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.
D-separation implies independence (Pearl, 1988)
D-connection does not imply dependence
p(Y=0|X=0) = p(Y=0|X=1)
WSampling Local Distributions
If we randomly sample local parameters, how
often will the joint be perfect?
Probability of sampling non-perfect
distribution is measure zero
(Meek, 1995 and Spirtes et al., 2001)
Extend to finite-bit representations:
upper bound on probability of non-perfect
Required for “gold standard” experimental
evaluation of structure learning
Renormalized Delta Functions
YIndependence = 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!
For a fixed X, at most three roots (3 roots per column)
For a fixed Y, at most two roots (2 roots per row)
Y has three states: two independent parameters
Theorem does not apply.
Sample new parameters independently from Beta distribution
Sample “normal” parameters from a Dirichlet distribution
Simple Extension to S-Z:
Bound is a function of the maximum density height
16 vars, 4 states, 64-bit numbers p(non-perfect) ≤ 1/232