## 1. Introduction



The *Common Cause Principle* (RCCP) was introduced by Hans
Reichenbach, in *The Direction of Time*, which was published
posthumously in 1956. The principle posits a connection between causal
structure and probabilistic correlations between events. After
presenting the principle in
 Section 2,
 we will provide some historical background in
 Section 3.
 The following two sections present some illustrations and alleged
counterexamples to RCCP.
 Section 6
 discusses Reichenbach's *fork asymmetry*, a (putative)
temporal asymmetry in macrostatistical patterns that he associated
with RCCP.
 Section 7
 presents the *Causal Markov Condition*, which plays a central
role in causal modeling methods. The Causal Markov Condition is often
seen as a modern successor of RCCP, and its relationship to RCCP will
be examined.
 Section 8
 will develop RCCP in a formal setting that is suitable for examining
the status of RCCP in quantum mechanics.
 Section 9
 and
 Section 10
 will then consider whether RCCP is compatible with quantum
physics.


## 2. Reichenbach's Common Cause Principle



Let *A* and *B* be events. A storm, a person getting sick, a
soccer player scoring a goal, and a scientific measurement yielding a
specific result are all examples of events. Assume we can meaningfully
assign probabilities to these events occurring. Reichenbach himself
developed a sophisticated frequency interpretation of probability
(Reichenbach 1949), but we will assume only that some kind of
objective probability can be meaningfully applied. Suppose that events
*A* and *B* are positively probabilistically correlated:





\[\tag{1} \label{introcorr}
p(A\cap B)>p(A)p(B).\]



That is, the probability that *both* *A* and *B*
occur is greater than the product of the individual probabilities.
Reichenbach's Common Cause Principle says that when such a
probabilistic correlation between *A* and *B* exists, this
is because one of the following causal relations exists: *A* is a
cause of *B*; *B* is a cause of *A*; or *A* and
*B* are both caused by a third factor, *C*. In the last
case, the common cause *C* occurs prior to *A* and *B*,
and must satisfy the following four independent conditions: 




\[
\begin{align}
\tag{2} p(A\cap B|C) &= p(A|C)p(B|C) \label{off1}\\
\tag{3} p(A\cap B|\overline{C})&= p(A|\overline{C})p(B|\overline{C}) \label{off2}\\
\tag{4} p(A|C)&> p(A|\overline{C}) \label{nagy1}\\
\tag{5} p(B|C)&> p(B|\overline{C})\label{nagy2}
\end{align}\]



where 

 

\[
p(X|Y)\doteq\frac{p(X\cap Y)}{p(Y)}
\]


denotes the conditional probability of *X* on condition *Y*,
\(\overline{C}\) denotes the absence of event *C* (the negation
of the proposition that *C* happens) and it is assumed that
neither *C* nor \(\overline{C}\) has probability zero. Line (2)
says that *A* and *B* are *conditionally
independent*, given *C*. In Reichenbach's terminology,
*C* screens *A* off from *B*. Line (3) says that
\(\overline{C}\) also screens *A* off from *B*. Lines (4)
and (5) say that *A* and *B* are more probable, conditional
on *C*, than conditional on the absence of *C*. These
inequalities are natural consequences of *C* being a cause of
*A* and of *B*. Together, conditions (2) through (5)
mathematically entail
 (1).
 The common cause can thus be understood to explain the correlation in
(1). The probability relations described in lines (1) through (3)
exhibit a version of *Simpson's Paradox*. For more about
this probabilistic phenomenon, see the entry on
 Simpson's Paradox.



It will often be helpful to represent such a common cause structure
diagrammatically, as in
 Figure 1.
 Arrows indicate causal relationships, and the vertical dimension
represents time (with later times appearing higher up).



![on the left the word 'time' with an arrow pointing up; on the right the letter C with an arrow pointing from it up and to the left to the letter A and another arrow pointing from C up and to the right to the letter B](figure1.svg)

Figure 1: A common cause structure.
*C* occurs earlier than *A* and *B*, and *C* is a
cause of *A* and *B*.




RCCP says that probabilistic correlations are ultimately derived from
causal relationships. That is, if \(p(A\cap B)>p(A)p(B)\), that is
either because one of these events causes the other, or else the
inequality can be derived from other inequalities \(p(A|C) >
p(A|\overline{C})\) and \(p(B|C) > p(B|\overline{C})\) where
*C* is a cause of *A* and *B*. The principle is
significant because it posits a connection between causal structure
and probabilistic correlations that licenses inferences to causal
relationships from empirically observable correlations.



As formulated above, RCCP assumes that there are only two possible
states of the common cause: *C* and \(\overline{C}\). That is,
*C* is a binary event that is either present or absent. A natural
extension of RCCP would be to allow the common cause to be a random
variable *Z* with many possible values \(z\_1,\dots,z\_n\). In this
case we would expect that \(Z = z\_i\) screens off *A* from
*B* for \(i=1,\dots,n\). However, this does not imply that we can
divide the values of *Z* into two sets \(\mathbf{S}\) and
\(\mathbf{S^\prime}\) such that \(Z\in \mathbf{S}\) (corresponding to
*C*) and \(Z \in \mathbf{S^\prime}\) (corresponding to
\(\overline{C}\)) each screen off *A* and *B*. More
generally, we would not expect coarsenings of *Z* to yield events
that screen off *A* and *B*.



One corollary of this generalization is that if *A* and *B*
have two distinct binary common causes *C* and *D*, we would
expect each of \(CD, C\overline{D}, \overline{C}D, \overline{CD}\) to
screen off *A* and *B*, but would not expect *C* and
*D* to screen off *A* and *B*. For example, even if we
condition on the common cause *C*, we would expect *A* and
*B* to be correlated because of the further common cause
*D*.



We can further generalize to allow the correlated effects to be
random variables instead of binary events. The fully general version
of RCCP then reads as follows:


* Suppose *X* and *Y* are random variables that are
correlated; i.e., there exist values \(x\_i\) and \(y\_j\) such that


 

\[p(X=x\_i \cap Y=y\_j) \neq p(X=x\_i)p(Y=y\_j).\]


Then there exists a set of variables \(Z\_1,\dots,Z\_m\), such that each
variable is a cause of *X* and *Y*, and

 

\[\begin{align}
p(X=x\_i \cap Y=y\_j |& Z\_1 = z\_{k\_1},\dots,Z\_m = z\_{k\_m}) =\\
&p(X=x\_i | Z\_1 = z\_{k\_1},\dots,Z\_m = z\_{k\_m})\\
&\times p(Y=y\_j | Z\_1 = z\_{k\_1},\dots,Z\_m = z\_{k\_m})
\end{align}\]


for all \(i,j,k\_1,\dots,k\_m\)



With this generalization of RCCP, it is trickier to formulate analogs
for conditions 4 and 5 that capture in probabilistic terms the idea
that each variable \(Z\_k\) is a cause of both *X* and *Y*.
In addition, this set of conditions will not strictly imply that
*X* and *Y* are probabilistically correlated. We will return
to these issues in
 Section 7.


## 3. Historical Background



*The Direction of Time* (Reichenbach 1956) was unfinished at
the time of Reichenbach's death in 1953. The manuscript was
edited by his wife Maria Reichenbach, and published posthumously in
1956. The book was concerned with temporally asymmetric
phenomena--phenomena that we associate with the distinction
between the past and the future. It included major sections on the
role of time in classical physics, thermodynamics, statistical
mechanics, and quantum mechanics. It also included a section on
temporal asymmetries in macrostatistics. Reichenbach had plans to
include a final section on the subjective experience of time.



The book makes a number of important and original contributions. It
contains a detailed investigation into the philosophical foundations
of statistical mechanics, especially examining the status of the
second law of thermodynamics, which states that the entropy of a
closed system can increase, but never decrease. Reichenbach explored
connections between statistical mechanics and the new field of
information theory--Shannon and Weaver's classic book
(1949) on the topic had been published just a few years earlier. In
addition to the asymmetry of entropy increase described by the second
law of thermodynamics, the book examined other temporally asymmetric
phenomena. One of these was the asymmetry of records: We have detailed
records of past events, including human memories and human made
documents such as records and history books, but also including
natural phenomena such as fossils, tree rings, and geological strata.
These records provide us with a rich source of information about the
past. We have no comparable source of information about the future,
although we can reliably predict certain events, such as solar
eclipses.



A third important temporal asymmetry is the asymmetry of cause and
effect. Causes invariably precede their effects in time. Reichenbach
attempted to analyze causal direction in terms of macrostatistics or
probability. He appealed to his notion of screening-off to define a
relation of causal betweenness, and also to define causal direction.
It is in this latter context that the RCCP was introduced. The
connection between RCCP and causal direction will be discussed in
 Section 6
 below. Reichenbach also formulated a probabilistic theory of
causation, and explored the connections between these new ideas and
the *mark method* for distinguishing causal direction, which he
had proposed much earlier (Reichenbach 1958).



RCCP connects with a number of threads in Reichenbach's thought.
Reichenbach had developed and defended a frequency interpretation of
probability (Reichenbach 1949), as well as a thoroughgoing
probabilistic epistemology (Reichenbach 1938) He had also explored the
connection between probability and causation in earlier works
(Reichenbach 1925 [1978], 1930 [1978]). *The Direction of Time*
takes the project further in developing a probabilistic metaphysics.
Reichenbach had also explored the connection between causation and the
direction of time in Reichenbach (1925 [1978]) and Reichenbach (1958).
The latter work developed a causal theory of time in the context of
relativity theory.


## 4. Illustrations




**Example 1.** The barometer and the storm (Jeffrey
1969).



![The letter A with an arrow pointing from it up and to the left to the letter B and another arrow pointing from A up and to the right to the letter S](figure2.svg)

Figure 2: The barometer and the storm.
\(A =\) drop in atmospheric pressure; \(B =\) drop in mercury level in
barometer; \(S =\) storm.




A drop in the level of mercury in a barometer is frequently followed
by a storm. Call these events *B* and *S*, respectively.
Since storms in general are not so frequent, these events are
probabilistically correlated: \(p(B \cap S) > p(B)p(S)\). The
behavior of one barometer doesn't affect the weather, so
*B* is not a cause of *S*; rather, *B* and *S*
have a common cause: a drop in atmospheric pressure *A* (see
 Figure 2).
 *A* increases the probability of both *B* and *S*:


 

\[p(B|A) > p(B|\overline{A}) \textrm{ and }p(S|A) > p(S|\overline{A}). \]


Moreover, *A* will screen off *B* from *S*:



 \[p(B \cap S|A) = p(B|A)p(S|A)\] 


and 

 

\[p(B \cap S|\overline{A}) = p(B|\overline{A})p(S|\overline{A}).\]


For example, if the atmospheric pressure drops, but the column of
mercury in the barometer does not drop because the barometer is
malfunctioning, the probability of a storm is the same as it would be
if the barometer were functioning properly.





**Example 2.** *The theatre troupe* (Reichenbach
1956).



A small theatre troupe travels around the country putting on
performances. Occasionally, the leading man becomes seriously
ill--call this event *M*--and an understudy must take
his place. The same thing sometimes happens to the leading
lady--*L*. Although both events are rare, they tend to occur
together: \(p(L \cap M) > p(L)p(M)\). The reason is that the actors
usually eat together at the same restaurants, where they occasionally
share tainted food--*T*. See
 Figure 3.



![The letter T with an arrow pointing from it up and to the left to the letter L and another arrow pointing from T up and to the right to the letter M](figure3.svg)

Figure 3: The theatre troupe. \(T =\)
food tainted at restaurant patronized by actors; \(L =\) leading lady
gets sick; \(M =\) leading man gets sick.




Suppose the probabilities are as follows:

 

\[\begin{align}
p(T) & = .1\\
p(L|T) = p(M|T) & = .8\\
p(L|\overline{T}) = p(M|\overline{T}) & = .1\\
\end{align}\]


Then we can compute:

 

\[\begin{align}
p(L \cap M|T) = p(L|T)p(M|T) &= .64\\
p(L \cap M|\overline{T}) = p(L|\overline{T})p(M|\overline{T}) &= .01\\
\end{align}\]


These calculations make use of the fact that *T* and
\(\overline{T}\) screen *L* off from *M*. We can also
compute:

 

\[\begin{align}
p(L) = p(L|T)p(T) + p(L|\overline{T})p(\overline{T}) & = .17 = p(M)\\
p(L \cap M) = p(L \cap M|T)p(T) + p(L \cap M|\overline{T})p(\overline{T}) & = .073\\
\end{align}\]


Thus

 

\[.073 = p(L \cap M) > p(L)p(M) = .17^{2} = .0289.\]


*L* and *M* are probabilistically correlated.



Suppose that, on a given night, both the leading man and the leading
lady are seriously ill. Can we infer that they ate tainted food? From
the probabilities above, we can compute

 

\[p(T|L \cap M) = \frac{.064}{.073} \cong .877.\]


While it is probable that they ate tainted food, it is by no means a
certainty. This example shows that the Common Cause Principle, by
itself, does not license *token-level* causal inferences. That
is, it does *not* tell us that when two effects *A* and
*B* occur on a particular occasion, then their common cause
*C* also occurred on this occasion. Instead, the Common Cause
Principle licenses the inference from a probabilistic correlation to
the existence of a *type-level* common cause.





**Example 3.** *Language descent.*



Common words in English that start with the letter 'F'
often have counterparts in Spanish that start with the letter
'P': 'foot'/'pie',
'fish'/'pez',
'father'/'padre', etc. One could quantify this
by looking at canonical wordlists from both languages, and seeing how
often words begin with 'F' in English and with
'P' in Spanish. By treating the relative frequencies in
this list as probabilities, one would discover that there is indeed a
probabilistic correlation between the English word-initial
'F' and Spanish word-initial 'P'.



The explanation for this correlation is that English and Spanish are
descended from a common language, called *Proto-Indo-European*.
Some, but not all, words in Proto-Indo-European began with a consonant
we can label [P/F], which evolved into 'P' in Romance
languages (including Spanish), and 'F' in Germanic
languages (including English). (Note that it is the phonetic
pronunciation, rather than the spelling that is of interest here; for
example, many German words whose spelling begins with 'V'
are counted for this purpose.) In the separate lineages leading from
Proto-Indo-European to English and to Spanish, Proto-Indo-European
roots were retained for some words, but replaced for others. The roots
were retained often enough that the correlation can still be detected.
Moreover, the two lineages evolved more or less independently after
splitting from Proto-Indo-European. Note that the common cause is not
the Proto-Indo-European language as a whole, and the effects are not
English and Spanish. It would make little sense to assign
probabilities to these languages (let alone a joint probability to
English and Spanish). Rather, the common cause is an initial consonant
[P/F] in Proto-Indo-European; the effects are initial 'F'
in English and initial 'P' in Spanish. It makes sense to
assign probabilities to these sounds, since we can count the frequency
with which words in these languages start with these sounds. See
 Figure 4.



![The phrase 'PIE[P/F]' with an arrow pointing from it up and to the left to the phrase 'English F' and another arrow pointing up and to the right to the phrase 'Spanish P'](figure4.svg)

Figure 4: The descent of words with
initial consonant 'F' in English and initial consonant
'P' in Spanish, from words with initial consonant [P/F] in
Proto-Indo-European.




Note that the evidence of common descent between the two languages is
the *correlation* between the two sounds, rather than any
phonetic similarity between them. For example, there is a recognized
correlation between the Latin 'du' and Armenian
'erk', as in 'duo'/'erku'
('two'). Despite the differences in these sounds, the
correlation is evidence of common descent (both are also Indo-European
languages).



For further discussion of this example, see Hitchcock (1998).





**Example 4.** *Fried fish.*



Both the British and the Japanese eat battered seafood that has been
deep-fried in oil: the British in the form of fish and chips, and the
Japanese in the form of tempura. The technique of battering and
deep-frying seafood seems to have originated with Moors in the Iberian
Peninsula in the thirteenth century, where the dish was called
*mu'affar*. It spread to the Jewish and Christian
inhabitants of Spain and Portugal. In the sixteenth century, Sephardic
Jews fleeing persecution took the recipe to Britain, and Portuguese
traders carried it to Japan.



While it might be reasonable to call *mu'affar* the
common cause of fish and chips and tempura, this is not an instance of
the Common Cause Principle. There are no probabilities involved in
this example, and there is no clear sense in which the British and
Japanese both eating fried seafood is a probabilistic correlation.
Perhaps one could construct a probabilistic model of global food
distribution that would supply such probabilities, but in the absence
of such a model, we do not have an instance of RCCP.



## 5. (Putative) Counterexamples



A number of authors have proposed counterexamples to the Common Cause
Principle. A counterexample would involve two events *A* and
*B* such that


* *A* and *B* are probabilistically correlated,
* neither event is a cause of the other, and
* the correlation between *A* and *B* cannot be explained by a
common cause, either because they have no common cause, or because
their common cause does not screen them off from one another.



One type of case that is often said to violate the Common Cause
Principle involves entangled states in quantum mechanics, such as
those found in Einstein-Podolski-Rosen (EPR) thought experiment. We
will discuss this case in detail in
 Section 10
 below. The present section will consider several other examples. As
we shall see, these examples typically raise questions about the
proper scope and interpretation of the Common Cause Principle, rather
than refuting the principle outright.




**Example 5.** *Cartwright's factory.*



Cartwright (1999: 108-109) asks us to imagine a factory that
produces a chemical, *C*, that is used in sewage treatment. The
factory employs a genuinely indeterministic process, so that when the
process is initiated, *I*, there is a probability of .8 that the
chemical is produced. However, whenever the chemical is produced, the
factory also releases a pollutant, *P* as a by-product. Thus



\[.8 = p(C \cap P|I) > p(C|I)p(P|I) = .64\]


Neither *C* nor *P* cause the other.




Glymour (1999) objects that such a factory is nowhere to be found.
This raises a question about the status of the Common Cause Principle.
If the principle is intended to be a conceptual truth about the
relationship between causation and probability, then we could
undermine the principle by showing that a causal structure that
violates it can be clearly conceived. On the other hand, if the
principle is intended to be an empirical generalization about the
relationship between causation and probability in the actual world,
then Glymour is right to demand more than a hypothetical example.




**Example 6.** *Random darts.*



Suppose that darts are shot at a dart board using an indeterministic
process that can hit any part of the board. (If the reader is not
comfortable with indeterministic darts, she may imagine photons
hitting a scintillation screen after passing through a narrow slit.)
Suppose that A and B
are two regions of the dart board such that
A is fully contained within B,
and B does
not fill the entire board (see
 Figure 5(a)).
 Let *A* be the event corresponding to the dart landing in region
A, and analogously for *B*. Then
we will have:

 

\[p(A \cap B) = p(A) > p(A)p(B).\]


This follows because \(A \cap B = A\) and \(p(B) < 1\). While the
throw of the dart (or whatever process launches the dart) is a common
cause of *A* and *B*, this will not screen them off. And if
the process is genuinely indeterministic, there will be no cause that
screens them off.




![Two large circles, the dartboards, one is (a) and the other (b). Dartboard (a), has two concentric circles in it with the inner labeled 'A' and the outer labeled 'B'. Dartboard (b) has two mostly overlapping circles in it with the left one labeled 'A' and the right one 'B'.](figure5.svg)

Figure 5: The large circle represents
the dart board. Circles A and B
are regions of the dartboard. In (a) the
region A is entirely contained in the
region B. In (b) the regions almost
(but not quite) completely overlap. 




David Lewis famously tried to analyze causation in terms of
counterfactuals (Lewis 1973). He recognized that in order for his
theory to succeed, he needed to restrict the analysis to
counterfactual relations among *distinct* events (see
especially Lewis 1986). For instance, we typed the word "Lewis"
near the beginning of this paragraph. If we had not typed the letters
L-e-w, then we would not have typed the word "Lewis". But typing
the letters L-e-w did not *cause* us to type "Lewis".
Typing L-e-w was *part* of the act of typing "Lewis". The
events *typing L-e-w* and *typing "Lewis"* are not
*distinct* from one another in the right kind of way to stand
in causal relations to each other. Lewis's account of
distinctness is somewhat involved, but it specifically excludes
relations of logical entailment and spatiotemporal inclusion.



It appears that the Common Cause Principle requires a similar
restriction to distinct events. In our example of the dart board, the
region A is spatially included within
the region B; hence, the events
*A* and *B* will not be distinct, in Lewis's sense.
This means that there may be correlations between these events that
are entirely due to their spatiotemporal relationship, and don't
have any distinctively causal basis.



Now suppose that the regions A and
B almost completely overlap, but
neither is contained within the other (see
 Figure 5(b)).
 Again, the corresponding events *A* and *B* will be
correlated, and the earlier common cause may not screen them off.
(Arntzenius (1999 [2010: section 2.4]) has an example that has
essentially this structure.) It seems that this case, too, is one in
which the events *A* and *B* are insufficiently distinct.
But now it becomes difficult to formulate a notion of distinctness
that is sufficiently general, without ruling out too much. To see the
problem, imagine that instead of a literal dartboard, we have a Venn
diagram representing the possible states of a system that evolves
indeterministically. The spatial regions A
and B now
represent sets of possible states, corresponding to the states in
which events *A* and *B* occur. And the area corresponds to
probability. Thus the set of states in which *A* occurs almost
completely overlaps with the set of states in which *B* occurs,
as measured by the probability distribution over states. Is this a
reason to think that *A* and *B* are not distinct? It had
better not be, because this is just a representation of the generic
case where events *A* and *B* are probabilistically
correlated. Thus spelling out the precise notion of
*distinctness* at work remains a challenge.




**Example 7.** *Conserved quantities.*



Salmon (1984) and Schurz (2017) argue that systems governed by
probabilistic dynamics together with conservation laws will give rise
to violations of RCCP. Suppose that a brick weighing 2 kg falls onto a
hard, peaked surface. The brick cracks and breaks into two pieces,
A and B.
Suppose that this process is genuinely chancy. Perhaps the precise
point at which the brick strikes the surface, or the precise process
of crack formation in the brick, is not determined by earlier states
of the system. Let *A* be the event corresponding to piece A
weighing some specific amount, say 1.2 kg,
and let *B* be the event corresponding to piece B
weighing the complementary amount, say 0.8
kg. Since the combined mass must be 2 kg, *A* will occur if and
only if *B* occurs. Since the process is chancy, \(p(A) = p(B) =
r < 1\). However, 

 

\[p(A \cap B) = r > r^2 = p(A)p(B),\]


so *A* and *B* are correlated. No earlier event can screen
off this correlation unless it determines that *A* and *B*
will occur. Since the process is genuinely chancy, there is no such
event. Hence RCCP is violated.



Whether there are any actual violations of RCCP having this form
depends on whether there are actual processes governed by
probabilistic dynamics and conservation laws. We don't find such
processes described by classical physics. We do encounter this
combination already in quantum mechanics, at least on some
interpretations (so-called collapse interpretations). However, these
cases are further complicated by the role of quantum
*entanglement*. We discuss the status of the Common Cause
Principle in quantum mechanics in greater detail in sections 9 and 10
below.





**Example 8.** *Time series.*



Sober (2001) notes that sea levels in Venice and bread prices in
London have both been rising over the past few centuries. Let *V*
represent Venetian sea levels higher than some specified level, and
*L* London bread prices higher than a given mark. If we sample
Venetian sea levels and London bread prices over time, we will find
that *V* and *L* are correlated: In years where *V*
obtains, *L* tends to obtain as well (since these will tend to be
more recent years). However, we have no reason to think that these
phenomena share a common cause. They appear to be causally
independent.




To understand what is going on in this example, we need to look more
closely at the relationship between sample statistics and underlying
probability distributions. We frequently think about this in terms of
a classic urn model. An urn contains a certain proportion of black and
white balls. We draw balls from the urn, and use the frequency of
black balls drawn to estimate the proportion of black balls in the
urn. Using statistical frequencies to estimate probabilities in this
way typically requires an assumption that the samples are
probabilistically independent: drawing a black ball does not affect
the probability that the next ball drawn will be black. Sober's
example involves what statisticians call a *time series*. When
we sample Venetian sea levels over the course of years, we are not
drawing probabilistically independent samples from a stable
probability distribution. If the sea level is high in a particular
year, we can predict that the sea level will be similarly high the
following year (they tend not to change dramatically from one year to
the next). For this reason, we cannot interpret the relative
frequencies that we obtain as estimates of an underlying probability
distribution. Thus, even though there is a correlation between
*V* and *L* in our samples, it is impossible to interpret
this as a probabilistic correlation with \(p(V \cap L) > p(V)p(L)\). Not all correlations in
statistical samples bespeak *probabilistic* correlations. (See
also Hoover 2003 and Steel 2003 for further discussion of
Sober's example.)



This defense of the Common Cause Principle raises a worry, however.
Many cases where we would like to apply the Common Cause Principle may
also turn out to behave like time series. For example, consider
Reichenbach's example of the traveling theatre troupe
 (Example 2
 above). It is reasonable to expect that if the leading man is sick on
a particular day, then he will be more likely to be sick on the
following day--some illnesses last more than a day. And if he is
exposed to a particular pathogen at one time, he may gain immunity
against that pathogen in the future. If we exclude all such examples,
then we run the risk of excluding many of the standard examples that
are supposed to lend support to the principle.


## 6. The Fork Asymmetry



Reichenbach's *Direction of Time* (1956) was centrally
concerned with temporally asymmetric phenomena. Much of the work is
devoted to the status of the Second Law of Thermodynamics, which says
that in a closed system, entropy can increase but will never decrease.
But Reichenbach also tried to define a macroscopic statistical
asymmetry using the Common Cause Principle. Suppose that events
*A* and *B* are correlated, i.e., that \(p(A \cap B) >
p(A)p(B)\). If there is an event *C* that satisfies conditions
 (2)-(5)
 above, Reichenbach called the trio *ACB* a *conjunctive
fork*. If *C* occurs earlier than *A* and *B*, and
there is no event satisfying (2)-(5) that occurs later than
*A* and *B*, then *ACB* is said to form a conjunctive
fork *open to the future* (see
 Figure 6(a)).




![a diagram: link to extended description below](figure6.svg)

Figure 6: (a) Conjunctive fork open to
the future. (b) Conjunctive fork open to the past. (c) Closed fork.
[An
 extended description of figure 6
 is in the supplement.] 




 Analogously, if there is a later event satisfying (2)-(5), but
no earlier event, we have a conjunctive fork *open to the past*
 (Figure 6(b)).
 If an earlier event *C* and a later event *D* both satisfy
(2)-(5), then *ACBD* forms a *closed fork*
 (Figure 6(c)).
 Reichenbach claimed that in our world, there are a great many forks
open to the future, but few or none open to the past. Moreover, he
proposed that the direction from cause to effect could be grounded in
this statistical asymmetry.



Reichenbach saw his fork asymmetry as a macro-statistical analog of
the second law of thermodynamics. Suppose we have a system such as a
gas that is made up of a large number of particles. Each particle can
be in one of many possible states \(s\_1,\dots,s\_n\). Let \(p\_i\) be
proportion of the particles that are in state \(s\_i\). Then one
expression for the *entropy* of the system is \(S = {-}\sum\_i
p\_i log(p\_i)\). This sum will reach a maximum when the particles are
evenly distributed among the *n* states (or distributed as evenly
as possible given constraints on the system). The second law of
thermodynamics states that the entropy of a closed system can increase
but never decrease; hence it will evolve toward a state of maximum
entropy. Reichenbach claims that if we find a closed system in a state
of low entropy, we can infer that it was recently prepared in that
state, before being closed off from the rest of the environment.



Now suppose that we have two events *A* and *B*, and a
probability distribution *p* over the four states \(AB\),
\(A\overline{B}\), \(\overline{A}B\), \(\overline{A} \overline{B}\).
We can apply the formula for entropy to this probability distribution;
holding the probability of *A* and *B* fixed, the entropy
will be highest when *A* and *B* are probabilistically
independent. Thus, when we find a correlation between *A* and
*B*, this is like finding a system in a state of low entropy, and
we should look for some event in the past of *A* and *B* to
explain this correlation. This is only a formal analogy, but
Reichenbach attempted to show that the same basic principles that give
rise to the second law of thermodynamics would also give rise to the
Common Cause Principle. For more discussion, see the entry on
 thermodynamic asymmetry in time.



Reichenbach's fork asymmetry has been subjected to numerous
criticisms. Horwich (1987) argues that our willingness to infer a past
interaction often has little to do with entropy. We may infer from a
pile of rubble that a city has been bombed. A pile of rubble
corresponds to higher entropy than intact buildings; but it is the
former, rather than the latter, that leads us to infer that a certain
kind of interaction with the city has taken place.



![a diagram: link to extended description below](figure7.svg)

Figure 7:  \(\mathcal{S}\) is the entire
state space. Three copies of \(\mathcal{S}\) are shown, corresponding
to times 0, *t*, and \(t^\prime > t\). If the system is in state
*s* at time 0, it will be in state \(U\_t(s)\) at time *t*,
and state \(U\_{t^\prime}(s)\) at time \(t^\prime\). The set of states
corresponding to event *C* will evolve into \(U\_t(C)\) and
\(U\_{t^\prime}(C)\). [An
 extended description of figure 7
 is in the supplement.]




Arntzenius (1990) points out that the fork asymmetry is highly
problematic in the context of classical statistical mechanics within
which Reichenbach was working. Let \(\mathcal{S}\) be a state-space
containing all possible states of a physical system (see
 Figure 7).
 The state of a system *s* at a particular time *t* will
involve the specification of parameters, such as the positions and
momenta of all the particles in the system. The system evolves
deterministically, so that for every time time *t*, if the system
was in state *s* at time 0, it will be in state \(U\_t(s)\) at
time *t*, where each \(U\_t\) is a one-one function from
\(\mathcal{S}\) into itself. An event such as *C* corresponds to
a subset of \(\mathcal{S}\): the set of states in which *C*
occurs. We can then define \(U\_t(C)\) to be the set of states of the
form \(U\_t(s)\), where \(s \in C\).



A probability function *p* on \(\mathcal{S}\) at time 0 can then
be extended to a probability distribution over trajectories through
\(\mathcal{S}\), yielding probabilities for events at later times.



Suppose that *C* occurs at time 0, and *A* and *B*
occur at later time *t*. Suppose, moreover, that *ACB* forms
a conjunctive fork. Then we can evolve the set of microstates in
*C* forward to time \(t^\prime > t\). Then \(D =
U\_{t^\prime}(C)\) will occur after *A* and *B*, but stand in
the same probabilistic relationship to *A* and *B* that
*C* does (see
 Figure 8).
 Thus *ACBD* will form a closed fork. Since this recipe is
perfectly general, it seems that every conjunctive fork *ACB*
with *C* earlier than *A* and *B* must belong to a
closed fork *ABCD*.



![a diagram: link to extended description below](figure8.svg)

Figure 8: Event *D* at time
\(t^\prime\) results from evolving *C* forward from time 0 to
\(t^\prime\). It will stand in the same probabilistic relationship to
events *A* and *B* at time *t* that *C* did at
time 0. [An
 extended description of figure 8
 is in the supplement.] 




A natural response to this worry is to propose that the earlier set of
states *C* will be sufficiently coherent to form a natural event,
while the later set of states *D* will be a heterogeneous
collection, having nothing in common except for the fact that they all
evolved states in *C*. Such a heterogeneous collection of states
would not qualify as an *event*--see, e.g., Lewis (1986)
for discussion of what qualifies as a genuine event. Hence, there is
an earlier event that screens off *A* and *B*, but no later
event that does so. However, the framework of statistical mechanics
gives us no principled reason for thinking that in all such cases, the
earlier screening off event will be coherent enough to be a proper
event, while the later screener will not be. Moreover, Arntzenius
(1990) offers a counterexample to one way of making this proposal
precise.



We will also see in Section 7 that the modern theory of causal
modeling does not use conjunctive forks to determine the direction of
causation, but rather uses a probabilistic pattern that is essentially
the exact opposite of a conjunctive fork.


## 7. The Causal Markov Condition



Reichenbach's Common Cause Principle is closely related to the
*Causal Markov Condition* that is commonly employed in causal
modeling. A version of the *Causal Markov Condition* was first
proposed by Kiiveri, Speed, and Carlin (1984). Detailed programs in
causal modeling featuring the Causal Markov Condition have been
developed by Pearl (2009) and Spirtes, Glymour and Scheines
(2000).



We will describe here just one kind of causal model, the *causal
Bayes net*. A causal Bayes net uses a *directed acyclic
graph* (DAG) to represent causal relations among a set of
variables, and pairs it with a probability distribution over the set
of variables. More precisely, a causal Bayes net is a triple
\((\mathbf{V}, \mathcal{G}, p)\), where \(\mathbf{V}\) is a set of
variables, \(\mathcal{G}\) is a directed acyclic graph over
\(\mathbf{V}\), and *p* is a probability distribution over the
field of events generated by \(\mathbf{V}\). The variables in
\(\mathbf{V}\) correspond to the properties of individuals in some
population. For example, in a population of American adults, we might
have variables representing an individual's education level,
work experience, and present income. A variable can be binary,
representing the presence or absence of some property, but it can also
be multiple-valued or continuous.



\(\mathcal{G}\) is a set of ordered pairs of variables in
\(\mathbf{V}\). If \((X, Y) \in \mathcal{G}\), we represent this
graphically by drawing an arrow from *X* to *Y*, and we say
that *X* is a *parent* of *Y*. If there are arrows
from \(X\_{1}\) to \(X\_{2}\), \(X\_{2}\) to \(X\_{3}\),... and
\(X\_{n-1}\) to \(X\_{n}\), then there is a *directed path* from
\(X\_{1}\) to \(X\_{n}\). In this case, we say that \(X\_{n}\) is a
*descendant* of \(X\_{1}\). \(\mathcal{G}\) is acyclic if there
is no directed path from any variable to itself.
 Figure 9
 shows a DAG on the variable set \(\{U, V, W, X, Y, Z\}\). In this
DAG, there is a directed path from *U* to *Z*, and *Z*
is a descendant of *U*. \(\PA(X)\) is the set of all parents of
*X* and \(\ND(X)\) is the set of all variables in \(\mathbf{V}\)
other than *X* and its descendants. In
 Figure 9,
 \(\PA(W) = \{U, V\}\) and \(\ND(W) = \{U, V, X\}\).



A DAG represents the qualitative causal structure among the set of
variables in an individual from the relevant population. In
particular, an arrow from *X* to *Y* indicates that *X*
has a causal influence on *Y* that is not mediated by any other
variables in the set. In this case, we say that *X* is a direct
cause of *Y*.



![a diagram: link to extended description below](figure9.svg)

Figure 9: A DAG on the variable set
\(\{U, V, W, X, Y, Z\}\). \(\PA(W) = \{U, V\}\) is the set of parents
of *W* and \(\ND(W) = \{U, V, X\}\) is the set of non-descendants
of *W*. [An
 extended description of figure 9
 is in the supplement.] 




The Causal Markov Condition connects the graph \(\mathcal{G}\) with
the probability distribution *p* over the algebra generated by
the variables. It says that *p* must exhibit specific relations
of conditional probabilistic independence that are implied by the
graph. In the special case where \(\mathcal{G}\) is a DAG, the
following three formulations of the Causal Markov Condition are
equivalent:


1. **Causal Markov Condition: Screening off version.**



For every variable *X* in \(\mathbf{V}\), and every set of
variables \(\mathbf{Y} \subseteq \ND(X):\)

 

\[p(X | \PA(X) \cap \mathbf{Y}) = p(X | \PA(X))\]


We employ a notational convention where statements involving variables
are understood to involve universal quantification over values of
those variables (or over measurable sets of values). This version of
the Causal Markov Condition says that the parents of variable *X*
screen *X* off from all other variables, except for *X*
itself and the descendants of *X*. Given the values of the
variables that are parents of *X*, the values of the variables in
\(\mathbf{Y}\) make no further difference to the probability that
*X* will take on any given value. This version of the Causal
Markov Condition is closest in form to Reichenbach's Common
Cause Principle, although it is formulated in terms of the parents of
*X*, rather than the common cause(s) of *X* and
*Y*.
2. **Causal Markov Condition: Factorization version.**



Let \(\mathbf{V} = \{X\_{1}, X\_{2}, \dots, X\_{n}\}\). Then:



\[p(X\_{1} \cap X\_{2} \cap \dots \cap X\_{n}) = \prod\_{i} p(X\_{i}|\PA(X\_{i})).\]


This version tells us that once we know the conditional probability
distribution of each variable given its parents,
\(p(X\_{i}|\PA(X\_{i}))\), we can compute the complete joint
distribution over all of the variables. This captures
Reichenbach's idea that probabilistic correlations between
events can ultimately be derived from probabilistic correlations
resulting from cause and effect relationships.
3. **Causal Markov Condition: *d*-separation
version.**



Let \(\mathbf{X}, \mathbf{Y},\) and \(\mathbf{Z}\) be disjoint subsets
of \(\mathbf{V}\). Then:

 

\[p(\mathbf{X} \cap \mathbf{Y}|\mathbf{Z}) = p(\mathbf{X}|\mathbf{Z}) \times p(\mathbf{Y}|\mathbf{Z})\] 


if \(\mathbf{Z}\) *d*-separates \(\mathbf{X}\) and
\(\mathbf{Y}\) in \(\mathcal{G}.\)



This version introduces the graphical notion of *d*-separation.
A path from *X* to *Y* is a sequence of variables \((X =
X\_{1},\dots, X\_{k} = Y)\) such that for each \(X\_{i}\), \(X\_{i+1},\) there
is either an arrow from \(X\_{i}\) to \(X\_{i+1}\) or an arrow from
\(X\_{i+1}\) to \(X\_{i}\) in \(\mathcal{G}\). For example, in
 Figure 9,
 \(\{U, Y, W, V, X\}\) forms a path. A variable \(X\_{i}\), \(1 < i <
k\) is a *collider* on the path just in case there is an arrow
from \(X\_{i-1}\) to \(X\_{i}\) and from \(X\_{i+1}\) to \(X\_{i}\). In
other words, \(X\_{i}\) is a collider just in case the arrows converge
on \(X\_{i}\) in the path: 



\[X\_{i-1} \rightarrow X\_i \leftarrow X\_{i+1}.\]

in
 Figure 9,
 *Y* is a collider on the path \(\{U, Y, W, V, X\}.\) Let
\(\mathbf{X}, \mathbf{Y}\), and \(\mathbf{Z}\) be disjoint subsets of
\(\mathbf{V}\). \(\mathbf{Z}\) *d*-separates \(\mathbf{X}\) and
\(\mathbf{Y}\) just in case every path \((X\_{1}, \dots, X\_{k})\) from
a variable in \(\mathbf{X}\) to a variable in \(\mathbf{Y}\) contains
at least one variable \(X\_{i}\) such that either: (i) \(X\_{i}\) is a
collider, and neither \(X\_{i}\) nor any descendant of \(X\_{i}\) is in
\(\mathbf{Z}\); or (ii) \(X\_{i}\) is not a collider, and \(X\_{i}\) is
in \(\mathbf{Z}\). In
 Figure 9,
 \(\{U, V\}\) *d*-separates \(\{Y\}\) and \(\{X\}\). The path
\(\{Y, W, Z, X\}\) contains a collider at *Z*, and all other
paths from *Y* to *X* include *U* or *V* as a
non-collider.



The Causal Markov Condition implies the generalized version of RCCP
presented in
 Section 2.
 This is most easily seen using the *d*-separation version. If
variables *X* and *Y* are correlated, then \(\{X\}\) and
\(\{Y\}\) are not *d*-separated by the empty set. This means
that at least one path between *X* and *Y* must be without a
collider. If we further assume that neither variable is a cause of the
other, then there is no directed path between them. It then follows
that the collider-free path must contain a common cause: a variable
*Z* that has both *X* and *Y* as descendants.
Furthermore, the set of all such common causes *d*-separates
\(\{X\}\) and \(\{Y\}\). (The proof is a bit fussy, since a common
cause on one path can be a collider on another path.) Hence the set of
common causes of *X* and *Y* will screen them off from one
another.



Note that the Causal Markov Condition tells us that certain causal
structures give rise to relations of conditional probabilistic
*in*dependence. The Causal Markov Condition never entails
probabilistic *de*pendence. For purposes of causal inference,
the Causal Markov Condition is often supplemented by a further
principle governing when probabilistic dependence is to be expected.
One such principle is the *Faithfulness Condition* (Spirtes et
al. 2000) which says that *only* the probabilistic
independencies entailed by the Causal Markov Condition are present.
Looked at another way, the Faithfulness Condition says that when we
find a relation of conditional probabilistic independence, we should
infer a causal structure that entails that independence relation
rather than one that doesn't.



One justification for assuming the Causal Markov Condition is a
theorem due to Pearl and Verma (1991). Suppose that we have a variable
set \(\mathbf{V}\), and DAG \(\mathcal{G}\) representing the causal
relations among the variables in \(\mathbf{V}\). Suppose, in addition,
that the value of each variable *X* in \(\mathbf{V}\) is a
deterministic function of its parents in \(\mathbf{V}\), together with
an *error variable* \(U\_{X}\), which represents the influence
of any variables that are not included in \(\mathbf{V}\). In other
words, \(X = f\_{X}(\PA(X), U\_{X})\). Then the values of all of the
error variables will uniquely determine the value of all of the
variables in \(\mathbf{V}\), and a probability distribution \(p^{\*}\)
over the error variables will induce a probability distribution
*p* over the variables in \(\mathbf{V}\). If the error variables
are independent in \(p^{\*}\), then the induced probability
distribution *p* will satisfy the Causal Markov Condition with
respect to \(\mathcal{G}\). The idea is that if we include enough
variables in \(\mathbf{V}\) so that any remaining causal influences
are probabilistically independent of one another, then the probability
distribution over \(\mathbf{V}\) will satisfy the Causal Markov
Condition.



A set of variables \(\mathbf{V}\) is *causally sufficient* if
there is no variable *W* omitted from \(\mathbf{V}\), such that
if it were added to \(\mathbf{V}\), it would be a direct cause of two
variables in \(\mathbf{V}\). In
 Figure 9,
 \(\{U, W, Y\}\) is causally sufficient (assuming the original DAG
is), but \(\{U, W, Y, Z\}\) is not, since *W* and *Z* have a
common cause, *V*, that is left out of this set.



It is usually assumed that if a variable set is causally sufficient,
then the error variables will be probabilistically independent, and
the probability distribution over \(\mathbf{V}\) will satisfy the
Causal Markov Condition with respect to the true causal graph. Note
that this assumption is very similar to the Common Cause Principle
itself. If *X* and *Y* are variables included in a causally
sufficient DAG, and \(U\_X\) and \(U\_Y\) are their corresponding error
variables, then neither \(U\_X\) nor \(U\_Y\) is a cause of the other,
and they do not have a common cause. If they had a common cause, this
would be a common cause of *X* and *Y*; if \(U\_X\) is a
cause of \(U\_Y\), then \(U\_X\) is a common cause of *X* and
*Y*. So the causal sufficiency of the variable set implies that
\(U\_X\) and \(U\_Y\) are causally unrelated. The Common Cause Principle
would then imply that they are probabilistically independent. Thus it
would be inaccurate to say that the Causal Markov Condition can be
used to *justify* the Common Cause Principle: both involve
comparable assumptions about the relationship between causation and
probability.



Examining the *d*-separation version of the Causal Markov
Condition, we see that it is not common causes but rather colliders
that give rise to distinctive probabilistic relationships. Suppose
that we have a variable set with three variables \(\{X, Y, Z\}\), and
that the probability distribution *p* satisfies the Causal Markov
Condition with respect to the true causal graph. Finally, suppose that
*X* and *Z* are correlated, but screened off by *Y*.
Then there are three different causal graphs that all imply just this
set of probabilistic independence relations:

 

\[\begin{align}
X & \rightarrow Y \rightarrow Z\\
X & \leftarrow Y \leftarrow Z\\
X & \leftarrow Y \rightarrow Z\\
\end{align}\]


The last of these, of course, is the Common Cause Structure
characterized by Reichenbach. (However, Reichenbach also postulated
that intermediate causes would screen off distal causes from their
effects, as indicated in the first two diagrams.) On the other hand,
suppose the relations of probabilistic (in)dependence are exactly the
opposite. That is: *X* and *Z* are probabilistically
independent, but dependent conditional on *Y*. In this case,
there is only one causal structure that implies just this set of
probabilistic independence relations:

 

\[X \rightarrow Y \leftarrow Z\]


Indeed, algorithms for inferring causal structure from relations of
conditional probabilistic dependence and independence, such as the
*PC* algorithm of Spirtes et al. (2000), proceed by searching
for this type of probabilistic signature. Thus Reichenbach was
mistaken in looking to conjunctive forks to define the direction of
causation (see
 Section 6
 above). He would have done better to look to colliders.


## 8. Technical Results on Common Causes Relevant for Assessing the Status of RCCP



A classical probability measure space is a triplet \((X,{\cal S},p)\),
where *X* is the set of elementary random events, \({\cal S}\) is
a Boolean algebra of some subsets of *X* and *p* is the
additive probability measure from \({\cal S}\) into the unit interval
\([0,1]\). The probability measure space \((X,{\cal S},p)\) is called
*common cause complete* if there exists a common cause *C*
in \({\cal S}\) of every correlated pair \((A,B)\) of elements \(A,B\)
in \({\cal S}\), common cause *incomplete* otherwise. One can
show that common cause (in)completeness can be characterized in terms
of the measure theoretic (non)atomicity of probability measure spaces:
\((X,{\cal S},p)\) is common cause complete if and only if \((X,{\cal
S},p)\) contains at most one measure theoretic atom (Gyenis &
Redei 2011). A measure theoretic atom is an element \(A\_0\) in
\({\cal S}\) such that \(p(A\_0)\not=0\), and, if \(B\subset A\_0\),
then \(p(B)=0\). In particular measure theoretically purely non-atomic
probability measure spaces (i.e., probability spaces that do not
contain any measure theoretic atom) are common cause complete (Gyenis
& Redei 2011; Hofer-Szabo et al. 2013; Marczyk &
Wronski 2015). An example of a measure theoretically purely
non-atomic probability measure space is the unit interval \([0,1]\)
with the uniform probability (Lebesgue measure) on the Lebesgue
measurable subsets of \([0,1]\); a class of examples are probability
measures on the (Lebesgue measurable subsets of the) real line where
the probability is given by a density function with respect to the
Lebesgue measure on the real line.



A common cause incomplete probability space \((X,{\cal S},p)\) is
called *common cause completable* if it can be embedded into a
larger probability space \((X',{\cal S}',p')\) that is
common cause complete. An embedding of \((X,{\cal S},p)\) into
\((X',{\cal S}',p')\) is an injective Boolean algebra
homomorphism *h* from \({\cal S}\) into \({\cal S}'\) such
that \(p'(h(A))=p(A)\) for all *A* in \({\cal S}\). One can
show that every probability space \((X,{\cal S},p)\) can be embedded
into a purely nonatomic probability space. This entails that every
common cause incomplete probability space is common cause completable
(Gyenis & Redei 2011; Hofer-Szabo et al. 2013;
Marczyk & Wronski 2015).



In quantum theory one uses non-classical (quantum) probability spaces
to describe quantum physical systems. A general quantum probability
space is a pair \(({\cal P} ({\cal N}),\phi)\), where \({\cal P}
({\cal N})\) is the orthocomplemented, orthomodular lattice of
projections of a non-commutative von Neumann algebra \({\cal N}\) and
\(\phi\) is a countably additive probability measure on \({\cal P}
({\cal N})\), which is the restriction to \({\cal P} ({\cal N})\) of a
normal state on \({\cal N}\). Redei & Summers (2007a)
provides a brief description of non-commutative probability theory in
terms of von Neumann algebras; Landsman (2017) contains an
encyclopedic treatment of the algebraic structures relevant for
quantum theory; the SEP entry on
 quantum theory and mathematical rigor
 gives a very brief informal introduction to some basic facts on von
Neumann algebras, including their Murray-von Neumann classification,
which is relevant from the perspective of the Common Cause
Principle--see below). A special example of a quantum probability
space is obtained by taking the set of all bounded operators \({\cal
B} ({\cal H} )\) on a (possibly infinite dimensional) Hilbert space
\({\cal H}\) as the von Neumann algebra and \(\phi\) a quantum state
given by a density matrix. The resulting specific quantum probability
space \(({\cal P}({\cal B}({\cal H})),\phi)\) is known as the
"Hilbert space formalism" of quantum mechanics; this space
describes quantum systems of finite degrees of freedom. The
orthomodular lattice \({\cal P}({\cal B}({\cal H}))\), frequently
denoted as \({\cal P} ({\cal H})\), is the set of all projections on
\({\cal H}\), this lattice is also called "Hilbert
lattice".



The concept of correlation between commuting projections \(A,B\) in
\({\cal P} ({\cal N})\) and the notion of common cause of such
correlations as a projection *C* commuting with both *A* and
*B* and satisfying the (analogues) of the
 four Reichenbachian conditions (2)-(5)
 make perfect sense in quantum probability spaces. So do the concepts
of common cause (in)completeness, common cause completability (with a
suitably defined embedding of orthomodular lattices), measure
theoretic atoms and measure theoretic non-atomicity.



One can show that common cause completeness of such quantum
probability spaces can be characterized in terms of measure theoretic
atomicity of \(({\cal P} ({\cal N}),\phi)\) in complete analogy with
the classical case: \(({\cal P} ({\cal N}),\phi)\) is common cause
complete if it contains at most one measure theoretic atom (Kitajima
2008; Gyenis & Redei 2014; Kitajima & Redei
2015). This entails that measure theoretically purely non-atomic
quantum probability spaces are common cause complete. Specifically,
quantum probability spaces \(({\cal P} ({\cal N}),\phi)\) with a type
III or type II von Neumann algebra \({\cal N}\) are common cause
complete because the quantum probability spaces defined by these types
of von Neumann algebras are purely measure theoretically non-atomic.
The quantum probability space \(({\cal P} ({\cal H}),\phi)\) is not
purely nonatomic, it contains a lot of measure theoretic atoms: all
the rank-one projections (equivalently: one dimensional linear spaces
spanned by vectors in \({\cal H}\)) are atoms of the Hilbert lattice
\({\cal P} ({\cal H})\), hence all rank-one projections that are
assigned non-zero probability by state \(\phi\) are also measure
theoretic atoms in \(({\cal P} ({\cal N}),\phi)\). Thus \(({\cal P}
({\cal H}),\phi)\) is not common cause complete. Whether common cause
incomplete quantum probability spaces are common cause completable, is
not known; results are available on common cause
*extendability* of quantum probability spaces only: Every
quantum probability space \(({\cal P} ({\cal N}),\phi)\) is embeddable
into a larger quantum probability space \(({\cal P}({\cal
N}'),\phi')\) in such a way that every correlation in \(({\cal
P} ({\cal N}),\phi)\) has a common cause in \(({\cal P}({\cal
N}'),\phi')\) (Hofer-Szabo, Redei, &
Szabo 1999; 2013: 62, Proposition 6.3).



The philosophical significance of common cause completability and
common cause extendability of (classical and quantum) probability
spaces is that they show that it is always possible in principle to
explain any correlation, hence also any correlation between causally
independent events, in terms of (possibly hidden) common
causes--hidden in the sense that the common causes need not be in
the probability space that contains the correlation: the common causes
can be "hiding" in a larger probability space. Thus these
common cause completability and common cause extendability results
constrain the possible ways of attempts to falsify the Common Cause
Principle: Any attempt at falsification should impose some further
conditions on the common causes in addition to the four defining
 conditions (2)-(5),
 otherwise falsification can be evaded by referring to hidden common
causes. Formulated differently: The Common Cause Principle, as
formulated exclusively in terms of the notion of a Reichenbachian
common cause, either in a classical or in a quantum probability
theory, is not falsifiable. This does not mean that the common cause
extendability results can be regarded as proof that the Common Cause
Principle is true: Even the weaker question of whether a common cause
extendability result can be taken as confirming evidence for the
Common Cause Principle depends on whether the extended probability
measure space, existence of which is warranted by the mathematical
common cause extendability results, is (part of) an empirically
confirmed scientific theory.



The significance of common cause complete spaces is that they display
a very strong compliance with the Common Cause Principle: An
equivalent formulation of the Common Cause Principle is that
probability theories describing the world are causally closed in the
following sense: if *A* and *B* are correlated and are
causally independent, \(R\_{ind}(A,B)\) in notation, then the
correlation has to have a common cause that explains the correlation.
Common cause complete spaces are causally closed in the extremely
strong sense that the causal independence relation \(R\_{ind}(A,B)\)
can be taken to be the strongest one, containing all correlated
events. Thus, if a scientific theory describing some aspect of reality
uses a measure theoretically purely non-atomic probability space, such
a theory can be regarded as evidence in favor of the Common Cause
Principle because that theory contains common causes of all
correlations, including correlations that are between events that the
theory might regard as causally unrelated (independent). In view of
the interpretation of common cause completeness, common cause
completability and common cause extendability, it is a relevant
question whether our well-confirmed scientific theories are causally
closed or common cause extendable. Quantum field theory and standard
non-relativistic quantum mechanics are two theories for which this
questions has been analyzed extensively.


## 9. Quantum Field Theory and the Common Cause Principle



Relativistic quantum field theory (see entry on
 quantum field theory)
 predicts an abundance of correlations between observables that are
localized in spacelike separated (hence causally independent)
spacetime regions. This is a consequence of prevalence of entanglement
and of violation of Bell's inequality in quantum field theory,
which is even more striking than violation of Bell's inequality
by standard, non-relativistic quantum mechanics (Summers 1997; Clifton
& Halvorson 2001; Halvorson & Clifton 2000; entry on
 Bell's theorem).
 If relativistic quantum field theory is causally closed in the sense
of providing common causes of these correlations, then this theory is
confirming evidence for the truth of the Common Cause Principle.



The quantum probability spaces describing relativistic quantum field
theory are based on type III von Neumann algebras (Haag 1992; Horuzhy
1986 [1990]; Araki 1993 [1999]; entry on
 quantum field theory;
 consequently, these quantum probability spaces are measure
theoretically purely non-atomic. It follows then from the common cause
completeness of such quantum probability spaces that there exists
common causes of the correlations between spacelike separated
observables. Yet, one cannot conclude from this that quantum field
theory satisfies the Common Cause Principle, for the following reason:
observables in quantum field theory are explicitly associated with
specific spacetime regions. So the common causes also should be
required to belong to specific spacetime regions. Given a correlation
between observable *A* localized in spacetime region \(V\_1\) and
observable *B* localized in spacetime region \(V\_2\) spacelike
separated from \(V\_1\), the common cause for this correlation should
be localized in a region *V* that is in the intersection of the
backward light cones of the spacelike separated spacetime regions
\(V\_1\) and \(V\_2\). But the mere pure non-atomicity of the quantum
probability space of quantum field theory does not entail any such
locality of the common causes that exist as a consequence of the
non-atomicity of the quantum probability space describing quantum
fields. When one imposes this additional localizability constraint on
the common causes in quantum field theory, the question of whether
such *properly localized* common causes exist according to the
theory, becomes a highly non-trivial problem (Redei 1997). So
non-trivial that the answer to it is still not known--the problem
of status of causal completeness of quantum field theory is an open
question.



It is known however that quantum field theory is causally complete in
the weaker sense of containing common causes localized in the
*union* (as opposed to the *intersection*) of the
backward light cones of the spacelike separated spacetime regions
\(V\_1\) and \(V\_2\) that contain the correlated observables
(Redei & Summers 2002, 2007b; Hofer-Szabo et al.
2013). It also is possible logically that the status of the Common
Cause Principle in relativistic quantum field theory cannot be
decided: The quantum field theory in which the abundance of
correlations between spacelike separated observables has been proven
is defined by axioms (Haag-Kastler axiomatization). The axioms might
just be too weak to give an answer to the question whether all
correlations between spacelike separated observables have properly
localized common causes. Proving such an independence result would be
very interesting. Given how difficult it is to create any model of the
Haag-Kastler axioms, attempting such a proof also appears to be
extremely difficult because it requires displaying two models, one in
which the Common Cause Principle holds (with properly localized common
causes), one in which it does not.



One also can consider the problem of the status of the Common Cause
Principle in discrete (lattice) quantum field theory. This is a
simplified model of quantum field theory in which calculations are
frequently easier to carry out. One of the simplifications of this
theory is that the algebras of local observables are finite
dimensional. The quantum probability spaces determined by such
observables are not purely non-atomic and this prevents common
causes--localized or not--to exist in these theories
(Hofer-Szabo & Vecsernyes 2012a). If, in the quantum
context, one allows common causes not to commute with the correlated
observables, then "non-commutative" common causes can be
shown to exist in lattice quantum field theory (Hofer-Szabo
& Vecsernyes 2012b, 2013, 2018). One also can raise the
problem of the status of the Common Cause Principle in categorial
quantum field theory (Brunetti, Fredenhagen, & Verch 2003;
Redei 2014a). This requires a reformulation of the concept of
common cause in terms of the notions of the categories used in
categorial quantum field theory, which is done in Redei
(2014a). This then leads to a number of specific questions about the
status of the suitable reformulated Common Cause Principle. Most of
these questions are open--this is an active area of research.


## 10. EPR Correlations and RCCP



Standard non-relativistic quantum mechanics of finite degrees of
freedom predicts the famous EPR correlations (see entry on
 the Einstein-Podolsky-Rosen argument in quantum theory),
 which have been observed in sophisticated physical measurements (see
entry on
 Bell's theorem).
 These correlations are between events that are spacelike separated
(hence causally independent according the special theory of
relativity). Thus, they can serve as evidence to assess the status of
the Common Cause Principle: These EPR correlations should have common
causes if the Common Cause Principle describes the physical world
correctly.



The quantum probability space that predicts the EPR correlations does
not contain common causes for these correlations because this quantum
probability space is defined on a finite dimensional Hilbert space and
such a quantum probability space is not purely non-atomic hence not
common cause complete. The question of whether "hidden"
common causes for such correlations can in principle exist has been
extensively discussed in a large literature. The first informal
discussion seems to be von Neumann's letter to Schrodinger
in 1935 (Von Neumann 2005: 211-213). In this letter von Neumann
takes the position that no correlation, including EPR-type quantum
correlations, is problematic as long as one can find a common cause
for it--although von Neumann does not use the term "common
cause", nor does he define any notion of common cause explicitly
(Redei 2006). The first technically explicit no-go theorem on
common causes of EPR correlations is due to van Fraassen (1982). In
this paper it is assumed however that a single common cause explains
different correlations--i.e., that common causes are
*common* common causes. It turns out however that, given a set
\((A\_i,B\_i)\) (\(i=1,2,\ldots\)) of pairs of correlated events in a
classical probability measure space, requiring the existence of a
common cause \(C\_i\) for each correlated pair \((A\_i,B\_i)\) is a much
weaker assumption than requiring the existence of a single *C*
that is the common cause of all correlated pairs \((A\_i,B\_i)\)
(\(i=1,2,\ldots\)): one can give examples of different correlations in
a classical probability measure space that cannot have the same common
cause (Hofer-Szabo, Redei, & Szabo 2002).
Thus if one aims at deriving no-go propositions on the existence of
common causes of EPR correlations then one has to either argue that
the different correlations in the EPR-situation should have the same
common cause or one should impose further conditions on the common
causes in addition to those in
 Reichenbach's definition (2)-(5).



There are two types of such additional conditions, both motivated by
the specific features of the EPR-scenario: "Locality
conditions" and requirements expressing what is called
"no-conspiracy". Furthermore, both locality and
no-conspiracy come in two varieties: *surface* and
*hidden* locality conditions on one hand and *weak* and
*strong* no-conspiracy conditions on the other. The content of
the surface locality condition is that "... the probability
of the outcome in one wing of the EPR correlation experiment is the
same no matter in which direction the measurement is carried out in
the other wing of the EPR experiment." (Hofer-Szabo et
al. 2013: 143). The hidden locality conditions require that 



> 
> 
> ... given a pair of correlated outcomes of an EPR
> correlation experiment, the probability of the outcome in one wing of
> the EPR correlation experiment be the same no matter in which
> direction the measurement is carried out in the other wing of the EPR
> experiment *if* the hypothetical common cause of this
> correlation also has happened. (Hofer-Szabo et al. 2013:
> 144). 
> 
> 
> 
> The *weak* no-conspiracy condition requires that any choice of
> the direction in which a measurement is decided to be carried out in
> any wing of the EPR experimental setup is probabilistically
> independent of the hypothetical common cause explaining the
> correlation in the chosen direction. The *strong no-conspiracy*
> condition requires that any Boolean *combination* of choices of
> measurement directions in the two wings of the EPR experiment are
> probabilistically independent of any Boolean *combination* of
> the hypothetical common causes. (Hofer-Szabo et al. 2013:
> 178). 
> 
> 
> 



The differences between the different versions of the locality and
no-nonspiracy conditions are conceptually significant and they are
also crucial from the perspective of whether hypothetical common
causes of EPR correlations can be ruled out by no-go theorems: The
surface locality conditions are empirically testable due to the fact
that all the events involved in it are observable. The hidden locality
conditions involve the unobserved, hypothetical common cause events
and, as a consequence, these locality conditions are not testable
independently of the observation of the common causes. The
interpretation of the no-conspiracy conditions is somewhat
controversial: it declares as an implausible conspiracy on the part of
nature if common causes influenced not only the outcomes of the EPR
correlation experiments but also the choices of measuring in
particular directions in the wings of the EPR experiment. These
choices, we feel, are free, depending on the experimenter's
decision.



One can show that *weakly* non-conspiratorial common cause
explanations of EPR correlations satisfying surface locality are
possible. But *strongly* non-conspiratorial, *hidden*
local common cause explanations of EPR correlations are ruled out
(Hofer-Szabo et al. 2013: 152, 163). These results show
that the EPR correlations cannot be regarded as strictly empirical
evidence against the Common Cause Principle because the type of common
causes that can be ruled out have features by assumption that are not
empirically testable (hidden locality and no-conspiracy)--they
are metaphysical in character.


## 11. Further Reading



Other accessible overviews of Reichenbach's Common Cause
Principle include Chapter 6 of Salmon (1984) and Arntzenius
(1999 [2010]), which is a previous version of the present
entry. The entry on
 Hans Reichenbach
 provides an overview of Reichenbach's philosophy.
Reichenbach's probabilistic theory of causation is discussed in
the entry on
 causation, probabilistic.
 Issues related to temporal asymmetry in thermodynamics are discussed
in the entry on
 thermodynamic asymmetry in time.
 The Causal Markov Condition is discussed at greater length in the
entry on
 causal models.
 This encyclopedia contains numerous entries on philosophical issues
related to quantum mechanics. Of particular relevance are
 the Einstein-Podolski-Rosen argument in quantum theory,
 philosophical issues in quantum theory,
 quantum logic and probability theory,
 and
 Quantum Mechanics.



Two recent book-length treatments of RCCP are Hofer-Szabo et
al. (2013) and Wronski (2014). Further good discussions of the
Common Cause Principle include Arntzenius (1990, 1992); Cartwright
(1988); Van Fraassen (1982b); Gyenis and Redei (2011, 2014); Henson
(2005); Hofer-Szabo et al. (1999, 2002); Marczyk and
Wronski (2015); Mazzola (2012, 2013); Mazzola and Evans (2017);
Redei (2014b); San Pedro (2008); Sober (1984, 1988 1989,
2008); Spohn (1994); Suppes (1970, 1984); Uffink (1999); and
Wronski (2010).



Discussion of counterexamples to RCCP are found in Cartwright (1994,
2007); Hoover (2003); Sober (2001); and Steel (2003). 



Discussions of RCCP in the context of quantum theory include
Butterfield (1989, 2007); Chang and Cartwright (1992) Kowalski and
Placek (1999); Penrose and Percival (1962); Placek (2000a,b); van
Fraassen (1982a, 1991); Wuthrich (2004); and Wiseman and
Cavalcanti (2017).