Causal Criterion for Multivariate Correlation under Postselection
Keywords: Causal Inference, postselection, perfect correlation, multivariate correlation
TL;DR: What is the criterion to state that some set of N nodes is able to be perfectly correlated when one allows for postselection on some variables? In this work, we answer this question.
Abstract: The field of causal inference examines the relationship between statistical correlations and causal connections among
a set of variables. The causal structure is depicted using
a directed acyclic graph (DAG), which can include both
observed and latent (unobserved) nodes. It is particularly
interesting to study if some variables are allowed by the
causal structure to show correlation among them, i.e., if one
knows the value of one variable, one is able to gain some
knowledge about the value of all the others. The interest in
this question arises because correlation is useful for different
information processing tasks.
In the simplest scenario with two observed variables, it is
easy to see that correlation among them is only possible if
the corresponding nodes are connected by either a direct
causal influence, a latent common cause, or both. These
causal structures are represented in Fig. 1. Notice that in
both scenarios the two observed nodes share a common
ancestor, i.e., there is a node in their common causal past. In
the first case, Fig. 1a), the common ancestor is the node A
itself (notice that one node can be its own ancestor according
to our definition); in the second case, Fig. 1b), it is the latent
common cause C and, in the third case, Fig. 1c), both A and
C can take that role.
There is a third way through which correlation between
these two variables can be established, which is through
postselection (or selection bias). This way requires that the
two variables have a common effect, as in Fig. 2. Then, conditioning on the value of the effect, one generates correlation
among the initially independent nodes.
The most useful graphical tool to address the question of
whether some nodes can be correlated at all is the concept
of d-separation. This is a ternary relation between three
set of nodes, X, Y, Z, that serves as a criterion to answer
whether X can be correlated with Y given the value of
Z. The concept is to associate stastistical dependence with
connection in the causal graph (i.e., the existence of a connecting path) and statistical independence with separation
in the causal graph. We can see that in the examples with 2
variables and no postselection given in Fig. 1, A and B are
always d-connected given the empty set, i.e., conditioning
on nobody.Then, in the example with postselection of Fig. 2,
again we check that the 2 variables are d-connected when
conditioning on S.
Now, the interesting question is how to know if a set of N
arbitrary nodes in a given causal structure can be correlated
or not. As a first step, we consider the case of perfect correlation among all the nodes which means that one learns
with certainty the value of all the nodes from knowing the
value of one of them. The case of not having the possibility of postselecting on any node was studied by Steudel
and Ay in Steudel and Ay [2015]. They used the idea of
common ancestors and showed that to be able to observe
perfect correlation among a set of N variables, there must
be a common ancestor shared by all of them.
Then, the remaining question is what is the criterion to state
that some set of N nodes is able to be perfectly correlated
when one allows for postselection on some variables. In this
work, we answer this question.
To understand the answer, first look at the DAG of Fig. 3.
There, it is possible to achieve perfect correlation among the
nodes A, B, C, D, E, F, G when we condition on $S_1$ and
$S_2$. This can be done through the following choice of causal
parameters:
• The root nodes (i.e., nodes without causal ancestors),
A, D, F, G, are completely random bits.
• B = 0 if F = G = 0; B = 1 if F = G = 1 and
B = 2 otherwise
• $S_1$ = 0 if A = B and $S_1$ = 1 otherwise
• $S_2$ = 0 if B = C and $S_2$ = 1 otherwise
This leads us to observe perfect correlation among A,B,C,D,E,F,G when condition on $(S_1,S_2)=(0,0)$, that is,
$$ P(a,b,c,d,e,f,g|s_1=0,s_2=0)=\frac{1}{2}[0000000]+\frac{1}{2}[1111111]. $$
However, in the DAG of Fig.4, it is impossible to observe
perfect correlation among the nodes A, B, C, D, E, F even
when we condition on $S_1$. This is because the node E is
d-separated of A, B and C given $S_1$. Hence, the causal structure imposes that E is statistically independent of A, B, C
conditioning on $S_1$,
$$p(a,b,c,e|s_1)= p(a,b,c|s_1)p(e|s_1).$$
With these two examples in mind, we can illustrate the general result that we found in this work: we showed that perfect correlation among the nodes of a set X when conditioning on another set of nodes S is possible if and only if there is a root node R that is d-connected to every node in the set X given the nodes of
S. In the example of Fig. 3, all of the root nodes of the DAG satisfy this condition while in the example of Fig. 4, there is no root node that is d-connected to A,B,C,D,E,F given $S_1$.
**Please note that the figures named in the text can be found in the PDF file**
Submission Number: 24
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