Causal completeness of probability theories –
results and open problems
Miklós Rédei
Department of Philosophy, Logic and Scientific Method
London School of Economics and Political Science
Houghton Street
London WC2A 2AE, UK
e-mail: [email protected]
Balázs Gyenis
Department of History and Philosophy of Science
University of Pittsburgh
1017 Cathedral of Learning
Pittsburgh, PA 15260, U.S.A.
e-mail: [email protected]
Forthcoming in
“Causality in the Sciences”, ed. by F. Russo and J. Williamson
1
The Common Cause Principle and causal completeness informally
The aim of this paper is to give a review of the known results and open problems
concerning causal completeness (closedness) of classical and non-classical (quantum) probability spaces. Causal closedness of a probabilistic theory means that
the theory is causally rich enough to be able to explain causally all the correlations it predicts. It is natural to ask whether probabilistic theories are causally
closed if one assumes that Reichenbach’s Common Cause Principle holds: this
principle states that if two events A and B are probabilistically correlated then
either the correlation is due to a causal interaction between A and B, or, if A
and B are causally independent, Rind (A, B), then there exists a third event C,
a so-called common cause that explains the correlation by being related to A
and B probabilistically in a specific way (see Definition 1). Causal closedness of
a probabilistic theory is intended to express that the theory complies with the
Common Cause Principle; accordingly, and more precisely, a probability theory
is defined to be causally closed with respect to a causal independence relation
1
Rind defined between pairs of random events if for any correlation between elements A and B such that Rind (A, B) holds, there exists a common cause C
in that theory of the correlation between A and B (Definition 3). The problem is then under what conditions on the probability space and on Rind is the
probability space causally closed.
We will see that causal closedness is non-trivial and not impossible in classical probability spaces, not even if the probability space contains a finite number
of random events only – but causal completeness is not typical either. There
does not seem to be any regular and easily characterizable behavior of probability spaces from the perspective of causal closedness; one has to check in each
and every case by brute force whether the causal closedness holds.
The notion of causal closedness of classical probability spaces was introduced
in [1] but the notion of common cause can be naturally defined in non-classical
(quantum) probability spaces as well; hence the notion of causal closedness also
makes perfect sense for such probability theories. Very little known about causal
closedness of such general probability spaces. The known results are summarized
in Section 4.
2
The notion of common cause and some terminology
In what follows (X, S, p) denotes a classical (Kolmogorovian) probability space
with Boolean algebra S of subsets of a set X (with respect to the set theoretic
operations ∩, ∪ and A⊥ = X \ A as Boolean algebra operations) and with the
probability measure p on S. To simplify notation, occasionally we write (S, p)
instead of (X, S, p), when the precise nature of X is not important. For instance,
when S has a final number of elements, then S is the set of all subsets of a finite
set having n < ∞ elements, in which case we write (Sn , p).
Given (S, X, p), the quantity Corrp (A, B) defined by
Corrp (A, B) ≡ p(A ∩ B) − p(A)p(B)
(1)
is called the correlation of A, B in p. Events A and B are said to be positively correlated if Corrp (A, B) > 0. A correlation Corrp (A, B) 6= 0 is called
non-degenerate (and (A, B) a non-degenerate correlated pair) if A 6= B. A
correlation is called maximal if
p(A|B) = p(B|A) = 1
(2)
The next definition specifies the notion of common cause. Since the definition
was first given by Reichenbach in [13], this type of common cause is called
“Reichenbachian”; however, since in this paper only Reichenbachian common
causes feature, the qualifier “Reichenbachian” will be omitted.
2
Definition 1. C is a common cause of the correlation (1) if the following
(independent) conditions hold:
p(A ∩ B|C) = p(A|C)p(B|C)
p(A ∩ B|C ⊥ ) = p(A|C ⊥ )p(B|C ⊥ )
p(A|C) > p(A|C ⊥ )
p(B|C) > p(B|C ⊥ )
where
(3)
(4)
(5)
(6)
p(X ∩ Y )
p(Y )
p(X|Y ) =
denotes the conditional probability of X on condition Y , C ⊥ denotes the complement of C and it is assumed that none of the probabilities p(X) (X =
A, B, C, C ⊥ ) is equal to zero.
Since the notion of common cause is a measure theoretic one, measure zero
sets have to be dealt with. The next definition takes care of this and summarizes
some terminology used later.
Definition 2.
1. Let
∆(X, Y ) ≡ (X \ Y ) ∪ (Y \ X)
be the symmetric difference of sets X, Y . A common cause C of the
correlation between A, B is called proper if
p (∆(B, C)) 6= 0 6= p (∆(A, C))
(7)
That is to say, a common cause C of the correlation Corrp (A, B) > 0 is
proper if the common cause differs from the correlated events by more
than a measure zero event. Otherwise C is called improper.
2. It can happen that, in addition to being a probabilistic common cause, the
common cause event C logically implies both A and B, i.e. C ⊆ A ∩ B. If
this is the case then we call C a strong common cause. If C is a common
cause such that C 6⊆ A and C 6⊆ B then C is called a genuinely probabilistic
common cause.
3. A common cause C will be called deterministic if
p(A|C) =
⊥
p(A|C ) =
3
1 = p(B|C)
⊥
0 = p(B|C )
(8)
(9)
3
Causal closedness of classical probability theories
Given the notion of common cause one can define the concept of common cause
closedness in a very natural manner:
Definition 3. Let (X, S, p) be a probability space and Rind be a two-place
causal independence relation between elements of S. The probability space
(X, S, p) is called common cause closed with respect to Rind , if for every correlation Corrp (A, B) > 0 with A ∈ S and B ∈ S such that Rind (A, B) holds,
there exists a common cause C in S. If there are no elements A, B in S that
are positively correlated, then (X, S, p) is called trivially common cause closed.
Proposition 1 ([1]). Let (Sn , p) be a finite probability space. If Rind contains
all the pairs of events A, B in Sn that are correlated in p, then (Sn , p) is not
non-trivially common cause closed with respect to Rind .
Proposition 1 shows that a probability space containing a finite number of
random events contains more correlations than it can account for exclusively
in terms of common causes. But this is not surprising because common cause
closedness with respect to a causal independence relation that leaves no room for
causal dependence is unreasonably strong. One can of course make a probability
space (X, S, p) causally closed by stipulating that Rind (A, B) does not hold (i.e.
that A and B are causally related) whenever A and B are correlated but there
exists no common cause C ∈ S of the correlation. But this is unacceptable in
general since this move makes the notion of causal closedness trivial and the
causal dependence so defined (and the causal independence relation Rind so
defined) may turn out not to have reasonable features. One needs a disciplined,
independent definition of the causal independence relation.
Intuitively, causal independence of A and B should imply that from the
presence or absence of A one should not be able to infer either the occurrence
or non-occurrence of B, and conversely: presence or absence of B should not
entail occurrence or non-occurrence of A. Taking, as it is common, the partial
ordering ⊆ in the Boolean algebra S as the implication relation between events
(equivalently: between propositions that the corresponding events occur), this
requirement about Rind can be expressed by the demand that Rind (A, B) should
imply all of the following relations
A 6⊆ B,
B 6⊆ A,
A⊥ 6⊆ B
B ⊥ 6⊆ A
, A 6⊆ B ⊥ ,
, B⊆
6 A⊥ ,
A⊥ 6⊆ B ⊥
B ⊥ 6⊆ A⊥
This requirement can be expressed compactly by saying that Rind (A, B) implies
that A and B are logically independent; equivalently, that
{∅, A, A⊥ , X} and {∅, B, B ⊥ , X}
are logically independent Boolean subalgebras of S in the sense of the following
Definition 4:
4
Definition 4. Two Boolean subalgebras L1 , L2 of the Boolean algebra S are
called logically independent if
A ∩ B 6= ∅
whenever ∅, X 6= A ∈ L1 and ∅, X 6= B ∈ L2
(10)
The pair (L01 , L02 ) of Boolean subalgebras of Boolean algebra S is called a maximal logically independent pair, if logical independence of Boolean subalgebras
L1 and L2 containing respectively L01 and L02 as Boolean subalgebras implies
L01 = L1 and L02 = L2 .
For later purposes we also need the following notions:
Definition 5. The pair (A, B) is called logically independent modulo zero probability if there exist A0 , B 0 such that
p(A0 ) = p(B 0 ) = 0
(11)
and (A \ A0 ) and (B \ B 0 ) are logically independent.
This motivates the following definition, which formulates a natural notion
of causal closedness.
Definition 6. (X, S, p) is called common cause closed with respect to the pair
(L1 , L2 ) of logically independent Boolean subalgebras of S, if for every A ∈ L1
and B ∈ L2 that are correlated in p, there exists a common cause C in S of the
correlation between A and B.
Proposition 2 ([1]). Let (S5 , pu ) be the probability space with the Boolean algebra S5 generated by 5 atoms and with pu being the probability measure defined
by the uniform distribution on atoms of S5 . Then (S5 , pu ) is common cause
closed with respect to every pair of logically independent Boolean subalgebras
(L1 , L2 ) of S5 .
The next proposition shows that the behavior of the probability space (S5 , pu )
described in Proposition 2 is exceptional.
Proposition 3 ([1]). If the probability space (Sn , p) is not (S5 , pu ), then it
is not non-trivially common cause closed with respect to every pair of logically
independent Boolean subalgebras.
But causally not closed probability spaces can be extended in such a manner
that the extension contains common causes of a finite number of correlations in
a given pair of logically independent sublattices:
Proposition 4 ([1],[5]). If (X, S, p) with finite S is not common cause closed
with respect to a logically independent pair (L1 , L2 ), then it can be extended into
a (X 0 , S 0 , p0 ), with S 0 being also finite, in such a manner that (X 0 , S 0 , p0 ) is common cause closed with respect to the logically independent pair (h(L1 ), h(L2 )),
where h(Li ) is the homomorphic image in S 0 of Li (i = 1, 2).
5
By an extension is meant here that there exists a Boolean algebra embedding
h of S into S 0 that preserves the probability in the sense that p(X) = p0 (h(X))
for all X ∈ S. Note that the images of L1 , L2 under h will not necessarily
be maximally logically independent, not even if L1 , L2 is a maximally logically
independent pair; so we have the following Problem, which is open:
Problem 1. Does Proposition 4 remain true if “logically independent” means
maximal logically independent?
The following problem is also open:
Problem 2. Let (S, p) be a probability space with S having an infinite number
of elements and assume that (L1 , L2 ) is a logically independent pair of Boolean
subalgebras of S such that there exist an infinite number of pairs (Ai , Bi ) of
events Ai ∈ L1 and Bi ∈ L2 that are correlated in p. Does there exist and
extension (S 0 , p0 ) of (S, p) such that (S 0 , p0 ) is common cause closed with respect
to (h(L1 ), h(L2 ))? Does there exist such an extension so that (h(L1 ), h(L2 )) is
a maximal pair of logically independent sub-Boolean algebras in S 0 ?
Probability spaces with infinite Boolean algebras can however be causally
closed as the next proposition shows. Before stating the proposition recall that
a probability space (X, S, p) is called atomless if for any A ∈ S, p(A) 6= 0 there
exists B ⊆ A, B ∈ S such that 0 < p(B) < p(A).
Proposition 5 ([1]). If (X, S, p) is an atomless probability space, then it contains uncountably many proper common causes of every non-degenerate correlation in it. Moreover if A and B are correlated, logically independent modulo
measure zero events, then S contains both uncountably many strong and uncountably many genuinely probabilistic common causes of the correlation between
A and B.
The notion of common cause can be naturally generalized to cover the case
when the correlation is not explainable by a single common cause but by system
of common cause like events. One such generalization was given in [3, 4]:
Definition 7. Let (X, S, p) be a probability space and A, B be two events in S.
The partition {Ci }i∈I of S is said to be a Reichenbachian common cause system
(RCCS for short) for the pair (A, B) if the following two conditions are satisfied
p(A ∩ B|Ci ) = p(A|Ci )p(B|Ci )
for all i ∈ I
[p(A|Ci ) − p(A|Cj )][p(B|Ci ) − p(B|Cj )] > 0 (i 6= j)
(12)
(13)
The cardinality of the index set I (i.e. the number of events in the partition) is
called the size of the RCCS. Since C, C ⊥ with a Reichenbachian common cause
C is a RCCS of size 2, we call a RCCS proper if its size is greater than 2.
It was shown in [3, 4] that Reichenbachian Common Cause systems of arbitrary finite size exist for any non-maximal correlation in the sense that for
any such correlation in any probability space there exists an extension of that
6
probability space that contains a Reichenbachian Common Cause system of the
prescribed size.
In view of this, a natural refinement of the definition of causal closedness
of (X, S, p) is obtained if one replaces the notion of common cause with the
concept of common cause system:
Definition 8. (X, S, p) is called causally N -closed with respect to a causal
independence relation Rind if for any correlation Corrp (A, B) > 0 such that
Rind (A, B) holds, there exists in (X, S, p) a Reichenbachian common cause system of size N for the correlation.
There are a number of questions one can ask in connection with causal N closedness:
Problem 3. On what condition on (X, S, p) and Rind is (X, S, p) causally N closed for a fixed N ?
We have seen that probability spaces may or may not be causally 2-closed
– causal 2-closedness depends sensitively on how Rind is defined. This leads to
the following open problems:
Problem 4. Can a probability space which is not causally 2-closed be causally
N -closed for some fixed N > 2 (with respect to some non-trivial causal independence relation Rind )?
Problem 5. Can a probability space be causally N -closed for every N (with
respect to some non-trivial causal independence relation Rind )?
We conjecture that atomless probability spaces are causally N -closed for
every N with respect to every pair of logically independent Boolean subalgebras.
4
Causal closedness of non-classical probability
spaces
The notion of common cause can be defined in non-classical (quantum) probability spaces (L, φ), where an orthomodular lattice L takes the place of the Boolean
algebra and φ is an additive (or σ additive) map from L into [0, 1] (generalized
probability measure), replacing a classical probability measure. Special examples of such spaces are the quantum probability spaces (P(N ), φ) where P(N )
is the projection lattice of a von Neumann algebra N and φ is a (normal) state
on N (see [21], [6] for the theory of von Neumann algebras). An even more
specific example of the latter is the probability space when P(N ) is the von
Neumann lattice of all projections on a Hilbert space H (in this case we write
(H, P(H), φ); this latter non-classical probability space describes standard, nonrelativistic quantum systems.
Two elements A, B ∈ L are called compatible if they belong to the same
distributive sublattice of L. This condition is equivalent to
A = (A ∧ B) ∨ (A ∨ B ⊥ )
7
If A, B are compatible and
Corrφ (A, B) ≡ φ(A ∧ B) − φ(A)φ(B) > 0
(14)
then A and B are called (positively) correlated with respect to the state φ.
Definition 9. If A and B are positively correlated, then C ∈ L is called a
common cause of the correlation (14) if C is compatible with both A and B and
the following conditions hold.
φ(A ∧ B|C)
φ(A ∧ B|C ⊥ )
φ(A|C)
φ(B|C)
=
=
>
>
φ(A|C)φ(B|C)
φ(A|C ⊥ )φ(B|C ⊥ )
φ(A|C ⊥ )
φ(B|C ⊥ )
where
φ(X|Y ) =
(15)
(16)
(17)
(18)
φ(X ∧ Y )
φ(Y )
denotes the conditional probability of X on condition Y and it is assumed that
none of the probabilities φ(X), (X = A, B, C, C ⊥ ) is equal to zero.
Extension of (L, φ), logical independence of events in L and causal independence relation Rind on L can all be defined in complete analogy with the
classical definitions, which makes it possible to define causal completeness as
well in complete analogy with the classical case:
Definition 10. Let (L, φ) be a non-classical probability space and Rind be a
two-place causal independence relation between elements of L. The probability
space (L, φ) is called common cause closed with respect to Rind , if for every
correlation Corrφ (A, B) > 0 with A ∈ L and B ∈ L such that Rind (A, B)
holds, there exists a common cause C in L (in the sense of Definition 9). If
there are no compatible elements A, B in L that are positively correlated, then
(L, p) is called trivially common cause closed.
Problem 6. On what conditions on (L, φ) and Rind is the probability space
(L, φ) common cause closed with respect to Rind ?
This problem is largely open in this generality. The only general result
known is
Proposition 6 ([7]). If L is an atomless, complete, orthomodular lattice and
φ is a faithful state then (L, φ) is causally closed with respect to every pair of
logically independent sublattices.
The above result is the quantum counterpart of Proposition 5; and the key
fact that it rests on is that if L is an atomless lattice and φ is a faithful state
on L then (L, φ) is atomless as a measure space in the sense that for any 0 6=
A ∈ L, and for any real number 0 6= r < p(A) there exists B ≤ A, B ∈ L
8
such that p(B) = r. This latter fact was proved in [11] (see also [12]) for
the specific quantum probability space (P(N ), φ) where N is a type III von
Neumann algebra and φ is a faithful normal state on N , and Kitajima [7]
showed that the proof of can be carried over from the von Neumann algebra
framework to more general non-classical probability spaces. It is known that
the projection lattices of type II von Neumann algebras are also atomless; so
one has as a specific case of Proposition 6 the following
Proposition 7. Let (P(N ), φ) be a quantum probability space with N as a type
III or type II von Neumann algebra and φ as a faithful normal state on N . Then
(P(N ), φ) is causally closed with respect to every pair of logically independent
sublattices.
Note that the lattice P(H) of all projections on a Hilbert space H is not
atomless (it is atomic) irrespective of the dimensionality of the Hilbert space H
[9]; moreover, the quantum probability spaces (H, P(H), φ) are not atomless in a
measure theoretic sense; consequently, Propositions 6 and 7 do not say anything
about the causal closedness of the quantum probability space (H, P(H), φ) and
it is not known under what conditions such quantum probability spaces are
causally closed (with respect to some Rind ).
Just like in the classical case, the notion of a (Reichenbachian) common
cause system also can be formulated in a non-classical probability space, and
one can define naturally the more general notion of causal N -closedness of a
non-classical probability space: The set {Ci , i ∈ J} of elements (J being an
index set, Ci ∈ L) is called a partition in L if ∨i Ci = I and Ci and Cj are
orthogonal whenever i 6= j; i.e. Ci ≤ Cj⊥ for i 6= j.
Definition 11. A partition {Ci , i ∈ J} is a (Reichenbachian) common cause
system for the correlation (14) between compatible elements A and B if Ci is
compatible with both A and B for every i ∈ J and the following conditions
(analogous to (12)-(13)) hold
φ(A ∧ B|Ci ) = φ(A|Ci )φ(B|Ci )
for all i ∈ J
[φ(A|Ci ) − φ(A|Cj )][φ(B|Ci ) − φ(B|Cj )] > 0 (i 6= j)
(19)
(20)
The cardinality of the index set J is called the size of the common cause system.
Definition 12. The probability space (L, φ) is called causally N -closed (with
respect to some causal independence relation Rind ) if for any correlation between
elements that stand in the causal independence relation there exists in (L, φ) a
Reichenbachian common cause system of size N .
There are a number of open problems in connection with Reichenbachian
common cause systems in non-classical probability spaces and causal N -closedness
of such probability theories:
Problem 7. Given a correlation in a general probability space (L, φ) that does
not have a common cause system of a given size N > 2 of the correlation, does
there exist an extension (L0 , φ0 ) of (L, φ) such that there exists a Reichenbachian
common cause system of size N of the correlation in the extension (L0 , φ0 )?
9
We conjecture a positive answer to the above question.
Problem 8.
1. Do there exist non-classical probability spaces that are causally N -closed
for some fixed N (with respect to some nontrivial Rind )?
2. Do there exist non-classical probability spaces that are causally N -closed
for every N (with respect to some decent Rind )?
3. Do there exist non-classical probability spaces that are causally closed (with
respect to some non-trivial causal independence relation Rind ) in such a
way that every correlation in the space has a common cause system of
countably infinite size?
These questions have not been investigated.
5
Closing comments on causal closedness
Further generalization can be achieved by treating the specific form the correlation measure Corr takes as a variable of the notion of the common cause.
By allowing Corr to measure correlation between pairs of ordered partitions it
becomes possible to handle the case of common cause-type explanations of correlating variables, not just that of events. This allows a more detailed analysis of
causal closedness and of falsification attempts against the Common Cause Principle. It can be shown that Reichenbachian common cause systems are special
cases of the resulting notion of a generalized Reichenbachian common cause. By
imposing mild conditions on Corr extension theorems analogous to Proposition
4 can be proven. However the question of common cause closedness of general
probability spaces with respect to generalized Reichenbachian common causes
is still open. For further details the Reader is referred to [2].
One can strengthen Reichenbach’s notion of common cause by requiring the
common cause to satisfy some additional conditions. The additional conditions
can be motivated by physics: after all, the probability measure spaces in terms
of which the concept of common cause is formulated are not just abstract mathematical entities in physics but physically interpreted structures. Being organic
parts of specific physical theories, these measure spaces offer means to express
a possibly large variety of physical facts and principles. Two of such important
principles are locality and causality. Both locality and causality are rich and
many-layered concepts and there is no unique way of expressing them in terms
of probability measure spaces. But it can happen that a physical theory entails both some additional conditions as necessary for the common cause C of
a correlation between A and B to be “local” and a causal dependence relation
between random events. In such a situation the problem of causal closedness
should be reformulated by taking into account these further restrictions.
This happens in local, algebraic, relativistic quantum field theory (AQFT).
The theory predicts correlations between localized, causally independent (specelike separated) observables [16, 17, 18, 19], [20, 14, 15], and the common causes
10
of these correlations have to be localized [8]. It turns out that there is no unique
way of defining locality of the common cause and, consequently, causal closedness of AQFT can also be specified in different ways [12]: strong, weak and right
(”desirable”) localizability of common causes lead to the concepts of “strong”,
“weak” and “desirable” causal closedness of AQFT. To decide which of these
causal closedness hold for AQFT is a difficult matter. While it is easy to see
that strong causal closedness is violated in AQFT [12], and it could be shown
that AQFT is weakly causally closed [11] (see also the review [10]), it remains
an open problem whether AQFT is causally closed in the most desirable sense.
Acknowledgement: Work supported in part by the Hungarian Scientific Research Found (OTKA), contract number: K68043.
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