Takahiro Kawashima
December 04, 2018
930

# 論文解説 Bayesian Computing with INLA: A Review

ベイズ事後分布の近似手法INLAのレビュー論文 https://arxiv.org/abs/1604.00860 の解説．

## Takahiro Kawashima

December 04, 2018

## Transcript

1. ### ࿦จྠߨ Bayesian Computing with INLA: A Review ઒ౡوେ December 4,

2018 ిؾ௨৴େֶ ঙ໺ݚڀࣨ B4
2. ### ͍ΜͱΖ ΞϒσϡϥԦཱ޻Պେֶ (α΢δΞϥϏΞ) ͷ H˚ avard Rue ʹΑΔ ࿦จ ม෼ਪ࿦ɾMCMC

ʹ͙࣍ (ʁ) ϕΠζࣄޙ෼෍ͷۙࣅख๏ INLA ͷ review 2
3. ### ͍ΜͱΖ original paper(2009) ͸ͬͪ͜ɽஶऀ͸ review ͱಉ͡ H˚ avard Rueɽ Journal

of Royal Statistical Society, Series B (Statistical Methodology) ʹܝࡌ͞Ε͍ͯΔ (Ҿ༻਺΋͍͢͝ɼ͔͠͠ 70 ϖʔδҎ্͋ΓͭΒ͍) original ͷࣥච࣌ͷॴଐ͸ϊϧ΢ΣʔՊֶٕज़େֶͱͷ͜ͱ (ͪͳ Έʹ˚ a ͸ϊϧ΢Σʔޠͷจࣈ) 3
4. ### ͍ΜͱΖ INLA(Integrated Nested Laplace Approximation) ໊લͷ௨ΓϥϓϥεۙࣅΛ֦ுͨ͠ϕΠζࣄޙ෼෍ͷۙࣅख๏ LGM(Latent Gaussian Model) ͱΑ͹ΕΔΫϥεʹଐ͢ΔϞσϦϯ

ά͕ͳ͞Εͨܥʹର͠ɼߴ଎͔ͭߴਫ਼౓ͳࣄޙ෼෍ͷۙࣅΛ༩ ͑Δ (ۭؒ౷ܭͱ͔ͷਓ͸݁ߏ஌͍ͬͯΔ͔Μ͕͋͡Δʁ) 4
5. ### ४උฤ - LGM; Latent Gaussian Model LGM(Latent Gaussian Model) GMRF(Gaussian

Markov Random Field) ʹै͏જࡏม਺͔Β؍ ଌม਺͕ੜ͑ͯ͘ΔϞσϧ xɿજࡏม਺ɼyɿ؍ଌม਺ ৭෇͖ͷϊʔυ ͸؍ଌม਺ 5
6. ### ४උฤ - LGM; Latent Gaussian Model ·͡ΊʹఆࣜԽ͍͖ͯ͠·͠ΐ͏ɽ σʔλͷΠϯσοΫεͷू߹Λ I ͱදه͢Δͱɼ؍ଌม਺

y = {yi;i ∈ I} ͷಉ࣌෼෍͸ɼ y∣x,θ1 = ∏ i∈I π(yi∣xi,θ1) ͱͳΔɽθ1 ͸ yi ʹؔ͢ΔϋΠύʔύϥϝʔλɽ જࡏม਺͸ GMRF ʹै͏ͷͰͦΕΒͷಉ࣌෼෍΋ Gaussianɼ Αͬͯ x∣θ2 = N(x∣µ(θ2),Q−1(θ2)) ͱॻ͚Δɽµ(⋅),Q(⋅) ͸ GMRF ͷฏۉϕΫτϧ͓Αͼਫ਼౓ߦྻɽ θ2 ͸ x ʹؔ͢ΔϋΠύʔύϥϝʔλ (Q(⋅) ͸Ұൠʹૄ) 6
7. ### ४උฤ - LGM; Latent Gaussian Model જࡏม਺ x ͓ΑͼϋΠύʔύϥϝʔλू߹ θ

= (θ1,θ2)⊺ ͷࣄޙ෼ ෍͸ π(x,θ∣y) = π(y∣x,θ)π(x∣θ)π(θ) π(y) ∝ π(θ)π(x∣θ)∏ i∈I π(yi∣xi,θ) (1) ͜͜ͰࠓޙͷܭࢉͷརศੑͷͨΊʹ 3 ͭͷॏཁͳԾఆΛ͓͘ 1. ϋΠύʔύϥϝʔλͷ਺ ∣θ∣ ͸ 2 ∼ 5 ఔ౓ͱখ͍͞ɽ20 Ҏ্ ͱ͔ʹͳΔͱμϝμϝ 2. x∣θ ͸ Gaussianɼ͞Βʹ x ؒͷ૬ޓ࡞༻΋ GMRF(΋͘͠͸ ΄ͱΜͲ GMRF ͱΈͳͤΔܗ) ͰೖΔ 3. y ͷ֤ཁૉ͸ x,θ ͕ॴ༩ͷ΋ͱͰ৚݅෇͖ಠཱɽ͢ͳΘͪ yi yj∣x,θ (i ≠ j) (҉ʹ π(yi∣x,θ) = π(yi∣xi,θ) ΛԾఆ) 7
8. ### ४උฤ - Additive Model LGM ͸ͦΕͳΓʹҰൠతͳϞσϧͰɼՃ๏Ϟσϧ (Additive Model) ͳͲΛؚΉɽ͢ͳΘͪ AR

Ϟσϧ΍ GLM ͳͲ΋දͤΔ ྫͱͯ͠જࡏม਺ {xi;i ∈ I} Λઢܗ༧ଌࢠ ηi ͱղऍͯ͠ΈΔ µɿ੾ยɼzɿڞมྔɼβɿڞมྔʹର͢ΔҰ࣍ޮՌ fk,jk(i) ɿj ൪໨ͷཁૉ͔Β ηi ΁ k ൪໨ͷ૬ޓ࡞༻ޮՌΛ௨ͯ͠ೖ Δ࡞༻ (model component ͱ໊෇͚ΒΕ͍ͯΔ) ηi = µ + ∑ j βjzij + ∑ k fk,jk(i) (2) AR Ϟσϧͱ͔ϥϯμϜޮՌϞσϧͱ͔͸͜ͷܗࣜʹؚ·ΕΔ 8
9. ### ४උฤ - Additive Model ηi = µ + ∑ j

βjzij + f1,j1(i) + f2,j2(i) + ϵi, i = 1,...,n ۩ମྫΛڍ͛ͯΈΔ X f1,j1(i) = null, f2,j2(i) = null ˠ GLM X ∑j βjzij = null, f1,j1(i) ∼ AR(1), f2,j2(i) ∼ Season ˠ قઅௐ੔ೖΓ AR(1) Ϟσϧ 9
10. ### ४උฤ - Additive Model ࣜ (2) ʹ͓͍ͯ π(µ) ΍ π(β)

͕ Gaussian ͩͱɼ x = (η,µ,β,f1 ,f2 ,...) (3) ΋ GMRF ʹै͏ɽ͞Βʹ͜͜ʹద౰ͳ؍ଌϊΠζ͕ೖΔͱߟ͑Δ ͱ LGM ͷܗࣜͱͳΔɽ ϋΠύʔύϥϝʔλ θ ͸໬౓ p(x∣θ) ͚ͩͰͳ͘ɼmodel component ͷύϥϝʔλ΋ؚΉɽ 10
11. ### ४උฤ - Additive Models and GMRFs INLA ͸ࣜ (3) ͷಉ࣌෼෍͕

GMRF ʹै͏͜ͱΛར༻͢Δɽ ∣θ∣ ͕খ͍͞ (GMRF ͕ sparse) ͳΒಉ࣌෼෍ͷਫ਼౓ߦྻ΋ sparse ʹͳΓɼܭࢉޮ཰͕େ͖͘޲্ ྫͱͯ͠ ηi = µ + βzi + f1,j1(i) + f2,j2(i) + ϵi, i = 1,...,n (4) ͱ͍͏ϞσϧΛߟ͑ΔɽϕΫτϧදݱ͢Δͱ η = µ1 + βz + A1f1 + A2f2 + ϵ ͱͳΔɽA1,A2 ͸ͦΕͧΕ n × m1,n × m2 ͷૄߦྻͰɼ֤ཁૉ͸ 0,1 ͷೋ஋ (૬ޓ࡞༻͕͋Δ߲ͷΈ 1)ɽ 11
12. ### ४උฤ - Additive Models and GMRFs ͞Βʹ f1 ,f2 ͕ͦΕͧΕਫ਼౓ߦྻͱͯ͠

Q1 (θ) ∈ Rm1×m1 ,Q2 (θ) ∈ Rm2×m2 Λ΋ͪɼ µ ∼ N(0,τ−1 µ ) β ∼ N(0,τ−1 β ) ϵ ∼ N(0,τ−1 ϵ ) ͱ͢Δɽ ͜ͷͱ͖ͷಉ࣌෼෍ π(η,f1 ,f2 ,β,µ) ͷਫ਼౓ߦྻ Qjoint (θ) Λٻ Ί͍ͯ͘ 12
13. ### ४උฤ - Additive Models and GMRFs exp (− τϵ 2

(η − (µ1 + βz + A1 f1 + A2 f2 ))⊺(η − (µ1 + βz + A1 f1 + A2 f2 ) − τµ 2 µ2 − τβ 2 β2 − 1 2 f⊺ 1 Q1 (θ)f1 − 1 2 f⊺ 2 Q2 (θ)f2 ) = exp (− 1 2 (η, f1 , f2 , β, µ)⊺Qjoint (θ)(η, f1 , f2 , β, µ)) ͜͜Ͱਫ਼౓ߦྻ͸ Qjoint (θ) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ τϵI τϵA1 τϵA2 τϵIz τϵI1 Q1 (θ) + τϵA1A⊺ 1 τϵA1A⊺ 2 τϵA1z τϵA11 Q2 (θ) + τϵA2A⊺ 2 τϵA2z τϵA21 sym. τβ + τϵz⊺z τϵz⊺1 τµ + τϵ1⊺1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ͷ n + m1 + m2 + 2 ࣍ݩ 13
14. ### ४උฤ - Additive Models and GMRFs Qjoint (θ) = ⎡

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ τϵI τϵA1 τϵA2 τϵIz τϵI1 Q1 (θ) + τϵA1A⊺ 1 τϵA1A⊺ 2 τϵA1z τϵA11 Q2 (θ) + τϵA2A⊺ 2 τϵA2z τϵA21 sym. τβ + τϵz⊺z τϵz⊺1 τµ + τϵ1⊺1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ॏཁͳ఺͸ X Q⋅ (θ),A1,A2 ͕ૄߦྻͩͬͨͨΊɼQjoint (θ) ΋ૄ X θ ʹґଘ͢Δͷ͸ Q1 (θ),Q2 (θ) ͷΈͳͷͰɼθ ͷ஋͕มΘͬ ͯ΋࠶ܭࢉͷίετ͸Θ͔ͣ 14
15. ### ४උฤ - ϥϓϥεۙࣅ ϥϓϥεۙࣅ (ͱ͘ʹ n ͕େ͖͍ͱ͖) ੵ෼ In =

∫ x exp(nf(x))dx ͷ f(x) ΛϞʔυͷपΓͰೋ࣍ۙࣅ͠ɼΨ΢εੵ෼ʹΑΓ In ͷ ۙࣅ஋ΛಘΔํ๏ ۩ମతʹ਺ࣜͰॻ͘ͱɼf′(x0) = 0 ΑΓ In ≈ ∫ x exp ⎛ ⎝ n ⎛ ⎝ f(x0) + 1 2 (x − x0)2f′′(x0) ⎞ ⎠ ⎞ ⎠ dx (5) = exp(nf(x0)) 2π −nf′′(x0) = ˜ In (6) 15
16. ### ४උฤ - ϥϓϥεۙࣅ x ͕த৺ۃݶఆཧ͕੒Γཱͭ෼෍ʹै͏ͳΒɼnf(x) Λର਺໬౓ ͷ࿨ͱ͢Δͱɼn → ∞ Ͱ͜ͷۙࣅ͸

exactɽ ଟ࣍ݩʹ֦ுͯ͠΋ϥϓϥεۙࣅޡࠩ͸ In = ˜ In(1 + O(n−1)) ˣ த৺ۃݶఆཧʹجͮ͘઴ۙ࿦ͳΜ͔͸େମऩଋϨʔτ͕ n−1/2 ͳ ͷͰɼ͜ͷลΛߟ͑Δͱ݁ߏྑ͍ɽ 16
17. ### ४උฤ - ϥϓϥεۙࣅ ͜͜Ͱपล෼෍ π(γ1) Λಉ࣌෼෍ π(γ) ͔ΒٻΊΔ໰୊Λߟ͑Δɽ π(γ1) =

π(γ) π(γ−1 ∣γ1) ≈ π(γ) πG(γ−1 ;µ(γ1),Q(γ1)) γ−1 =µ(γ1) (7) γ−1 ͸ γ ͔Βཁૉ γ1 ͷΈΛআ͍ͨू߹ɽ LGM ͷจ຺Ͱ͸ γ = (x,θ)ɽ Ϟʔυ͔Β O(n−1/2) ͷൣғͰ͸ࣜ (7) ͷ૬ରޡࠩ͸ O(n−3/2) ʹ ͳΔ͜ͱ͕ࣔ͞Ε͍ͯΔ (͞Βʹخ͍͠ʂ)ɽ 17
18. ### ४උฤ - ϥϓϥεۙࣅ ͱ͍͏Θ͚Ͱϥϓϥεۙࣅ͸Ҋ֎ѱ͘ͳ͍͕ɼ҉ʹ X ͻͱͭͷ෼෍͔Β͢΂ͯͷ؍ଌ஋͕ग़ͯ͘Δ X n → ∞

ΛԾఆ͍ͯ͠ΔͨΊɼϥϯμϜޮՌϞσϧͷΑ͏ͳจ຺Ͱ͸μϝ μϝ ΍ͬͱຊฤ΁ʜʜ 18
19. ### INLA; Integrated Nested Laplace Approximation INLA(Integrated Nested Laplace Approximation) ͷͶΒ͍͸

X LGM X ϥϓϥεۙࣅ X ਺஋ੵ෼ ͷ஌ݟΛ͏·͘૊Έ߹Θͤͯɼߴ଎͔ͭߴਫ਼౓ͳࣄޙ෼෍ͷۙࣅ ΛಘΔ͜ͱɽ ؾ࣋ͪͱͯ͠͸ 1. LGM ͷ࿮૊ΈͰϞσϦϯά͠ 2. ࣄޙ෼෍Λѻ͍΍͍͢෼෍ͷ૊Έ߹Θͤʹ෼ղ 3. ѻ͍ͮΒͦ͏ͳ෼෍͕ग़͖ͯͨΒϥϓϥεۙࣅ 4. ߴ࣍ݩͷੵ෼͕ඞཁͳͱ͜Ζ͸਺஋ੵ෼ ͱ͍͏͔Μ͡ 19
20. ### INLA; Integrated Nested Laplace Approximation ࣄޙ෼෍͸ π(x,θ∣y) ͕ͩɼ͜͜Ͱ஌Γ͍ͨͷ͸पลԽࣄޙ෼෍ π(θ∣y) ͱ

π(xi∣y)ɽ ·ͣ π(θ∣y) Λߟ͑Δͱ π(θ∣y) = π(x,θ∣y) π(x∣θ,y) = π(y∣x,θ)π(x,θ) π(y) 1 π(x∣θ,y) = π(y∣x,θ)π(x∣θ)π(θ) π(y) 1 π(x∣θ,y) ∝ π(θ)π(x∣θ)π(y∣x,θ) π(x∣θ,y) (13) 20
21. ### INLA; Integrated Nested Laplace Approximation π(θ∣y) ∝ π(θ)π(x∣θ)π(y∣x,θ) π(x∣θ,y) ͜͜Ͱ෼฼

π(x∣θ,y) Λ͞Βʹ෼ղ͢Δͱ π(x∣θ,y) = π(x∣θ)π(y∣x,θ) π(y∣θ) ∝ π(x∣θ)π(y∣x,θ) = N(x∣µ(θ),Q−1(θ))∏ i π(yi∣xi,θ) ∝ exp ⎛ ⎝ − 1 2 x⊺Q(θ)x + ∑ i log π(yi∣xi,θ) ⎞ ⎠ (14) 21
22. ### INLA; Integrated Nested Laplace Approximation π(x∣θ,y) ∝ exp ⎛ ⎝

− 1 2 x⊺Q(θ)x + ∑ i log π(yi∣xi,θ) ⎞ ⎠ exp ͷݞͷୈೋ߲ʹϥϓϥεۙࣅΛ͔͚ΔͳͲ͕ͯ͠Μ͹Δͱ πG (x∣θ, y) ∝ (2π)−n/2∣P (θ)∣1/2exp ⎛ ⎝ − 1 2 (x − µ(θ))⊺P (θ)(x − µ(θ)) ⎞ ⎠ (15) ͜͜Ͱ P (θ) = Q(θ) + diag(c(θ))ɼµ(θ) ͸ϞʔυͷҐஔɼ c(θ) ͸֤ xi ʹΑΔର਺໬౓ͷϞʔυपΓͰͷೋ࣍ඍ෼ɽ 22
23. ### INLA; Integrated Nested Laplace Approximation πG (x∣θ, y) ∝ (2π)−n/2∣P

(θ)∣1/2exp ⎛ ⎝ − 1 2 (x − µ(θ))⊺P (θ)(x − µ(θ)) ⎞ ⎠ where P (θ) = Q(θ) + diag(c(θ)) ॏཁͳ఺͸ X y ͷ؍ଌલޙͰɼਫ਼౓ߦྻ͸ର֯੒෼͚͔ͩ͠มԽ͠ͳ͍ ˠ ࠶ܭࢉίετ͸Θ͔ͣ X ۙࣅͷӨڹ͕ର֯੒෼ʹ͔͠ೖΒͳ͍ͨΊߴਫ਼౓ ͜ΕͰύϥϝʔλͷपลԽࣄޙ෼෍Λ π(θ∣y) ∝ ∼ π(θ)π(x∣θ)π(y∣x,θ) πG(x∣θ,y) ͱͯ͠ධՁͰ͖ͨ (ΊͰ͍ͨ)ɽ 23
24. ### INLA; Integrated Nested Laplace Approximation ࠷ޙʹ π(xi∣y) = ∫ π(xi∣θ,y)π(θ∣y)dθ

(16) Λߟ͍͑ͨɽ͔͠͠ʜʜ 1. θ ͷ marginalization ͷࡍɼθ ͕ߴ࣍ݩͩͱͭΒ͍ 2. ࣜ (16) ͸ n ճධՁ͠ͳ͚Ε͹ͳΒͳ͍ɽϥϓϥεۙࣅͰ͸ (n − 1) × (n − 1) ߦྻͷ factorization ͕ඞཁ 1 ͳͷͰɼn ͕େ ͖͍ͱͭΒ͍ ͱ͍͏ͭΒΈμϒϧύϯν͕ଘࡏɽ ͜ͷ͏ͪ 1 ʹؔͯ͠͸ɼθ Λ௿࣍ݩʹ੍ݶ͢Δ͜ͱͰղܾࡁ (ʁ) 1࣍ϖʔδิ଍ 24
25. ### INLA; Integrated Nested Laplace Approximation ิ଍ɿજࡏม਺ͷपลԽࣄޙ෼෍ʹର͢Δϥϓϥεۙࣅ ࣜ (16) ͷ͏ͪ π(xi∣θ,y)

ΛͳΜͱ͔͍ͨ͠ π(xi∣θ,y) = π(xi,x−i∣θ,y) π(x−i∣xi,θ,y) = π(xi,x−i,θ∣y) π(θ∣y) 1 π(x−i∣xi,θ,y) ∝ π(x,θ∣y) π(x−i∣xi,θ,y) ∝ π(θ)π(x∣θ)π(y∣x) π(x−i∣xi,θ,y) ≈ π(θ)π(x∣θ)π(y∣x) πG(x−i∣xi,θ,y) x−i ͷ৚݅෇͖෼෍ʹ͍ͭͯͷϥϓϥεۙࣅʹͳΔɽ (n − 1) × (n − 1) ߦྻΛѻΘͳ͍ͱ͍͚ͳ͍ͯͨ͘΁Μɽ (LU ෼ղΛ༻͍ͨߦྻࣜͷܭࢉྔ͸ O(n3)) 25
26. ### INLA; Integrated Nested Laplace Approximation ͜ͷϥϓϥεۙࣅΛ͕Μ͹ΕΔ৔߹ ˠ logπ(xi∣θ,y) Λ 3

࣍·Ͱల։͠ɼ࿪ਖ਼ن෼෍ͱͯ͠ ﬁtting ͦ͏Ͱͳ͍৔߹ ˠ޻෉ͨ۠͠෼ٻੵ๏Ͱ਺஋ੵ෼ ࿦จ͕͢͜͠Θ͔ΓͮΒ͍ͨΊɼҎԼ͸จݙ [3] ʹԊͬͯղઆ 26
27. ### INLA; Integrated Nested Laplace Approximation ؆୯ͷͨΊɼθ = (θ1,θ2)⊺ ͷ৔߹Ͱߟ͑Δ طʹಘͨ

logπ(θ∣y) ʹ͍ͭͯ 1. ४χϡʔτϯ๏ͳͲͰϞʔ υ ˆ θ Λ୳ࡧ 2. (θ1,θ2) ࠲ඪ͔Β (z1,z2) ࠲ ඪʹม׵ 3. H ݸͷ఺ {θh}H h=1 ʹ͍ͭͯɼ π(θh∣y) ΛධՁɽಉ࣌ʹର Ԡ͢ΔॏΈ ∆h ΋ಘΔ 27
28. ### INLA; Integrated Nested Laplace Approximation ؆୯ͷͨΊɼθ = (θ1,θ2)⊺ ͷ৔߹Ͱߟ͑Δ طʹಘͨ

logπ(θ∣y) ʹ͍ͭͯ 1. ४χϡʔτϯ๏ͳͲͰϞʔ υ ˆ θ Λ୳ࡧ 2. (θ1,θ2) ࠲ඪ͔Β (z1,z2) ࠲ ඪʹม׵ 3. H ݸͷ఺ {θh}H h=1 ʹ͍ͭͯɼ π(θh∣y) ΛධՁɽಉ࣌ʹର Ԡ͢ΔॏΈ ∆h ΋ಘΔ 28
29. ### INLA; Integrated Nested Laplace Approximation ؆୯ͷͨΊɼθ = (θ1,θ2)⊺ ͷ৔߹Ͱߟ͑Δ طʹಘͨ

logπ(θ∣y) ʹ͍ͭͯ 1. ४χϡʔτϯ๏ͳͲͰϞʔ υ ˆ θ Λ୳ࡧ 2. (θ1,θ2) ࠲ඪ͔Β (z1,z2) ࠲ ඪʹม׵ 3. H ݸͷ఺ {θh}H h=1 ʹ͍ͭͯɼ π(θh∣y) ΛධՁɽಉ࣌ʹର Ԡ͢ΔॏΈ ∆h ΋ಘΔ 29
30. ### INLA; Integrated Nested Laplace Approximation ಘΒΕͨ {θh}H h=1 Λ༻͍ͯॴ๬ͷੵ෼Λ۠෼ٻੵ๏Ͱܭࢉ π(xi∣y)

= ∫ π(xi∣θ,y)π(θ∣y)dθ ≈ ∑ h π(xi∣θh,y)π(θh∣y)∆h Ҏ্Ͱ໨తͷपลԽࣄޙ෼෍ π(θ∣y),π(xi∣y) ΛධՁͰ͖ͨɽ ͋ͱ͸࣮ݧͷ࿩ɽ خ͍͜͠ͱʹ R-INLA ύοέʔδ͕ఏڙ͞Ε͍ͯΔ [5]ɽ - ΞϧΰϦζϜͷίΞͳ෦෼͸ C Ͱॻ͔Ε͍ͯΔͬΆ͍ʁ R-INLA ͷ঺հ͕ϝΠϯͳͷͰ࣮ݧͷߟ࡯ͳͲ͸͔ͳΓࡶɽ 30
31. ### ࣮ݧ - ਓ޻σʔλ γϛϡϨʔγϣϯσʔλʹରͯ͠ͷ࣮ݧɽ y∣η ∼ Poisson(exp(η)) ηi = µ

+ βwi + uj(i) u ∼ Nm(0,τ−1I) µ = 0,β = 1 ͱ͠ɼj(i) ͸ 1 ∶ n ͔ Β 1 ∶ m ΁ͷ (ϥϯμϜͳ) ࣸ૾ ͜͜Ͱ͸ n = 50,m = 10 ਪఆ͢Δύϥϝʔλ͸ µ,β,u ・・・ 31
32. ### ࣮ݧ - ਓ޻σʔλ u1 ∼ N (0,(1 3 )2 )

ͷࣄޙ෼෍ ࣮ઢɿINLAɼഁઢɿ୯७ͳϥϓϥεۙࣅɼώετάϥϜɿMCMC MCMC ͸ 100,000 ճΠςϨʔγϣϯɽ͍͍ײ͡Ͱ͢Ͷɽ 32

34. ### ࣮ݧ - ۭؒσʔλ Πϯάϥϯυ๺੢ΤϦΞͷന݂පʹؔ͢Δݚڀσʔλɽ 1. ΤϦΞΛϝογϡʹ۠੾ͬͯ 2. ֬཰ภඍ෼ํఔࣜΛཱͯͯ 3. ஍ҬޮՌΛਪఆ

ͱ͍͏͜ͱΛ͍ͯ͠ΔΒ͍͠ MCMC ΍ઌߦݚڀͱͷ݁ՌൺֱͳͲ͸͍ͯ͠ͳ͍ͷͰɼ݁Ռͷղ ऍ͕··ͳΒͳ͍͕ʜʜ ͨͿΜʮR-INLA ύοέʔδͩͱ͜͏͍͏λεΫ͕؆୯ʹͰ͖Δ ͥʂʯͱ͍͏એ఻ 34

36. ### ·ͱΊͱॴײ ·ͱΊ X ϥϓϥεۙࣅʹجͮ͘ࣄޙ෼෍ͷۙࣅख๏ INLA Λ঺հ X LGM ͷܗࣜͰද͞ΕΔϞσϧʹରͯ͠༗ޮ X

R ύοέʔδ R-INLA ͔Β͓खܰʹ࢖͑Δ ॴײ X MCMC ͷଟ͘ͷख๏΍ม෼ਪ࿦ͱ͸ҟͳΓɼߋ৽ࣜͷಋग़ ΍৬ਓతͳύϥϝʔλௐ੔͸ෆཁͬΆ͍ͷ͸ར఺ X ͲΕ͘Β͍ෳࡶͳϞσϧ·Ͱ࢖͑ͦ͏͔͸ཁݕ౼ X ը૾ॲཧͱ͔ʹ͸ͦΕͳΓʹ͍͚Δؾ͕͢Δ 36
37. ### References i [1] H. Rue, A. Riebler, S. H. Sørbye,

J. B. Illian, D. P. Simpson, and F. K. Lindgren,ʠ Bayesian Computing with INLA: A Review, ʡarXiv:1604.00860 [stat], Apr. 2016. [2] H. Rue, S. Martino, and N. Chopin,ʠ Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations, ʡJournal of the Royal Statistical Society: Series B (Statistical Methodology), vol. 71, no. 2, pp. 319392, Apr. 2009. [3] G. Baio, An introduction to INLA with a comparison to JAGS, http://www.statistica.it/gianluca/Talks/INLA.pdf
38. ### References ii [4] D. Bolin, The R-INLA package, https://www.stat.washington.edu/peter/ PASI/pasi

practical intro.pdf [5] The R-INLA Project, http://www.r-inla.org/ [6] jsta/r-inla - GitHub, https://github.com/jsta/r-inla