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CONFOUNDING EQUIVALENCE

CONFOUNDING EQUIVALENCE. Judea Pearl – UCLA, USA Azaria Paz – Technion, Israel (www.cs.ucla.edu/~judea/). L. T. Z. W 1. W 4. ?. Z  T. W 2. W 3. V 1. V 2. X. Y. Bias =. CONFOUNDING EQUIVALENCE. PROBLEM: When would two measurements be

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CONFOUNDING EQUIVALENCE

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  1. CONFOUNDING EQUIVALENCE Judea Pearl – UCLA, USA Azaria Paz – Technion, Israel (www.cs.ucla.edu/~judea/)

  2. L T Z W1 W4 ? Z T W2 W3 V1 V2 X Y Bias = CONFOUNDING EQUIVALENCE PROBLEM:When would two measurements be equally effective for reducing confounding bias?

  3. CONFOUNDING EQUIVALENCE DEFINITION AND SOLUTION • DEFINITION: • T and Z are c-equivalent if • DEFINITION (Markov boundary): • Markov boundary Sm of S (relative to X) is the minimal subset of S that d-separates X from all other members of S. • THEOREM: • Z and T are c-equivalent iff • Zm=Tm, or • Z and T are admissible (i.e., satisfy the back-door • condition)

  4. T Z Z T ARE Z AND T CONFOUNDING EQUIVALET? ANSWER:Yes! Because both are admissible W1 W4 W2 W3 V1 V2 X Y

  5. L T Z Z T ARE Z AND T CONFOUNDING EQUIVALET? ANSWER:No! None is admissible and W1 W4 W2 W3 V1 V2 X Y

  6. A PUZZLE: The definition of c-equivalence is purely statistical. The graphical criterion we found invokes “admissibility,” which is causal – why? Corollary 1: c-equivalence is an invariant property of M-equivalent graphs Corollary 2: There ought to be a graphical criterion that does not invoke “admissibility,” based ond-separation only. Indeed, Theorem:Z~ T iff and

  7. EXAMPLE: TESTING IF Z~T WITHOUT INVOKING ADMISSIBILITY L Z T Z T Z T W1 W4 W2 W3 V1 V2 X Y Z T W1 W4 W2 W3 V1 V2 X Y

  8. V W1 W2 V W1 W2 Z1 Z2 Z1 Z2 X Y X Y APPLICATIONS TO MODEL TESTING Are these two models Markov-equivalent? (a) (b) Answer:No,  a CI that holds in (b) but not in (a) Answer:No, because {Z1,W1,W2}~ {W1,W2,Z2} holds in (b) but not in (a) Which is easier to detect? Which is easier to test?

  9. WHICH IS EASIER TO TEST WHEN Z IS HIGH DIMENSIONAL? V W1 W2 V W1 W2 Z1 Z2 Z1 Z2 X Y X Y    (a) (b) Conditional independence? or c-equivalence? c-equivalence can invoke propensity scores (Scalars): And we can ignore the outcome process

  10. CONCLUSIONS • c-equivalence provides: • A simple, polynomial time test for deciding • whether one set of measurements has the • same bias-reducing potential as another. • A powerful tool for: • Testing from sampled data, the validity of a • given graphical model and, eventually, and • systematic search for graph structures.

  11. CONCLUSIONS • c-equivalence provides: • A simple, polynomial time test for deciding • whether one set of measurements has the • same bias-reducing potential as another. • A powerful tool for: • Testing from sampled data, the validity of a • given graphical model and, eventually, and • systematic search for graph structures.

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