Preliminaries: Factors

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Presentation transcript:

Preliminaries: Factors Probabilistic Graphical Models Introduction Preliminaries: Factors

Factors A factor (X1,…,Xk) Scope = {X1,…,Xk}  : Val(X1,…,Xk) → R

Joint Distribution P(I,D,G) I D G Prob. i0 d0 g1 0.126 g2 0.168 g3 d1 0.009 0.045 i1 0.252 0.0224 0.0056 0.06 0.036 0.024 P(I,D,G)

Unnormalized measure P(I,D,g1) Prob. i0 d0 g1 0.126 d1 0.009 i1 0.252 0.06 P(I,D,g1)

Conditional Probability Distribution (CPD) 0.3 0.08 0.25 0.4 g2 0.02 0.9 i1,d0 0.7 0.05 i0,d1 0.5 g1 g3 0.2 i1,d1 i0,d0 P(G | I,D) Explain intuition Conditioning variables Conditioned variables

General factors A B  a0 b0 30 b1 5 a1 1 10

Factor Product a1 b1 c1 0.5·0.5 = 0.25 c2 0.5·0.7 = 0.35 b2 0.8·0.1 = 0.08 0.8·0.2 = 0.16 a2 0.1·0.5 = 0.05 0.1·0.7 = 0.07 0·0.1 = 0 0·0.2 = 0 a3 0.3·0.5 = 0.15 0.3·0.7 = 0.21 0.9·0.1 = 0.09 0.9·0.2 = 0.18 a1 b1 0.5 b2 0.8 a2 0.1 a3 0.3 0.9 b1 c1 0.5 c2 0.7 b2 0.1 0.2

Factor Marginalization b1 c1 0.25 c2 0.35 b2 0.08 0.16 a2 0.05 0.07 a3 0.15 0.21 0.09 0.18 a1 c1 0.33 c2 0.51 a2 0.05 0.07 a3 0.24 0.39

Factor Reduction a1 b1 c1 0.25 c2 0.35 b2 0.08 0.16 a2 0.05 0.07 a3 a3 0.15 0.21 0.09 0.18 a1 b1 c1 0.25 c2 0.35 b2 0.08 0.16 a2 0.05 0.07 a3 0.15 0.21 0.09 0.18 a1 b1 c1 0.25 b2 0.08 a2 0.05 a3 0.15 0.09

Why factors? Fundamental building block for defining distributions in high-dimensional spaces Set of basic operations for manipulating these probability distributions