Lecture 24, Causal Discovery

ly, the algorithm works as follows: • Start with a complete undirected graph on all variables. • For each pair of variables, see if conditioning on some set of variables makes them conditionally independent; if so, remove their edge. • Identify all colliders by checking for conditional dependence; orient the edges of colliders. • Try to orient undirected edges by consistency with already-oriented edges; do this recursively until no more edges can be oriented. Pseudo-code is in the appendix. Call the result of the SGS algorithm Ĝ. If all of the assumptions above hold, and the algorithm is correct in its guesses about when variables are conditionally independent, then Ĝ = G. In practice, of course, conditional independence guesses are really statistical tests based on finite data, so we should write the output as Ĝn, to indicate that it is based on only n samples. If the conditional independence test is consistent, then lim n→∞ Pr ( Ĝn 6= G ) = 0 In other words, the SGS algorithm converges in probability on the correct causal structure; it is consistent for all graphs G. Of course, at finite n, the probability of error — of having the wrong structure — is (generally!) not zero, but this Pearl (2009); Janzing (2007) makes a related suggestion). Arguably then using order in time to orient edges in a causal graph begs the question, or commits the fallacy of petitio principii. But of course every syllogism does, so this isn’t a distinctively statistical issue. (Take the classic: “All men are mortal; Socrates is a man; therefore Socrates is mortal.” How can we know that all men are mortal until we know about the mortality of this particular man, Socrates? Isn’t this just like asserting that tomatoes and peppers must be poisonous, because they belong to the nightshade family of plants, all of which are poisonous?) While these philosophical issues are genuinely fascinating, this footnote has gone on long enough, and it is time to return to the main text.