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ベイズ深層学習(4.1)
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catla
February 07, 2020
Science
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ベイズ深層学習(4.1)
catla
February 07, 2020
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Transcript
ϕΠζਂֶश αϯϓϦϯάʹجͮ͘ਪख๏ ܡɹঘً
αϯϓϦϯάʹجͮ͘ਪख๏ ˞ʮύλʔϯೝࣝͱػցֶशɹԼרʯষࢀߟ
αϯϓϦϯάʹجͮ͘ਪख๏ ɹ؍ଌσʔλΛ ɼඇ؍ଌͷมͷू߹ʢFHύϥϝʔλɼજࡏม Λ ͱͨ͠ͱ ͖ɼϕΠζਪʹΑΔ౷ܭղੳͰɼ֬Ϟσϧ Λઃܭ͢Δඞཁ͕͋Δɽ ɹ࣮ࡍʹɼ֬ϞσϧΛ༻ֶ͍ͯश༧ଌɼࣄޙ Λܭࢉͯ͠ߦΘΕΔɽ X
Z p(X, Z) p(Z|X) ղܾࡦ ɹෳࡶͳϞσϧʢFHχϡʔϥϧωοτʣɼ ͕ղੳతʹٻΊΒΕͳ͍͜ͱ͕ ଟ͍ɽ p(Z|X) ɹ ΛղੳతʹٻΊΔΘΓʹɼ͜ͷ͔ΒෳͷαϯϓϧΛಘΔ͜ͱͰɼ ͷಛੑΛௐΔɽ ɹͱ͍͏͜ͱ͔ΒɼαϯϓϦϯά͢Δํ๏Λࠓճษڧ͢ΔΑʂ p(Z|X)
ຊͷ༰ ୯७ϞϯςΧϧϩ๏ غ٫αϯϓϦϯά ࣗݾਖ਼نԽॏαϯϓϦϯά ʢϚϧίϑ࿈ϞϯςΧϧϩ๏ʣ ϝτϩϙϦεɾϔΠεςΟϯάε๏ ϝτϩϙϦε๏ ϋϛϧτχΞϯϞϯςΧϧϩ๏ ϥϯδϡόϯϞϯςΧϧϩ๏ ΪϒεαϯϓϦϯά
ຊͷ༰ ୯७ϞϯςΧϧϩ๏ غ٫αϯϓϦϯά ࣗݾਖ਼نԽॏ৺αϯϓϦϯά ʢϚϧίϑ࿈ϞϯςΧϧϩ๏ʣ ϝτϩϙϦεɾϔΠεςΟϯάε๏ ϝτϩϙϦε๏ ϋϛϧτχΞϯϞϯςΧϧϩ๏ ϥϯδϡόϯϞϯςΧϧϩ๏ ΪϒεαϯϓϦϯά
.$.$
୯७ϞϯςΧϧϩ๏ ɹ ʹର͢Δؔ ͷظɹ ΛٻΊ͍ͨɽɹ p(z) f(z) p(z) [ f(z)]
= ∫ f(z)p(z)dz త ঢ়گ ɹظ ͷղੳతͳੵܭࢉ͕ࠔɽ ɹ ͔ΒͷαϯϓϦϯά༰қɽ ∫ f(z)p(z)dz p(z) ख๏ ɹ Λेେ͖ͳͱͨ͠ͱ͖ɼ T z(1), z(2), …, z(T) ∼ p(z) ∫ f(z)p(z)dz ≈ 1 T T ∑ t=1 f(z(t)) ͔Β ݸαϯϓϦϯά ⟵ p(z) T
ɹύϥϝʔλ Λ࣋ͭϞσϧ ͷपล Λܭࢉ͢Δࡍʹ༻͢ Δ߹ɼ θ p(X, θ) =
p(X|θ)p(θ) p(X) p(X) = ∫ p(X|θ)p(θ)dθ = ∫ N ∏ n=1 p(xn |θ)p(θ)dθ = p(θ) [p(X|θ)] ≈ 1 T T ∑ t=1 N ∏ n=1 p(xn |θ(t)), (θ(1), …, θ(T) ∼ p(θ)) ɹظ ʹ͓͍ͯɼ ͔Βͷαϯϓϧ ͷൣғ෯͘ͱΔඞཁ͕͋ΓɼҰํ Ͱɼ ڱ͍ ͷൣғͰ͔͠େ͖ͳΛऔΒͳ͍έʔε͕ଟ͍ɽ ɹ ൚༻త͚ͩͲɼܭࢉޮ͕ѱ͍ɽ p(z) [ f(z)] p(z) z f(z) z ⟹ ୯७ϞϯςΧϧϩ๏
غ٫αϯϓϦϯά ɹີܭࢉ͕ࠔͳ֬ ͔ΒαϯϓϧΛಘΔɽɹ p(z) z(1), z(2), … ∼ p(z) త
ঢ়گ ɹਖ਼نԽ͞Ε͍ͯͳ͍ؔ ܭࢉՄೳɽͭ·Γɼ ɽ ˜ p(z)( = Zp ⋅ p(z)) ∫ ˜ p(z)dz ≠ 1 ख๏ ɹఏҊ Λઃఆ͢Δɽҙͷ ʹରͯ͠ɼ ɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹ ͱͳΔΑ͏ʹɼਖ਼ͷఆ ΛఆΊΔɽ q(z) z kq(z) > ˜ p(z) k ఏҊ ɹαϯϓϦϯά͕؆୯ʹߦ͑ΔΑ͏ͳ Ծͷɽ
غ٫αϯϓϦϯά ख๏ ͖ͭͮʣ ɹఏҊ ͔ΒαϯϓϧΛಘΔɽ ɹҰ༷ ͔ΒͷαϯϓϧΛಘΔɽ
ɹαϯϓϧ ͷड༰ʢBDDFQUʣغ٫ʢSFKFDUʣબɽ ɹɹɹ q(z) z(t) ∼ q(z) Uni(0,kq(z)) ˜ u ∼ Uni (0,kq(z(t))) z(t) if ˜ u > ˜ p(z(t)) then SFKFDU else BDDFQU ड༰ ∫ q(z) ˜ p(z) kq(z) dz = 1 k ∫ ˜ p(z)dz ߴ࣍ݩͷมͷαϯϓϦϯά͕ඞཁͳ߹ɼड༰͕ඇৗʹ͘ͳΔɽ
غ٫αϯϓϦϯά z ˜ p(z) ͷαϯϓϧΛغ٫αϯϓϧϦϯάͰ֫ಘ͢Δɽ ະɽ طɽ p(z) p(z) ˜
p(z) p(z)
غ٫αϯϓϦϯά z ˜ p(z) αϯϓϧ͕༰қͳఏҊ Λઃఆɽ p(z) q(z)
غ٫αϯϓϦϯά z ˜ p(z) kq(z) Λ෴͍͔Ϳ͞ΔΑ͏ʹ Λઃఆɽ ˜ p(z) k
kq(z) > ˜ p(z) q(z) × k
غ٫αϯϓϦϯά z ˜ p(z) kq(z) z(t) ఏҊ ͔ΒαϯϓϧΛಘΔɽ q(z) z(t)
∼ q(z) q(z)
غ٫αϯϓϦϯά z ˜ p(z) kq(z) z(t) kq(z(t)) ˜ u Ұ༷
͔ΒͷαϯϓϧΛಘΔ Uni(0,kq(z)) ˜ u ∼ Uni (0,kq(z(t)))
غ٫αϯϓϦϯά z ˜ p(z) kq(z) z(t) kq(z(t)) ड༰ غ٫ ˜
u αϯϓϧ ͷड༰ʢBDDFQUʣغ٫ʢSFKFDUʣબɽ z(t) if ˜ u > ˜ p(z(t)) then SFKFDU else BDDFQU ˜ p(z(t))
ࣗݾਖ਼نԽॏ৺αϯϓϦϯά ɹ ʹର͢Δؔ ͷظɹ Λ୯७ϞϯςΧϧϩ๏Α ΓޮతʹٻΊ͍ͨɽɹ p(z) f(z) p(z) [
f(z)] = ∫ f(z)p(z)dz త ঢ়گ ɹظ ͷղੳతͳੵܭࢉ͕ࠔɽ ɹ ͔ΒαϯϓϦϯάΛಘΒΕͳ͍ɽ ɹਖ਼نԽ͞Ε͍ͯͳ͍ؔ ܭࢉՄೳɽ ∫ f(z)p(z)dz p(z) ˜ p(z)( = Zp ⋅ p(z)) എܠ ɹغ٫αϯϓϦϯάΛ༻͍ͯɼ ΛΘͣʹαϯϓϧΛऔಘ͠ɼظ Λٻ ΊΔ͜ͱͰ͖Δ͕ɼ ͷ͕খ͞ͳྖҬʹαϯϓϧ͕ूத͢ΔՄೳੑ͕͋Δɽ ୯७ϞϯςΧϧϩ๏ͷܭࢉͷد༩͕গͳ͍ɽ ͷ͕େ͖͘ͳΔΑ͏ͳ ྖҬΛॏతʹαϯϓϧͨ͠ํ͕ޮ͕͍͍ɽ p(z) p(z) [ f(z)] f(z) ⟹ f(z)p(z)
ࣗݾਖ਼نԽॏ৺αϯϓϦϯά ɹ·ͣɼఏҊ Λઃఆ͢Δɽਖ਼نԽ͞Ε͍ͯͳ͍ؔ ɼҎԼͷΑ͏ʹද ͞ΕΔɽ ɹظͷܭࢉɼҎԼͷΑ͏ʹมܗͰ͖Δɽ q(z)
˜ p(z), ˜ q(z) p(z) = 1 Zp ˜ p(z), q(z) = 1 Zq ˜ q(z) ∫ f(z)p(z)dz = ∫ f(z) p(z) q(z) q(z)dz = q(z) [ f(z) p(z) q(z) ] = ∫ f(z) 1 Zp ˜ p(z) 1 Zq ˜ q(z) q(z)dz = Zq Zp q(z) [ f(z) ˜ p(z) ˜ q(z) ] ≈ Zq Zp 1 T T ∑ t=1 f(z(t)) ˜ p(z(t)) ˜ q(z(t)) = Zq Zp 1 T T ∑ t=1 f(z(t))w(t), w(t) = ˜ p(z(t)) ˜ q(z(t)) ख๏
ࣗݾਖ਼نԽॏ৺αϯϓϦϯά ɹ ະͷͳͷͰɼҎԼͷΑ͏ʹਖ਼نԽ߲ͷൺΛۙࣅ͢Δɽ ɹΑͬͯɼؔ ͷظ͕ۙࣅతʹಘΒΕΔɽ Zp
Zp Zq = ∫ ˜ p(z) Zq dz = ∫ ˜ p(z) ˜ q(z) q(z)dz = q(z) [ ˜ p(z) ˜ q(z) ] ≈ 1 T T ∑ t=1 w(t), z(1), …, z(T) ∼ q(z) f(z) ख๏ ͖ͭͮʣ ( ∵ p(z) = 1 Zp ˜ p(z) ) ( ∵ ∫ p(z)dz = 1 Zp ∫ ˜ p(z)dz = 1 )
ࣗݾਖ਼نԽॏ৺αϯϓϦϯά ɹ ະͷͳͷͰɼҎԼͷΑ͏ʹਖ਼نԽ߲ͷൺΛۙࣅ͢Δɽ ɹΑͬͯɼؔ ͷظ͕ۙࣅతʹಘΒΕΔɽ Zp
Zp Zq = ∫ ˜ p(z) Zq dz = ∫ ˜ p(z) ˜ q(z) q(z)dz = q(z) [ ˜ p(z) ˜ q(z) ] ≈ 1 T T ∑ t=1 w(t), z(1), …, z(T) ∼ q(z) f(z) ख๏ ͖ͭͮʣ ( ∵ p(z) = 1 Zp ˜ p(z) ) ( ∵ ∫ p(z)dz = 1 Zp ∫ ˜ p(z)dz = 1 ) ͳΜͷͨΊʹ Λ ͖࣋ͬͯͨΜͩΖ͏ʜ ˜ q(z)
ࣗݾਖ਼نԽॏαϯϓϦϯάʢޡهͷՄೳੑʣ ʮϕΠζਂֶशʯQ ɹࣗݾਖ਼نԽॏαϯϓϦϯάͷஈམͷ࠷ॳ ޡΓ ɹغ٫αϯϓϦϯάͱҟͳΔɼ ͔Βͷαϯϓϧʜʜ p(z) ɹ୯७ϞϯςΧϧϩ๏ͱҟͳΔɼ ͔Βͷαϯϓϧʜʜ p(z)
Ϛϧίϑ࿈ϞϯςΧϧϩ๏ʢ.$.$ʣ ɹغ٫αϯϓϦϯάͷɼߴ࣍ݩʹͳΔͱड༰͕ඇৗʹখ͘͞ͳΔ͜ͱɽ࣮ࡍ ʹɼ࣍ݩఔͷ؆୯ͳੵۙࣅʹ͔͠ద༻Ͱ͖ͳ͍ɽ ɹͰɼߴ࣍ݩۭؒͰޮతʹαϯϓϦϯά͢Δʹʜʜ ɹɹɹ Ϛϧίϑ࿈ϞϯςΧϧϩ๏ʢ.$.$ʣ͕ఏҊ͞Ε͍ͯΔɽ ⟹ ࣍Ϛϧίϑ࿈ ɹ֬มͷܥྻ
ʹରͯ͠ ͕Γཱͭͱ͖ͷܥྻ ͷ͜ͱɽ z(1), z(2), … p(z(t) |z(1), z(2), …, z(t−1)) = p(z(t) |z(t−1)) z(1), z(2), … άϥϑΟΧϧϞσϧ z(1) z(2) z(t−1) z(t) ⋯ ɹભҠ֬ɹΛ ͱ͓͍ͨͱ͖ɼ ͕Γཱͭͱ͖ɼ Λɹఆৗɹͱ͍͏ɽ (z(t−1), z(t)) = p(z(t) |z(t−1)) p* (z) = ∫ (z′ , z)p* (z′ )dz′ p* (z)
Ϛϧίϑ࿈ϞϯςΧϧϩ๏ʢ.$.$ʣ ɹఆৗ ʹऩଋ͢ΔΑ͏ͳભҠ֬ Λઃܭ͢Δͱɼ ͔Βαϯϓϧ ΛಘΔ͜ͱ͕Ͱ͖Δɽ ఆৗʹີܭࢉ͕ࠔͳ֬Λ͓͘ɽ p* (z) (z(t−1),
z(t)) p* (z) ⟹ ख๏ͷΩϞ ৄࡉΓ߹͍݅ p* (z)(z, z′ ) = p* (z′ )(z′ , z) ʲे݅ʳৄࡉΓ߹͍͕݅Γཱͭ ఆৗͱͳΔɽ ⟹ p* (z) p* (z)(z, z′ ) = p* (z′ )(z′ , z) ⇒ ∫ p* (z)(z, z′ )dz′ = ∫ p* (z′ )(z′ , z)dz′ ⇔ p* (z) ∫ p(z′ |z)dz′ = ∫ p* (z′ )(z′ , z)dz′ ⇔ p* (z) = ∫ p* (z′ )(z′ , z)dz′
Ϛϧίϑ࿈ϞϯςΧϧϩ๏ʢ.$.$ʣ ɹΓ߹͍݅ʹՃ͑ͯɼαϯϓϧ͕ ͱͨ͠ͱ͖ɼॳظঢ়ଶ ʹ͔͔ΘΒ ͣɼ ͕ఆৗ ʹऩଋ͢Δඞཁ͕͋Δɽ Τϧΰʔυੑ t →
∞ p(z(1)) p(z(t)) p* (z) ⟹ Τϧΰʔυੑ w طੑɹɿҙͷঢ়ଶ͔Βҙͷঢ়ଶ༗ݶճͰભҠՄೳɽ w ඇपظੑɿͯ͢ͷঢ়ଶ͕ݻఆͷपظੑΛͨͳ͍ɽ w ਖ਼࠶ؼੑɿಉ͡ঢ়ଶ͕༗ݶճͰΔ͜ͱ͕Մೳɽ
ϝτϩϙϦεɾϔΠεςΟϯάε๏ʢ.)๏ʣ త ɹະͷ֬ ͔ΒαϯϓϦϯάΛಘΔɽ p(z) લఏ ɹ ͱͳΔਖ਼نԽ͞Ε͍ͯͳ͍ؔ طͰ͋Δɽ p(z)
∝ ˜ p(z) ˜ p(z) ख๏ ɹભҠ֬ ͕ઃܭ͕͍͠߹ɼભҠͷఏҊ Λ͑Δɽ (z′ , z) q(z|z′ ) ɽఏҊ ͔Β࣍ͷαϯϓϧͷީิ ΛαϯϓϦϯά͢Δɽ ɽ࣍ͷൺ Λܭࢉ͢Δɽ ɽ Λ֬ ʹΑͬͯ ͱͯ͠ड༰͠ɼͦ͏Ͱͳ͍߹ ɼ ͱ͢Δɽ q( ⋅ |z(t)) z* r r = ˜ p(z* )q(z(t) |z* ) ˜ p(z(t))q(z* |z(t)) z* min(1,r) z(t+1) ⟵ z* z(t+1) ⟵ z(t) ΞϧΰϦζϜͷྲྀΕ
ϝτϩϙϦεɾϔΠεςΟϯάε๏ʢ.)๏ʣ ৄࡉΓ߹͍݅ͷূ໌ ɹભҠ֬ɼҎԼͷΑ͏ʹͳΔɽ (z, z′ ) = q(z′ |z) min
(1, ˜ p(z′ )q(z|z′ ) ˜ p(z)q(z′ |z) ) p(z)(z, z′ ) = p(z)q(z′ |z) min (1, ˜ p(z′ )q(z|z′ ) ˜ p(z)q(z′ |z) ) = p(z)q(z′ |z) min (1, p(z′ )q(z|z′ ) p(z)q(z′ |z) ) = min (p(z)q(z′ |z), p(z′ )q(z|z′ )) = min (p(z′ )q(z|z′ ), p(z)q(z′ |z)) = p(z′ )q(z|z′ ) min (1, p(z)q(z′ |z) p(z′ )q(z|z′ )) = p(z′ )q(z|z′ ) min (1, ˜ p(z)q(z′ |z) ˜ p(z′ )q(z|z′ )) = p(z′ )(z′ , z) ɹ ͷ߹ɼϝτϩϙϦε๏ͱݺΕΔɽ q(z′ |z) = q(z|z′ )
ϝτϩϙϦεɾϔΠεςΟϯάε๏ʢ.)๏ʣ ʮϕΠζਂֶशʯQࣜʢʣ ޡΓ (z, z′ ) = q(z′ |z) min
(1, ˜ p(z′ )q(z|z′ ) ˜ p(z)q(z′ |z) ) (z, z′ ) = q(z|z′ ) min (1, ˜ p(z′ )q(z|z′ ) ˜ p(z)q(z′ |z) )
ϝτϩϙϦεɾϔΠεςΟϯάε๏ʢ.)๏ʣ ۩ମྫͰֶͿ ඪʢະʣ ɹɹɹɹɹɹɹ ਖ਼نԽ͞Ε͍ͯͳ͍ؔʢطʣ ఏҊ
ͭ·Γ p(z) = (z|μ, Σ) = 1 (2π)D |Σ| exp {− 1 2 (z − μ)TΣ−1(z − μ)} ˜ p(z) = exp {− 1 2 (z − μ)TΣ−1(z − μ)} q(z* |z) = (z′ |z, I) z* ∼ (z, I) https://drive.google.com/open?id=1vcBZWp9HPzfCzBjj2INdkH_CJUDL-JA3
ɹαϯϓϦϯάͷલʹϋϛϧτχΞϯΛར༻ͨ͠ղੳֶతͳγϛϡϨʔγϣϯΛղ આɽຎࡲʹΑΔΤωϧΪʔͷݮগ͕ͳ͍ͱԾఆ͢ΔͱɼϋϛϧτχΞϯҎԼͷΑ͏ʹ ද͞ΕΔɽ ℋ(z, p) = (z) + (p), (p)
= 1 2m pTp, z ∈ ℝD: ମͷҐஔϕΫτϧ, p ∈ ℝD: ମͷӡಈྔϕΫτϧ, m ∈ ℝ: ମͷ࣭ྔ, ℋ(z, p): ϋϛϧτχΞϯ, (z): ϙςϯγϟϧΤωϧΪʔ, (p): ӡಈΤωϧΪʔ . ϋϛϧτχΞϯϞϯςΧϧϩ๏ʢ).$๏ʣ ).$๏ʢϋΠϒϦοτϞϯςΧϧϩ๏ʣʹମͷيಓͷझຯϨʔγϣϯ .)๏ × ɹ).$๏ɼϥϯμϜΥʔΫతͳ.)ͱൺͯɼޮతʹۭؒΛ୳ࡧՄೳɽ ϋϛϧτχΞϯͷγϛϡϨʔγϣϯ
ϋϛϧτχΞϯϞϯςΧϧϩ๏ʢ).$๏ʣ ɹ࣭ྔΛͱ͠ɼ ͱ ͷ࣌ؒ ʹؔ͢ΔڍಈɼϋϛϧτχΞϯͷภඍʹΑܾͬͯఆɽ ͜ͷඍํఔ͕ࣜղੳతʹղ͚ͳ͍ͷͱ͠ɼγϛϡϨʔγϣϯʹΑͬͯيಓΛܭ ࢉ͢Δɽ z
p τ dpi dτ = − dℋ dzi = − d dzi , dzi dτ = dℋ dpi = d dpi . ΦΠϥʔ๏ ࣌ࠁ ઌͷڍಈΛۙࣅతʹ༧ଌɽ ϵ > 0 pi (τ + ϵ) = pi (τ) + ϵ dpi dτ τ = pi (τ) − ϵ d dzi zi (τ) , zi (τ + ϵ) = zi (τ) + ϵ dzi dτ τ = zi (τ) + ϵpi (τ) ࢄԽʹΑΔޡ͕ࠩେ͖͍ɽ Ϧʔϓϑϩοά๏ ⟹
ϋϛϧτχΞϯϞϯςΧϧϩ๏ʢ).$๏ʣ Ϧʔϓϑϩοά๏ pi (τ + ϵ 2 ) = pi
(τ) − ϵ 2 d dzi zi (τ) , zi (τ + ϵ) = zi (τ) + ϵpi (τ + ϵ 2 ), pi (τ + ϵ) = pi (τ + ϵ 2 ) − ϵ 2 d dzi zi (τ + ϵ) . ͜ΕΛ ճ܁Γฦ͢͜ͱͰ࣌ࠁ ઌͷମͷҐஔ ͱӡಈྔ ΛۙࣅతʹܭࢉͰ͖Δɽ L ϵL z* p* ϋϛϧτχΞϯͷੑ࣭ ɽ ࣌ؒ ʹΑͬͯෆมɽ ɽՄٯੑɿ ͔Β ͷભҠҰରҰɽ ɽମੵอଘ ℋ τ (z, p) (z* , p* )
ϋϛϧτχΞϯϞϯςΧϧϩ๏ʢ).$๏ʣ αϯϓϦϯάΞϧΰϦζϜͷద༻ త ɹະͷ֬ ͔ΒαϯϓϦϯάΛಘΔɽ p(z) લఏ ɹ ͱͳΔਖ਼نԽ͞Ε͍ͯͳ͍ؔ طͰ͋Δɽ
ɹ ͱ֦ு͢Δͱɼ पล ͔Βαϯϓϧ͕ಘΒΕΔɽ ɹ p(z) ∝ ˜ p(z) ˜ p(z) p(z, p) = p(z)p(p) z p(z) p(p) = (p|0, I) (z) = − log (˜ p(z)) (p) = 1 2 pTp
ϋϛϧτχΞϯϞϯςΧϧϩ๏ʢ).$๏ʣ ɹಉ࣌Λܭࢉ͢ΔͱɼҎԼͷΑ͏ʹͳΔɽ ϝτϩϙϦε๏ͰΘΕΔൺ ɼҎԼͷΑ͏ʹͳΔɽ p(z, p) = p(z)p(p) =
exp (log p(z) + log p(p)) ∝ exp (log ˜ p(z) − 1 2 pTIp ) = exp (−(z) − (p)) = exp (−ℋ(z, p)) r r = p(z* , p* ) p(z, p) = exp (−ℋ(z* , p* ) + ℋ(z, p))
ϋϛϧτχΞϯϞϯςΧϧϩ๏ʢ).$๏ʣ ख๏ ɽӡಈྔΛαϯϓϦϯά ɽϦʔϓϑϩοά๏Ͱݱࡏͷ ͔Βީิ ΛಘΔɽ ɽ࣍ͷൺ Λܭࢉ͢Δɽ
ɽ Λ֬ ʹΑͬͯ ͱͯ͠ड༰͠ɼͦ͏Ͱͳ͍߹ ɼ ͱ͢Δɽ p ∼ (0, I) (z(t), p) (z* , p* ) r r = p(z* , p* ) p(z, p) z* min(1,r) z(t+1) ⟵ z* z(t+1) ⟵ z(t) ΞϧΰϦζϜͷྲྀΕ
ϋϛϧτχΞϯϞϯςΧϧϩ๏ʢ).$๏ʣ Ϧʔϓϑϩοάͷύϥϝʔλʹؔͯ͠ɼҎԼͷΑ͏ͳτϨʔυΦϑ͕͋Δɽ ɹ).$๏ɼࣄޙͷඍ͑͞ܭࢉͰ͖Εద༻Ͱ͖ɼඇৗʹ൚༻తɽҰൠతͳ χϡʔϥϧωοτϫʔΫ࿈ଓͳજࡏมͷΈͰΓཱ͍ͬͯΔ͜ͱ͕ଟ͍ͷͰɼ).$ ๏χϡʔϥϧωοτϫʔΫͷϕΠζֶशʹΘΕ͖ͯͨɽ େ͖͍ εςοϓαΠζ ϵ εςοϓ L
খ͍͞ খ͍͞ େ͖͍ ड༰ ड༰ ୳ࡧޮ ܭࢉྔ ߴ͍ ߴ͍ ͍ ͍ େ͖͍ খ͍͞ ߴ͍ ͍
ϋϛϧτχΞϯϞϯςΧϧϩ๏ʢ).$๏ʣ ϥϯδϡόϯಈྗֶ๏ ɹ ͱͨ͠߹ɼϥϯδϡόϯϞϯςΧϧϩ๏ɹ·ͨɹϥϯδϡόϯಈྗֶ๏ɹͱ ݺΕΔɽ ɹਂֶश͚ʹϛχόονֶश͕ߦ͑ΔΑ͏ʹͨ͠ɹ֬తޯϥϯδϡόϯಈྗֶ ๏ɹʹల։͞ΕΔɽ L =
1 z*i = zi (τ + ϵ) = zi (τ) + ϵ pi (τ) − ϵ 2 d dzi zi (τ) = zi (τ) − ϵ2 2 d dzi zi (τ) + ϵpi (τ)
ඪʢະʣ ɹɹɹɹɹɹɹ ਖ਼نԽ͞Ε͍ͯͳ͍ؔʢطʣ ӡಈྔͷαϯϓϦϯάɿ ӡಈΤωϧΪʔɿ
ҐஔΤωϧΪʔɿ ҐஔΤωϧΪʔͷภඍɿ p(z) = (z|μ, Σ) = 1 (2π)D |Σ| exp {− 1 2 (z − μ)TΣ−1(z − μ)} ˜ p(z) = exp {− 1 2 (z − μ)TΣ−1(z − μ)} p ∼ (0, I) (p) = 1 2 pTp (z) = − log (˜ p(z)) = − 1 2 (z − μ)TΣ−1(z − μ) ∂ ∂z = − (z − μ)TΣ−1 ϋϛϧτχΞϯϞϯςΧϧϩ๏ʢ).$๏ʣ ۩ମྫͰֶͿ https://drive.google.com/open?id=11zWctTbECXEhlHm7AqPiXC_MErAYl7hJ
ΪϒεαϯϓϦϯά త ɹ֬ ͔Β શମΛαϯϓϦϯά͢Δ͜ͱ͕͍͠ͱ͖ͷ୳ࡧɽ p(Z) Z લఏ ɹ֬ طɽ
p(Z) ख๏ ɹɽม Λ ݸͷ෦ू߹ʹ͚Δɽ ɹɽ෦ू߹Λஞ࣍తʹ୳ࡧ͢Δɽ ɹ Z M Z = {Z1 , Z2 , …, ZM } Z1 ∼ p(Z1 |Z2 , Z3 , …, ZM−1 , ZM ) Z2 ∼ p(Z2 |Z1 , Z3 , …, ZM−1 , ZM ) ⋮ ZM ∼ p(ZM |Z1 , Z2 , …, ZM−2 , ZM−1 )
ΪϒεαϯϓϦϯά ɹΪϒεαϯϓϦϯάͷଥੑɼαϯϓϦϯάͷखଓ͖͕.)๏ͷҰछͱͯ͠ղऍͰ ͖Δ͜ͱ͕อূ͞Ε͍ͯΔɽ ͷΑ͏ʹ͚ɼ Λ͚݅ͨ͠ͱͰ ͷα ϯϓϦϯάΛ͢Δ͜ͱΛߟ͑ͨ߹ɼ ͔ͭ ɹൺ
Λܭࢉ͢ΔͱҎԼͷΑ͏ʹͳΔɽ Αͬͯɼશͯड༰͞ΕΔɽ ಉ༷ɽ Z = {Z1 , Z2 } Z2 Z1 q(Z* |Z) = p(Z1* |Z2* ) Z2 = Z2* r r = p(Z* )q(Z|Z* ) p(Z)q(Z* |Z) = p(Z1* , Z2* )p(Z1 |Z2* ) p(Z1 , Z2 )p(Z1* |Z2 ) = p(Z1* |Z2* )p(Z2* )p(Z1 |Z2* ) p(Z1 |Z2 )p(Z2 )p(Z1* |Z2 ) = 1 Z2
ΪϒεαϯϓϦϯά ۩ମྫͰֶͿ ඪʢطʣ ɹɹɹɹ ͱ͢Δɽ p(z) = (z|μ,
Σ) = 1 (2π)D |Σ| exp {− 1 2 (z − μ)TΣ−1(z − μ)} z = ( z1 z2 ), μ = ( μ1 μ2 ), Σ = ( Σ11 Σ12 Σ21 Σ22 ), Λ = Σ−1 = ( Λ11 Λ12 Λ21 Λ22 ) log p(z) = log p(z1 , z2 ) = − 1 2 (z1 − μ1 )TΛ11 (z1 − μ1 ) + (z1 − μ1 )TΛ12 (z2 − μ2 )) − 1 2 (z2 − μ2 )TΛ22 (z2 − μ2 ) + (z2 − μ2 )TΛ21 (z1 − μ1 )) = − 1 2 ( zT 1 Λ11 z1 − 2z1 {Λ11 μ1 − 1 2 Λ12 (z2 − μ2 )}) + C1 − 1 2 ( zT 2 Λ22 z2 − 2z2 {Λ22 μ2 − 1 2 Λ21 (z1 − μ1 )}) + C2
ΪϒεαϯϓϦϯά ۩ମྫͰֶͿ Αͬͯɼ͖݅֬ͷରɼ ͱͳΔͷͰɼ͖݅֬ΨεͰ͋Δɼ log p(zi |zj
) = − 1 2 ( zT i Λii zi − 2zi { Λii μi − 1 2 Λij (zj − μj )}) + C p(zi |zj ) = (zi |μi , Σi ), Σ−1 i = Λii , Σ−1 i μi = Λii μi − 1 2 Λij (zj − μj ), ⇔ μi = Σi ( Λii μi − 1 2 Λij (zj − μj )) . https://drive.google.com/open?id=1ReYNvvH-NgtsuRiDDV-lz1779sps2pT0