SlowGuess/NeurIPS_2023_OCR huggingface dataset:
We express our sincere gratitude to Robin Evans at the University of Oxford, Michel de Lara at École des Ponts ParisTech and Wojciech Niemiro at the University of Warsaw for fruitful discussions and providing valuable feedback on earlier drafts. We also thank anonymous reviewers for their suggestions for improvements.
This work was supported by the Tübingen AI Center.
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A Mathematical Preliminaries
In this section, we recall some basic facts about measure and probability theory that we need for the development in the main body of the paper. We follow Činlar [14].
A.1 Measure Theory
Suppose that  $E$  is a set. We first define the notion of a  $\sigma$ -algebra. A non-empty collection  $\mathcal{E}$  of  $E$  is called a  $\sigma$ -algebra on  $E$  if it is closed under complements and countable unions, that is, if
(i)  $A\in \mathcal{E}\Rightarrow E\backslash A\in \mathcal{E};$
(ii)  $A_{1},A_{2},\ldots \in \mathcal{E}\Rightarrow \cup_{n = 1}^{\infty}A_{n}\in \mathcal{E}$
[14, p.2]. We call  $\{\emptyset, E\}$  the trivial  $\sigma$ -algebra of  $E$ . If  $\mathcal{C}$  is an arbitrary collection of subsets of  $E$ , then the smallest  $\sigma$ -algebra that contains  $\mathcal{C}$ , or equivalently, the intersection of all  $\sigma$ -algebras that contain  $\mathcal{C}$ , is called the  $\sigma$ -algebra generated by  $\mathcal{C}$ , and is denoted  $\sigma \mathcal{C}$ .
A measurable space is a pair  $(E, \mathcal{E})$ , where  $E$  is a set and  $\mathcal{E}$  is a  $\sigma$ -algebra on  $E$  [14, p.4].
Suppose  $(E, \mathcal{E})$  and  $(F, \mathcal{F})$  are measurable spaces. For  $A \in \mathcal{E}$  and  $B \in \mathcal{F}$ , we define the measurable rectangle  $A \times B$  as the set of all pairs  $(x, y)$  with  $x \in A$  and  $y \in B$ . We define the product  $\sigma$ -algebra  $\mathcal{E} \otimes \mathcal{F}$  on  $E \times F$  as the  $\sigma$ -algebra generated by the collection of all measurable rectangles. The measurable space  $(E \times F, \mathcal{E} \otimes \mathcal{F})$  is the product of  $(E, \mathcal{E})$  and  $(F, \mathcal{F})$  [14, p.4]. More generally, if  $(E_1, \mathcal{E}_1), \ldots, (E_n, \mathcal{E}_n)$  are measurable spaces, their product is
$$
\bigotimes_ {i = 1} ^ {n} (E _ {i}, \mathcal {E} _ {i}) = (\bigotimes_ {i = 1} ^ {n} E _ {i}, \bigotimes_ {i = 1} ^ {n} \mathcal {E} _ {i}),
$$
where  $E_1 \times \ldots \times E_n$  is the set of all  $n$ -tuples  $(x_1, \ldots, x_n)$  with  $x_i$  in  $E_i$  for  $i = 1, \ldots, n$  and  $\mathcal{E}_1 \otimes \ldots \otimes \mathcal{E}_n$  is the  $\sigma$ -algebra generated by the measurable rectangles  $A_1 \times \ldots \times A_n$  with  $A_i$  in  $\mathcal{E}_i$  for  $i = 1, \ldots, n$  [14, p.44]. If  $T$  is an arbitrary (countable or uncountable) index set and  $(E_t, \mathcal{E}_t)$  is a measurable space for each  $t \in T$ , the product space of  $\{E_t : t \in T\}$  is the set  $\times_{t \in T} E_t$  of all collections  $(x_t)_{t \in T}$  with  $x_t \in E_t$  for each  $t \in T$ . A rectangle in  $\times_{t \in T} E_t$  is a subset of the form
$$
\underset {t \in T} {\times} A _ {t} = \left\{x = \left(x _ {t}\right) _ {t \in T} \in \underset {t \in T} {\times} E _ {t}: x _ {t} \in A _ {t} \text {f o r e a c h} t \text {i n} T \right\}
$$
where  $A_{t}$  differs from  $E_{t}$  for only a finite number of  $t$ . It is said to be measurable if  $A_{t} \in \mathcal{E}_{t}$  for every  $t$  (for which  $A_{t}$  differs from  $E_{t}$ ). The  $\sigma$ -algebra on  $\times_{t \in T} E_{t}$  generated by the collection of all measurable rectangles is called the product  $\sigma$ -algebra and is denoted by  $\bigotimes_{t \in T} \mathcal{E}_{t}$  [14, p.45].
A collection  $\mathcal{C}$  of subsets of  $E$  is called a p-system if it is closed under intersections [14, p.2]. If two measures  $\mu$  and  $\nu$  on a measurable space  $(E,\mathcal{E})$  with  $\mu (E) = \nu (E) < \infty$  agree on a p-system generating  $\mathcal{E}$ , then  $\mu$  and  $\nu$  are identical [14, p.16, Proposition 3.7].
Let  $(E, \mathcal{E})$  and  $(F, \mathcal{F})$  be measurable spaces. A mapping  $f: E \to F$  is measurable if  $f^{-1}B \in \mathcal{E}$  for every  $B \in \mathcal{F}$  [14, p.6].
Let  $(E, \mathcal{E})$  and  $(F, \mathcal{F})$  be measurable spaces. Let  $f$  be a bijection between  $E$  and  $F$ , and let  $\hat{f}$  denote its functional inverse. Then,  $f$  is an isomorphism if  $f$  is measurable relative to  $\mathcal{E}$  and  $\mathcal{F}$ , and  $\hat{f}$  is measurable with respect to  $\mathcal{F}$  and  $\mathcal{E}$ . The measurable spaces  $(E, \mathcal{E})$  and  $(F, \mathcal{F})$  are isomorphic if there exists an isomorphism between them [14, p.11].
A measurable space  $(E, \mathcal{E})$  is a standard measurable space if it is isomorphic to  $(F, \mathcal{B}_F)$  for some Borel subset  $F$  of  $\mathbb{R}$ . Polish spaces with their Borel  $\sigma$ -algebra are standard measurable spaces [14, p.11].
Let  $A \subset E$ . Its indicator, denoted by  $1_A$ , is the function defined by
$$
1 _ {A} (x) = \left\{ \begin{array}{l l} 1 & \text {i f} x \in A \\ 0 & \text {i f} x \notin A \end{array} \right.
$$
[14, p.8]. Obviously,  $1_A$  is  $\mathcal{E}$ -measurable if and only if  $A \in \mathcal{E}$ . A function  $f: E \to \mathbb{R}$  is said to be simple if it is of the form
$$
f = \sum_ {i = 1} ^ {n} a _ {i} 1 _ {A _ {i}}
$$
for some  $n \in \mathbb{N}$ ,  $a_1, \ldots, a_n \in \mathbb{R}$  and  $A_1, \ldots, A_n \in \mathcal{E}$  [14, p.8]. The  $A_1, \ldots, A_n \in \mathcal{E}$  can be chosen to be a measurable partition of  $E$ , and is then called the canonical form of the simple function  $f$ . A positive function on  $E$  is  $\mathcal{E}$ -measurable if and only if it is the limit of an increasing sequence of positive simple functions [14, p.10, Theorem 2.17].
A measure on a measurable space  $(E, \mathcal{E})$  is a mapping  $\mu : \mathcal{E} \to [0, \infty]$  such that
(i)  $\mu (\emptyset) = 0$
(ii)  $\mu (\cup_{n = 1}^{\infty}A_n) = \sum_{n = 1}^{\infty}\mu (A_n)$  for every disjoint sequence  $(A_{n})$  in  $\mathcal{E}$
[14, p.14]. A measure space is a triplet  $(E, \mathcal{E}, \mu)$ , where  $(E, \mathcal{E})$  is a measurable space and  $\mu$  is a measure on it.
A measurable set  $B$  is said to be negligible if  $\mu(B) = 0$ , and an arbitrary subset of  $E$  is said to be negligible if it is contained in a measurable negligible set. The measure space is said to be complete if every negligible set is measurable [14, p.17].
Next, we review the notion of integration of a real-valued function  $f: E \to \mathbb{R}$  with respect to  $\mu$  [14, p.20, Definition 4.3].
(a) Let  $f: E \to [0, \infty]$  be simple. If its canonical form is  $f = \sum_{i=1}^{n} a_i 1_{A_i}$  with  $a_i \in \mathbb{R}$ , then we define
$$
\int f d \mu = \sum_ {i = 1} ^ {n} a _ {i} \mu \left(A _ {i}\right).
$$
(b) Suppose  $f: E \to [0, \infty]$  is measurable. Then by above, we have a sequence  $(f_n)$  of positive simple functions such that  $f_n \to f$  pointwise. Then we define
$$
\int f d \mu = \lim  _ {n \rightarrow \infty} \int f _ {n} d \mu ,
$$
where  $\int f_n d\mu$  is defined for each  $n$  by (a).
(c) Suppose  $f: E \to [-\infty, \infty]$  is measurable. Then  $f^{+} = \max \{f, 0\}$  and  $f^{-} = -\min \{f, 0\}$  are measurable and positive, so we can define  $\int f^{+} d\mu$  and  $\int f^{-} d\mu$  as in (b). Then we define
$$
\int f d \mu = \int f ^ {+} d \mu - \int f ^ {-} d \mu
$$
provided that at least one term on the right be positive. Otherwise,  $\int f d\mu$  is undefined. If  $\int f^{+}d\mu < \infty$  and  $\int f^{-}d\mu < \infty$ , then we say that  $f$  is integrable.
Finally, we review the notion of transition kernels, which are crucial in the consideration of conditional distributions. Let  $(E,\mathcal{E})$  and  $(F,\mathcal{F})$  be measurable spaces. Let  $K$  be a mapping  $E\times \mathcal{F}\to [0,\infty ]$ . Then,  $K$  is called a transition kernel from  $(E,\mathcal{E})$  into  $(F,\mathcal{F})$  if
(a) the mapping  $x\mapsto K(x,B)$  is measurable for every set  $B\in \mathcal{F}$ ; and
(b) the mapping  $B\mapsto K(x,B)$  is a measure on  $(F,\mathcal{F})$  for every  $x\in E$
A transition kernel from  $(E, \mathcal{E})$  into  $(F, \mathcal{F})$  is called a probability transition kernel if  $K(x, F) = 1$  for all  $x \in E$ . A probability transition kernel  $K$  from  $(E, \mathcal{E})$  into  $(E, \mathcal{E})$  is called a Markov kernel on  $(E, \mathcal{E})$  [14, p.37,39,40].
A.2 Probability Theory
Now we translate the above measure-theoretic notions into the language of probability theory, and introduce some additional concepts. A probability space is a measure space  $(\Omega ,\mathcal{H},\mathbb{P})$  such that
$\mathbb{P}(\Omega) = 1$  [14, p.49]. We call  $\Omega$  the sample space, and each element  $\omega \in \Omega$  an outcome. We call  $\mathcal{H}$  a collection of events, and for any  $A \in \mathcal{H}$ , we read  $\mathbb{P}(A)$  as the probability that the event  $A$  occurs [14, p.50].
A random variable taking values in a measurable space  $(E, \mathcal{E})$  is a function  $X: \Omega \to E$ , measurable with respect to  $\mathcal{H}$  and  $\mathcal{E}$ . The distribution of  $X$  is the measure  $\mu$  on  $(E, \mathcal{E})$  defined by  $\mu(A) = \mathbb{P}(X^{-1}A)$  [14, p.51]. For an arbitrary set  $T$ , let  $X_{t}$  be a random variable taking values in  $(E, \mathcal{E})$  for each  $t \in T$ . Then the collection  $\{X_{t}: t \in T\}$  is called a stochastic process with state space  $(E, \mathcal{E})$  and parameter set  $T$  [14, p.53].
Henceforth, random variables are defined on  $(\Omega ,\mathcal{H},\mathbb{P})$  and take values in  $[- \infty ,\infty ]$  . We define the expectation of a random variable  $X:\Omega \to [-\infty ,\infty ]$  as  $\mathbb{E}[X] = \int_{\Omega}Xd\mathbb{P}$  [14, p.57-58]. We also define the conditional expectation [14, p.140, Definition 1.3]. Suppose  $\mathcal{F}$  is a sub- $\sigma$  -algebra of  $\mathcal{H}$
(a) Suppose  $X$  is a positive random variable. Then the conditional expectation of  $X$  given  $\mathcal{F}$  is any positive random variable  $\mathbb{E}_{\mathcal{F}}X$  satisfying
$$
\mathbb {E} [ V X ] = \mathbb {E} \left[ V \mathbb {E} _ {\mathcal {F}} X \right]
$$
for all  $V:\Omega \to [0,\infty ]$  measurable with respect to  $\mathcal{F}$
(b) Suppose  $X:\Omega \to [-\infty ,\infty ]$  is a random variable. If  $\mathbb{E}[X]$  exists, then we define
$$
\mathbb {E} _ {\mathcal {F}} X = \mathbb {E} _ {\mathcal {F}} X ^ {+} - \mathbb {E} _ {\mathcal {F}} X ^ {-},
$$
where  $\mathbb{E}_{\mathcal{F}}X^{+}$  and  $\mathbb{E}_{\mathcal{F}}X^{-}$  are defined in (a).
Next, we define conditional probabilities, and regular versions thereof [14, pp.149-151]. Suppose  $H \in \mathcal{H}$ , and let  $\mathcal{F}$  be a sub- $\sigma$ -algebra of  $\mathcal{H}$ . Then the conditional probability of  $H$  given  $\mathcal{F}$  is defined as
$$
\mathbb {P} _ {\mathcal {F}} H = \mathbb {E} _ {\mathcal {F}} 1 _ {H}.
$$
Let  $Q(H)$  be a version of  $\mathbb{P}_{\mathcal{F}}H$  for every  $H \in \mathcal{H}$ . Then  $Q: (\omega, H) \mapsto Q_{\omega}(H)$  is said to be a regular version of the conditional probability  $\mathbb{P}_{\mathcal{F}}$  provided that  $Q$  be a probability transition kernel from  $(\Omega, \mathcal{F})$  into  $(\Omega, \mathcal{H})$ . Regular versions exist if  $(\Omega, \mathcal{H})$  is a standard measurable space [14, p.151, Theorem 2.7].
The conditional distribution of a random variable  $X$  given  $\mathcal{F}$  is any transition probability kernel  $L: (\omega, B) \mapsto L_{\omega}(B)$  from  $(\Omega, \mathcal{F})$  into  $(E, \mathcal{E})$  such that
$$
P _ {\mathcal {F}} \{Y \in B \} = L (B) \quad \text {f o r a l l} B \in \mathcal {E}.
$$
If  $(E, \mathcal{E})$  is a standard measurable space, then a version of the conditional distribution of  $X$  given  $\mathcal{F}$  exists [14, p.151].
Suppose that  $T$  is a totally ordered set, i.e. whenever  $r,s,t\in T$  with  $r < s$  and  $s < t$ , we have  $r < t$  and for any  $s,t\in T$ , exactly one of  $s < t$ ,  $s = t$  and  $t < s$  holds [18, p.62]. For each  $t\in T$ , let  $\mathcal{F}_t$  be a sub- $\sigma$ -algebra of  $\mathcal{H}$ . The family  $\mathcal{F} = \{\mathcal{F}_t:t\in T\}$  is called a filtration provided that  $\mathcal{F}_s\subset \mathcal{F}_t$  for  $s < t$  [14, p.79]. A filtered probability space  $(\Omega ,\mathcal{H},\mathcal{F},\mathbb{P})$  is a probability space  $(\Omega ,\mathcal{H},\mathbb{P})$  endowed with a filtration  $\mathcal{F}$ .
Finally, we review the notion of independence and conditional independence. For a fixed integer  $n \geq 2$ , let  $\mathcal{F}_1, \ldots, \mathcal{F}_n$  be sub- $\sigma$ -algebras of  $\mathcal{H}$ . Then  $\{\mathcal{F}_1, \ldots, \mathcal{F}_n\}$  is called an independency if
$$
\mathbb {P} \left(H _ {1} \cap \dots \cap H _ {n}\right) = \mathbb {P} \left(H _ {1}\right) \dots \mathbb {P} \left(H _ {n}\right)
$$
for all  $H_{1}\in \mathcal{F}_{1},\dots,H_{n}\in \mathcal{F}_{n}$ . Let  $T$  be an arbitrary index set. Let  $\mathcal{F}_t$  be a sub- $\sigma$ -algebra of  $\mathcal{H}$  for each  $t\in T$ . The collection  $\{\mathcal{F}_t:t\in T\}$  is called an independency if its every finite subset is an independency [14, p.82].
Moreover,  $\mathcal{F}_1,\dots,\mathcal{F}_n$  are said to be conditional independent given  $\mathcal{F}$  if
$$
\mathbb {P} _ {\mathcal {F}} \left(H _ {1} \cap \dots \cap H _ {n}\right) = \mathbb {P} _ {\mathcal {F}} \left(H _ {1}\right) \dots \mathbb {P} _ {\mathcal {F}} \left(H _ {n}\right)
$$
for all  $H_{1}\in \mathcal{F}_{1},\dots,H_{n}\in \mathcal{F}_{n}$  [14,p.158].
B Causal Effect
In this section, we define what it means for a sub- $\sigma$ -algebra of the form  $\mathcal{H}_S$  to have a causal effect on an event  $A \in \mathcal{H}$ .
Definition B.1. Let  $(\Omega, \mathcal{H}, \mathbb{P}, \mathbb{K}) = (\times_{t \in T} E_t, \otimes_{t \in T} \mathcal{E}_t, \mathbb{P}, \mathbb{K})$  be a causal space,  $U \in \mathcal{P}(T)$ ,  $A \in \mathcal{H}$  an event and  $\mathcal{F}$  a sub- $\sigma$ -algebra of  $\mathcal{H}$  (not necessarily of the form  $\mathcal{H}_S$  for some  $S \in \mathcal{P}(T)$ ).
(i) If  $K_S(\omega, A) = K_{S \setminus U}(\omega, A)$  for all  $S \in \mathcal{P}(T)$  and all  $\omega \in \Omega$ , then we say that  $\mathcal{H}_U$  has no causal effect on  $A$ , or that  $\mathcal{H}_U$  is non-causal to  $A$ .
We say that  $\mathcal{H}_U$  has no causal effect on  $\mathcal{F}$ , or that  $\mathcal{H}_U$  is non-causal to  $\mathcal{F}$ , if, for all  $A \in \mathcal{F}$ ,  $\mathcal{H}_U$  has no causal effect on  $A$ .
(ii) If there exists  $\omega \in \Omega$  such that  $K_U(\omega, A) \neq \mathbb{P}(A)$ , then we say that  $\mathcal{H}_U$  has an active causal effect on  $A$ , or that  $\mathcal{H}_U$  is actively causal to  $A$ .