Upgrade to Pro
— share decks privately, control downloads, hide ads and more …
Speaker Deck
Features
Speaker Deck
PRO
Sign in
Sign up for free
Search
Search
ベイズ推論による機械学習入門 4章前半
Search
Takahiro Kawashima
October 01, 2018
Science
0
600
ベイズ推論による機械学習入門 4章前半
某所での輪読用資料
須山敦志『ベイズ推論による機械学習入門』4.1節〜4.3節
Takahiro Kawashima
October 01, 2018
Tweet
Share
More Decks by Takahiro Kawashima
See All by Takahiro Kawashima
引力・斥力を制御可能なランダム部分集合の確率分布
wasyro
0
180
集合間Bregmanダイバージェンスと置換不変NNによるその学習
wasyro
0
98
論文紹介:Precise Expressions for Random Projections
wasyro
0
400
ガウス過程入門
wasyro
0
500
論文紹介:Inter-domain Gaussian Processes
wasyro
0
170
論文紹介:Proximity Variational Inference (近接性変分推論)
wasyro
0
330
機械学習のための行列式点過程:概説
wasyro
0
1.7k
SOLVE-GP: ガウス過程の新しいスパース変分推論法
wasyro
1
1.3k
論文紹介:Stein Variational Gradient Descent
wasyro
0
1.3k
Other Decks in Science
See All in Science
LayerXにおける業務の完全自動運転化に向けたAI技術活用事例 / layerx-ai-jsai2025
shimacos
2
1.2k
Ignite の1年間の軌跡
ktombow
0
130
実力評価性能を考慮した弓道高校生全国大会の大会制度設計の提案 / (konakalab presentation at MSS 2025.03)
konakalab
2
180
academist Prize 4期生 研究トーク延長戦!「美は世界を救う」っていうけど、どうやって?
jimpe_hitsuwari
0
140
Factorized Diffusion: Perceptual Illusions by Noise Decomposition
tomoaki0705
0
390
データマイニング - ノードの中心性
trycycle
PRO
0
130
地表面抽出の方法であるSMRFについて紹介
kentaitakura
1
750
地質研究者が苦労しながら運用する情報公開システムの実例
naito2000
0
220
3次元点群を利用した植物の葉の自動セグメンテーションについて
kentaitakura
2
1.3k
データベース08: 実体関連モデルとは?
trycycle
PRO
0
700
学術講演会中央大学学員会府中支部
tagtag
0
270
統計学入門講座 第4回スライド
techmathproject
0
150
Featured
See All Featured
A designer walks into a library…
pauljervisheath
207
24k
Building Applications with DynamoDB
mza
95
6.5k
The World Runs on Bad Software
bkeepers
PRO
69
11k
Distributed Sagas: A Protocol for Coordinating Microservices
caitiem20
331
22k
Being A Developer After 40
akosma
90
590k
Visualizing Your Data: Incorporating Mongo into Loggly Infrastructure
mongodb
46
9.6k
RailsConf & Balkan Ruby 2019: The Past, Present, and Future of Rails at GitHub
eileencodes
138
34k
Practical Tips for Bootstrapping Information Extraction Pipelines
honnibal
PRO
20
1.3k
The Cost Of JavaScript in 2023
addyosmani
51
8.5k
YesSQL, Process and Tooling at Scale
rocio
173
14k
Fight the Zombie Pattern Library - RWD Summit 2016
marcelosomers
233
17k
Into the Great Unknown - MozCon
thekraken
39
1.9k
Transcript
ਢࢁຊ 4 ষલ ౡوେ October 1, 2018 ిؾ௨৴େֶ 4
࣍ 1. ࠞ߹Ϟσϧͱࣄޙͷਪ 2. ֬ͷۙࣅख๏ 3. ϙΞιϯࠞ߹Ϟσϧʹ͓͚Δਪ 2
ࠞ߹Ϟσϧͱࣄޙͷਪ
ࠞ߹Ϟσϧͷಈػ ෳͷͷ͋͠ΘͤͰΑΓෳࡶͳϞσϧΛ ˠࠞ߹Ϟσϧ ୯ҰͷΨεϞσϧͰઆ໌Ͱ͖ͳͦ͞͏ 3
ࠞ߹Ϟσϧͷσʔλੜաఔ Ϋϥελ K ط ੜσʔλ X = {x1, . .
. , xN } જࡏม (one-hot) S = {s1, . . . , sN } ࠞ߹ൺ π = (π1, . . . , πK)⊤ ֤Ϋϥελύϥϝʔλ Θ = (θ1, . . . , θK)⊤ 4
ࠞ߹Ϟσϧͷσʔλੜաఔ p(X, S, Θ, π) = p(X|S, Θ)p(S|π)p(Θ)p(π) = [
N ∏ n=1 p(xn|sn, Θ)p(sn|π) ] [ K ∏ k=1 p(θk) ] p(π) (4.5) sn ʹΧςΰϦΧϧɼͦͷύϥϝʔλ π ʹσΟϦΫϨͰ ڞࣄલ p(sn|π) = Cat(sn|π) (4.2) p(π) = Dir(π|α) (4.3) 5
ࠞ߹Ϟσϧͷࣄޙ ਪఆ͍ͨ͠ະมͷಉ࣌ࣄޙ p(S, Θ, π|X) = p(X, S, Θ, π)
p(X) (4.6) ͞ΒʹΫϥελΛਪఆ͢Δʹ p(S|X) = ∫∫ p(S, Θ, π|X)dΘdπ (4.7) ͷܭࢉ͕ඞཁ 6
ࠞ߹Ϟσϧͷࣄޙ ਖ਼نԽ߲ p(X) ΛཅʹಘΔʹ p(X) = ∑ S ∫∫ p(X,
S, Θ, π)dΘdπ = ∑ S p(X, S) (4.8) Λܭࢉ ੵڞࣄલΛ͑ղੳతʹධՁͰ͖Δ͕ʜʜ S ͷͯ͢ͷΈ߹Θͤʹର͢Δ͕ඞཁ ˠ MCMCɼมਪͳͲͰࣄޙΛۙࣅ 7
֬ͷۙࣅख๏
ΪϒεαϯϓϦϯά ѻ͍ͮΒ͍֬ p(z1, z2, z3) ͷ౷ܭྔΛಘ͍ͨ ˠ MCMC(Markov chain Monte
Carlo) Ͱ p(z1, z2, z3) ͔Βαϯϓ Ϧϯά ΪϒεαϯϓϦϯά ҎԼͷ full conditional ͔Β܁Γฦ͠αϯϓϦϯάͯ͠ p(z1, z2, z3) ͔ΒͷαϯϓϦϯάܥྻΛಘΔ z(i) 1 ∼ p(z1|z(i−1) 2 , z(i−1) 3 ) z(i) 2 ∼ p(z2|z(i) 1 , z(i−1) 3 ) (4.10) z(i) 3 ∼ p(z3|z(i) 1 , z(i) 2 ) 8
ΪϒεαϯϓϦϯά 2 ࣍ݩΨεʹରͯ͠ΪϒεαϯϓϦϯά (ਤ 4.4) ੨ઢɿਅͷɼઢɿαϯϓϧू߹͔Βಘͨۙࣅ 2 1 0 1
2 3 4 z1 2.5 2.0 1.5 1.0 0.5 0.0 0.5 1.0 z2 p(z) q(z) 2.5 2.0 1.5 1.0 0.5 0.0 0.5 1.0 z1 0.50 0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50 z2 p(z) q(z) มؒͷ૬͕ؔେ͖͍ͱո͘͠ͳΓ͕ͪ 9
ൃలख๏ 1ɿϒϩοΩϯάΪϒεαϯϓϦϯά ϒϩοΩϯάΪϒεαϯϓϦϯά z2, z3 ͷಉ࣌Λ༻͍ͯΪϒεαϯϓϦϯά z(i) 1 ∼ p(z1|z(i−1)
2 , z(i−1) 3 ) z(i) 2 , z(i) 3 ∼ p(z2, z3|z(i) 1 ) (4.11) • z2 ͱ z3 ͷ૬͕ؔڧͯ͘͏·͍͖͍͘͢ • p(z2, z3|z(i)) ͔ΒαϯϓϦϯά͍͢͠ඞཁ 10
ൃలख๏ 2ɿ่յܕΪϒεαϯϓϦϯά ่յܕΪϒεαϯϓϦϯά z3 ΛपลԽআڈޙɼp(z1, z2) ͔ΒΪϒεαϯϓϦϯά p(z1, z2) =
∫ p(z1, z2, z3)dz3 (4.12) z(i) 1 ∼ p(z1|z(i−1) 2 ) z(i) 2 ∼ p(z2|z(i) 1 ) (4.13) • ߴԽ͕ݟࠐΊΔ • पล͕ղੳతʹٻ·Δඞཁ • Γͷม͕αϯϓϦϯά͍͢͠ܗࣜͰ͋Δඞཁ 11
มਪ ֬ p(z1, z2, z3) Λѻ͍͍ۙ͢ࣅ q(z1, z2, z3) Ͱදݱ
ˠ KL ڑ࠷খԽ qopt.(z1, z2, z3) = arg min q KL[q(z1, z2, z3)∥p(z1, z2, z3)] (4.14) มਪ q ͷදݱೳྗΛݶఆͯ͠ KL ڑΛ࠷খԽ 12
มਪ ฏۉۙࣅ ֤֬มʹಠཱੑΛԾఆ p(z1, z2, z3) ≈ q(z1)q(z2)q(z3) (4.15) q(z1),
q(z2), q(z3) Λ KL ڑ͕খ͘͞ͳΔΑ͏ஞ࣍తʹमਖ਼ Notation ⟨·⟩q(z1)q(z2)q(z3) = ⟨·⟩1,2,3 13
มਪ q(z2), q(z3) Λॴ༩ͱͯ͠ q(z1) Λ࠷దԽ qopt.(z1) = arg min
q(z1) KL[q(z1)q(z2)q(z3)∥p(z1, z2, z3)] (4.16) KL[q(z1)q(z2)q(z3)∥p(z1, z2, z3)] = − ⟨ ln p(z1, z2, z3) q(z1)q(z2)q(z3) ⟩ 1,2,3 (4.18) = − ⟨⟨ ln p(z1, z2, z3) q(z1)q(z2)q(z3) ⟩ 2,3 ⟩ 1 (4.19) = − ⟨ ⟨ln p(z1, z2, z3)⟩2,3 − ⟨ln q(z1)⟩2,3 − ⟨ln q(z2)⟩2,3 − ⟨ln q(z3)⟩2,3 ⟩ 1 (4.20) 14
มਪ ⟨ln q(z1)⟩2,3 = ln q(z1)ɼq(z1) ͱແؔͳ෦Λఆʹཧ = − ⟨⟨ln
p(z1, z2, z3)⟩2,3 − ln q(z1)⟩ 1 + const. (4.21) = − ⟨ln [exp(⟨ln p(z1, z2, z3)⟩2,3)] − ln q(z1)⟩ 1 + const. = − ⟨ ln exp(⟨ln p(z1, z2, z3)⟩2,3) ln q(z1) ⟩ 1 + const. (4.22) = KL[q(z1)∥exp{⟨ln p(z1, z2, z3)⟩2,3}] + const. (4.23) ࠷ऴతʹࣜ (4.23) ͷ࠷খ ln q(z1) = ⟨ln p(z1, z2, z3)⟩q(z2)q(z3) + const. (4.24) ͰಘΒΕΔ (q(z2), q(z3) ʹ͍ͭͯಉ༷) 15
มਪ ฏۉۙࣅʹΑΔมਪ (ΞϧΰϦζϜ 4.1) q(z2), q(z3) ΛॳظԽ for i =
1, . . . , max iter do ln q(z1) = ⟨ln p(z1, z2, z3)⟩q(z2)q(z3) + const. ln q(z2) = ⟨ln p(z1, z2, z3)⟩q(z1)q(z3) + const. ln q(z3) = ⟨ln p(z1, z2, z3)⟩q(z1)q(z2) + const. end for ͏ͪΐ ͬͱ͔͍͜͠ऴྃ݅Λઃఆ͍ͨ͠ ˠͨͱ͑ ELBO(evidence lower bound) ΛධՁج४ʹ 16
มਪ ELBO(A.4, p.233) มਪʮपลͷԼݶʯͷ࠷େԽख๏ͱͯ͠ଊ͑ΒΕΔ Xɿ؍ଌσʔλɼZɿະ؍ଌม Z ∼ q(Z) ΛԾఆ ln
p(X) = ln ∫ p(X, Z)dZ = ln ∫ q(Z) p(X, Z) q(Z) dZ ≥ ∫ q(Z)ln p(X, Z) q(Z) dZ (Jensen ͷෆࣜ) =: L[q(Z)] (A.39) 17
มਪ ࢀߟɿJensen ͷෆࣜ ҙͷ “্ʹ” ತͳؔ fɼҙͷ֬ີؔ p ʹؔͯ͠ f
(∫ y(x)p(x)dx ) ≥ ∫ f(y(x))p(x)dx (A.40) 18
มਪ ELBO(A.4, p.233) पลͷԼݶ L[q(Z)] Λ q(Z) ͷ ELBO ͱΑͿ
ରपลͱ ELBO ͱͷࠩ q(Z) ͱ p(Z|X) ͱͷ KL ڑʹ ͍͠ KL[q(Z)∥p(Z|X)] = ∫ q(Z)ln q(Z) p(Z|X) dZ = ∫ q(Z)ln q(Z)p(X) p(X, Z) dZ = p(X) − ∫ q(Z)ln p(X, Z) q(Z) dZ = p(X) − L[q(Z)] (A.41) 19
มਪ ELBO(A.4, p.233) KL[q(Z)∥p(Z|X)] = p(X) − L[q(Z)] (A.41) ln
p(X) σʔλͱϞσϧॴ༩ͷͱఆ ˠ q(Z) ʹؔ͢Δ KL ڑ࠷খԽͱରपลͷԼݶ L[q(Z)] ͷ ࠷େԽՁ ELBO ͷมԽ͕ఆ ϵ ΑΓখ͘͞ͳͬͨͱ͖ʹมਪΞϧΰ ϦζϜΛࢭΊΔ 20
มਪ ߏԽมਪ ਅͷΛ෦తʹۙࣅؔʹղ p(z1, z2, z3) ≈ q(z1)q(z2, z3) (4.26)
21
มਪ (؆қ࣮ݧ) 2 ࣍ݩΨεʹมਪΛద༻ (ਤ 4.5) 1.0 0.5 0.0 0.5
0.50 0.25 0.00 0.25 0.50 1 of 10 0.5 0.0 0.5 0.50 0.25 0.00 0.25 0.50 2 of 10 0.5 0.0 0.5 0.50 0.25 0.00 0.25 0.50 3 of 10 0.5 0.0 0.5 0.50 0.25 0.00 0.25 0.50 4 of 10 0.5 0.0 0.5 0.50 0.25 0.00 0.25 0.50 5 of 10 0.5 0.0 0.5 0.50 0.25 0.00 0.25 0.50 6 of 10 0.5 0.0 0.5 0.50 0.25 0.00 0.25 0.50 7 of 10 0.5 0.0 0.5 0.50 0.25 0.00 0.25 0.50 8 of 10 0.5 0.0 0.5 0.50 0.25 0.00 0.25 0.50 9 of 10 0.5 0.0 0.5 0.50 0.25 0.00 0.25 0.50 10 of 10 ੨ઢɿਅͷ ઢɿۙࣅࣄޙ 22
มਪ (؆қ࣮ݧ) 2 ࣍ݩΨεʹมਪΛద༻ (ਤ 4.5) 2 4 6 8
10 iteration 0.46 0.48 0.50 0.52 0.54 KL divergence KL ڑ୯ௐݮগ 23
มਪ (؆қ࣮ݧ) 2 ࣍ݩΨεʹมਪΛద༻ (ਤ 4.5) • ͍ • ΠςϨʔγϣϯ͝ͱʹ
KL ڑ͕୯ௐݮগ • ڧ͍૬ؔΛଊ͑ΒΕͳ͍ 24
ϙΞιϯࠞ߹Ϟσϧʹ͓͚Δਪ
ϙΞιϯࠞ߹Ϟσϧ 1 ࣍ݩࢄඇෛσʔλͷΫϥελΛਪఆ (ਤ 4.6) 80 100 120 140 160
180 0 20 40 60 80 100 120 observation 25
ϙΞιϯࠞ߹Ϟσϧ p(xn|λk) = Poi(xn|λk) (4.27) ΑΓ p(xn|sn, λ) = K
∏ k=1 Poi(xn|λk)sn,k (4.28) λk ͷڞࣄલ p(λk) = Gamma(λk|a, b) (4.29) 26
ΪϒεαϯϓϦϯά ࠞ߹ͰજࡏมͱύϥϝʔλΛ͚ͯαϯϓϧ͢ΔͱΑ͍ S ∼ p(S|X, λ, π) (4.31) λ, π
∼ p(λ, π|X, S) (4.32) ม S ͷΈʹண p(S|X, λ, π) ∝ p(X|S, λ)p(S|π) = N ∏ n=1 p(xn|sn, λ)p(sn|π) (4.33) 27
ΪϒεαϯϓϦϯά p(xn|sn, λ), p(sn|π) ΛͦΕͧΕܭࢉ͢Δͱɼ࠷ऴతʹ sn ∼ Cat(sn|ηn ) (4.37)
ͨͩ͠ ηn,k ∼ exp{xnln λk − λk + ln πk} ( s.t. K ∑ k=1 ηn,k = 1 ) (4.38) ͕ಘΒΕΔ 28
ΪϒεαϯϓϦϯά p(λ, π|X, S) ∝ p(X, S, λ, π) =
p(X|S, λ)p(S|π)p(λ)p(π) (4.39) ˠ λ ͱ π ͷࣄޙಠཱ λ ʹؔͷ͋Δͱ͜Ζʹ͚ͩ p(λ|X, S) ∝ p(X|S, λ)p(λ) 29
ΪϒεαϯϓϦϯά ۩ମతʹܭࢉ͍ͯ͘͠ͱ λk ∼ Gam(λk|ˆ ak,ˆ bk) (4.41) ͨͩ͠ ˆ
ak = N ∑ n=1 sn,kxn + a ˆ bk = N ∑ n=1 sn,k + b (4.42) ͱͳΔ 30
ΪϒεαϯϓϦϯά π ʹؔͷ͋Δͱ͜Ζʹ͚ͩ p(π|X, S) ∝ p(S|π)p(π) ࠷ऴతʹ π ∼
Dir(π|ˆ α) (4.44) ͨͩ͠ ˆ αk = N ∑ n=1 sn,k + αk (4.45) 31
มਪ જࡏมͱύϥϝʔλʹղ (มϕΠζ EM ΞϧΰϦζϜ) p(S, λ, π|X) ≈ q(S)q(λ,
π) (4.46) มਪͷެࣜ ln q(z1) = ⟨ln p(z1, z2, z3)⟩q(z2)q(z3) + const. (4.24) Λ༻͍Δͱ q(S) ʹؔͯ͠ ln q(S) = ⟨ln p(X, S, λ, π)⟩q(λ,π) + const. = ⟨ln p(X|S, λ)p(S|π)p(λ)p(π)⟩q(λ,π) + const. = ⟨ln p(X|S, λ)⟩q(λ) + ⟨ln p(S|π)⟩q(π) + const. = [ N ∑ n=1 ⟨ln p(xn|sn, λ)⟩q(λ) + ⟨ln p(sn|π)⟩q(π) ] + const. (4.47) 32
มਪ (4.47) ࣜ૯ͷୈ 1 ߲ ⟨ln p(xn|sn, λ)⟩q(λ) = K
∑ k=1 ⟨sn,k ln Poi(xn|λk)⟩qk = K ∑ k=1 sn,k(xn⟨ln λk⟩ − ⟨λk⟩) + const. (4.48) ୈ 2 ߲ ⟨ln p(sn|π)⟩q(π) = ⟨ln Cat(sn|π)⟩q(π) = K ∑ k=1 sn,k⟨ln πk⟩ (4.49) 33
มਪ ࣜ (4.47),(4.48),(4.49) ͔Β ln q(sn) = ⟨ln p(xn|sn, λ)⟩q(λ)
+ ⟨ln p(sn|π)⟩q(π) + const. = K ∑ k=1 sn,k(xn⟨ln λk⟩ − ⟨λk⟩ + ⟨ln πk⟩ + const.) ͜͜Ͱ ln Cat(s|π) = ∑ K k=1 sn,k ln πk ΑΓ q(sn) = Cat(sn|ηn ) (4.50) ͨͩ͠ ηn,k ∝ exp{xn⟨ln λk⟩ − ⟨λk⟩ + ⟨ln πk⟩} ( s.t. K ∑ k=1 ηn,k = 1 ) (4.51) λ, π ͷظܭࢉҰ୴͋ͱ·Θ͠ 34
มਪ ଓ͍ͯύϥϝʔλͷۙࣅ ln q(λ, π) = ⟨ln p(X, S, λ,
π)⟩q(S) + const. = ⟨ln p(X|S, λ)⟩q(S) + ln p(λ) + ⟨ln p(S|π)⟩q(S) + ln p(π) + const. ΑΓɼλ, π ͕ಠཱʹղ͞Ε͍ͯΔ͜ͱ͕Θ͔Δ ˠ q(λ, π) ͷΘΓʹ q(λ), q(π) ΛͦΕͧΕٻΊΕΑ͍ 35
มਪ q(sn) ͷͱ͖ͱಉ༷ʹܭࢉ͍ͯ͘͠ͱɼ݁Ռͱͯ͠ q(λk) = Gam(λk|ˆ ak,ˆ bk) (4.54) ͨͩ͠
ˆ ak = N ∑ n=1 ⟨sn,k⟩xn + a ˆ bk = N ∑ n=1 ⟨sn,k⟩ + b (4.55) ͓Αͼ q(π) = Dir(π|ˆ α) (4.56) ͨͩ͠ ˆ αk = N ∑ n=1 ⟨sn,k⟩ + αk (4.57) ͕ಘΒΕΔ 36
มਪ ࣜ (4.57) ͷظ ⟨sn,k⟩ = ⟨sn,k⟩q(S) ɼ q(sn) =
Cat(sn|ηn ) (4.50) ΑΓɼ ⟨sn,k⟩q(S) = ηn,k 37
มਪ q(λk) = Gam(λk|ˆ ak,ˆ bk), q(π) = Dir(π|ˆ α)
͕Θ͔ͬͨͷͰɼ ͋ͱ·Θ͠ʹ͍ͯͨ͠ q(sn) ͷظ ⟨λ⟩, ⟨ln λ⟩, ⟨ln π⟩ Λܭࢉ ͜͜Ͱ Eλ∼Gam(λ|a,b) [λ] = a b (2.59) Eλ∼Gam(λ|a,b) [ln λ] = ψ(a) − ln b (2.60) Eπ∼Dir(π|α) [ln πk] = ψ(αk) − ψ ( K ∑ l=1 αk ) (2.52) ψ(x) σΟΨϯϚؔ ψ(x) = d dx ln Γ(x) (A.26) 38
มਪ ࣜ (2.59), (2.60), (2.52) Λ༻͍ΔͱɼٻΊ͍ͨظ ⟨λk⟩ = ˆ ak
ˆ bk (4.60) ⟨ln λk⟩ = ψ(ˆ ak) − ln ˆ bk (4.61) ⟨πk⟩ = ψ(ˆ αk) − ψ ( K ∑ l=1 ˆ αk ) (4.62) ͱಘΒΕΔ 39
่յܕΪϒεαϯϓϦϯά ࠞ߹Ϟσϧͷ่յܕΪϒεαϯϓϦϯάͰಉ͔࣌Βύϥ ϝʔλΛपลԽআڈ p(X, S) = ∫∫ p(X, S, λ,
π)dλdπ (4.63) ͋ͱ p(S|X) ͔ΒαϯϓϦϯάͰ͖ΕΑ͍͕ʜʜ 40
่յܕΪϒεαϯϓϦϯά पลԽલޙͷάϥϑΟΧϧϞσϧ (ਤ 4.7) sn ͕΄͔ͷશͯͷ S ͷཁૉͱґଘؔ (શάϥϑ) 41
่յܕΪϒεαϯϓϦϯά p(S|X) = p(X|S)p(S) ∑ S p(X|S)p(S) ΑΓɼp(S|X) ͔ΒαϯϓϦϯά͢ΔʹɼؔͷධՁ ʹ
KN ճͷܭࢉ͕ඞཁ ˠ S ͷ֤ཁૉʹΪϒεαϯϓϦϯάΛద༻ p(sn|X, S\n ) ∝ p(xn, X\n , sn, S\n ) (4.64) = p(xn|X\n , sn, S\n )p(X\n |sn, S\n ) × p(sn|S\n )p(S\n ) (4.65) ∝ p(xn|X\n , sn, S\n )p(sn|S\n ) (4.66) 42
่յܕΪϒεαϯϓϦϯά (4.66) ࣜӈଆ p(sn|S\n ) = ∫ p(sn|π)p(π|S\n )dπ (4.70)
= Cat(sn|η\n ) (4.74) η\n,k ∝ ∑ n′̸=n sn′,k + αk (4.75) α ࣄલ p(π) = Dir(π|α) ͷύϥϝʔλ 43
่յܕΪϒεαϯϓϦϯά (4.66) ࣜࠨଆ p(xn|X\n , sn, S\n ) = ∫
p(xn|sn, λ)p(λ|X\n , S\n )dλ (4.76) ͜Ε sn,k = 1 Ͱ͚݅Δͱղੳతʹ࣮ߦͰ͖ͯ p(xn|X\n , sn,k = 1, S\n ) = NB ( xn ˆ a\n,k , 1 ˆ b\n,k + 1 ) (4.81) ˆ a\n,k = ∑ n′̸=n sn′,kxn′ + ak (4.80) ˆ b\n,k = ∑ n′̸=n sn′,k + bk (4.81) ak, bk ࣄલ p(λk) = Gam(λk|ak, bk) ͷύϥϝʔλ 44
่յܕΪϒεαϯϓϦϯά ۩ମతͳ p(sn|S\n ) ͔ΒͷαϯϓϦϯάखॱ 1. sn ͷ࣮ݱͱͯ͠ (1, 0,
. . . , 0)⊤ ͔Β (0, 0, . . . , 1)⊤ Λ༻ҙ 2. ͦΕͧΕʹରͯ͠ p(sn|S\n ) = Cat(sn|η\n ) (4.74) p(xn|X\n , sn,k = 1, S\n ) = NB ( xn ˆ a\n,k , 1 ˆ b\n,k + 1 ) (4.81) ΛධՁ 3. ͜ͷ K ݸͷΛਖ਼نԽ͢Δͱɼp(sn|X) Λࣔ͢ΧςΰϦΧ ϧ͕ಘΒΕΔ 4. ಘΒΕͨ p(sn|X) ͔ΒαϯϓϦϯά 45
؆қ࣮ݧ 1 ࣍ݩࢄඇෛσʔλͷΫϥελਪఆ݁Ռ (มਪ) 80 100 120 140 160 180
0 20 40 60 80 100 120 observation 80 100 120 140 160 180 0 20 40 60 80 100 120 estimation ͱ੨ͷ 2 Ϋϥελʹ Ϋϥελॴଐ֬Λதؒ৭Ͱදݱ 46
؆қ࣮ݧ ELBO ͷऩଋ࣌ؒ (ਤ 4.10) ॎ࣠ɿELBOɼԣ࣠ (ର)ɿܭࢉ࣌ؒ [µs] 10 5
10 4 10 3 computation time( s) 5400 5200 5000 4800 4600 4400 ELBO VI GS CGS ؆୯ͳͳͷͰ࠷ऴతͳਫ਼ʹ͕ࠩͳ͍ 47
؆қ࣮ݧ େ·͔ͳͱͯ͠ • ͍ͷมਪ • ࠷ऴతʹਫ਼͕ྑ͍ͷ่յܕ GS • ่յܕ GS
ΠςϨʔγϣϯॳظ͔Βߴਫ਼ ΦεεϝɿͱΓ͋͑ͣ GS Λࢼ͠ɼਫ਼ʹೲಘ͕͍͔ͳ͚ Εมਪɾ่յܕ GS ಋग़ͯ͠ΈΔ 48
·ͱΊ • ࣄޙͷۙࣅख๏ͱͯ͠ΪϒεαϯϓϦϯάɾϒϩοΩϯ άΪϒεαϯϓϦϯάɾ่յܕΪϒεαϯϓϦϯάɾมਪ Λհ • ϙΞιϯࠞ߹Ϟσϧʹରͯ͠ΪϒεαϯϓϦϯάɾ่յܕΪ ϒεαϯϓϦϯάɾมਪΛ۩ମతʹಋग़ • ܭࢉ͕͍࣌ؒͷมਪɼਫ਼͕ྑ͍ͷ่յܕΪϒε
αϯϓϦϯάɼಋग़ָ͕ͳͷΪϒεαϯϓϦϯά 49