Consider the following definition (Pearl 2009):
Definition (Pearl 2009)
Let \(V = \{V_1, V_2, \ldots \}\) be a finite set of variables.
Let \(P(\cdot)\) be a joint probability function over the variables in \(V\), and let \(X\), \(Y\), \(Z\) stand for any three subsets of the variables in \(V\).
The sets \(X\) and \(Y\) are said to be conditionally independent given \(Z\) if
\[P(x | y, z) = P(x|z)\] whenever \(P(y,z) > 0\).
Denoting this ternary relation as \[(X \mathrel{\unicode{x2AEB}} Y \vert Z)\], we have the following properties below.
The first property is symmetry.
\[(X \mathrel{\unicode{x2AEB}} Y \vert Z) \implies (Y \mathrel{\unicode{x2AEB}} X \vert Z)\]
While this is a form of symmetry, it is not the same as strongly symmetric when describing any given \(n\)-ary relation. If it were, we would have to accept propositions such as \[(X \mathrel{\unicode{x2AEB}} Y \vert Z) \implies (Y \mathrel{\unicode{x2AEB}} Z \vert Y)\]. But this cannot be so for the same type of reason as to why \(Pr(A\vert B) \neq Pr(B \vert A)\) for at least some probability measure \(Pr\) on events \(A\) and \(B\).
The second property is decomposition,
\[(X \mathrel{\unicode{x2AEB}} (Y, W) \vert Z) \implies (Y \mathrel{\unicode{x2AEB}} X \vert Z)\],
Pearl 2009 seems to use \(YW\) instead of \((Y,W)\), giving a false impression that we might be considering some kind of multiplication. See Notational confusion about conditional independence in Pearl 2009 for more information.
The third property is weak union.
\[(X \mathrel{\unicode{x2AEB}} (Y, W) \vert Z) \implies (Y \mathrel{\unicode{x2AEB}} X \vert (Z,W))\]
The fourth property is contraction.
\[(X \mathrel{\unicode{x2AEB}} Y | Z) \land (X \mathrel{\unicode{x2AEB}} W | (Z,Y)) \implies (X \mathrel{\unicode{x2AEB}} (Y,W) \vert Z)\]
Finally the fifth property is intersection.
\[(X \mathrel{\unicode{x2AEB}} W \vert (Z, Y)) \land (X \mathrel{\unicode{x2AEB}} Y \vert (Z, W)) \implies (X \mathrel{\unicode{x2AEB}} (Y,W) \vert Z)\]