Causal inference using the algorithmic Markov condition

Introduction to causal inference from statistical data

Causal inference from statistical data has attracted increasing interest in the past decade. In contrast to traditional statistics where statistical dependences are only taken to prove that some kind of relation between random variables exists, causal inference methods in machine learning are explicitly designed to generate hypotheses on causal directions automatically based upon statistical independence tests . The crucial assumption connecting statistics with causality is the causal Markov condition explained below after we have introduced some notations and terminology.

We denote random variables by capitals and their values by the corresponding lowercase letters. Let $X_{1},\dots,X_{n}$ be random variables and $G$ be a directed acyclic graph (DAG) representing the causal structure where an arrow from node $X_{i}$ to node $X_{j}$ indicates a direct causal effect. Here the term direct is understood with respect to the chosen set of variables in the sense that the information flow between the two variables considered is not performed via using one or more of the other variables as intermediate nodes. We will next briefly rephrase the postulates that are required in the statistical theory of inferred causation .

When we consider the causal structure that links $n$ random variables ${\cal V}:=\{X_{1},\dots,X_{n}\}$ we will implicitly assume that ${\cal V}$ is causally sufficient in the sense that all common causes of two variables in ${\cal V}$ are also in ${\cal V}$. Then a causal hypothesis $G$ is only acceptable as potential causal structure if the joint distribution $P$ of $X_{1},\dots,X_{n}$ satisfies the Markov condition with respect to $G$. There are several formulations of the Markov condition that are known to coincide under some technical condition (see Lemma 1). We will first introduce the following version which is sometimes referred to as the parental or the local Markov condition .

To this end, we introduce the following notations. $PA_{j}$ is the set of parents of $X_{j}$ and $ND_{j}$ the set of non-descendants of $X_{j}$ except itself. If $S,T,R$ are sets of random variables, S\mathchoice{\mathrel{\hbox to0.0pt{\displaystyle\perp\hss}\mkern 2.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{\textstyle\perp\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptstyle\perp\hss}\mkern 2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptscriptstyle\perp\hss}\mkern 2.0mu{\scriptscriptstyle\perp}}}T\,|R means $S$ is statistically independent of $T$, given $R$.

If a directed acyclic graph $G$ formalizes the causal structure among the random variables $X_{1},\dots,X_{n}$. Then

We call this postulate the statistical causal Markov condition because we will later introduce an algorithmic version. The fact that conditional irrelevance not only occurs in the context of statistical dependences has been emphasized in the literature (e.g. ) in the context of describing abstract properties (like semi-graphoid axioms) of the relation \cdot\mathchoice{\mathrel{\hbox to0.0pt{\displaystyle\perp\hss}\mkern 2.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{\textstyle\perp\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptstyle\perp\hss}\mkern 2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptscriptstyle\perp\hss}\mkern 2.0mu{\scriptscriptstyle\perp}}}\cdot\,|\cdot. We will therefore state the causal Markov condition also in an abstract form that does not refer to any specific notion of conditional informational irrelevance:

Given all the direct causes of an observable $O$, its non-effects provide no additional information on $O$.

Here, observables denote something in the real world that can be observed and the observation of which can be formalized in terms of a mathematical language. In this paper, observables will either be random variables (formalizing statistical quantities) or they will be strings (formalizing the description of objects). Accordingly, information will be statistical or algorithmic mutual information, respectively.

The importance of the causal Markov condition lies in the fact that it links causal terms like “direct causes” and “non-effects” to informational relevance of observables. The local Markov condition is rather intuitive because it echoes the fact that the information flows from direct causes to their effect and every dependence between a node and its non-descendants involves the direct causes. However, the independences postulated by the local Markov condition imply additional independences. It is therefore hard to decide whether an independence must hold for a Markovian distribution or not, solely on the basis of the local formulation. In contrast, the global Markov condition makes the complete set of independences obvious. To state it we first have to introduce the following graph-theoretical concept.

A path $p$ in a DAG is said to be d-separated (or blocked) by a set of nodes $Z$ if and only if

$p$ contains a chain $i\rightarrow m\rightarrow j$ or fork $i\leftarrow m\rightarrow j$ such that the middle node $m$ is in $Z$, or

$p$ contains an inverted fork (or collider) $i\rightarrow m\leftarrow j$ such that the middle node $m$ is not in $Z$ and such that no descendant of $m$ is in $Z$.

A set $Z$ is said to d-separate $X$ from $Y$ if and only if $Z$ blocks every (possibly undirected) path from a node in $X$ to a node in $Y$.

The following Lemma shows that d-separation is the correct condition for deciding whether an independence is implied by the local Markov condition , Theorem 3.27.

Let $P(X_{1},\dots,X_{n})$ have a density $P(x_{1},\dots,x_{n})$ with respect to a product measure. Then the following three statements are equivalent:

Recursive form: $P$ admits the factorization

where $P(.|pa_{j})$ is shorthand for the conditional probability density, given the values of all parents of $X_{j}$.

Local (or parental) Markov condition: for every node $X_{j}$ we have

i.e., it is conditionally independent of its non-descendants (except itself), given its parents.

for all three sets $S,T,R$ of nodes for which $S$ and $T$ are d-separated by $R$.

Moreover, the local and the global Markov condition are equivalent even if $P$ does not have a density with respect to a product measure.

The conditional densities $P(x_{j}|pa_{j})$ are also called the Markov kernels relative to the hypothetical causal graph $G$. It is important to note that every choice of Markov kernels define a Markovian density $P$, i.e., the Markov kernels define exactly the set of free parameters remaining after the causal structure has been specified.

To select graphs among all those that render $P$ Markovian, we also need an additional postulate:

Among all graphs $G$ for which $P$ is Markovian, prefer the ones for which all the observed conditional independences in the joint measure $P(X_{1},\dots,X_{n})$ are imposed by the Markov condition.

The idea is that the set of observed independences is typical for the causal structure under consideration rather than being the result of specific choices of the Markov kernels. This becomes even more intuitive when we restrict our attention to random variables with finite value set and observe that the values $P(x_{j}|pa_{j})$ then define a natural parameterization of the set of Markovian distributions in a finite dimensional space. The non-faithful distributions form a submanifold of lower dimension, i.e., a set of Lebesgue measure zero . They therefore almost surely don’t occur if we assume that “nature chooses” the Markov kernels for the different nodes independently according to some density on the parameter space.

The above “zero Lebesgue measure argument” is close to the spirit of Bayesian approaches , where priors on the set of Markov kernels are specified for every possible hypothetical causal DAG and causal inference is performed by maximizing posterior probabilities for hypothetical DAGs, given the observed data. This procedure leads to an implicit preference of faithful structures in the infinite sampling limit given some natural conditions for the priors on the parameter space. The assumption that “nature chooses Markov kernels independently”, which is also part of the Bayesian approach, will turn out to be closely related to the algorithmic Markov condition postulated in this paper.

We now discuss the justification of the statistical causal Markov condition because we will later justify the algorithmic Markov condition in a similar way. To this end, we introduce functional models :

If a directed acyclic graph $G$ formalizes the causal relation between the random variables $X_{1},\dots,X_{N}$ then every $X_{j}$ can be written as a deterministic function of $PA_{j}$ and a noise variable $N_{j}$ ,

where all $N_{j}$ are jointly independent.

Every joint distribution $P(X_{1},\dots,X_{n})$ generated according to the functional model in Postulate 4 satisfies the local and the global Markov condition relative to $G$.

We rephrase the proof in because our proof for the algorithmic version will rely on the same idea.

Functional models formalize the idea that the outcome of an experiment is completely determined by the values of all relevant parameters where the only uncertainty stems from the fact that some of these parameters are hidden. Even though this kind of determinism is in contrast with the commonly accepted interpretation of quantum mechanics , we still consider functional models as a helpful framework for discussing causality in real life since quantum mechanical laws refer mainly to phenomena in micro-physics.

Causal inference using the Markov condition and the faithfulness assumption has been implemented as causal learning algorithms . The following fundamental limitations of these methods deserve our further attention:

Markov equivalence: There are only few cases where the inference rules provide unique causal graphs. Often one ends up with a large class of Markov equivalent graphs, i.e., graphs that entail the same set of independences. For this reason, additional inference rules are desirable.

Dependence on i.i.d. sampling: the whole setting of causal inference relies on the ability to sample repeatedly and independently from the same joint distribution $P(X_{1},\dots,X_{n})$. As opposed to this assumption, causal inference in real life also deals with probability distributions that change in time and often one infers causal relations among single observations without referring to statistics at all.

The idea of this paper is to develop a theory of probability-free causal inference that helps to construct causal hypotheses based on similarities of single objects. Here, similarities will be defined by comparing the length of the shortest description of single objects to the length of their shortest joint description. Despite the analogy to causal inference from statistical data (which is due to known analogies between statistical and algorithmic information theory) our theory also implies new statistical inference rules. In other words, our approach to address weakness 2 also yields new methods to address 1.

The paper is structured as follows. In the remaining part of this Section, i.e., Subsection 1.2, we describe recent approaches from the literature to causal inference from statistical data that address problem 1 above. In Section 2 we develop the general theory on inferring causal relations among individual objects based on algorithmic information. This framework appears, at first sight, as a straightforward adaption of the statistical framework (using well-known correspondences between statistical and algorithmic information theory). However, Section 3 describes that this implies novel causal inference rules for statistical inference because non-statistical algorithmic dependences can even occur in data that were obtained from statistical sampling. In Section 4 we describe how to replace causal inference rules based on the uncomputable algorithmic information with decidable criteria that are still motivated by the uncomputable idealization.

The table in fig. 1 summarizes the analogies between the theory of statistical and the theory of algorithmic causal inference described in this paper. The differences, however, which are the main subject of Sections 3 to 4, can hardly be represented in the table.

2 Seeking for new statistical inference rules

In and we have proposed causal inference rules that are based on the idea that the factorization of $P({\rm cause},{\rm effect})$ into $P({\rm effect}|{\rm cause})$ and $P({\rm cause})$ typically leads to simpler terms than the “artificial” factorization into $P({\rm effect})P({\rm cause}|{\rm effect})$. The generalization of this principle reads: Among all graphs $G$ that render $P$ Markovian prefer the one for which the decomposition in eq. (1) yields the simplest Markov kernels. We have called this vague idea the “principle of plausible Markov kernels”.

where $\lambda$ determines the shift of the mean caused by switching between $x=1$ and $x=-1$.

One will prefer the causal structure $X\rightarrow Y$ compared to $Y\rightarrow X$ because the former explains in a natural way why $P(Y)$ is bimodal: the effect of $X$ on $Y$ is simply to shift the Gaussian distribution by $2\lambda$. In the latter model the bimodality of $P(Y)$ remains unexplained. To prefer one causal model to another one because the corresponding conditionals are simpler seems to be a natural application of Occam’s Razor. However, Section 3 will show that such an inference rule also follows from the theory developed in the present paper when simplicity is meant in the sense of low Kolmogorov complexity. In the remaining part of this section we will sketch some approaches to implement the “principle of plausible Markov kernels” in practical applications.

In we have defined a family of “plausible Markov kernels” by conditionals $P(X_{j}|PA_{j})$ that are second order exponential models, i.e., $\log P(x_{j}|pa_{j})$ is a polynomial of order two in the variables $\{X_{j}\}\cup\{PA_{j}\}$ up to some additive partition function (for normalization) that depends only on the variables $PA_{j}$. For every hypothetical causal graph, one thus obtains a family of “plausible joint distributions $P(X_{1},\dots,X_{n})$” that are products of the plausible Markov kernels. Then we prefer the causal direction for which the plausible joint distributions provide the best fit for the given observations.

In we have proposed the following principle for causal inference: Given a joint distribution of the random variables $X_{1},\dots,X_{n}$, prefer a causal structure for which

is minimal, where $C$ is some complexity measure on conditional probability densities.

There is also another recent proposal for new inference rules that refers to a related simplicity assumption, though formally quite different from the ones above. The authors of observe that there are joint distributions of $X_{1},\dots,X_{n}$ that can be explained by a linear model with additive non-Gaussian noise for one causal direction but require non-linear causal influence for the other causal directions. For real data they prefer the causal graph for which the observations are closer to the linear model.

To justify the belief that conditionals that correspond to the true causal direction tend to be simpler than non-causal conditionals (which is common to all the approaches above) is one of the main goals of this paper.

Inferring causal relations among individual objects

It has been emphasized that the application of causal inference principles often benefits from the non-determinism of causal relations between the observed random variables. In contrast, human learning in real-life often is about quite deterministic relations. Apart from that, the most important difference between human causal learning and the inference rules in is that the former is also about causal relations among single objects and does not necessarily require sampling. Assume, for instance, that the comparison of two texts show similarities (see e.g. ) such that the author of the text that appeared later is blamed to have copied it from the other one or both are blamed to have copied from a third one. The statement that the texts are similar could be based on a statistical analysis of the occurrences of certain words or letter sequences. However, such kind of simple statistical tests can fail in both directions: In Subsection 2.2 (before Theorem 3) we will discuss an example showing that they can erroneously infer causal relations even though they do not exist. This is because parts that are common two both objects, e.g., the two texts, are only suitable to prove a causal link if they are not “too straightforward” to come up with.

On the other hand, causal relations can generate similarities between texts for which every efficient statistical analysis is believed to fail. We will describe an idea from cryptography to show this. A cryptosystem is called ROR-CCA-secure (Real or Random under Chosen Ciphertext Attacks) if there is no efficient method to decide whether a text is random or the encrypted version of some known text without knowing the key . Given that there are ROR-CCA-secure schemes (which is unknown but believed by cryptographers) we have a causal relation leading to similarities that are not detected by any kind of simple counting statistics. However, once an attacker has found the key (maybe by exhaustive search), he recognizes similarities between the encrypted text and the plain text and infers a causal relation. This already suggests two things: (1) detecting similarities involves searching over potential rules how properties of one object can be algorithmically derived from the structure of the other. (2) It is likely that inferring causal relations therefore relies on computationally infeasible decisions (if computable at all) on whether two objects have information in common or not.

We will now describe how the information one object provides about the other can be measured in terms of Kolmogorov complexity. We start with some notation and terminology. Below, strings will always be binary strings since every description given in terms of a different alphabet can be converted into a binary word. The set of binary strings of arbitrary length will be denoted by $\{0,1\}^{*}$. Recall that the Kolmogorov complexity $K(s)$ of a string $s\in\{0,1\}^{*}$ is defined as the length of the shortest program that generates $s$ using a previously defined universal Turing machine . The conditional Kolmogorov complexity $K(t|s)$ of a string $t$ given another string $s$ is the length of the shortest program that can generate $t$ from $s$. In order to keep our notation simple we use $K(x,y)$ to refer to the complexity of the concatenation of $x,y$.

We will mostly have equations that are valid only up to additive constant terms in the sense that the difference between both sides does not depend on the strings involved in the equation (but it may depend on the Turing machines they refer to). To indicate such constants we denote the corresponding equality by $\stackrel{{\scriptstyle+}}{{=}}$ and likewise for inequalities. In this context it is important to note that the number $n$ of nodes of the causal graph is considered to be a constant. Moreover, for every string $s$ we define $s^{*}$ as its shortest description. If the latter is not unique, we consider the first one in an lexicographic order. It is necessary to distinguish between $K(\cdot|s)$ and $K(\cdot|s^{*})$. This is because there is a trivial algorithmic method to generate $s$ from $s^{*}$ (just apply the Turing machine to $s^{*}$), but there is no algorithm of length $O(1)$ that computes the shortest description $s^{*}$ from a general input $s$. One can show that $s^{*}\equiv(s,K(s))$. Here, the equivalence symbol $\equiv$ means that both sides can be obtained from each other by $O(1)$ programs. The following equation for the joint algorithmic information of two strings $x,y$ will be useful :

The most important notion in this paper will be the algorithmic mutual information measuring the amount of algorithmic information that two objects have in common. Following we define:

Let $x,y$ be two strings. Then the algorithmic mutual information of $x,y$ is

The mutual information is the number of bits that can be saved in the description of $y$ when the shortest description of $x$ is already known. The fact that one uses $x^{*}$ instead of $x$ ensures that it coincides with the symmetric expression :

In the following sections, non-vanishing mutual information will be taken as an indicator for causal relations, but more detailed information on the causal structure will be inferred from conditional mutual information. This is in contrast to approaches from the literature to measure similarity versus differences of single objects that we briefly review now. To measure differences between single objects, e.g. pictures , one defines the information distance $E(x,y)$ between the two corresponding strings as the length of the shortest program that computes $x$ from $y$ and $y$ from $x$. It can be shown that

where $\stackrel{{\scriptstyle{\rm log}}}{{=}}$ means equality up to a logarithmic term. However, whether $E(x,y)$ is small or large is not an appropriate condition for the existence and the strength of a causal link. Complex objects can have much information in common even though their distance is large. In order to obtain a measure that relates the amount of information that is disjoint for the two strings to the amount they share, Li et al. and Bennett et al. use the “normalized distance measure”

The intuitive meaning of $d_{s}(x,y)$ is obvious from its direct relation to mutual information, and $1-d(x,y)$ measures the fraction of the information of the more complex string that is shared with the other one. Bennett et al. propose to construct evolutionary histories of chain letters using such kinds of information distance measures. However, like in statistical causal inference, inferring adjacencies on the basis of strongest dependences is only possible for simple causal structures like trees. In the general case, non-adjacent nodes can share more information than adjacent ones when information is propagated via more than one path. Instead of constructing causal neighborhood relations by comparing information distances we will therefore use conditional mutual information.

In order to define its algorithmic version, we first observe that Definition 2 can be rewritten into the less concise form

This formula generalizes more naturally to the conditional analog :

Let $x,y,z$ be three strings. Then the conditional mutual algorithmic information of $x,y$, given $z$ is

As shown in (Remark II.3), the conditional mutual information also is symmetric up to a constant term:

Given three strings $x,y,z$, we call $x$ conditionally independent of $y$, given $z$ (denoted by x\mathchoice{\mathrel{\hbox to0.0pt{\displaystyle\perp\hss}\mkern 2.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{\textstyle\perp\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptstyle\perp\hss}\mkern 2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptscriptstyle\perp\hss}\mkern 2.0mu{\scriptscriptstyle\perp}}}y\,|z) if

In words: Given $z$, the additional knowledge of $y$ does not allow us a stronger compression of $x$. This remains true if we are given the Kolmogorov complexity of $y$, given $z$.

The theory developed below will describe laws where symbols like $x,y,z$ represent arbitrary strings. Then one can always think of sequences of strings of increasing complexity and statements like “the equation holds up to constant terms” are well-defined. We will then understand conditional independence in the sense of $I(x:y|z)\stackrel{{\scriptstyle+}}{{=}}0$. However, if we are talking about three fixed strings that represent objects in real-life, this does not make sense and the threshold for considering two strings dependent will heavily depend on the context. For this reason, we will not specify the symbol $\approx$ any further. This is the same arbitrariness as the cutoff rate for statistical dependence tests.

The definitions and lemmas presented so far were strongly motivated by the statistical analog. Now we want to focus on a theorem in that provides a mathematical relationship between algorithmic and statistical mutual information. First we rephrase the following theorem Theorem 7.3.1 of , showing that the Kolmogorov complexity of a random string is approximatively given by the entropy of the underlying probability distribution:

Let ${\bf x}=x_{1},x_{2},\cdots,x_{n}$ be a string whose symbols $x_{j}\in{\cal A}$ are drawn i.i.d. from a probability distribution $P(X)$ over the finite alphabet ${\cal A}$. Slightly overloading notation, set $P({\bf x}):=P(x_{1})\cdots P(x_{n})$. Let $H(.)$ denote the Shannon entropy of a probability distribution. Then there is a constant $c$ such that

where $E(.)$ is short hand for the expected value with respect to $P({\bf x})$. Hence

However, for our purpose, we need to see the relation between algorithmic and statistical mutual information. If ${\bf x}=x_{1},x_{2},\cdots,x_{n}$ and ${\bf y}=y_{1},y_{2},\cdots,y_{n}$ such that each pair $(x_{j},y_{j})$ is drawn i.i.d. from a joint distribution $P(X,Y)$, the theorem already shows that

This can be seen by writing statistical mutual information as

The above translations between entropy and algorithmic information refer to a particular setting and to special limits. The focus of this paper is mainly the situation where the above limits are not justified. Before we rephrase Theorem 5.3 in which provides insights into the general case, we recall that a function $f$ is called recursive if there is a program on a Turing machine that computes $f(x)$ from the input $x$, and halts on all possible inputs.

Given string-valued random variables $X,Y$ with a recursive probability mass function $P(x,y)$ over pairs $(x,y)$ of strings. We then have

where $K(P)$ is the length of the shortest prefix-free program that computes $P(x,y)$ from $(x,y)$.

We want to provide an intuition about various aspects of this theorem.

(1) If $I(X;Y)$ is large compared to $K(P)$ the expected algorithmic mutual information is dominated by the statistical mutual information.

(2) If $K(P)$ is no longer assumed to be small, statistical dependences do not necessarily ensure that the knowledge of $x$ allows us to compress $y$ further than without knowing $x$. It could be that the description of the statistical dependences requires more memory space than its knowledge would save.

(3) On the other hand, knowledge of $x$ could allow us to compress $y$ even in the case of a product measure on $x$ and $y$. Consider, for instance, the case that we have the point mass distribution on the pair $(x,y)$ with $x=y$. To describe a more sophisticated example generalizing this case we first have to introduce a family of product probability distributions on $\{0,1\}^{n}$ that we will need several times throughout the paper.

Let $P_{0},P_{1}$ be two probability distributions on $\{0,1\}$ and $c$ be a binary string of length $n$. Then

defines a distribution on $\{0,1\}^{n}$. We will later also need the following generalization: If $P_{00},P_{01},P_{10},P_{11}$ are four distributions on $\{0,1\}$, then

defines also a family of product measures on $\{0,1\}^{n}$ that is labeled by two strings.

Denote by ${\bf P}_{c}^{\otimes m}$ the $m$-fold copy of ${\bf P}_{c}$ from Definition 5. It describes a distribution on $\{0,1\}^{nm}$ assigning the probbaility ${\bf P}^{\otimes m}_{c}(x)$ to $x\in\{0,1\}^{nm}$. If

knowledge of $x$ in the typical case provides knowledge of $c$, provided $m$ is large enough. Then we can compress $y$ better than without knowing $x$ because we do not have to describe $c$ any more. Hence the algorithmic mutual information is large and the statistical mutual information is zero because $Q$ is by construction a product distribution. In other words, algorithmic dependences in a setting with i.i.d sampling can arise from statistical dependences and from algorithmic dependences between probability distributions.

2 Markov condition for algorithmic dependences among individual objects

Now we state the causal Markov condition for individual objects as a postulate that links algorithmic mutual dependences with causal structure:

Let $x_{1},\dots,x_{n}$ be $n$ strings representing descriptions of observations whose causal connections are formalized by a directed acyclic graph $G$ with $x_{1},\dots,x_{n}$ as nodes. Let $pa_{j}$ be the concatenation of all parents of $x_{j}$ and $nd_{j}$ the concatenation of all its non-descendants except $x_{j}$ itself. Then

As in Definition 4, the appropriate cut-off rate for rejecting $G$ when $I(x_{j}:nd_{j}|pa^{*}_{j})>0$ will not be specified here.

This formulation is a natural interpretation of Postulate 2 in terms of algorithmic independences. The only point that remains to be justified is why we condition on $pa_{j}^{*}$ instead of $pa_{j}$, i.e., why we are given the optimal joint compression of the parent strings. The main reason is that this turns out to yield nice statements on the equivalence of different Markov conditions (in analogy to Lemma 1). Since the differences between $I(x_{j}:nd_{j}|pa_{j})$ and $I(x_{j}:nd_{j}|pa_{j}^{*})$ can only be logarithmic in the string lengths this is because $K(x|y)-K(x|y^{*})=O(\log|y|)$, see we will not focus on this issue any further.

If we apply Postulate 5 to a trivial graph consisting of two unconnected nodes, we obtain the following statement.

If the mutual information $I(x:y)$ between two objects $x,y$ is significantly greater than zero they have some kind of common past.

Here, common past between two objects means that one has causally influenced the other or there is a third one influencing both. The statistical version of this principle is part of Reichenbach’s principle of the common cause stating that statistical dependences between random variables The original formulation considers actually dependences between events, i.e., binary variables. $X$ and $Y$ are always due at least one of the following three types of causal links: (1) $X$ is a cause of $Y$ or (2) vice versa or (3) there is a common cause $Z$. For objects, the term “common past” includes all three types of causal relations. For a text, for instance, it reads: similarities of two texts $x,y$ indicate that one author has been influenced by the other or that both have been influenced by a third one.

Before we construct a model of causality that makes it possible to prove the causal Markov condition we want to discuss some examples. If one discovers significant similarities in the genome of two sorts of animals one will try to explain the similarities by relatedness in the sense of evolution. Usually, one would, for instance, assume such a common history if one has identified long substrings that both animals have in common. However, the following scenario shows two observations that superficially look similar, but nevertheless we cannot infer a common past since their algorithmic complexity is low (implying that the algorithmic mutual information is low, too).

Assume two persons are instructed to write down a binary string of length $1000$ and both decide to write the same string $x=1100100100001111110...$. It seems straightforward to assume that the persons have communicated and agreed upon this choice. However, after observing that $x$ is just the binary representation of $\pi$, one can easily imagine that it was just a coincidence that both wrote the same sequence. In other words, the similarities are no longer significant after observing that they stem from a simple rule. This shows that the length of the pattern that is common to both observations, is not a reasonable criterion on whether the similarities are significant.

To understand the algorithmic causal Markov condition we will study its implications as well as its justification. In analogy to Lemma 1 we have

Given the strings $x_{1},\dots,x_{n}$ and a directed acyclic graph $G$. Then the following conditions are equivalent:

Recursive form: the joint complexity is given by the sum of complexities of each node, given the optimal compression of its parents:

Local Markov condition: Every node is independent of its non-descendants, given the optimal compression of its parents:

Below we will therefore no longer distinguish between the different versions and just refer to “the algorithmic Markov condition”. The intuitive meaning of eq. (6) is that the shortest description of all strings is given by describing how to generate every string from its direct causes. A similar kind of “modularity” of descriptions will also occur later in a different context when we consider description complexity of joint probability distributions.

For the proof of Theorem 3 we will need a Lemma that is an analogue of the observation that for any two random variables $X,Y$ the statistical mutual information satisfies $I(f(X);Y)\leq I(X;Y)$ for every measurable function $f$. The algorithmic analog is to consider two strings $x,y$ and one string $z$ that is derived from $x^{*}$ by a simple rule.

Let $x,y,z$ be three strings such that $K(z|x^{*})\stackrel{{\scriptstyle+}}{{=}}0$. Then

This lemma is a special case of Theorem II.7 in . We will also need the following result:

Note that $K(z|x^{*})\stackrel{{\scriptstyle+}}{{\geq}}K(z|x^{*},y)$ and $K(z|x^{*})\stackrel{{\scriptstyle+}}{{\geq}}K(z|x^{*},y*)$ is obvious but Lemma 7 is non-trivial because the star operation is jointly applied to $x$ and $y$.

Proof of Lemma 7: Clearly the string $x$ can be derived from $x,y$ by a program of length $O(1)$. Lemma 6 therefore implies

where $I(z:x,y)$ is shorthand for $I(z:(x,y))$. Hence

Then we obtain the statement by subtracting $K(z)$ and inverting the sign. $\Box$

The following lemma will only be used in Subsection 3.3. We state it here because it is closely related to the ones above.

The name “data processing inequality” is justified because the assumption x\mathchoice{\mathrel{\hbox to0.0pt{\displaystyle\perp\hss}\mkern 2.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{\textstyle\perp\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptstyle\perp\hss}\mkern 2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptscriptstyle\perp\hss}\mkern 2.0mu{\scriptscriptstyle\perp}}}y\,|z^{*} may arise from the typical data processing scenario where $y$ is obtained from $x$ via $z$.

where the second inequality holds because $K(z)+K(y|z^{*})$ can obviously be computed from the pair $(K(z),K(y|z^{*}))$ by an $O(1)$ program. The last equality uses, again, the equivalence of $z^{*}$ and $(z,K(z))$. Hence we obtain:

The first step is by Definition 2, the second one uses Lemma 7, the third step is a direct application of ineq. (7), the fourth one is due to Definition 3, and the last step is by assumption. $\Box$

Proof of Theorem 3: I $\Rightarrow$ III: Define a probability mass function $P$ on $(\{0,1\}^{*})^{\times n}$, i.e., the set of $n$-tuples of strings, as follows. Set

where $z_{j}$ is a normalization factor. In this context, it is important that the symbol $pa_{j}$ refers to conditioning on the $k$-tuple of strings $x_{i}$ that are parents of $x_{j}$ (in contrast to conditional complexities where we can interpret $K(.|pa^{*}_{j})$ equally well as conditioning on one string given by the concatenation of all those $x_{i}$). Note that Kraft’s inequality (see , Example 3.3.1) implies

for every $y$ entailing that the expression is indeed normalizable by $z_{j}\leq 1$. We have

i.e., $P$ is by construction recursive with respect to $G$. It is easy to see that $K(x_{1},\dots,x_{n})$ can be computed from $P$:

Remarkably, we can also compute Kolmogorov complexities of subsets of $\{x_{1},\dots,x_{n}\}$ from the corresponding marginal probabilities. We start by proving

where $\stackrel{{\scriptstyle\times}}{{=}}$ denotes equality up to a multiplicative constant. The equality follows from eq. (4) and the inequality is obtained by applying Kraft’s inequality to the conditional complexity $K(.|(x_{1},\dots,x_{n-1})^{*})$. On the other hand we have

since adding the $1$-bit string $x_{n}=0$ certainly can be performed by a program of length $O(1)$. Hence we have

Since the same argument holds for marginalizing over any other variable $x_{j}$ we conclude that

for every subset of strings of size $k$ with $k\leq n$. This follows by induction over $n-k$.

Now we can use the relation between marginal probabilities and Kolmogorov complexities to show that conditional complexities are also given by the corresponding conditional probabilities, i.e., for any two subsets $S,T\subset\{x_{1},\dots,x_{n}\}$ we have

Without loss of generality, set $S:=\{x_{1},\dots,x_{j}\}$ and $T:=\{x_{j+1},\dots,x_{k}\}$ for $j<k\leq n$. Using eqs. (4) and (13) we get

Let $S,T,R$ be three subsets of $\{x_{1},\dots,x_{n}\}$ such that $R$ d-separates $S$ and $T$. Then S\mathchoice{\mathrel{\hbox to0.0pt{\displaystyle\perp\hss}\mkern 2.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{\textstyle\perp\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptstyle\perp\hss}\mkern 2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptscriptstyle\perp\hss}\mkern 2.0mu{\scriptscriptstyle\perp}}}T\,|R with respect to $P$ because $P$ satisfies the recursion (9) (see Lemma 1) Since $P$ is, by construction, a discrete probability function, $P$ the density with respect to a product measure is directly given by the probability function itself.. Hence

This proves algorithmic independence of $S$ and $T$, given $R^{*}$ and thus I $\Rightarrow$ III.

To show that III $\Rightarrow$ II it suffices to recall that $nd_{j}$ and $x_{j}$ are d-separated by $pa_{j}$. Now we show II $\Rightarrow$ I in strong analogy to the proof for the statistical version of this statement in : Consider first a terminal node of $G$. Assume, without loss of generality, that it is $x_{n}$. Hence all strings $x_{1},\dots,x_{n-1}$ are non-descendants of $x_{n}$. We thus have $(nd_{n},pa_{n})\equiv(x_{1},\dots,x_{n-1})$ where $\equiv$ means that both strings coincide up to a permutation (on one side) and removing those strings that occur twice (on the other side). Due to eq. (4) we have

Using, again, the equivalence of $w^{*}\equiv(w,K(w))$ for any string $w$ we have

The second step follows from $K(nd_{n},pa_{n})=K(pa_{n})+K(nd_{n}|pa_{n}^{*})$. The inequality holds because $nd_{n},pa_{n},K(pa_{n})+K(nd_{n}|pa_{n}^{*})$ can be computed from $nd_{n},pa_{n}^{*},K(nd_{n}|pa^{*}_{n})$ via a program of length $O(1)$. The last step follows directly from the assumption x_{n}\mathchoice{\mathrel{\hbox to0.0pt{\displaystyle\perp\hss}\mkern 2.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{\textstyle\perp\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptstyle\perp\hss}\mkern 2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{\scriptscriptstyle\perp\hss}\mkern 2.0mu{\scriptscriptstyle\perp}}}nd_{n}\,|pa_{n}^{*}. Combining ineq. (15) with Lemma 7 yields

Then statement I follows by induction over $n$. $\Box$

To show that the algorithmic Markov condition can be derived from an algorithmic version of the functional model in Postulate 4 we introduce the following model of causal mechanisms.

Let $G$ be a DAG formalizing the causal structure among the strings $x_{1},\dots,x_{n}$. Then every $x_{j}$ is computed by a program $q_{j}$ with length $O(1)$ from its parents $pa_{j}$ and an additional input $n_{j}$. We write formally

meaning that the Turing machine computes $x_{j}$ from the input $pa_{j},n_{j}$ using the additional program $q_{j}$ and halts. The inputs $n_{j}$ are jointly independent in the sense

By defining new programs that contain $n_{j}$ we can, equivalently, drop the assumption that the programs $q_{j}$ are simple and assume that they are jointly independent instead.

We could also have assumed that $x_{j}$ is a function $f_{j}$ of all its parents, but our model is more general since the map defined by the input-output behavior of $q_{j}$ need not be a total function , i.e., the Turing machine simulating the process would not necessarily halt on all inputs $pa_{j},n_{j}$.

The idea to represent causal mechanisms by programs written for some universal Turing machine is basically in the spirit of various interpretations of the Church-Turing thesis. One formulation, given by Deutsch , states that every process taking place in the real world can be simulated by a Turing machine. Here we assume that the way different systems influence each other by physical signals can be simulated by computation processes that exchange messages of bit strings. Note, however, that sending quantum systems between the nodes could transmit a kind of information (“quantum information” ) that cannot be phrased in terms of bits. It is known that this enables completely new communication scenarios, e.g. quantum cryptography. The relevance of quantum information transfer for causal inference is not yet fully understood. It has, for instance, been shown that the violation of Bell’s inequality in quantum theory is also relevant for causal inference . This is because some causal inference rules between classical variables break down when the latent factors are represented by quantum states rather than being classical variables.

Note that mathematics also allows us to construct strings that are linked to each other in an uncomputable way. For instance, let $x$ be an arbitrary binary string and $y$ be defined by $y:=K(x)$. However, it is hard to believe that a real causal mechanism could create such kind of relations between objects given that one believes that real processes can always be simulated by algorithms. These remarks are intended to give sufficient motivation for our model.

Postulate 6 implies the algorithmic causal Markov condition:

Let $x_{1},\dots,x_{n}$ be generated by the model in Postulate 6. Then they satisfy the algorithmic Markov condition with respect to $G$.

It is trivial to construct examples where the causal Markov condition is violated if the programs $q_{j}$ are mutually dependent (for instance, the trivial graph with two nodes $x_{1},x_{2}$ and no edge would satisfy $I(x_{1}:x_{2})>0$ if the programs $q_{1},q_{2}$ computing $x_{1},x_{2}$ from an empty input are dependent).

The last sentence of Postulate 6 makes apparent that the mechanisms that generate causal relations are assumed to be independent. This is essential for the general philosophy of this paper. To see that such a mutual independence of mechanisms is a reasonable assumption we recall that the causal graph is meant to formalize all relevant causal links between the objects. If we observe, for instance, that two nodes are generated from their parents by the same complex rule we postulate another causal link between the nodes that explains the similarity of mechanisms. One could argue that this would be just the causal principle implying that similarities of the “machines” generating $x_{j}$ from $pa_{j}$ has to be explained by a causal relation, i.e., a common past of the machines. However, in the context of this paper, such an argument would be circular. We have argued that the causal principle is a special case of the Markov condition and derived the latter from the algorithmic model above. We will therefore consider the independence of mechanisms as a first principle.

3 Relative causality

This subsection explains why it is sensible to define algorithmic dependence and the existence or non-existence of causal links relative to some background information. To this end, we consider genetic sequences $s_{1},s_{2}$ of two persons that are not relatives. We certainly find high similarity that leads to a significant violation of $I(s_{1}:s_{2})=0$ due to the fact that both genes are taken from humans. However, given the background information “$s_{1}$ is a human genetic sequence”, $s_{1}$ can be further compressed. The same applies to $s_{2}$. Let $h$ be a code that is particularly adapted to the human genome in the sense that the expected conditional Kolmogorov complexity, given $h$, of a randomly chosen human genome is minimal. Then it would make sense to consider $I(s_{1}:s_{2}|h)>0$ as a hint for a relation that goes beyond the fact that both persons are human. In contrast, for the unconditional mutual information we expect $I(s_{1}:s_{2})\geq K(h)$. We will therefore infer some causal relation (here: common ancestors in the evolution) using the causal principle in Lemma 5 (cf. ).

The common properties between different and unrelated individuals of the same species can be screened off by providing the relevant background information. Given this causal background, we can detect further similarities in the genes by the conditional algorithmic mutual information and take them as an indicator for an additional causal relation that goes beyond the common evolutionary background. For this reason, every discussion on whether there exists a causal link between two objects (or individuals) requires a specification of the background information. In this sense, causality is a relative concept.

One may ask whether such a relativity of causality is also true for the statistical version of the causality principle, i.e., Reichenbach’s principle of the common cause. In the statistical version of the link between causality and dependence, the relevance of the background information is less obvious because it is evident that statistical methods are always applied to a given statistical ensemble. If we, for instance, ask whether there is a causal relation between the height and the income of a person without specifying whether we refer to people of a certain age, we observe the same relativity with respect to additionally specifying the “background information”, which is here given by referring to a specific ensemble.

In the following sections we will assume that the relevant background information has been specified and it has been clarified how to translate the relevant aspects of a real object into a binary string such that we can identify every object with its binary description.

Novel statistical inference rules from the algorithmic Markov condition

To describe the implications of the algorithmic Markov condition for statistical causal inference, we consider random variables $X$ and $Y$ where $X$ causally influences $Y$. We can think of $P(X)$ as describing a source $S$ that generates $x$-values and sends them to a “machine” $M$ that generates $y$-values according to $P(Y|X)$. Assume we observe that

Then we conclude that there must be a causal link between $S$ and $M$ that goes beyond transferring $x$-values from $S$ to $M$. This is because $P(X)$ and $P(Y|X)$ are inherent properties of $S$ and $M$, respectively which do not depend on the current value of $x$ that has been sent. Hence there must be a causal link that explains the similarities in the design of $S$ and $M$. Here we have assumed that we know that $X\rightarrow Y$ is the correct causal structure on the statistical level. Then we have to accept that a causal link on the level of the machine design is present.

If the causal structure on the statistical level is unknown, we would prefer causal hypotheses that explain the data without needing a causal connection on the higher level provided that they satisfy the statistical Markov condition. Given this principle, we thus will prefer causal graphs $G$ for which the Markov kernels $P(X_{j}|PA_{j})$ become algorithmically independent. This is equivalent to saying that the shortest description of $P(X_{1},\dots,X_{n})$ is given by concatenating the descriptions of the Markov kernels, a postulate that has already been formulated by Lemeire and Dirkx :

A causal hypothesis $G$ (i.e., a DAG) is only acceptable if the shortest description of the joint density $P$ is given by a concatenation of the shortest description of the Markov kernels, i.e.

If no such causal graph exists, we reject every possible DAG and assume that there is a causal relation of a different type, e.g., a latent common cause, selection bias, or a cyclic causal structure.

The sum on the right hand side of eq. (18) will be called the total complexity of the causal model $G$. Note that Postulate 7 implies that we have to reject every causal hypothesis for which the total complexity is not minimal because a model with shorter total complexity already provides a shorter description of the joint distribution. Inferring causal directions by minimizing this expression (or actually a computable modification) could also be interpreted in a Bayesian way if we consider $K(P(X_{j}|PA_{j}))$ as the negative log likelihood for the prior probability for having the conditional $P(X_{j}|PA_{j})$ (after appropriate normalization). However, Postulate 7 contains an idea that goes beyond known Bayesian approaches to causal discovery because it provides hints on the incompleteness of the class of models under consideration (in addition to providing rules for giving preference within the class).

Lemeire and Dirkx already show that the causal faithfulness principle (Postulate 3) follows from Postulate 7. Now we want to show that it also implies causal inference rules that go beyond the known ones.

To this end, we focus again on the example in Subsection 1.2 with a binary variable $X$ and a continuous variable $Y$. The hypothesis $X\rightarrow Y$ is not rejected on the basis of Postulate 7 because $I(P(X):P(Y|X))\stackrel{{\scriptstyle+}}{{=}}0$. For the equally weighted mixture of two Gaussians this already follows for the more general case $P(X=1)=p$ with $K(p)\gg 0$, this also follows if we assume that $p$ is algorithmically independent of the parameters that specify $P(Y|X)$ from $K(P(X))\stackrel{{\scriptstyle+}}{{=}}0$. On the other hand, $Y\rightarrow X$ violates Postulate 7. Elementary calculations show that the conditional $P(X|Y)$ is given by the sigmoid function

We observe that the same parameters $\sigma,\lambda,\mu$ that occur in $P(Y)$, also occur in $P(X|Y)$. This already shows that the two Markov kernels are algorithmically dependent. To be more explicit, we observe that $\mu$, $\lambda$, and $\sigma$ are required to specify $P(Y)$. To describe $P(X|Y)$, we need $\lambda/\sigma^{2}$ and $\mu$. Hence we have

where we have assumed that the strings $\mu,\lambda,\sigma$ are jointly independent. Note that the information that $P(Y)$ is a mixture of two Gaussians and that $P(X|Y)$ is a sigmoid counts as a constant because its description complexity does not depend on the parameters.

Therefore we reject the causal hypothesis $Y\rightarrow X$ due to Postulate 7. The interesting point is that we need not look at the alternative hypothesis $X\rightarrow Y$. In other words, we do not reject $Y\rightarrow X$ only because the converse direction leads to simpler expressions. We can reject it alone one the basis of observing algorithmic dependences between $P(Y)$ and $P(X|Y)$ making the causal model suspicious.

The fact that the set of mixtures of two Gaussians does not have six free parameters already shows that $P(X,Y)$ must be a more complex distribution than the one above. Fig. 3 shows an example of a joint distribution obtained for the “detuned” situation.

As already noted by , the independence of mechanisms is related to Pearl’s thoughts on the stability of causal statements: the causal mechanism $P(X_{j}|PA_{j})$ does not change if one changes the input distribution $P(PA_{j})$ by influencing the variables $PA_{j}$. The same conditional can therefore occur, under different background conditions, with different input distributions.

Postulate 7 naturally occurs in the probability-free version of the causal Markov condition. To explain this, assume we are given two strings ${\bf x}$ and ${\bf y}$ of length $n$ (describing two real-world observations) and noticed that ${\bf x}={\bf y}$. Now we consider two alternative scenarios:

(I) Assume that every pair $(x_{j},y_{j})$ of digits ($j=1,\dots,n$) has been independently drawn from the same joint distribution $P(X,Y)$ of the binary random variables $X$ and $Y$.

(II) Let ${\bf x}$ and ${\bf y}$ be single instances of string-valued random variables $X$ and $Y$.

The difference between (I) and (II) is crucial for statistical causal inference: In case (I), statistical independence is rejected with high confidence proving the existence of a causal link. In constrast, there is no evidence for statistical dependence in case (II) since the underlying joint distribution on $\{0,1\}^{n}\times\{0,1\}^{n}$ could, for instance, be the point mass on the pair $({\bf x},{\bf y})$, which is a product distribution, i.e.,

Hence, statistical causal inference would not infer a causal connection in case (II).

Algorithmic causal inference, on the other hand, infers a causal link in both cases because the equality ${\bf x}={\bf y}$ requires an explanation. The relevance of switching between (I) and (II) then consists merely in shifting the causal connection to another level: In the i.i.d setting, every $x_{j}$ must be causally linked to $y_{j}$. In case (II), there must be a connection between the two mechanisms that have generated the entire strings because $I(P(X):P(Y|X))=I(P(X):P(Y))\gg 0$. This can, for instance, be due to the fact that two machines emitting the same string were designed by the same engineer. A detailed discussion of the relevance of translating the i.i.d. assumption into the setting of algorithmic causal inference will be given in Subsection 3.2.

Examples with large probability spaces

In the preceding subsection we have ignored a serious problem with defining the Kolmogorov complexity of (conditional) probability distributions that even occurs in finite probability spaces. First of all the “true” probabilities may not be computable. For instance, a coin may produce “head” with probability $p$ where $p$ is some uncomputable number, i.e., $K(p)=\infty$. And even if it were some computable value $p$ with large $K(p)$ it would be quite artificial to call the probability distribution $(p,1-p)$ “complex” because $K(p)$ is high and “simple” if we have, for instance $p=1/\pi$. A more reasonable notion of complexity can be obtained by describing the probabilities only up to a certain accuracy $\epsilon$. If $\epsilon$ is not to small we obtain small complexity values for the distribution of a binary variable, and also low complexity for a distribution on a larger set that is $\epsilon$-close to the values of some simple analytical expression like a Gaussian distribution. There will still remain some unease about the concept of Kolmogorov complexity of “the true distribution”. We will subsequently develop a formalism that avoids this concept. However, Kolmogorov complexity of distributions is a useful idea to start with since it provides an intuitive understanding of the roots of the asymmetries between cause and effects that we will describe in Subsection 3.2.

Below, we will describe a gedankenexperiment with two random variables $X,Y$ linked by the causal structure $X\rightarrow Y$ where the total complexities of the causal models $X\rightarrow Y$ and $Y\rightarrow X$ both are well-defined and, in the generic case, different. First we will show that they can at most differ by factor two.

For every joint distribution $P(X,Y)$ we have

Proof: Since marginals and conditionals both can be computed from $P(X,Y)$ we have

Then the statement follows because $P(X,Y)$ can be computed from $P(X)$ and $P(Y|X)$. $\square$

To construct examples where the bound in Lemma 9 is attained we first introduce a method to construct conditionals with well-defined complexity:

Let $M_{0},M_{1}$ be two stochastic matrices that specify transition probabilities from $\{0,1\}$ to $\{0,1\}$. Then

defines transition probabilities from $\{0,1\}^{n}$ to $\{0,1\}^{n}$.

We also introduce the same construction for double indices: Let $M_{00},M_{01},M_{10},M_{11}$ be stochastic matrices describing transition probabilities from $\{0,1\}$ to $\{0,1\}$. Let $c,d\in\{0,1\}^{n}$ be two strings. Then

defines a transition matrix from $\{0,1\}^{n}$ to $\{0,1\}^{n}$. If the matrices $M_{j}$ or $M_{ij}$ denote joint distributions on $\{0,1\}\times\{0,1\}$ the objects ${\bf M}_{c}$ and ${\bf M}_{c,d}$ define joint distributions on $\{0,1\}^{n}\times\{0,1\}^{n}$ in a canonical way.

Let $X,Y$ be variables whose values are the set of strings in $\{0,1\}^{n}$. Define distributions $P_{0},P_{1}$ on $\{0,1\}$ and stochastic matrices $A_{0},A_{1}$ describing transition probabilities from $\{0,1\}$ to $\{0,1\}$. Then a string $c\in\{0,1\}^{n}$ defines a distribution $P(X):={\bf P}_{c}$ (using Definition 5) that has well-defined Kolmogorov complexity $K(c)$ if the description complexity of $P_{0}$ and $P_{1}$ is neglected. Furthermore, we set $P(Y|X):={\bf A}_{d}$ as in Definition 6, where we have used the canonical identification between stochastic matrices and conditional probabilities and $d\in\{0,1\}^{n}$ denotes some randomly chosen string. Let $R_{ij}$ denote the joint distribution on $\{0,1\}\times\{0,1\}$ induced by the marginal $P_{i}$ on the first component and the conditional $A_{j}$ for the second, given the first. Denote the corresponding marginal dsitribution on the right component by $Q_{ij}$, i.e.,

and let $B_{ij}$ be the stochastic matrix that describes the conditional probability for the first component, given the second.

Using these notations and the ones in Definition 6, we obtain

It is noteworthy that $P(Y)$ and $P(X|Y)$ are labeled by both strings while $P(X)$ and $P(Y|X)$ are described by only one string each. This already suggests that the latter are more complex in the generic case.

Now we compare the sum $K(P(X))+K(P(Y|X))$ to $K(P(Y))+K(P(X|Y))$ for the case$K(c)\stackrel{{\scriptstyle+}}{{=}}K(d)\stackrel{{\scriptstyle+}}{{=}}n$. We assume that $P_{i}$ and $A_{j}$ are computable and their complexity is counted as $O(1)$ because it does not depend on $n$. Nevertheless, we assume that $P_{i}$ and $A_{j}$ are “generic” in the following sense: All marginals $Q_{ij}$ and conditionals $B_{ij}$ are different whenever $P_{0}\neq P_{1}$ and $A_{0}\neq A_{1}$. If we impose one of the conditions $P_{0}=P_{1}$ and $A_{0}=A_{1}$ or both, we assume that only those marginals $Q_{ij}$ and conditionals $B_{ij}$ coincide for which the equality follows from the conditions imposed. Consider the following cases:

Case 1: $P_{0}=P_{1}$, $A_{0}=A_{1}$. Then all the complexities vanish because the joint distribution does not depend on the strings $c$ and $d$.

Case 2: $P_{0}\neq P_{1}$, $A_{0}=A_{1}$. Then the digits of $c$ are relevant, but the digits of $d$ are not. Those marginals and conditionals in table (20) that formally depend on $c$ and $d$, as well as those that depend on $c$, have complexity $n$. Those depending on $d$ have complexity $0$.

Case 3: $P_{0}=P_{1}$, $A_{0}\neq A_{1}$. Only the dependence on $d$ contributes to the complexity. This implies

Case 4: $P_{0}\neq P_{1}$ and $A_{0}\neq A_{1}$. Every formal dependence of the conditionals and marginals on $c$ and $d$ in table (20) is a proper dependence. Hence we obtain

The general principle of the above example is very simple. Given that $P(X)$ is taken from a model class that consists of $N$ different elements and $P(Y|X)$ is taken from a class with $M$ different elements. Then the class of possible $P(Y)$ and the class of possible $P(X|Y)$ both can contain $N\cdot M$ elements. If the simplicity of a model is quantified in terms of the size of the class it is taken from (within a hierarchy of more and more complex models), the statement that $P(Y)$ and $P(X|Y)$ are typically complex is just based on this simple counting argument.

Detecting common causes via dependent Markov kernels

The following model shows that latent common causes can yield joint distributions whose Kolmogorov complexity is smaller than $K(P(X))+K(P(Y|X))$ and $K(P(Y))+K(X|Y))$. Let $X,Y,Z$ have values in $\{0,1\}^{n}$ and let $P(Z):=\delta_{c}$ be the point mass on some random string $c\in\{0,1\}^{n}$. Let $P(X|Z)$ and $P(Y|Z)$ both be given by the stochastic matrix $A\otimes A\otimes\cdots\otimes A$. Let $P_{0}\neq P_{1}$ be the probability vectors given by the columns of $A$. Then

with ${\bf P}_{c}$ as in Definition 5. Since $P(Z)$ is supported by the singleton set $\{c\}$, we have $P(X|Y)=P(X)$ and $P(Y|X)=P(Y)$. Thus

By observing that there is a third variable $Z$ such that

we thus have obtained a hint that the latent model is the more appropriate causal hypothesis.

Analysis of the required sample size

The following arguments show that the above complexities of the Markov kernels become relevant already for moderate sample size. Readers who are not interested in technical details may skip the remaining part of the subsection.

Consider first the sampling required to estimate $c$ by drawing i.i.d. from ${\bf P}_{c}$ as in Definition 5. By counting the number of symbols $1$ that occur at position $j$ we can guess whether $c_{j}$ is $0$ or $1$ by choosing the distribution for which the relative frequency is closer to the corresponding probability. To bound the error probabilities from above set

Then the probability $q$ that the relative frequency deviates by more than $\mu/2$ decreases exponentially in the number of copies, i.e., $q\leq e^{-\mu m\alpha}$ where $\alpha$ is an appropriate constant. The probability to have no error for any digit is then bounded from below by $(1-e^{-\mu m\alpha})^{n}$. We want to increase $m$ such that the error probability tends to zero. To this end, choose $m$ such that $e^{-\mu m\alpha}\leq 1/n^{2}$, i.e., $m\geq\ln n^{2}/(\mu\alpha)$. Hence

The required sample size thus grows only logarithmically in $n$. In the same way, one shows that the sample size needed to distinguish between different conditionals $P(Y|X)={\bf A}_{c}$ increases only with the logarithm of $n$ provided that $P(X)$ is a strictly positive product distribution on $\{0,1\}^{n}$.

2 Resolving statistical ensembles into individual observations

The assumption of independent identically distributed random variables is one of the cornerstones of standard statistical reasoning. In this section we show that the independence assumption in a typical statistical sample is often due to prior knowledge on causal relations among single objects which can nicely represented by a DAG. We will see that the algorithmic causal Markov condition then leads to non-trivial implications.

Assume we describe a biased coin toss, $m$ times repeated, and obtain the binary string $x_{1},\dots,x_{m}$ as result. This is certainly one of the scenarios where the i.i.d. assumption is well justified because if we do not believe that the coin changes or that the result of one coin toss influences the other ones. The only relation between the coin tosses is that they refer to the same coin. We will thus draw a DAG representing the relevant causal relations for the scenario where $C$ (the coin) is the common cause of all $x_{j}$ (see fig. 4).

Given the relevant information on $C$ (i.e., given the probability $p$ for “head”), we have conditional algorithmic independence between the $x_{j}$ when applying the Markov condition to this causal graph. This is consistent with the following Bayesian interpretation: if we define a non-trivial prior on the possible values of $p$, the individual observations are statistically dependent when marginalizing over the prior, but knowing $p$ renders them independent. However, there are two problems: (1) it does not make sense to consider algorithmic mutual information among binary strings of length $1$. (2) Our theory developed so far (Theorems 3 and 4) considered the number of strings (which is $m+1$ here) as constant and thus even the complexity of $x_{1},\dots,x_{m}$ is considered as $O(1)$. To solve this problem, we define a new structure with three nodes as follows. For some arbitrary $k<m$ set ${\bf x}^{1}:=x_{1},\dots,x_{k}$ and ${\bf x}^{2}:=x_{k+1},\dots,x_{m}$. Then $C$ is the common cause of ${\bf x}^{1}$ and ${\bf x}^{2}$ and $I({\bf x}^{1};{\bf x}^{2}|C)=0$ because every similarity between ${\bf x}^{1}$ and ${\bf x}^{2}$ is due to their common source (note that the information that the strings ${\bf x}^{j}$ have been obtained by combining $k$ and $n-k$ results, respectively, is here implicitly considered as background information in the sense of relative causality in Subsection 2.3). We will later discuss examples where a source generates symbols from a larger probability space. Then every $x_{j}$ is a string and it is important to keep in mind the “format information”, i.e., the information how to read the concatenation $x_{1},x_{2},\cdots,x_{k}$ as a sample of $m$ strings. This format information will always be considered as background, too.

Of course, we may also consider partitions into more than two substrings keeping in mind that their number is considered as $O(1)$. When we consider causal relations between short strings we will thus always apply the algorithmic causal Markov condition to groups of strings rather than applying it to the “small objects” itself. The DAG that formalizes the causal relations between instances or groups of instances of a statistical ensemble and the source that determines the statistics in the above sense will be called the “resolution of statistical ensembles into individual observations”.

The resolution gets more interesting if we consider causal relations between two random variables $X$ and $Y$. Consider the following scenario where $X$ is the cause of $Y$. Let $S$ be a source generating $x$-values $x_{1},\dots,x_{m}$ according to a fixed probability distribution $P(X)$. Let $M$ be a machine that receives these values as inputs and generates $y$-values $y_{1},\dots,y_{m}$ according to the conditional $P(Y|X)$. Fig 5 (left) shows the causal graph for $m=4$.

In analogy to the procedure above, we divide the string ${\bf x}:=x_{1},\dots,x_{m}$ into ${\bf x}^{1}:=x_{1},\dots,x_{k}$ and ${\bf x}^{2}:=x_{k+1},\dots,x_{m}$ and use the same grouping for the $y$-values. We then draw the causal graph in fig. 5 (right) showing causal relations between ${\bf x}^{1},{\bf x}^{2},{\bf y}^{1},{\bf y}^{2},S,M$. Now we assume that $P(X)$ and $P(Y|X)$ are not known, i.e., we don’t have access to the relevant properties of $S$ and $M$. Thus we have to consider $S$ and $M$ as “hidden objects” (in analogy to hidden variables in the statistical setting). Therefore we have to apply the Markov condition to the causal structure in such a way that only the observed objects ${\bf x}^{1},{\bf x}^{2},{\bf y}^{1},{\bf y}^{2}$ occur. One checks easily that ${\bf x}^{2}$ d-separates ${\bf x}^{1}$ and ${\bf y}^{2}$ and ${\bf x}^{1}$ d-separates ${\bf x}^{2}$ and ${\bf y}^{1}$. Exhaustive search over all possible triples of subsets of ${\bf x}^{1},{\bf x}^{2},{\bf y}^{1},{\bf y}^{2}$ shows that these are the only non-trivial d-separation conditions. We conclude

The most remarkable property of eq. (21) is that it is asymmetric with respect to exchanging the roles of $X$ and $Y$ since, for instance, $I({\bf y}^{1};{\bf x}^{2}|{\bf y}^{2})\stackrel{{\scriptstyle+}}{{=}}0$ can be violated. Intuitively, the reason is that given ${\bf y}^{2}$, the knowledge of ${\bf x}^{2}$ provides better insights into the properties of $S$ and $M$ than knowledge of ${\bf x}^{1}$ would do, which can be an advantage when describing ${\bf y}^{1}$. The following example shows that this asymmetry can even be relevant for sample size $m=2$ provided that the probability space is large.

which correctly lets us prefer the causal direction $X\rightarrow Y$ because these dependences violate the global algorithmic Markov condition in Theorem 3 when applied to a hypothetical graph where ${\bf y}^{1}$ and ${\bf y}^{2}$ are the outputs of the source and ${\bf x}^{1}$ and ${\bf x}^{2}$ are the outputs of a machine that has received ${\bf y}^{1}$ and ${\bf y}^{2}$.

Even though the condition in eq. (21) does not explicitly contain the notion of complexities of Markov kernels it is closely related to the algorithmic independence of Markov kernels. To explain this, assume we would generate algorithmic dependences between $S$ and $M$ by adding an arrow $S\rightarrow M$ or $S\leftarrow M$ or by adding a common cause. Then ${\bf x}^{2}$ would no longer d-separate ${\bf x}^{1}$ from ${\bf y}^{2}$. The possible violation of eq. (21) could then be an observable result of the algorithmic dependences between the hidden objects $S$ and $M$ (and their statistical properties $P(X)$ and $P(Y|X)$, respectively).

3 Conditional density estimation on subsamples

Now we develop an inference rule that is even closer to the idea of checking algorithmic dependences of Markov kernels than condition (21), but still avoids the notion of Kolmogorov complexity of the “true” conditional distributions by using finite sample estimates instead. Before we explain the idea we mention two simpler approaches for doing so and describe their potential problems. It would be straightforward to apply Postulate 7 to the finite sample estimates of the conditionals. In particular, minimum description length (MDL) approaches appear promising from the theoretical point of view due to their close relation to Kolmogorov complexity. We rephrase the minimum complexity estimator described by Barron and Cover : Given a string-valued random variable $X$ and a sample $x_{1},\dots,x_{m}$ drawn from $P(X)$, set

where $Q$ runs over all probability densities on the probability space under consideration. If the data is sampled from a computable distribution, then $\hat{P}_{m}(X)$ converges in probability to $P(X)$ . Let us define a similar estimator $\hat{P}_{m}(Y|X)$ for the conditional density $P(Y|X)$. Could we reject the causal hypothesis $X\rightarrow Y$ after observing that $\hat{P}_{m}(X)$ and $\hat{P}_{m}(Y|X)$ are mutually dependent? In the context of the true probabilities, we have argued that $P(X)$ and $P(Y|X)$ represent independent mechanisms. However, for the estimators we do not see a justification for independence because the relative frequencies of the $x$-values influence the estimation of $\hat{P}_{m}(X)$ and $\hat{P}_{m}(Y|X)$. This counter-argument becomes irrelevant only if the sample size is such that the complexities of the estimators coincide with the complexities of the true distributions. If we assume that the latter are typically uncomputable (because generic real numbers are uncomputable) this sample size will never be attained.

The general idea of MDL also suggests the following causal inference principle: If we are given the data points $(x_{j},y_{j})$ with $j=1,\dots,m$, consider the MDL estimators $\hat{P}_{m}(X)$ and $\hat{P}_{m}(Y|X)$. They define a joint distribution that we denote by $\hat{P}_{X\rightarrow Y}(X,Y)$ (where we have dropped $m$ for convenience). The total description length

measures the complexity of the probabilistic model plus the complexity of the data, given the model. Then we compare $C_{X\rightarrow Y}$ to $C_{Y\rightarrow X}$ (defined correspondingly) and prefer the causal direction with the smaller value. However, it is not clear whether this kind of reasoning can be derived from the algorithmic Markov condition.

For this reason, we construct an inference rule that uses estimators in a more sophisticated way and whose justification is directly based on applying the algorithmic Markov condition to the resolution of ensembles. The idea of our strategy is that we do not use the full data set to estimate $P(Y|X)$. Instead, we apply the estimator to a subsample of $(x,y)$ pairs that no longer carries significant information about the relative frequencies of $x$-values in the full data set. As we will see below, this leads to algorithmically independent finite sample estimators for the Markov kernels if the causal hypothesis is correct.

Let $X\rightarrow Y$ be the causal structure that generated the data $({\bf x},{\bf y})$, with ${\bf x}:=x_{1},\dots,x_{m}$ and ${\bf y}:=y_{1},\dots,y_{m}$ after $m$-fold i.i.d. sampling from $P(X,Y)$. The resolution of the ensemble is the causal graph in fig. 7, left.