(At the request of many readers)

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".

**Example 1**

This graph contains one collider, at *t.*
The path *x-r-s-t* is unblocked, hence *x* and *t* are
*d*-connected. So is also the path *t-u-v-y,* hence
*t* and *y* are *d-*connected, as well as the pairs
*u* and *y, t* and *v, t* and *u, x* and *s* etc....
However, *x* and *y* are not *d-*connected; there is no
way of tracing a path from *x* to *y* without traversing
the collider at *t.*
Therefore, we conclude that *x* and *y* are *d-*separated,
as well as *x* and *v, s* and *u, r* and *u,* etc.
(The ramification is that the covariance terms corresponding
to these pairs of variables will be zero, for every choice
of model parameters).

**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.*

**Example 2**

Let *Z* be the set {*r, v*} (marked by circles 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.

**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.

**Example 3**

Let *Z* be the set {*r, p*} (again, marked with circles).
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*).

This completes the definition of *d-*separation, and the
reader is invited to try it on some more intricate graphs,
such as those shown in Figure 1.3

**Typical application:**

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

**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.*

Next Discussion (Bessler: *Bertrand Russell on Causality*)