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Learning BNs with Discrete and Continuous Variables 
Joe Suzuki 
Osaka University 
PGM 2014 @Utrecht 
Joe Suzuki (Os...
Road Map 
Road Map 
1. Learning BNs 
2. When a Density exists 
3. The General Case 
4. Practical BN Learning with Discrete...
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(...
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...
Learning BNs 
The Problem 
Identify the BN structures among (1)-(11) from n examples 
xn = (x1;    ; xn) ; yn = (y1;    ; ...
Learning BNs 
In any database, some
elds are discrete and others continuous 
Discrete Only: fMale,Femaleg fMarried,Unmarriedg Age 
Continous Only: Height Weig...
Learning BNs 
Previous Works 
Independent Testing PC Algorithm (Spirtes, 2000) etc. 
Bayesian the problem can be classi
ed into 
Factor Scores 
Structure Scores given Factor Scores to
nd the Best 
almost all assume 
Descrete only 
Gaussian only 
Descrete and Continuous are mixed: no performance guaranteed...
Learning BNs 
N = 2 (Bayesian Independence Test) 
w(): the Prior over  
p: the Prior of X ?? Y 
Qn(xn) := 
∫ 
Pn(xnj)w()d ...
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2014 9-16

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PGM 2014 @ Utrecht, Netherlands

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2014 9-16

  1. 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. 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. 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. 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. 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. 6. Learning BNs In any database, some
  7. 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. 8. Learning BNs Previous Works Independent Testing PC Algorithm (Spirtes, 2000) etc. Bayesian the problem can be classi
  9. 9. ed into Factor Scores Structure Scores given Factor Scores to
  10. 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. 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

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