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# Prml

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### Prml

1. 1. PRML 2.4 (id:syou6162) June 13, 2009 (id:syou6162) PRML 2.4
2. 2. 2.4 2 x η p(x|η) = h(x)g(η) exp {ηT µ(x)} x η (natural parameter) u(x) x (id:syou6162) PRML 2.4
3. 3. Figure: Figure: Figure: Figure: t (id:syou6162) PRML 2.4
4. 4. ? hoge ← (id:syou6162) PRML 2.4
5. 5. hoge ( ) η ηML ( ) (id:syou6162) PRML 2.4
6. 6. ∈ (1/2) : p(x|µ) = Bern(x|µ) = µ x (1 − µ)1−x → p(x|η) = h(x)g(η) exp {ηT µ(x)} p(x|µ) = (1 − µ) exp {log( 1−µ )x} µ natural parameter η η= log( 1−µ ) µ µ µ = σ(η) 1 σ(η) = 1+exp(−η) (id:syou6162) PRML 2.4
7. 7. ∈ (2/2) (2.194) p(x|η) = σ(−η) exp(ηx) p(x|η) = h(x)g(η) exp {ηT µ(x)} η = log( 1−µ ) µ µ(x) = x h(x) = 1 g(η) = σ(−η) (id:syou6162) PRML 2.4
8. 8. ∈ : M x M p(x|µ) = k=1 µk k = exp { k=1 xk log µk } (2.194) p(x|µ) = exp(ηT x) ηk = log uk η = (η1 , · · · , ηM )T p(x|η) = h(x)g(η) exp {ηT µ(x)} µ(x) = x h(x) = 1 g(η) = 1 (id:syou6162) PRML 2.4
9. 9. uk (k = 1, · · · , M) M k=1 uk =1 → uk M−1 M−1 (id:syou6162) PRML 2.4
10. 10. M     exp  xk log uk        k=1    M−1    = exp  xk log uk + x M log u M        k=1    M−1  M−1    M−1      = exp  xk log uk + 1 − xk  log 1 −      µk                     k=1 k=1 k=1   M−1 M−1  M−1   M−1       = exp  xk log uk − xk log 1 −  + log 1 −      µk  µk                   k=1 k=1 k=1 k=1   M−1    M−1   µk   = exp  xk log   + log 1 −        µk      M−1   1 − j=1 µ j            k=1 k=1  M−1    M−1      µk  = 1 − µk  exp  xk log            M−1   1 − j=1 µ j             k=1 k=1  (id:syou6162) PRML 2.4
11. 11. (1/2) M−1 M−1 1− k=1 µk exp k=1 xk log 1− µk M−1 µj j=1 log 1− µk M−1 µj = ηk j=1 k exp(ηk ) µk = 1+ M−1 exp(η j ) j=1 4 (id:syou6162) PRML 2.4
12. 12. (2/2) M−1 −1 p(x|η) = 1 + j=1 exp(η j ) exp(ηT x) natural parameter η = (η1 , · · · , ηM−1 )T p(x|η) = h(x)g(η) exp {ηT µ(x)} µ(x) = x h(x) = 1 M−1 −1 g(η) = 1 + j=1 exp(η j ) (id:syou6162) PRML 2.4
13. 13. ∈ : 1 p(x|µ, σ) = 1 exp {− 1 σ2(x − µ)2} 2 (2πσ2 ) 2 (2.194) 1 1 1 2 p(x|µ, σ) = 1 exp {− 2σ2 x2 + µ σ2 x − 2σ2 µ} (2πσ2 ) 2 p(x|η) = h(x)g(η) exp {ηT µ(x)} µ/σ2 η= −1/2σ2 x µ(x) = x2 1 h(x) = (2π)− 2 1 η2 g(η) = (−2η2 ) 2 exp ( 4η12 ) (id:syou6162) PRML 2.4
14. 14. 2.4.1 η p(x|η) = h(x)g(η) exp {ηT µ(x)} → g(η) h(x) exp {ηT u(x)}dx + g(η) h(x) exp {ηT u(x)}u(x)dx = 0 − log g(η) = E[u(x)] − log g(η) = cor[u(x)] (id:syou6162) PRML 2.4
15. 15. & i.i.d. X = (x1, · · · , xn ) : p(X|η) = L(η; X) = N N n=1 h(xn ) g(η)N exp ηT n=1 u(xn ) : 1 N − g(ηML ) = N n=1 u(xn ) N n=1 u(xn ) (sufﬁcient statistic) (id:syou6162) PRML 2.4
16. 16. u(x) = x N n=1 xn u(x) = (x, x2 )T N N ( n=1 xn, n=1 x2 )T n ( ) 8 (id:syou6162) PRML 2.4
17. 17. 2.4.2 : ( ) ( ) (id:syou6162) PRML 2.4
18. 18. p(η|χ) = f (χ, ν)g(η)ν exp {νηT χ} = × = p(η|χ) × p(X|η) = f (χ, ν)g(η)ν exp {νηT χ}  N   N   g(η)N exp ηT   h(xn) u(xn )      ×           n=1 n=1     N   T    ∝ g(η) µ+N exp η     u(xn) + νχ       n=1  (id:syou6162) PRML 2.4
19. 19. ν µ(x) χ (id:syou6162) PRML 2.4
20. 20. 2.4.3 ν ν → (id:syou6162) PRML 2.4
21. 21. 2.4.3 λ K 1 p(λ) = K x 1 p(x) = b−a constant 1 1 2 (id:syou6162) PRML 2.4
22. 22. 1: 1 → (improper prior) ( ) (id:syou6162) PRML 2.4
23. 23. (2.3.6 ) p(µ) = N(µ0 , σ2) 0 → µ0 = 0 → σ0 → ∞ → σ2 Nσ20 σ2 /σ2 0 N µN = µ Nσ2 +σ 0 + µ Nσ2 +σ ML = N+σ/σ2 0 µ + µ N+σ/σ2 ML → 0 0 0 0 µML 1 1 N N σ2 = σ2 + σ2 → σ2 N 0 (id:syou6162) PRML 2.4
24. 24. (1/2) h(λ) λ = η2 η = h(η2) ˆ pλ (λ) λ = η2 pη (η) = pλ (λ)| dλ | = pλ (η2 )2η ∝ η dη pλ (λ) pη (η) (id:syou6162) PRML 2.4
25. 25. (2/2) ? (translation invariance) (scale invariance) (id:syou6162) PRML 2.4
26. 26. p(x|µ) = f (x − µ) x x = x+c ˆ µ =µ+c ˆ p(x|µ) = f ( x − µ) = f ((x + c) − (µ + c)) = p( x|µ) ˆ ˆ ˆˆ (id:syou6162) PRML 2.4
27. 27. A≤µ≤B A−c ≤µ≤ B−c B B−c B A p(µ)dµ = p(µ)dµ = A−c A p(µ − c)dµ A B → p(µ − c) = p(µ) p(µ) ? (id:syou6162) PRML 2.4
28. 28. µ µ0 = 0 σ2 Nσ20 µN = µ Nσ2 +σ 0 + µ Nσ2 +σ ML = 0 0 σ2 /σ2 0 N N+σ/σ2 0 µ + µ N+σ/σ2 ML → µML 0 0 σ2 → 0 ∞ µ (id:syou6162) PRML 2.4
29. 29. 1 x σ>0 p(x|σ) = σ f (σ) x x = cx ˆ σ = cσ ˆ 1 x 1 cx 1 cx p(x|σ) = σ f ( σ ) = σ f ( cσ ) = σ f(σ) = ˆ 1 x ˆ 1 x ˆ cσ f ( σ ) = σ f ( σ )p( x|σ) ˆ ˆ ˆ ˆˆ (id:syou6162) PRML 2.4
30. 30. A≤σ≤B A/c ≤ µ ≤ B/c B B/c B A p(σ)dσ = p(σ)dσ = A/c A p( 1 σ) 1 dσ c c A B → p(σ) = p( 1 σ) 1 c c 1 p(σ) ∝ σ ? ? f (x) = 1/x x = 10 f (10/2) × 2 = f (5) × 1 = 0.2 × 1 2 1 2 = 0.1 = f (10) 1/x (id:syou6162) PRML 2.4
31. 31. µ σ N(x|µ, σ2 ) ∝ σ−1 exp{−( x/σ)2 } ˆ x= x−µ ˆ λ = 1/σ2 p(σ) ∝ 1/σ p(λ) ∝ 1/λ 3 3 p(λ) = p(σ) × | dσ | = p(σ) × | − λ− 2 /2| ∝ 1/σ × λ− 2 = dλ 1/2 −3 2 = 1/λ λ ×λ √ σ = 1/λ 3 dσ/dλ = −λ− 2 /2 λ (2.146) Gam(λ|a0 , b0 ) (id:syou6162) PRML 2.4
32. 32. Figure: (2.146) Gam(λ|a, b) a b a=b=0 (id:syou6162) PRML 2.4
33. 33. a0 = 0 b0 = 0 a N = a0 + N2 N bN = 1 n=1 (xn − µ)2 = b0 + N σ2 2 2 ML a0 = 0 b0 = 0 (id:syou6162) PRML 2.4