega^{2},\mathscr{H}^{2})$.

See, for example, [Cinlar, 2011, p.37, Section I.6] for details.

We remark that we can concatenate probability kernels and we can consider
product kernels.

We denote integration with respect to a (probability) measure $\mathbb{P}$ by

$\int\mathbb{P}(\mathrm{d}\omega)T(\omega)$

for a measurable map $T\mathrel{\mathop{\mathchar
58\relax}}(\Omega,\mathscr{H})\to(\mathbb{R},\mathscr{B}(\mathbb{R}))$.

For a measurable map $f\mathrel{\mathop{\mathchar
58\relax}}(\Omega^{1},\mathscr{H}^{1})\to(\Omega^{2},\mathscr{H}^{2})$ we
define the pushforward measure $f_{\ast}\mathbb{P}$ by
$f_{\ast}\mathbb{P}(A)=\mathbb{P}(f^{-1}(A))$. The transformation formula for
the pushforward measure reads

$\int
f_{\ast}\mathbb{P}(\mathrm{d}\omega)T(\omega)=\int\mathbb{P}(\mathrm{d}\omega^{\prime})T\circ
f(\omega^{\prime}).$

Finally we recall the Factorisation lemma [Cinlar, 2011, p.76, Theorem
II.4.4].

######  Lemma A.1 (Factorisation Lemma).

Let $T\mathrel{\mathop{\mathchar
58\relax}}\Omega^{1}\to(\Omega^{2},\mathscr{H}^{2})$ be a function. A function
$f\mathrel{\mathop{\mathchar 58\relax}}\Omega^{1}\to\mathbb{R}$ is
$\sigma(T)$-$\mathscr{B}(\mathbb{R})$ measurable if and only if there is a
measurable function $g\mathrel{\mathop{\mathchar
58\relax}}(\Omega^{2},\mathscr{H}^{2})\to(\mathbb{R},\mathscr{B}(\mathbb{R}))$
such that $f=g\circ T$.

##  Appendix B Proofs

In this appendix we collect the proofs of all the results in the paper.

###  B.1 Proofs for Section 3

See 3.2

###### Proof of Lemma 3.2.

It is a standard fact that $\mathbb{K}^{1}\otimes\mathbb{K}^{2}$ defines a
family of probability kernels444See, e.g.
math.stackexchange.com/questions/84078/product-of-two-probability-kernel-is-a-
probability-kernel. For the first axiom of causal kernels (Definition 2.1(i)),
we observe that

$\displaystyle(K^{1}\otimes
K^{2})_{\emptyset}((\omega_{1},\omega_{2}),A_{1}\times A_{2})$
$\displaystyle=K^{1}_{\emptyset}(\omega_{1},A_{1})K^{2}_{\emptyset}(\omega_{2},A_{2})$
$\displaystyle=\mathbb{P}^{1}(A_{1})\mathbb{P}^{2}(A_{2})$
$\displaystyle=\mathbb{P}^{1}\otimes\mathbb{P}^{2}(A_{1}\times A_{2}).$

By standard reasoning based on the monotone class theorem, this extends to
$A\in\mathscr{H}^{1}\otimes\mathscr{H}^{2}$ and therefore the first axiom of
causal spaces is satisfied. For the second axiom of causal spaces, for any
$S=S^{1}\cup S^{2}$, first fix arbitrary $A_{1}\in\mathscr{H}^{1}_{S^{1}}$ and
$A_{2}\in\mathscr{H}^{1}_{S^{2}}$. Then, for all $B_{1}\in\mathscr{H}^{1}$ and
$B_{2}\in\mathscr{H}^{2}$, we find that for all
$\omega=(\omega_{1},\omega_{2})$,

$\displaystyle L_{S}(\omega,(A_{1}\times A_{2})\cap(B_{1}\times B_{2}))$
$\displaystyle=K^{1}_{S_{1}}(\omega_{1},A_{1}\cap
B_{1})K^{2}_{S^{2}}(\omega_{2},A_{2}\cap B_{2})$
$\displaystyle=\mathbf{1}_{A_{1}}(\omega_{1})K^{1}_{S^{1}}(\omega_{1},B_{1})\mathbf{1}_{A_{2}}(\omega_{2})K^{2}_{S^{2}}(\omega_{2},B_{2})$
$\displaystyle=\mathbf{1}_{A_{1}\times A_{2}}(\omega)L_{S}(\omega,B_{1}\times
B_{2}).$

Hence, for this fixed pair $A_{1}$, $A_{2}$ and this $\omega$, the measures
$B\mapsto L_{S}(\omega,(A_{1}\times A_{2})\cap B)$ and
$B\mapsto\mathbf{1}_{A_{1}\times A_{2}}(\omega)L_{S}(\omega,B)$ are identical
on the generating rectangles $B_{1}\times B_{2}$, hence they are identical on
all of $\mathscr{H}^{1}\otimes\mathscr{H}^{2}$ by the standard monotone class
theorem reasoning. Now, since this is true for arbitrary rectangles
$A_{1}\times A_{2}$ with $A_{1}\in\mathscr{H}^{1}_{S^{1}}$ and
$A_{2}\in\mathscr{H}^{2}_{S^{2}}$, if we now fix
$B\in\mathscr{H}^{1}\otimes\mathscr{H}^{2}$, we have that the two measures
$A\mapsto L_{S}(\omega,A\cap B)$ and
$A\mapsto\mathbf{1}_{A}(\omega)L_{S}(\omega,B)$ on
$\mathscr{H}^{1}_{S^{1}}\otimes\mathscr{H}^{2}_{S^{2}}$ are identical on the
generating rectangles $A_{1}\times A_{2}$, hence they are identical on all of
$\mathscr{H}^{1}_{S^{1}}\otimes\mathscr{H}^{2}_{S^{2}}$. Now both $A$ and $B$
are arbitrary elements of $\mathscr{H}^{1}\otimes\mathscr{H}^{2}$ and
$\mathscr{H}^{1}_{S^{1}}\otimes\mathscr{H}^{2}_{S^{2}}$ respectively. To
conclude, we have, for all $\omega$,
$A\in\mathscr{H}^{1}\otimes\mathscr{H}^{2}$ and
$B\in\mathscr{H}^{1}_{S^{1}}\otimes\mathscr{H}^{2}_{S^{2}}$,

$L_{S}(\omega,A\cap B)=\mathbf{1}_{A}(\omega)L_{S}(\omega,B),$

confirming the second axiom of causal spaces. ∎

See 3.3

###### Proof of Lemma 3.3.

  1. (i)

Denote the causal kernels on the product space by $K^{p}$. Take any event
$A\in\mathscr{H}_{T^{2}}$, and any $S\subseteq T^{1}\cup T^{2}$. Note that $S$
can be written as a union $S=S^{1}\cup S^{2}$ for some $S^{1}\subseteq T^{1}$
and $S^{2}\subseteq T^{2}$. Then see that, by writing $A=\Omega^{1}\times
A^{\prime}\in\mathscr{H}_{T^{1}}\otimes\mathscr{H}_{T^{2}}$ with
$A^{\prime}\subseteq\Omega^{2}$,

$\displaystyle K_{S}^{p}(\omega,A)$ $\displaystyle=K^{p}_{S^{1}\cup
S^{2}}(\omega,A)$
$\displaystyle=K_{S^{1}}^{1}(\omega,\Omega_{1})K_{S^{2}}^{2}(\omega,A^{\prime})$
$\displaystyle=K_{\emptyset}^{1}(\omega,\Omega_{1})K_{S^{2}}(\omega,A^{\prime})$
$\displaystyle=K_{S\setminus T^{1}}(\omega,A).$

Here we used that
$K_{S^{1}}^{1}(\omega,\Omega_{1})=1=K_{\emptyset}^{1}(\omega,\Omega_{1})$
because $K(\omega,\cdot)$ is a probability measure for a probability kernel.
So $\mathscr{H}_{T^{1}}$ has no causal effect on $A$. Implication in the other
direction follows the same argument.

  2. (ii)

Take any $A\in\mathscr{H}_{T^{2}}$. By (i), $\mathscr{H}_{T^{1}}$ has no
causal effect on $A$, so

$K_{T^{1}}(\omega,A)=K_{T^{1}\setminus
T^{1}}(\omega,A)=K_{\emptyset}(\omega,A)=\mathbb{P}(A).$

But since $\mathscr{H}_{T^{1}}$ and $\mathscr{H}_{T^{2}}$ are
probabilistically independent, $\mathbb{P}_{T^{1}}(A)=\mathbb{P}(A)$. Hence,
$\mathbb{P}_{T^{1}}(A)=K_{T^{1}}(\omega,A)$, meaning $\mathscr{H}_{T_{1}}$ is
a source of $A$. Since $A\in\mathscr{H}_{T_{2}}$ was arbitrary,
$\mathscr{H}_{T_{1}}$ is a source of $\mathscr{H}_{T_{2}}$. The implication in
the other direction follows the same argument.

∎

###  B.2 Proofs for Section 6.1

See 6.1

###### Proof of Lemma 6.1.

First, we claim that the pair
$(\kappa_{3},\rho_{3})=(\kappa_{1}\circ\kappa_{2},\rho_{1}\circ\rho_{2})$ is
admissible. We have to show that, for any $S^{3}\subset\rho_{3}(T^{1})$ and
$A\in\mathscr{H}^{3}_{S^{3}}$, the map $\kappa_{3}(\cdot,A)$ is measurable
with respect to $\mathscr{H}^{1}_{\rho_{3}^{-1}(S^{3})}$.

Let us call $\rho_{2}^{-1}(S^{3})=S^{2}$. Note that, since
$(\kappa_{2},\rho_{2})\mathrel{\mathop{\mathchar
58\relax}}\mathscr{C}^{2}\to\mathscr{C}^{3}$ is a causal transformation,
$\kappa_{2}(\cdot,A)$ is measurable with respect to $\mathscr{H}^{2}_{S^{2}}$.
Since we assume that the first map is an abstraction, we find that
$S^{2}\subset\rho^{1}(T^{1})=T^{2}$, and thus by Definition 4.1 that for
$B\in\mathscr{H}_{S^{2}}^{2}$ the function $\kappa_{1}(\cdot,B)$ is measurable
with respect to $\mathscr{H}^{1}_{\rho^{-1}_{3}(S^{3})}$, where we used
$\rho^{-1}_{3}(S^{3})=\rho_{1}^{-1}(S^{2})$. We now use the relation
$\kappa_{3}(\omega,A)=\int\kappa_{1}(\omega,\mathrm{d}\omega^{\prime})\kappa_{2}(\omega^{\prime},A)$.
Since $\kappa_{2}(\cdot,A)$ is measurable with respect to
$\mathscr{H}^{2}_{S^{2}}$, we conclude that we can approximate
$\kappa_{2}(\cdot,A)$ by a simple function
$\sum\alpha_{i}\boldsymbol{1}_{B_{i}}(\cdot)$ with
$B_{i}\in\mathscr{H}^{2}_{S^{2}}$. But for such a simple function, we find

$\int\kappa_{1}(\omega,\mathrm{d}\omega^{\prime})\sum_{i}\alpha_{i}\boldsymbol{1}_{B_{i}}(\omega^{\prime})=\sum_{i}\alpha_{i}\kappa_{1}(\omega,B_{i}),$

which is measurable with respect to $\mathscr{H}^{1}_{S^{1}}$ as a sum of
measurable functions because $(\kappa_{1},\rho_{1})$ is admissible. By passing
to the limit $(\kappa_{3},\rho_{3})$ is admissible.

Next we show that distributional consistency holds, which follows directly
from distributional consistency of $(\kappa_{1},\rho_{1})$ and
$(\kappa_{2},\rho_{2})$:

$\displaystyle\int\mathbb{P}^{1}(\mathrm{d}\omega)\kappa_{3}(\omega,A)$
$\displaystyle=\int\mathbb{P}^{1}(\mathrm{d}\omega)\kappa_{1}(\omega,\mathrm{d}\omega_{2})\kappa_{2}(\omega_{2},A)$
$\displaystyle=\int\mathbb{P}^{2}(\mathrm{d}\omega_{2})\kappa_{2}(\omega_{2},A)$
$\displaystyle=\mathbb{P}^{3}(A).$

Next we consider interventional consistency. Let $S^{3}\subset\rho_{3}(T^{1})$
and define $S^{2}=\rho_{2}^{-1}(S^{3})$ and
$S^{1}=\rho^{-1}_{3}(S^{1})=\rho_{1}^{-1}(S^{2})$. Note that, since
$(\kappa_{1},\rho_{1})$ is an abstraction, i.e., $\rho_{1}$ is surjective, we
have $S^{2}\subset\rho_{1}(T^{1})=T^{2}$. Now we find that, for
$\omega_{1}\in\Omega^{1}$ and $A\in\mathscr{H}^{3}$,

$\displaystyle\int\kappa_{3}(\omega,\mathrm{d}\omega^{\prime})K_{S_{3}}^{3}(\omega^{\prime},A)$
$\displaystyle=\int\kappa_{1}(\omega,\mathrm{d}\omega_{2})\kappa_{2}(\omega_{2},\mathrm{d}\omega^{\prime})K_{S_{3}}^{3}(\omega^{\prime},A)$
$\displaystyle=\int\kappa_{1}(\omega,\mathrm{d}\omega_{2})K_{S_{2}}^{2}(\omega_{2},\mathrm{d}\omega^{\prime})\kappa_{2}(\omega^{\prime},A)$
$\displaystyle=\int
K_{S_{1}}^{1}(\omega,\mathrm{d}\omega^{\prime})\kappa_{1}(\omega^{\prime},\mathrm{d}\omega_{2})\kappa_{2}(\omega_{2},A)$
$\displaystyle=\int
K_{S_{1}}^{1}(\omega,\mathrm{d}\omega^{\prime})\kappa_{3}(\omega^{\prime},A).$

This ends the proof as we have shown that $(\kappa_{3},\rho_{3})$ is a causal
transformation. ∎

See 6.3

###### Proof of Lemma 6.3.

First we note that as in Example 4.5 it is clear that $(\kappa,\rho)$ is
admissible and

$\int\kappa(x_{S},A)\mathbb{P}^{S}(dx_{S})=\int\mathbb{P}_{\mathscr{H}^{1}}(A)d\mathbb{P}^{S}=\mathbb{P}(A),$

so we have distributional consistency.

For interventional consistency, let $A\in\mathscr{H}^{1}$,
$S^{\prime}\subseteq S$ and $\omega\in\Omega^{1}$ be arbitrary. Then see that

$\displaystyle\int
K_{S^{\prime}}(\omega,d\omega^{\prime})\kappa(\omega^{\prime},A)$
$\displaystyle=\int
K_{S^{\prime}}(\omega,d\omega^{\prime})\mathbb{P}_{\mathscr{H}^{1}}(\omega^{\prime},A)$
$\displaystyle=\int
K_{S^{\prime}}(\omega,d\omega^{\prime})\mathbf{1}_{A}(\omega^{\prime})$
$\displaystyle\text{since }A\in\mathscr{H}^{1}$
$\displaystyle=K_{S^{\prime}}(\omega,A).$

On the other hand, see that

$\displaystyle\int\kappa(\omega,d\omega^{\prime})L_{S^{\prime}}(\omega^{\prime},A)$
$\displaystyle=\int\mathbb{P}_{\mathscr{H}^{1}}(\omega,d\omega^{\prime})L_{S^{\prime}}(\omega^{\prime},A)$
$\displaystyle=\int\mathbf{1}_{d\omega^{\prime}}(\omega)L_{S^{\prime}}(\omega^{\prime},A)$
$\displaystyle\text{since }L_{S^{\prime}}(\cdot,A)\text{ is measurable with
respect to }\mathscr{H}^{1}$ $\displaystyle=L_{S^{\prime}}(\omega,A).$

But by the marginalisation condition on the causal mechanisms $\mathbb{K}$ and
$\mathbb{L}$, we have that $L_{S^{\prime}}(\omega,A)=K_{S^{\prime}}(\omega,A)$
for all $\omega\in\Omega^{1}$. This proves interventional consistency.

∎

See 6.4

###### Proof of Lemma 6.4.

Applying distributional consistency of $\varphi$ and $\tilde{\varphi}$, we
find, for all $A\in\mathscr{H}^{2}$,

$\displaystyle\mathbb{P}^{2}(A)=\int\mathbb{P}^{1}(\mathrm{d}\omega)\kappa(\omega,A)=\tilde{P}^{2}(A)$
(4)

and thus $\mathbb{P}^{2}=\tilde{\mathbb{P}}^{2}$. Next, we consider
$A\in\mathscr{H}^{2}_{\rho(T^{1})}$ and $S\subset\rho(T^{1})$. Let us define

$\displaystyle B=\\{\omega\in\Omega^{2}\mathrel{\mathop{\mathchar
58\relax}}K^{2}_{S}(\omega,A)<\tilde{K}_{S}^{2}(\omega,A)\\}.$ (5)

Since $K^{2}_{S}(\cdot,A)$ and $\tilde{K}^{2}_{S}(\cdot,A)$ are
$\mathscr{H}^{2}_{S}$ measurable we find
$B\in\mathscr{H}^{2}_{S}\subset\mathscr{H}^{2}_{\rho(T^{1})}$. Then the
definition of causal spaces (see Definition 2.1) implies that

$\displaystyle K^{2}_{S}(\omega^{\prime},A\cap
B)=\boldsymbol{1}_{B}(\omega^{\prime})K^{2}_{S}(\omega^{\prime},A).$ (6)

Note that $A\cap B\in\mathscr{H}^{2}_{\rho(T^{1})}$ so we can apply
interventional consistency (2) for $\mathscr{C}^{2}$ and
$\tilde{\mathscr{C}}^{2}$ and obtain for any $\omega$

$\displaystyle\begin{split}\int\kappa(\omega,\mathrm{d}\omega^{\prime})\boldsymbol{1}_{B}(\omega^{\prime})K^{2}_{S}(\omega^{\prime},A)&=\int\kappa(\omega,\mathrm{d}\omega^{\prime})\boldsymbol{1}_{B}(\omega^{\prime})K^{2}_{S}(\omega^{\prime},A\cap
B)=\int
K^{1}_{\rho^{-1}(S)}(\omega,\mathrm{d}\omega^{\prime})\kappa(\omega^{\prime},A)\\\
&=\int\kappa(\omega,\mathrm{d}\omega^{\prime})\boldsymbol{1}_{B}(\omega^{\prime})\tilde{K}^{2}_{S}(\omega^{\prime},A\cap
B)=\int\kappa(\omega,\mathrm{d}\omega^{\prime})\boldsymbol{1}_{B}(\omega^{\prime})\tilde{K}^{2}_{S}(\omega^{\prime},A)\end{split}$
(7)

We integrate this relation with respect to $\mathbb{P}^{1}(\mathrm{d}\omega)$
and then apply distributional consistency and get

$\displaystyle\begin{split}0&=\int\mathbb{P}^{1}(\mathrm{d}\omega)\kappa(\omega,\mathrm{d}\omega^{\prime})\boldsymbol{1}_{B}(\omega^{\prime})(\tilde{K}^{2}_{S}(\omega^{\prime},A)-K^{2}_{S}(\omega^{\prime},A))\\\
&=\int\mathbb{P}^{2}(\mathrm{d}\omega^{\prime})\boldsymbol{1}_{B}(\omega^{\prime})(\tilde{K}^{2}_{S}(\omega^{\prime},A)-K^{2}_{S}(\omega^{\prime},A))\\\
&=\int_{B}\mathbb{P}^{2}(\mathrm{d}\omega^{\prime})\,(\tilde{K}^{2}_{S}(\omega^{\prime},A)-K^{2}_{S}(\omega^{\prime},A)).\end{split}$
(8)

On $B$ the last term is strictly positive by definition. Thus we conclude that
$\mathbb{P}^{2}(B)=0$ and thus
$\tilde{K}^{2}_{S}(\omega^{\prime},A)\leq{K}^{2}_{S}(\omega^{\prime},A)$ holds
almost surely. The same reasoning implies the reverse bound and we conclude
that $\mathbb{P}^{2}$ almost surely the relation

$\displaystyle\tilde{K}^{2}_{S}(\omega^{\prime},A)={K}^{2}_{S}(\omega^{\prime},A)$
(9)

holds. ∎

See 6.5

###### Proof of Lemma 6.5.

We first prove uniqueness. The relation
$f_{\ast}\mathbb{P}^{1}=\mathbb{P}^{2}$ that are necessarily true for
deterministic maps (see Section 4) implies that $\mathbb{P}^{2}$ is
predetermined. Moreover, we find that, by (3), for any $A\in\mathscr{H}^{2}$,
$S\subset T^{2}$ and any $\omega\in\Omega^{1}$,

$K^{1}_{\rho^{-1}(S)}(\omega,f^{-1}(A))=K^{2}_{S}(f(\omega),A).$

But since $f$ is surjective we conclude that due to interventional consistency
$K^{2}_{S}(\omega^{\prime},A)$ for $\omega^{\prime}\in\Omega^{2}$ is unique.

To prove the existence we note that by assumption for fixed
$A\in\mathscr{H}^{2}$ the function $K^{1}_{\rho^{-1}(S^{2})}(\cdot,f^{-1}(A))$
is measurable with respect to $f^{-1}(\mathscr{H}^{2}_{S^{2}})$. Now by the
Factorisation Lemma (see Lemma A.1 in Appendix A) there is a measurable
function $g\mathrel{\mathop{\mathchar
58\relax}}(\Omega^{2},\mathscr{H}^{2}_{S^{2}})\to\mathbb{R}$ such that

$K^{1}_{\rho^{-1}(S^{2})}(\omega,f^{-1}(A))=g\circ f(\omega).$

We define $K^{2}_{S^{2}}(\omega^{\prime},A)=g(\omega^{\prime})$. By
surjectivity this defines $K^{2}_{S^{2}}$ everywhere and this defines a
probability kernel because $g$ is measurable.

It remains to verify that the resulting $\mathscr{C}^{2}$ is indeed a causal
space. Using interventional and distributional consistency we obtain

$\displaystyle K^{2}_{\emptyset}(f(\omega),A)$
$\displaystyle=K^{1}_{\emptyset}(\omega,f^{-1}(A))$
$\displaystyle=\mathbb{P}^{1}(f^{-1}(A))$
$\displaystyle=f_{\ast}\mathbb{P}^{1}(A)$ $\displaystyle=\mathbb{P}^{2}(A).$

This verifies the first property of causal spaces. For the second property we
observe that for $A\in\mathscr{H}^{2}_{S^{2}}$, $S^{1}=\pi^{-1}(S^{2})$ using
causal consistency

$\displaystyle K^{2}_{S^{2}}(f(\omega),A\cap B)$
$\displaystyle=K^{1}_{S^{1}}(\omega,f^{-1}(A\cap B))$
$\displaystyle=K^{1}_{S^{1}}(\omega,f^{-1}(A)\cap f^{-1}(B))$
$\displaystyle=\boldsymbol{1}_{f^{-1}(A)}(\omega)K^{1}_{S^{1}}(\omega,f^{-1}(B))$
$\displaystyle=\boldsymbol{1}_{A}(f(\omega))K^{2}_{S^{2}}(f(\omega),B).$

Here we used that $\mathscr{C}^{1}$ is a causal space and
$f^{-1}(A)\in\mathscr{H}^{1}_{S^{1}}$. Thus, we conclude that we obtained a
causal space $\mathscr{C}^{2}$. ∎

See 6.6

###### Proof of Lemma 6.6.

First, we note that by Lemma 6.5 $\mathbb{L}^{2}$ exists and is unique. Thus,
we need to verify distributional consistency and interventional consistency.

Let us first show
$f_{\ast}(\mathbb{P}^{1})^{\text{do}(U^{1},\mathbb{Q}^{1})}=(\mathbb{P}^{2})^{\text{do}(U^{2},\mathbb{Q}^{2})}$.
Since $(f,\rho)$ is a causal transformation (i.e., interventional consistency
as in (2) holds), we find that, for $A\in\mathscr{H}^{2}$,

$\displaystyle f_{\ast}(\mathbb{P}^{1})^{\text{do}(U^{1},\mathbb{Q}^{1})}(A)$
$\displaystyle=\int\mathbb{Q}^{1}(\mathrm{d}\omega)K_{U^{1}}^{1}(\omega,f^{-1}(A))$
$\displaystyle=\int\mathbb{Q}^{1}(\mathrm{d}\omega)K_{U^{2}}^{2}(f(\omega),A)$
$\displaystyle=\int(f_{\ast}\mathbb{Q}^{1})(\mathrm{d}\omega^{\prime})K_{U^{2}}^{2}(\omega^{\prime},A)$
$\displaystyle=\int\mathbb{Q}^{2}(\mathrm{d}\omega^{\prime})K_{U^{2}}^{2}(\omega^{\prime},A)$
$\displaystyle=(\mathbb{P}^{2})^{\text{do}(U^{2},\mathbb{Q}^{2})}(A).$

Here we used the change of variable for pushforward-measures.

Next, we show interventional consistency of
$(f,\rho)\mathrel{\mathop{\mathchar
58\relax}}\mathscr{C}^{1}_{I}\to\mathscr{C}^{2}_{I}$. For this, we introduce
the shorthand $f_{S}=\pi_{S}\circ f$. Note that since $f_{S}$ is measurable
with respect to $\mathscr{H}^{1}_{\rho^{-1}(S)}$ we can find $\tilde{f}_{S}$
such that $f_{S}(\omega)=\tilde{f}_{S}(\omega_{\rho^{-1}(S)})$. Note that, by
the interventional consistency of $(f,\rho)\mathrel{\mathop{\mathchar
58\relax}}\mathscr{C}^{1}\to\mathscr{C}^{2}$, we have

$K_{\rho^{-1}(S)}^{1}(\omega,f^{-1}(S))=K^{2}_{S}(f(\omega),A)=K^{2}_{S}(\tilde{f}_{S}(\omega_{S}),A).$

We can now show for $A\in\mathscr{H}^{2}$ and $S^{1}=\rho^{-1}(S^{2})$ that

$\displaystyle(K^{1})_{S^{1}}^{(U^{1},\mathbb{Q}^{1},\mathbb{L}^{1})}(\omega,f^{-1}(A))$
$\displaystyle=\int L^{1}_{S^{1}\cap U^{1}}(\omega_{S^{1}\cap
U^{1}},\mathrm{d}\omega^{\prime}_{U^{1}})K^{1}_{S^{1}\cup
U^{1}}((\omega_{S^{1}\setminus U^{1}},\omega^{\prime}_{U^{1}}),f^{-1}(A))$
$\displaystyle=\int L^{1}_{S^{1}\cap U^{1}}(\omega_{S^{1}\cap
U^{1}},\mathrm{d}\omega^{\prime}_{U^{1}})K^{2}_{S^{2}\cup
U^{2}}(\tilde{f}_{S^{2}\setminus U^{2}}(\omega_{S^{1}\setminus
U^{1}}),\tilde{f}_{U^{2}}(\omega^{\prime}_{U^{1}}),A)$
$\displaystyle=\int\left((\tilde{f}_{U^{1}})_{\ast}(L^{1}_{S^{1}\cap
U^{1}}(\omega_{S^{1}\cap
U^{1}},\cdot)\right)(\mathrm{d}\overline{\omega}_{U^{2}})K^{2}_{S^{2}\cup
U^{2}}(\tilde{f}_{S^{2}\setminus U^{2}}(\omega_{S^{1}\setminus
U^{1}}),\overline{\omega}_{U^{2}},A)$ $\displaystyle=\int L^{2}_{S^{2}\cap
U^{2}}(f(\omega)_{S^{2}\cap
U^{2}}),\mathrm{d}\overline{\omega}_{U^{2}})K^{2}_{S^{2}\cup
U^{2}}(f(\omega)_{S^{2}\setminus U^{2}}),\overline{\omega}_{U^{2}},A)$
$\displaystyle=(K^{2})_{S^{2}}^{(U^{2},\mathbb{Q}^{2},\mathbb{L}^{2})}(f(\omega),A).$

This ends the proof. ∎

###  B.3 Proofs for Section 6.2

See 6.7

###### Proof of Lemma 6.7.

Consider $A\in\mathscr{H}^{2}_{V^{2}}$ and any $S^{2}\subset T^{2}$. Then for
any $\omega^{\prime}\in\Omega^{2}$ we find an $\omega\in\Omega^{1}$ such that
$f(\omega)=\omega^{\prime}$. Using interventional consistency and
$f^{-1}(A)\in\mathscr{H}^{1}_{V^{1}}$ we conclude

$\displaystyle K^{2}_{S^{2}}(\omega^{\prime},A)$
$\displaystyle=K^{1}_{\rho^{-1}(S^{2})}(\omega,f^{-1}(A))$
$\displaystyle=K^{1}_{\rho^{-1}(S^{2})\setminus\rho^{-1}(U^{2})}(\omega,f^{-1}(A))$
$\displaystyle=K^{1}_{\rho^{-1}(S^{2}\setminus U^{2})}(\omega,f^{-1}(A))$
$\displaystyle=K^{2}_{S^{2}\setminus U^{2}}(\omega^{\prime},A).$

This ends the proof. ∎

See 6.8

###### Proof of Lemma 6.8.

Since $\mathscr{H}^{2}_{U^{2}}$ has an active causal effect on
$\mathscr{H}^{2}_{V^{2}}$ in $\mathscr{C}^{2}$, we find that there is an
$\omega^{\prime}\in\Omega^{2}$ and an $A\in\mathscr{H}^{2}_{V^{1}}$ such that

$K_{U^{2}}^{2}(\omega^{\prime},A)\neq\mathbb{P}^{2}(A).$

By surjectivity there is $\omega\in\Omega^{1}$ such that
$\omega^{\prime}=f(\omega)$ and thus

$\displaystyle K^{1}_{U^{1}}(\omega,f^{-1}(A))$
$\displaystyle=K_{U^{2}}^{2}(\omega^{\prime},A)$
$\displaystyle\neq\mathbb{P}^{2}(A)$
$\displaystyle=\mathbb{P}^{1}(f^{-1}(A)).$

The claim follows because $f^{-1}(A)\in\mathscr{H}^{1}_{U^{1}}$. ∎

See 6.10

###### Proof of Theorem 6.10.

Our goal is to show that $K_{U^{2}}^{2}(\cdot,A)$ is a version of the
conditional probability $\mathbb{P}^{2}_{\mathscr{H}^{2}_{U^{2}}}(A)$ for
$A\in\mathscr{H}^{2}_{V^{2}}$. It is sufficient to show that for all
$B\in\mathscr{H}_{U^{2}}^{2}$ the following relation holds

$\int\mathbb{P}^{2}(\mathrm{d}\omega^{\prime})\boldsymbol{1}_{A}(\omega^{\prime})\boldsymbol{1}_{B}(\omega^{\prime})=\int\mathbb{P}^{2}(\mathrm{d}\omega^{\prime})\boldsymbol{1}_{B}(\omega^{\prime})K_{U^{2}}^{2}(\omega^{\prime},A).$
(10)

Using that $(f,\rho)$ is a perfect abstraction,
$f^{-1}(A)\in\mathscr{H}^{1}_{V^{1}}$, $f^{-1}(B)\in\mathscr{H}^{1}_{U^{1}}$,
and that $\mathscr{H}^{1}_{U^{1}}$ is a local source of
$\mathscr{H}^{1}_{V^{1}}$ we find

$\displaystyle\int\mathbb{P}^{2}(\mathrm{d}\omega^{\prime})\boldsymbol{1}_{A}(\omega^{\prime})\boldsymbol{1}_{B}(\omega^{\prime})$
$\displaystyle=\int
f_{\ast}\mathbb{P}^{1}(\mathrm{d}\omega^{\prime})\boldsymbol{1}_{A}(\omega^{\prime})\boldsymbol{1}_{B}(\omega^{\prime})$
$\displaystyle=\int\mathbb{P}^{1}(\mathrm{d}\omega)\boldsymbol{1}_{A}(f(\omega))\boldsymbol{1}_{B}(f(\omega))$
$\displaystyle=\int\mathbb{P}^{1}(\mathrm{d}\omega)\boldsymbol{1}_{f^{-1}(A)}(\omega)\boldsymbol{1}_{f^{-1}(B)}(\omega)$
$\displaystyle=\int\mathbb{P}^{1}(\mathrm{d}\omega)K^{1}_{U^{1}}(\omega,f^{-1}(A))\boldsymbol{1}_{f^{-1}(B)}(\omega)$
$\displaystyle=\int\mathbb{P}^{1}(\mathrm{d}\omega)K^{2}_{U^{2}}(f(\omega),A)\boldsymbol{1}_{B}(f(\omega))$
$\displaystyle=\int
f_{\ast}\mathbb{P}^{1}(\mathrm{d}\omega^{\prime})K^{2}_{U^{2}}(\omega^{\prime},A)\boldsymbol{1}_{B}(\omega^{\prime})$
$\displaystyle=\int\mathbb{P}^{2}(\mathrm{d}\omega^{\prime})K^{2}_{U^{2}}(\omega^{\prime},A)\boldsymbol{1}_{B}(\omega^{\prime}).$

Thus we have shown that (10) holds and the proof is completed. ∎

##  Appendix C Equivalence of causal mechanism definitions

Let us here briefly comment on the notation and equivalence of the notions
$K_{S}(\omega,A)$ to $K_{S}(\omega_{S},A)$. To clarify the equivalence let us
recall the factorization lemma.

######  Lemma C.1 (Factorization Lemma).

Let $T\mathrel{\mathop{\mathchar 58\relax}}\Omega\to\Omega^{\prime}$ be a map
and $(\Omega^{\prime},\mathscr{A}^{\prime})$ a measurable space. A function
$f\mathrel{\mathop{\mathchar 58\relax}}\Omega\to[0,1]$ is
$\sigma(T)/\mathscr{B}([0,1])$ measurable if and only if $f=g\circ T$ for some
$\mathscr{A}^{\prime}/\mathscr{B}([0,1])$ measurable function
$g\mathrel{\mathop{\mathchar 58\relax}}\Omega^{\prime}\to[0,1]$.

We apply this where for $T$ we use the projection
$\pi_{S}\mathrel{\mathop{\mathchar
58\relax}}(\Omega,\mathscr{H})\to(\Omega_{S},\mathscr{H}_{S})$. Then, by the
factorization lemma, a $\mathscr{H}_{S}$ measurable map
$K_{S}(\cdot,A)\mathrel{\mathop{\mathchar 58\relax}}\Omega\to[0,1]$ gives rise
to a map $K_{S}^{\prime}(\cdot,A)\mathrel{\mathop{\mathchar
58\relax}}\Omega_{S}\to[0,1]$ such that

$\displaystyle
K_{S}(\omega,A)=K_{S}^{\prime}(\pi_{S}\omega,A)=K_{S}^{\prime}(\omega_{S},A).$
(11)

Moreover, this maps is unique because $\pi_{S}$ is surjective. On the other
hand, give $K_{S}^{\prime}(\cdot,A)\mathrel{\mathop{\mathchar
58\relax}}\Omega_{S}\to[0,1]$ we can just define
$K_{S}(\cdot,A)\mathrel{\mathop{\mathchar 58\relax}}\Omega\to[0,1]$ by (11).
For this reason $K_{S}(\omega,A)$ and $K_{S}(\omega_{S},A)$ can be used
equivalently when identified as explained here and with a slight abuse of
notation we will use both depending on the context without indicating the
different domains.