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- 1. . . Learning BNs with Discrete and Continuous Variables Joe Suzuki Osaka University PGM 2014 @Utrecht Joe Suzuki (Osaka University) Learning BNs with Discrete and Continuous Variables PGM 2014 @Utrecht 1 / 27
- 2. Road Map Road Map 1. Learning BNs 2. When a Density exists 3. The General Case 4. Practical BN Learning with Discrete and Continuous Variables 5. Conclusion Joe Suzuki (Osaka University) Learning BNs with Discrete and Continuous Variables PGM 2014 @Utrecht 2 / 27
- 3. Learning BNs Factor P(X; Y ; Z) P(X)P(Y )P(Z) P(X)P(Y ; Z) P(Y )P(Z; X) P(X; Y )P(X; Z) P(Z)P(X; Y ) P(X) P(X; Y )P(Y ; Z) P(Y ) P(X; Z)P(Y ; Z) P(Z) P(Y )P(Z)P(X; Y ; Z) P(Y ; Z) P(Z)P(X)P(X; Y ; Z) P(Z;X) P(X)P(Y )P(X; Y ; Z) P(X; Y ) P(X;Y ; Z) Joe Suzuki (Osaka University) Learning BNs with Discrete and Continuous Variables PGM 2014 @Utrecht 3 / 27
- 4. Learning BNs BNs for X; Y ; Z X m Xm (1) (2) (3) (4) m X m X AAU Ym- Zm m X A AK m Y m Z m X AAU Y m- Z m m X AKA Y m Z m m X m X AAU Y m Z m X Y m Z m m Y m Z m X m m - Y m Z m Y m Z Ym- Zm m AKA Y m Z (5) (6) (7) (8) (9) (10) (11) Markov Equivalence (5) (8) m X AAU Y m Z m m X AAU Y m Z m m X AKA Y m Z m m X AKA Y m Z m Joe Suzuki (Osaka University) Learning BNs with Discrete and Continuous Variables PGM 2014 @Utrecht 4 / 27
- 5. Learning BNs The Problem Identify the BN structures among (1)-(11) from n examples xn = (x1; ; xn) ; yn = (y1; ; yn) ; zn = (z1; ; zn) X = x1 Y = y1 Z = z1 X = x2 Y = y2 Z = z2 ... ... ... X = xn Y = yn Z = zn 9= ; i:i:d: (N̸= 3 variables will be considered) Joe Suzuki (Osaka University) Learning BNs with Discrete and Continuous Variables PGM 2014 @Utrecht 5 / 27
- 6. Learning BNs In any database, some
- 7. elds are discrete and others continuous Discrete Only: fMale,Femaleg fMarried,Unmarriedg Age Continous Only: Height Weight Footsize Discrete/Continous: Height Weight Age . BN Structure Learning with both Discrete and Continuous Variables. . .Why do you solve the easiest but unrealistic problems ? Joe Suzuki (Osaka University) Learning BNs with Discrete and Continuous Variables PGM 2014 @Utrecht 6 / 27
- 8. Learning BNs Previous Works Independent Testing PC Algorithm (Spirtes, 2000) etc. Bayesian the problem can be classi
- 9. ed into Factor Scores Structure Scores given Factor Scores to
- 10. nd the Best almost all assume Descrete only Gaussian only Descrete and Continuous are mixed: no performance guaranteed 1. Friedman and Goldszmidt (UAI-97) decretizing continous vaiables 2. the R package by Bottcher (2003) assuming Gaussian 3. Monti and Cooper (NIPS-96) approxmating neural networks Shenoy (PGM-12): mxtures of polynomials only for density estimation Joe Suzuki (Osaka University) Learning BNs with Discrete and Continuous Variables PGM 2014 @Utrecht 7 / 27
- 11. Learning BNs N = 2 (Bayesian Independence Test) w(): the Prior over p: the Prior of X ?? Y Qn(xn) := ∫ Pn(xnj)w()d ; Qn(yn) := ∫ Pn(ynj)w()d ; Qn(xn; yn) := ∫ Pn(xn; ynj)w()d ; The Posterior Prob. of X ?? Y given xn; yn: P(X ?? Y jxn; yn) = pQn(xn)Q(yn) pQn(xn)Q(yn) + (1