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
580
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
論文紹介:Precise Expressions for Random Projections
wasyro
0
290
ガウス過程入門
wasyro
0
380
論文紹介:Inter-domain Gaussian Processes
wasyro
0
140
論文紹介:Proximity Variational Inference (近接性変分推論)
wasyro
0
310
機械学習のための行列式点過程:概説
wasyro
0
1.5k
SOLVE-GP: ガウス過程の新しいスパース変分推論法
wasyro
1
1.2k
論文紹介:Stein Variational Gradient Descent
wasyro
0
1.1k
次元削減(主成分分析・線形判別分析・カーネル主成分分析)
wasyro
0
740
論文紹介: Supervised Principal Component Analysis
wasyro
1
850
Other Decks in Science
See All in Science
ベイズ最適化をゼロから
brainpadpr
2
1.1k
小杉考司(専修大学)
kosugitti
2
600
私たちのプロダクトにとってのよいテスト/good test for our products
camel_404
0
250
Cross-Media Information Spaces and Architectures (CISA)
signer
PRO
3
30k
[第62回 CV勉強会@関東] Long-CLIP: Unlocking the Long-Text Capability of CLIP / kantoCV 62th ECCV 2024
lychee1223
1
830
Introduction to Image Processing: 2.Frequ
hachama
0
460
MoveItを使った産業用ロボット向け動作作成方法の紹介 / Introduction to creating motion for industrial robots using MoveIt
ry0_ka
0
290
山形とさくらんぼに関するレクチャー(YG-900)
07jp27
1
260
ほたるのひかり/RayTracingCamp10
kugimasa
1
510
はじめての「相関と因果とエビデンス」入門:“動機づけられた推論” に抗うために
takehikoihayashi
17
7.2k
学術講演会中央大学学員会いわき支部
tagtag
0
130
最適化超入門
tkm2261
14
3.4k
Featured
See All Featured
ReactJS: Keep Simple. Everything can be a component!
pedronauck
666
120k
Understanding Cognitive Biases in Performance Measurement
bluesmoon
27
1.5k
Designing Dashboards & Data Visualisations in Web Apps
destraynor
231
53k
Art, The Web, and Tiny UX
lynnandtonic
298
20k
Distributed Sagas: A Protocol for Coordinating Microservices
caitiem20
330
21k
Agile that works and the tools we love
rasmusluckow
328
21k
The Illustrated Children's Guide to Kubernetes
chrisshort
48
49k
Making Projects Easy
brettharned
116
6k
GraphQLの誤解/rethinking-graphql
sonatard
68
10k
The Pragmatic Product Professional
lauravandoore
32
6.4k
Practical Tips for Bootstrapping Information Extraction Pipelines
honnibal
PRO
12
950
Java REST API Framework Comparison - PWX 2021
mraible
28
8.4k
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