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Practically Perfect

Practically Perfect. Chris Meek Max Chickering. X. Y. Perfectness. D-separation implies independence (Pearl, 1988) but… D-connection does not imply dependence. p ( Y =0| X =0) = p ( Y =0| X= 1) (parameter cancellation). X. Z. Y. W. Sampling Local Distributions.

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Practically Perfect

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  1. Practically Perfect Chris Meek Max Chickering

  2. X Y Perfectness D-separation implies independence (Pearl, 1988) but… D-connection does not imply dependence p(Y=0|X=0) = p(Y=0|X=1) (parameter cancellation)

  3. X Z Y W Sampling Local Distributions If we randomly sample local parameters, how often will the joint be perfect?

  4. Previous Result Probability of sampling non-perfect distribution is measure zero (Meek, 1995 and Spirtes et al., 2001) This Paper Extend to finite-bit representations: upper bound on probability of non-perfect distribution

  5. Why do we care? • Previous theoretical result not applicable to real-world software • Correctness of learning: typically need perfectness Required for “gold standard” experimental evaluation of structure learning

  6. Continuous to Finite Sampling probability 0 1 Renormalized Delta Functions Sampling probability 11 00 01 10

  7. X Y Independence = Polynomial Root Base polynomial for “X independent of Y”: p(X=0,Y=0) · p(X=1,Y=1) – p(X=0,Y=1) · p(X=1,Y=0) Perfect Polynomial: Enumerate every d-connection in the model, and multiply corresponding base polynomials. Non-perfect only if perfect polynomial is zero!

  8. Schwartz-Zippel Theorem • Every polynomial has a finite number of roots • Sample each variable in the polynomial independently from a uniform over fixed set of values • Bound on the probability that we sample a root!

  9. Example For a fixed X, at most three roots (3 roots per column) For a fixed Y, at most two roots (2 roots per row) 1 Y 0 0 1 X

  10. Problem 1: Variational Dependence 1 p(Y=0) 0 0 1 p(Y=1) Y has three states: two independent parameters Theorem does not apply.

  11. Solution: Change Parameterization 1 0 0 1 Sample new parameters independently from Beta distribution = Sample “normal” parameters from a Dirichlet distribution

  12. Problem 2: Non-uniform Sampling 1 Solution: Simple Extension to S-Z: Bound is a function of the maximum density height 0 0 1

  13. Example • b-bit representation • Sample parameters from a uniform Dirichlet • m variables • No variable has more than rmax states 16 vars, 4 states, 64-bit numbers  p(non-perfect) ≤ 1/232

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