Conditional independence notation

Ten years ago I wrote a blog post that concludes with this observation: The ideas of being relatively prime, independent, and perpendicular are all related, and so it makes sense to use a common symbol to denote each.

This post returns to that theme, particularly looking at independence of random variables.

History Graham, Knuth, and Patashnik proposed using ⊥ for relatively prime numbers in their book Concrete Mathematics, at least by the second edition (1994).

Maybe it was in their first edition (1988), but I don’t have that edition.

Philip Dawid proposed a similar symbol ⫫ for (conditionally) independent random variables in 1979 [1].

As explained here, independent random variables really are orthogonal in some sense, so it’s a good notation.

Typography The symbol ⫫ (Unicode 2AEB, DOUBLE TACK UP) may or may not show up in your browser; it’s an uncommon character and your font may not have a glyph for it.

There’s no command in basic LaTeX for the symbol.

You can enter the Unicode character in XeTeX, or you can use import the symbol for various packages as discussed here.

A simple work-around is to use perp!!!perp This says to take two perpendicular symbols, and kern them together by inserting three negative spaces between them.

Graphoid axioms As an example of where you might see the ⫫ symbol used for conditional independence, the table below gives the graphoid axioms for conditional independence.

(They’re theorems, not axioms, but they’re called axioms because you could think of them as axioms for working with conditional independence at a higher level of abstraction.

) Note that the independence symbol ⫫ has higher precedence than the conditional symbol |.

That is, X ⫫ Y | Z means X is independent of Y, once you condition on Z.

The axioms above are awfully dense, but they make sense when expanded into words.

For example, the symmetry axiom says that if knowledge of Z makes Y irrelevant to X, it also makes X irrelevant to Y.

The decomposition axiom says that if knowing Z makes the combination of Y and W irrelevant to X, then knowing Z makes Y alone irrelevant to X.

The intersection axiom requires strictly positive probability distributions, i.


you can’t have events with probability zero.

*** [1] AP Dawid.

Conditional Independence in Statistical Theory.

Journal of the Royal Statistical Society.

Series B (Methodological), Vol.

41, No.

1 (1979), pp.


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