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
620
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
260
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
150
論文紹介:Precise Expressions for Random Projections
wasyro
0
460
ガウス過程入門
wasyro
0
590
論文紹介:Inter-domain Gaussian Processes
wasyro
0
180
論文紹介:Proximity Variational Inference (近接性変分推論)
wasyro
0
350
機械学習のための行列式点過程:概説
wasyro
0
1.8k
SOLVE-GP: ガウス過程の新しいスパース変分推論法
wasyro
1
1.4k
論文紹介:Stein Variational Gradient Descent
wasyro
0
1.4k
Other Decks in Science
See All in Science
データベース03: 関係データモデル
trycycle
PRO
1
270
Accelerated Computing for Climate forecast
inureyes
0
120
会社でMLモデルを作るとは @電気通信大学 データアントレプレナーフェロープログラム
yuto16
1
270
データベース10: 拡張実体関連モデル
trycycle
PRO
0
990
蔵本モデルが解き明かす同期と相転移の秘密 〜拍手のリズムはなぜ揃うのか?〜
syotasasaki593876
0
100
研究って何だっけ / What is Research?
ks91
PRO
1
130
Ignite の1年間の軌跡
ktombow
0
160
02_西村訓弘_プログラムディレクター_人口減少を機にひらく未来社会.pdf
sip3ristex
0
640
データマイニング - グラフ構造の諸指標
trycycle
PRO
0
190
機械学習 - pandas入門
trycycle
PRO
0
320
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
1.9k
AI(人工知能)の過去・現在・未来 —AIは人間を超えるのか—
tagtag
1
140
Featured
See All Featured
VelocityConf: Rendering Performance Case Studies
addyosmani
332
24k
Learning to Love Humans: Emotional Interface Design
aarron
274
41k
Imperfection Machines: The Place of Print at Facebook
scottboms
269
13k
How To Stay Up To Date on Web Technology
chriscoyier
791
250k
Cheating the UX When There Is Nothing More to Optimize - PixelPioneers
stephaniewalter
285
14k
How STYLIGHT went responsive
nonsquared
100
5.8k
Let's Do A Bunch of Simple Stuff to Make Websites Faster
chriscoyier
507
140k
Improving Core Web Vitals using Speculation Rules API
sergeychernyshev
19
1.2k
The World Runs on Bad Software
bkeepers
PRO
71
11k
Optimizing for Happiness
mojombo
379
70k
CoffeeScript is Beautiful & I Never Want to Write Plain JavaScript Again
sstephenson
162
15k
How GitHub (no longer) Works
holman
315
140k
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