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201810݄25 Watanebe, 6.3 @unaoya
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౷ܭతֶश 1. ਅͷq(x)·͔ͨͦ͜Βੜ͞ΕΔαϯ ϓϧΛ༧ଌ͍ͨ͠ 2. Ϟσϧp(x|w)ͱύϥϝʔλۭؒW Λઃఆ 3. ༩͑ΒΕͨαϯϓϧ͔Βύϥϝʔλ্ۭؒͷଌ ͓Αͼ༧ଌ ˆ p(x)Λܾఆ ϞσϧΛධՁ͍ͨ͠
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Bayes quartet ࢛ͭͷϞσϧͷධՁج४ Bayesਪଌ Gibbsਪଌ Bg Gg αϯϓϧ Bt Gt ͜ΕΒαϯϓϧDn ʹґଘͨ֬͠ม
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6.3ͷඪ • αϯϓϧn → ∞ʹ͓͚Δۙڍಈ • αϯϓϧʹ͍ͭͯͷظ • ͜ΕΒͷ4ͭͷؒͷؔ ʹ͍ͭͯௐΔɻ
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ࠓGg ʹ͍ͭͯ
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Gibbsਪଌ ࣄޙʹैͬͯύϥϝʔλ ˆ wΛαϯϓϦϯά ͠ɺˆ p(x) = p(x| ˆ w)Λ༧ଌͱ͢Δɻ ൚ԽޡࠩGg q(x)ͱ ˆ p(x)ͷKL divergenceΛwʹ͍ͭͯࣄޙ p(w|Dn )Ͱੵͨ͠ͷ Gg = ∫ W K(w)p(w|Dn )dw
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n → ∞ͰnGg ͕ͲͷΑ͏ͳ֬มʹऩଋ͢ Δ͔ʁ ओཁ߲ Gg (ϵ) = ∫ K(w)≤ϵ K(w)p(w|Dn )dw ∫ K(w)≤ϵ p(w|Dn )dw ิ 1 (Lemma 6.3). nGg − nGg (ϵ)0ʹ֬ ऩଋ͢Δɻ
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ओཁ߲ͷධՁ Gg (ϵ) = ∫ K(w)≤ϵ K(w)p(w|Dn )dw ∫ K(w)≤ϵ p(w|Dn )dw = E[K(w)|K(w)≤ϵ ] ͷධՁΛ͍͕͍ͨ͠͠ɻ ಛҟղফΛ͏
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ඪ४ܗ f(x, g(u)) = log( q(x) p(x|g(u)) ) = a(x, u)uk ͱ͠ K(g(u)) = u2k Kn (g(u)) = u2k − 1 √ n ukξn (u) ξn (u) = 1 √ n n ∑ i=1 {a(Xi , u) − EX [a(X, u)]}
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ξn (u)αϯϓϧDn ʹґଘͨ֬͠աఔɻξn Gaussաఔξʹ๏ଇऩଋ͢Δɻ ิ 2 (6.51). G∗ g (ξn ) = Ey,t [t|ξn ] ͱఆٛ͢Δͱ nGg (ϵ) − G∗ g (ξn ) →P 0
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ಛҟղফM ্ͰͷੵEy,t ͱEu • ξ(u): M ্ͷC1 ڃؔʢαϯϓϧͷ֬աఔʣ • f(u): M ্ͷؔʢK(w)ʹର͠f(u) = u2kʣ • 0 ≤ σ ≤ 1 Eσ u [f(u)|ξ] = ∑ α∈A ∫ [0,b]d f(u)Z(u, ξ)du ∑ α∈A ∫ [0,b]d Z(u, ξ)du A࠲ඪۙͷʢ༗ݶʣू߹ɻ
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Z(u, ξ) uhϕ∗(u) exp(−βnu2k+β √ nukξ(u)−σuka(X, u)) ࣄޙp(w|Dn )ͱͷؔɻ • uhϕ∗(u)͕ࣄલϕ(w)ʹରԠɻ • σ = 0ͱͯ͠ Z0 n p(w|Dn ) = exp(−nβKn (w)) = exp(−βnu2k + β √ nukξn (u))
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ิ 3 (6.41). Gg (ϵ) = E0 u [u2k|ξn ] u2k = K(g(u))Ͱ͋ͬͨɻ
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ຊ࣭త෦ ࠲ඪu = (x, y)ͱຊ࣭త෦A∗ ⊂ AʢK(w)ͷ ಛҟղফ͔Βܾ·Δʣ Ey,t [f(y, t)|ξ] = ∑ α∈A∗ ∫ dt ∫ [0,b]d−m f(y, t)Z0 (y, t, ξ)du ∑ α∈A∗ ∫ dt ∫ [0,b]d−m Z0 (y, t, ξ)du
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Z0 (y, t, ξ) = γb yµtλ−1 exp(−βt+β √ tξ0 (y))ϕ∗ 0 (y) ิ 4 (Lemma 6.6, p = 1, f = 1, ξ = ξn ). |E0 u [nu2k|ξn ] − Ey,t [t|ξn ]| ≤ D(ξn , 1, ϕ∗) log n ͜Εͷূ໌ʹ4ষͰͷؔͷܭࢉΛ༻͍Δɻ
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ऩଋઌͷߏ ఆٛ 1 (6.46). M ্ͷؔψ(u)ʹର͠ G∗ g (ψ) = Ey,t [t|ψ] ͜ΕΛ͖ͬͯͬ͞ͷิΛॻ͖͢ͱ ิ 5. |nGg (ϵ) − G∗ g (ξn )| ≤ D(ξn , 1, ϕ∗) log n
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݁ ξn ͕ξʹ๏ଇऩଋ͢Δ͜ͱ͔Β • ิ4Λ༻͍ͯnGg (ϵ) − G∗ g (ξn ) → 0 • G∗ g (ξn ) − G∗ g (ξ) → 0 ͕ݴ͑Δɻ શͯ߹ΘͤͯnGg − G∗ g (ξ) → 0͕ূ໌Ͱ͖ͨɻ G∗ g (ξ)ξʹґଘͨ֬͠มͰ͋Δɻ
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4ষͷ෮श θʔλؔ ։ू߹U ⊂ Rd ্ͷඇෛղੳతؔK(w)ͱίϯ ύΫτC∞ ؔϕ(w)ʹର͠ɺ ζ(z) = ∫ K(w)zϕ(w)dw ͱఆٛ͢Δɻ͜ΕͷۃͷҐஔͱͦͷҐͲͷΑ ͏ͳใΛ͔࣋ͭʁ
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ঢ়ଶີؔ ζ(z)ͷٯMellinมঢ়ଶີؔ v(t) = ∫ δ(t − K(w))ϕ(w)dw Ͱ͋ΔɻMellinมͷཧʹΑΓɺ͜Εͷൃࢄͷ Φʔμʔ͕ζ(z)ͷۃͷҐஔͱରԠɻ
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ؔ ঢ়ଶີؔv(t)ͷLaplaceม Z(n) = ∫ exp(−nK(w))ϕ(w)dw Λؔͱ͍͏ɻ͜Ε͕6ষલͰௐ͍ͯͨ ͷɻ
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ಛҟֶशཧ Remark 4.4ʹ͋ΔΑ͏ʹ Z = ∫ exp(−nβK(w)+β √ nK(w)ξ(w))ϕ(w)dw ͷn → ∞ͰͷڍಈΛௐ͍ͨɻ
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K ʹ͍ͭͯͷಛҟղফʹΑΓɺnormal crossing ͷ߹ͷੵZ(n, ξ, ϕ)Λ༻͍ͯ Z = ∑ α Z(n, ξ ◦ gα , ϕ ◦ gα |g′ α |) ͱॻ͚ΔͷͰɺZ(n, ξ, ϕ)ʹ͍ͭͯௐΔͷ͕4.4 ͷඪɻ
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Zp(n, ξ, ϕ) = ∫ [0,b]r dx ∫ [0,b]s dyK(X, y)pxhyh′ ϕ(x, y) exp(−nβK(x, y)2 + √ nβK(x, y)ξ(x, y)) ͱఆٛ͢Δɻ
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͞Βʹ͜ΕͰξ = 0, ϕ = 1ͱஔ͍ͨͷΛ Zp(n) = ∫ [0,b]r dx ∫ [0,r]s dy K(x, y)pxh, yh′ exp(−nβK(x, y)2) ͱॻ͘͜ͱʹ͢Δɻ
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ఆཧ 1 (Theorem 4.7). hi + 1 2ki = λ ͕ҰఆͰ h′ j + 1 2k′ j > λ ͱ͢ΔɻK(x, y) = xkyk′ ͷͱ͖ʹɺ͋Δ a1 , a2 > 0͕ଘࡏͯ͠ҙͷnʹରͯ͠ a1 (log n)r−1 nλ+p ≤ Zp(n) ≤ a2 (log n)r−1 nλ+p
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ฏۉޡࠩؔK(w)ͱࣄલϕ(w)ʹରͯ͠ θʔλؔ ζ(z) = ∫ K(w)zϕ(w)dw Λఆٛ͢Δɻ͜Εͷۃͷใ͔ΒK ͷʁಛҟ ͷ࣮ରᮢ͕ͱ·Δɻ͜Ε͕ࣗ༝ΤωϧΪʔ ͓Αͼ൚Խଛࣦͷཧ͕໌Β͔ʹͳΔɻ
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0.1 ٙ ਖ਼ଇͳ߹ͷܭࢉΛΔɻಛʹ͜ͷ࣌ਖ਼ଇੑΛͲ ͜Ͱ͏͔ɻਖ਼ଇͳ߹ɺαϯϓϧ͕ଟ͍͜ͱ͕ Ծఆ͞ΕΔʁ