Adapted from the original in Judea Pearl's book "Causality" with his permission.

*d-*separation is a criterion for deciding, from a given a causal graph,
whether a set *X* of variables is independent of another
set *Y,* given a third set *Z.* The idea is to associate
"dependence" with "connectedness" (i.e., the existence of
a connecting path) and "independence" with "unconnected-ness"
or "separation". The only twist on this simple idea is to define
what we mean by "connecting path", given that we are dealing
with a system of directed arrows in which some
vertices (those residing in *Z*) correspond to
measured variables, whose values are known precisely.
To account for the orientations of the arrows we use the
terms "*d-*separated" and "*d-*connected" (*d* connotes
"directional").

We start by considering separation between two singleton variables,
*x* and *y;* the extension to sets of variables is
straightforward (i.e., two sets are separated if and only if
each element in one set is separated from every element
in the other).

**Rule 1:** *x* and *y* are *d-*connected if there is an unblocked path
between them.

By a "path" we mean any consecutive sequence of edges, disregarding their directionalities. By "unblocked path" we mean a path that can be traced without traversing a pair of arrows that collide "head-to-head". In other words, arrows that meet head-to-head do not constitute a connection for the purpose of passing information, such a meeting will be called a "collider".

All graphs on this page are clickable; clicking on a variable shows the effect of conditioning on it.

*x* and *y* are *d*-connected (shown as thick edges)
because *r*
is not a collider. This path represents an indirect causal effect.

x E @0,0 y O @6,6 r 1 @3,3 x r r y

*x* and *y* are not *d*-connected (shown as thin edges)
because
*r* is a collider. This path does not transmit any
correlation or other statistical relationship.

x E @0,0 y O @6,6 r 1 @3,3 x r y r

*x* and *y* are *d*-connected because
there is a path between them that does not contain a
collider. This path transmits a correlation that does not
reflect a causal effect (biasing path).

x E @0,0 r 1 @1,1 s 1 @2,2 t 1 @3,3 u 1 @4,4 v 1 @5,5 y O @6,6 s r t s u v t v y r x

*x* and *y* are not *d*-connected because
*t* is a collider. Again, no statistical correlation can
result from this path.

x E @0,0 r 1 @1,1 s 1 @2,2 t 1 @3,3 u 1 @4,4 v 1 @5,5 y O @6,6 r s s t u v t v y x rThis graph contains one collider, at

**Motivation:** When we measure a set *Z* of variables, and take their
values as given, the conditional distribution of the
remaining variables changes character; some dependent
variables become independent, and some independent variables
become dependent. To represent this dynamics in the
graph, we need the notion of "conditional *d-*connectedness" or,
more concretely,
"*d-*connectedness, conditioned on a set *Z* of measurements".

**Rule 2:** *x* and *y* are *d-*connected,
conditioned on a set *Z* of nodes,
if there is a collider-tree path between *x* and *y*
that traverses no member of *Z.*
If no such path exists, we say that *x* and *y* are
*d-*separated by *Z,*
We also say then that every path between *x* and *y* is
"blocked" by *Z.*

x E @0,0 r A @1,1 s 1 @2,2 t 1 @3,3 u 1 @4,4 v A @5,5 y O @6,6 r s s t u v t v y x r

Let *Z* be the set {*r, v*} (marked in gray in the figure).
Rule 2 tells us that *x* and *y* are *d-*separated by *Z,*
and so are also *x* and *s, u* and *y, s* and *u* etc.
The path *x-r-s* is blocked by *Z,* and so are also
the paths *u-v-y* and *s-t-u.*
The only pairs of unmeasured nodes that remain *d-*connected in
this example, conditioned on *Z,* are *s* and *t* and
*u* and *t.*
Note that, although *t* is not in *Z,* the path
*s-t-u* is nevertheless blocked by *Z,* since *t* is a collider,
and is blocked by Rule 1.

**Exercise:** Is it necessary to condition for *r* and *v* in the above
example to *d*-separate
*x* and *y*? If not, which is the smallest set *Z* that *d*-separates
*x* and *y* in the graph above?

**Motivation:** When we measure a common effect of two independent
causes, the causes becomes dependent, because finding
the truth of one makes the other less likely (or "explained
away"), and refuting one implies the truth of the other.
This phenomenon (known as Berkson paradox, or "explaining
away") requires a slightly special treatment when we condition
on colliders (representing common effects) or their
descendants (representing effects of common effects).

**Rule 3:** If a collider is a member of the conditioning set *Z,*
or has a descendant in *Z,* then it no longer blocks
any path that traces this collider.

x E @0,0 r A @1,0 w 1 @1,1 s 1 @2,0 t 1 @3,0 p A @3,1 u 1 @4,0 v 1 @5,0 q 1 @5,1 y O @6,0 r s w s t u v t v y x r t p v q

Let *Z* be the set {*r, p*} (again, marked in gray).
Rule 3 tells us that *s* and *y* are *d*-connected by *Z,*
because the collider at *t* has a descendant (*p*) in *Z,*
which unblocks the path *s-t-u-v-y.* However,
*x* and *u* are still *d*-separated by *Z,* because although
the linkage at *t* is unblocked, the one at *r* is
blocked by Rule 2 (since *r* is in *Z*).

**Exercise:** Click on *r* in the above graph
to remove the conditioning. This will render
*x* and *y* *d*-connected.
Then find another node that you can condition on to
*d*-separate *x* and *y* again.

This completes the definition of *d-*separation, and the
reader is invited to try it on some more intricate graphs.

**Typical application:**
Suppose we consider the regression
of *y* on *p, r* and *x,*

and suppose we wish to predict which coefficient in this regression
is zero. From the discussion above we can conclude immediately
that *c _{3}* is zero, because

**Remark on correlated errors:**
Correlated exogenous variables (or error terms)
need no special treatment. These are represented
by bi-directed arcs (double-arrowed) and their arrowheads are treated
as any other arrowhead for the purpose of path tracing.
For example, if we add to the graph above a bi-directed arc
between *x* and *t,* then *y* and *x* will no longer
be *d-*separated (by *Z*={*r, p*}), because the path
*x-t-u-v-y* is *d-*connected — the collider at *t* is
unblocked by virtue of having a descendant, *p,* in *Z.*