Juliaで学ぶ Hamiltonian Monte Carlo (NUTS 入り)

1. Julia Tokyo #2 Julias»‚ Hamiltonian Monte Carlo (NUTSâ–) ,‡ W+ (Kenta Sato) @bicycle1885 1 / 55

2. ²òÅòÃ ß…Ãu Ýê²Ô×7áòÅªêì] (MCMC) Metropolis-Hastings Hamiltonian Monte Carlo (HMC) No-U-Turn Sampler (NUTS) JuliasžT—MCMCÐÂ°–· 2 / 55

3. _ŸvÑÇzkzP¼se MCMCw”—´òÖéò¯žÕèÂ®ÛÂ®¸wck[vP Stanvuz´òÖè–zqvž£fkP JuliasžT—´òÖè–ž—–kP 3 / 55

4. ¶ ÔzZÖ|MCMCß9{

5. zw÷˜— _se D»0wrÃvø{?cvPsAaP Æ?w{Qz¶žorPŠeX|š“’T«PXN— WŽc˜ŠgŸ izÔ{izÝs®,crPkl]—tˆW–Še 4 / 55

6. ß…Ãu / ÐÂ°–·Ãu 5 / 55

7. ß…Ãu ,‡ W+ Twitter/GitHub: @bicycle1885 }: rRR»Ç»ó š: Bioinformatics JuliaÉ: ñz|›xW ¾Yv{§: Julia / Haskell ”[žR{§: Python / R 6 / 55

8. HaskellŽ”™c[V,PcŠe ²Haskellw”— Ê€ íÖì¯èÞò¯³z™ëÒä–žcŠc k} 7 / 55

9. ÐÂ°–·Ãu DocOpt.jl - https://github.com/docopt/DocOpt.jl ×êÖàÂº–·žÐ–¸c|²ÝòÈè£òÂDzÐ–¸že— RandomForests.jl - https://github.com/bicycle1885/RandomForests.jl A†»?¡ê³é¹ßRandom ForestzJuliaö GeneOntology.jl - https://github.com/bicycle1885/GeneOntology.jl f»zGene OntologyzÃ–ê¬ÂÇžö®cr— 8 / 55

10. Ýê²Ô×7áòÅªêì] (MCMC) 9 / 55

11. MCMC]t{ 1 Y {›ft—˜k|± Ú W•z´òÖêžÝê²Ô×7žJ Pr¤—´òÖéò¯]z~tp} ¤•˜k´òÖêß9{ Úz?– Œvuö“v–ž4‘e— zwž›˜—} Ýê²Ô×7t{|xWzæAz‹sÕzæAz± ÚXEŠ—± f z_tžPR} 1 9U

12. Y 9 Y 2 9U YU 1 9U

13. Y 9U YU Y Y 5 Y Y _ztY|æA W• „·¤e—±ž t„Y|·¤± t‚} 10 / 55

14. Ýê²Ô×7z. ·¤± 5 Y Y 1 9U

15. Y 9U Y 11 / 55

16. ccPMCMC Y ö0z± Ú W•´òÖéò¯e—w{|·¤±Xºka v]˜|v•vP[õXN—} ÚzÓ/[: §ê³–È[: Y D 5 Y Y Y EY 1 U Y ³ Y BT U ³ 7 GPSBOZ 1 Y ©´òÖè–X_˜•žºketPR_tzÚvu{à{cvP (sYvP)} 12 / 55

17. ûÕpósz´òÖéò¯{ÕcP Q Y Q Y ±Ã_D iz”cLBÃ_D XN—te—} ûÕpósz´òÖéò¯zÕca ýP¥°{|óÖz`[´ówK¯crP— Q Y cWci˜Xu_W{´òÖéò¯¼w{ W•vP 2pzƒ 1. izÝW•ýP[„ýP[„t‹ 2. ýPt_™žfp]k•i_W•9Pt_™„{NŠ–íWvP ➠ MCMC{Šawiz”Rvƒžt— 13 / 55

18. MetropolisHastings Night View with Tokyo Tower Special Lightup (Shibakouen, Tokyo, Japan) by t-mizo is licensed under CC BY 2.0 14 / 55

19. MetropolisHastings MCMC´òÖéò¯z~tps|Ž ÚtPR´òÖéò¯ckP Út{*z ÚW•°BFž–cc| P–v•izFž*v c|iRsv]˜|izÝwtuŠ—} R ú °BFžfe—Ž Ú {özvPcL Úvu|´ò Öéò¯ceP Úwe—} ú ú N ú N NJO Q ú R N ú °BF {’Az± s*va˜—: Q N R ú N Q Q __s| {´òÖéò¯ckP Ú z”cLBÃ_D 15 / 55

20. MetropolisHastings z¡ê³é¹ß ”cLB± ÚD W•´òÖéò¯e— 1. |æA žE| we— 2. Ž Ú W•kvF žt— 3. ± s ž´òÖêtcr*vc|iRsv]˜| @qe— 4. *va˜kÝ{ tc|@qa˜kÝ{ te— 5. tcr|2~4ž z´òÖêX¤•˜—Šs– 4e Q N ± R ú N ú ú N ú N

21. ± ú N

22. ± N N ± N

23. . 16 / 55

24. Ž Ú{cL Ú( randn) # p: (unnormalized) probability density function # θ₀: initial state # M: number of samples # ϵ: step size function metropolis(p::Function, θ₀::Vector{Float64}, M::Int, ϵ::Float64) d = length(θ₀) # allocate samples' holder samples = Array(typeof(θ₀), M) # set the current state to the initial state θ = θ₀ for m in 1:M # generate a candidate sample from # the proposal distribution (normal distribution) θ̃ = randn(d) * ϵ + θ if rand() min(1.0, p(θ̃) / p(θ)) # accept the proposal θ = θ̃ end samples[m] = θ print_sample(θ) end samples end metropolis.jl 17 / 55

25. cc 2/Dz/DwözN—cL Ú # mean μ = [0.0, 0.0] # covariance matrix Σ = [1.0 0.8; 0.8 1.0] # precision matrix Λ = inv(Σ) # unnormalized multivariate normal distribution normal = x - exp(-0.5 * ((x - μ)' * Λ * (x - μ))[1]) |–x₀|´òÖêDM|¸ÅÂÖüϵž®cr´òÖéò¯ samples = metropolis(normal, x₀, M, ϵ) 18 / 55

26. HQ created with Gadfly.jl -4 -2 0 2 4 x Iteration 500 400 300 200 100 1 4 2 0 -2 -4 y Metropolis (ϵ = 1.0) 19 / 55

27. HQ created with Gadfly.jl -3 -2 -1 0 1 x Iteration 500 400 300 200 100 1 0.5 0.0 -0.5 -1.0 -1.5 -2.0 y Metropolis (ϵ = 0.1) -3 -2 -1 0 1 2 x Iteration 500 400 300 200 100 1 3 2 1 0 -1 -2 -3 y Metropolis (ϵ = 0.5) -4 -2 0 2 4 x Iteration 500 400 300 200 100 1 4 2 0 -2 -4 y Metropolis (ϵ = 1.0) -3 -2 -1 0 1 2 3 x Iteration 500 400 300 200 100 1 3 2 1 0 -1 -2 -3 y Metropolis (ϵ = 2.0) 20 / 55

28. MetropolisHastings zþUF 1. @qzÇë–È©Ô 1 ¸ÅÂÖ´£¹ z–s|@qt[ÿzÇë–È©ÔXN— 2. èò¿ß¥¨–® ´òÖêzÊXèò¿ß¥¨–®že— 21 / 55

29. þU1: @qzÇë–È©Ô ± Úz–XK¯cr—z{`[´ól]} 1 1 ¸ÅÂÖ´£¹ R ➠ RY[”]—X|@qXÖX— ¸ÅÂÖ´£¹ Ÿ ➠ @q{WT•˜—X|NŠ–”]vP MCMCW•v—…[¬€v´òÖêž¤—w{¸ÅÂÖ´£¹žRY [ckPX|@qXÖX—k´òÖéò¯zµXu[v—Çë –È©ÔXN—} 1 Õp(´òÖéò¯e—/D)w”orÐèà–¾ z P–X£v –|ÛXÕcP} 22 / 55

30. þU2: èò¿ß¥¨–® Metropolis-HastingsW•¤•˜k´òÖêÊ{|èò¿ß¥¨–®ž crP— ú N Ž ÚXÜe—°BF {|xWz– W•‹r[0 X¤”ck–W•ewûorYrcŠR_tXN— N èò¿ß¥¨–®s{(VVŠWw{or)`ûxDz[$w‡º ck—|cWvP óž…W•…ŠsU—zwWv–`ûxDX«Pwv— 23 / 55

31. Hamiltonian Mote Carlo (HMC) Hamiltonian circuit on a small rhombicosidodecahedron by fdecomite is licensed under CC BY 2.0 24 / 55

32. Hamiltonian Monte Carlo Hamiltonian Monte Carlo](HMC){|ÎÞêÇò¦»(Hamiltonian dynamics)žpwæŽa˜kMCMC]z~tp} ±Ã_Dz¶žqJe— (|Œ0v± Ú{sYvP) ószŠ¨zå”žðor´òÖêž¤— )zMCMCz¡ê³é¹ßt‡¶cr|öz vP P´òÖ êX¤•˜— _z]žPagkNo-U-Turn Sampler (NUTS){StantPR Ø£¹4zkzÖì¯èÞò¯{§wöa˜rP— 25 / 55

33. Boltzmann Ú Y Y 1 Y æA z§Ìê– t± Ú {Õz”Rw×Ô]• ˜—} ; 1 Y FYQ + Y ; __s| {± ÚzcLBDsN—} _˜žuwžT||± Úz§Ìê–X4‘sY—} Y + MPH 1 Y + MPH ; 26 / 55

34. ÎÞêÇò¦» S Š¨zå”žæT—} žŠ¨z”ŸØ®Çê| žå”£Ø®Çê tckÔzŠ¨zå”žE—ÎÞêÇò[ ä: EJ EU ESJ EU ) SJ + ) J ) S 6 __s|ÎÞêÇÊ¡ò {ÜÅò¶âê§Ìê– t å”§Ìê– z÷tcraa˜—} , S ) S 6

35. , S 27 / 55

36. ´òÖéò¯„z%J /D ”ŸØ®Çê : ´òÖéò¯ckP±/D å”£Ø®Çê : å”zBˆ0v/D §Ìê– ÜÅò¶âê§Ìê– : Boltzmann Úž2w å”§Ìê– : 2|vå”§Ìê–w ”ŸØ®Çêtå”£Ø®Çêz•Ô Ú { ”–’Az”Rw wsY—} S 6 , S Q S ) S 6

37. , S Q S FYQ +) S FYQ +6 FYQ +, S ; ; 28 / 55

38. HMCz*?± Üa˜k°BFwe—*?±{’Az”Rwv—} NJO ú FYQ ) S + ) ú S ú ) ) S + ) ú S ú võ0w{| z–{Ó/vzs ’T«f *va˜— ( ) {flX|²òÓä–¾sD–0wÎÞêÇò[ äž|ŒBcrw[t«f¨1XPfe—kxö0w{@q{ »ìsvP} 29 / 55

39. Leapfrog|ŒB ÎÞêÇò[ ä{w}0ww[z{ÕcPzs|D– žíR} i_s{|Leapfrog|ŒBtPR’AzÒäžpWR} SJ U

40. 1 J U

41. 1 SJ U

42. 1 SJ U + 1 6 U J J U

43. 1SJ U

44. 1 SJ U

45. 1 + 1 6 U

46. 1 J 30 / 55

47. vhLeapfrog|ŒBvzW Q S ) S •Ô Ú žÓ/we—kw{| z9žÓ/w cv]˜|v•vP cWc|Euler]vus{(h_zuažÏÁcrŽ)9X/Bc rcŠRzs| Q S XÓ/wv•vP Leapfrog|ŒBs{|3pzÒä{i˜j˜¹ (shear mapping)vzs|i˜j˜2JcrŽ9X/BcvP VerticalShear m=1.25 by RobHar - Own work using Inkscape. Licensed under Public domain via Wikimedia Commons - http://commons.wikimedia.org/wiki/File:VerticalShear_m%3D1.25.svg#mediaviewer/File:VerticalShear_m%3D1.25.svg 31 / 55

48. HMCw”—´òÖéò¯¡ê³é¹ß 1. |æA žE| we— 2. å”£žcL ÚvuW•´òÖéò¯e— 3. W•¸ÅÂÖ´£¹ sLeapfrog|ŒBw”—Òž x –4c| ž¤— 4. ± s*vc|iRsv] ˜|@qe— 5. *va˜kÝ{ tc|@qa˜kÝ{ te— 6. tcr|2~5ž z´òÖêX¤•˜—Šs– 4e N ± N 1 - ú NJO FYQ ) S + ) ú S ú N

49. ± ú N

50. ± N N ± N

51. . 32 / 55

52. # U : potential energy function # ∇U : gradient of the potential energy function # θ₀ : initial state # M : number of samples # ϵ : step size # L : number of steps function hmc(U::Function, ∇U::Function, θ₀::Vector{Float64}, M::Int, ϵ::Float64, L::Int) d = length(θ₀) # allocate sampels' holder samples = Array(typeof(θ₀), M) # set the current sate to the initail state θ = θ₀ for m in 1:M # sample momentum variable p = randn(d) H = U(θ) + p ⋅ p / 2 θ̃ = θ for l in 1:L p -= ϵ / 2 * ∇U(θ̃) # half step in momentum variable θ̃ += ϵ * p # full step in location variable p -= ϵ / 2 * ∇U(θ̃) # half step in momentum variable again end H̃ = U(θ̃) + p ⋅ p / 2 if randn() min(1.0, exp(H - H̃)) # accept the proposal θ = θ̃ end samples[m] = θ print_sample(θ) end samples end hmc.jl 33 / 55

53. HQ created with Gadfly.jl -4 -2 0 2 4 x Iteration 500 400 300 200 100 1 4 2 0 -2 -4 y HMC (ϵ = 0.1, L = 10) - 34 / 55

54. HQ created with Gadfly.jl -2 -1 0 1 x Iteration 500 400 300 200 100 1 1 0 -1 -2 y HMC (ϵ = 0.01, L = 10) -4 -2 0 2 4 x Iteration 500 400 300 200 100 1 3 2 1 0 -1 -2 -3 y HMC (ϵ = 0.05, L = 10) -4 -2 0 2 4 x Iteration 500 400 300 200 100 1 4 2 0 -2 -4 y HMC (ϵ = 0.1, L = 10) -4 -2 0 2 4 x Iteration 500 400 300 200 100 1 4 2 0 -2 -4 y HMC (ϵ = 0.5, L = 10) - 35 / 55

55. HQ created with Gadfly.jl -0.3 -0.2 -0.1 0.0 0.1 x Iteration 500 400 300 200 100 1 0.1 0.0 -0.1 -0.2 -0.3 y HMC (ϵ = 0.01, L = 1) -1.0 -0.5 0.0 0.5 x Iteration 500 400 300 200 100 1 0.5 0.0 -0.5 -1.0 -1.5 y HMC (ϵ = 0.05, L = 1) -2 -1 0 1 x Iteration 500 400 300 200 100 1 1 0 -1 -2 y HMC (ϵ = 0.1, L = 1) -4 -2 0 2 4 x Iteration 500 400 300 200 100 1 3 2 1 0 -1 -2 -3 y HMC (ϵ = 0.5, L = 1) - 36 / 55

56. HQ created with Gadfly.jl -4 -2 0 2 4 x Iteration 500 400 300 200 100 1 3 2 1 0 -1 -2 -3 y HMC (ϵ = 0.01, L = 50) -3 -2 -1 0 1 2 3 x Iteration 500 400 300 200 100 1 4 2 0 -2 -4 y HMC (ϵ = 0.05, L = 50) -4 -2 0 2 4 x Iteration 500 400 300 200 100 1 4 2 0 -2 -4 y HMC (ϵ = 0.1, L = 50) -2 -1 0 1 2 3 x Iteration 500 400 300 200 100 1 3 2 1 0 -1 -2 y HMC (ϵ = 0.5, L = 50) - 37 / 55

57. HMCXwEck_t §Ìê–z¶áQžžR_ts|Iÿwvok_t: 1 ¸ÅÂÖ´£¹ žP Ÿa[t˜|Leapfrogz ¨1XŸa [v–|@qž[WT•˜— Š¨X - ¸ÅÂÖ×crÍ•Ww¤”e—k|èò¿ß¥¨ –®t‡…r[Šs”]— @qžWTpp¼z”Ÿ”–[Šs”[_tXsY—”Rwv –|¤•˜—´òÖêX”–¬€v´òÖêwßqPk} 38 / 55

58. HMCzÕca HMCzqF{|å”žÛe—2pzÐèà–¾ ¸ÅÂÖ´£¹ ¸ÅÂÖD 1 - z–Xm“Ru P–wa˜rP—tPR_tw•#crP—} Úzæw”orm“Ru P–X/›—k|_˜•z–žuŸ v ÚwŽRŠ[P[”R6e—z{ÓIÿ} 39 / 55

59. 1 - t zÛ^ž«T—tuRv—W 1 ¸ÅÂÖ´£¹ : XŸaeZ— ➠ Š¨XNŠ–”WvP XRYeZ— ➠ leapfrog|ŒBXìeZr@qXÖX— 1 1 1 - žŸa[e—t žRY[cvPtP]vPk|4‘²¸ÇŽW W—} - ¸ÅÂÖD : XŸaeZ— ➠ èò¿ß¥¨–®žcrcŠR XRYeZ— ➠ Š¨XÂY4e (U¾–ò) - - ➠ ß”0wÐèà–¾žÛckP 40 / 55

60. NoUTurn Sampler (NUTS) No Turning Back by Pak Gwei is licensed under CC BY-NC-SA 2.0 41 / 55

61. NoUTurn Sampler HMC{¸ÅÂÖ´£¹ 1 t¸ÅÂÖD - z2pzÐèà–¾wÑ÷l okX| No-U-Turn Sampler (NUTS)s{_˜•zÐèà–¾(§w )žRŠP_tÛcr[˜—} zÛ ➠ ´òÖéò¯¼zdual averagingw”–?2B zÛ ➠ ´òÖéò¯¯zŠ¨z}cPå”žccrµŠ— - 1 - ÑÇXsÐèà–¾zÀä–Êò¯e—_tv[?2vHMC´òÖ è–t•d[•õz P´òÖêX¤•˜—”RwvorP—} 42 / 55

62. ±¿seXNUTSz¡ê³é¹ßse (Hoffman Gelman, 2014) 43 / 55

63. NUTSzPF aeXwÀóžÃue—z{rcPzsPFžÃue—t| ´òÖêzP…{|0|vp—sàP[XPP vzs6 - žgf|P…žuŸuŸü|crP[ ü|ceZrå”XU¾–òž¥k•|P…žü|ezžµ— izP…wN—Æ–¾FW•|cP´òÖêž¤— P…W•z´òÖéò¯{|ÍQ–PžVavP”Rwe— ÍQ–P(detailed balance)❏ t{| ÚXÓ/wv—kzP ãL 44 / 55

64. No! UTurn!! ú P…zàazÔ/B{|¥F W•xWzF ŠszØ®Çêt å”£Ø®Çê Sú zw‡ºe—: ú 5 E ú ú 5Sú + + + E EU ú+ 5 ú+ EU _z–X ’Awvok•|P…XU¾–òžc¥k_twv—} 45 / 55

65. àPzsDwv—[{´òÖê²–Èznuts.jlžˆ¹crAaP} # L: logarithm of the joint density θ # ∇L: gradient of L # θ₀: initial state # M: number of samples # ϵ: step size function nuts(L::Function, ∇L::Function, θ₀::Vector{Float64}, M::Int, ϵ::Float64) d = length(θ₀) samples = Array(typeof(θ₀), M) θ = θ₀ for m in 1:M r₀ = randn(d) u = rand() * exp(L(θ) - r₀ ⋅ r₀ / 2) θ⁻ = θ⁺ = θ r⁻ = r⁺ = r₀ C = Set([(θ, r₀)]) j = 0 s = 1 while s == 1 v = randbool() ? -1 : 1 if v == -1 θ⁻, r⁻, _, _, C′, s′ = build_tree(L, ∇L, θ⁻, r⁻, u, v, j, ϵ) else _, _, θ⁺, r⁺, C′, s′ = build_tree(L, ∇L, θ⁺, r⁺, u, v, j, ϵ) end if s′ == 1 C = C ∪ C′ end s = s′ * ((θ⁺ - θ⁻) ⋅ r⁻ ≥ 0) * ((θ⁺ - θ⁻) ⋅ r⁺ ≥ 0) j += 1 end θ, _ = rand(C) samples[m] = θ print_sample(θ) end samples end nuts.jl 46 / 55

66. HQ created with Gadfly.jl -3 -2 -1 0 1 2 3 x Iteration 500 400 300 200 100 1 3 2 1 0 -1 -2 -3 y NUTS (ϵ = 0.1) 47 / 55

67. HQ created with Gadfly.jl -3 -2 -1 0 1 2 3 x Iteration 500 400 300 200 100 1 4 2 0 -2 -4 y NUTS (ϵ = 0.01) -3 -2 -1 0 1 2 3 x Iteration 500 400 300 200 100 1 3 2 1 0 -1 -2 -3 y NUTS (ϵ = 0.05) -3 -2 -1 0 1 2 3 x Iteration 500 400 300 200 100 1 3 2 1 0 -1 -2 -3 y NUTS (ϵ = 0.1) -3 -2 -1 0 1 2 3 x Iteration 500 400 300 200 100 1 3 2 1 0 -1 -2 -3 y NUTS (ϵ = 0.5) 48 / 55

68. JuliasžT— MCMCÐÂ°–· 49 / 55

69. JuliasžT—MCMCÐÂ°–· MCMC.jl - https://github.com/JuliaStats/MCMC.jl ´òÖè–zÆÐ–Ç (12$·!) È¬äàòÇXvP Stan.jl - https://github.com/goedman/Stan.jl StanÏ£òÆ¢ò¯ (via CmdStan) izRmMCMC.jlw–Š˜—o‰P? Mamba.jl - https://github.com/brian-j-smith/Mamba.jl Wv–˜Do‰PJuliakzöJ0vMCMCÔë–ßî–® OözÈ¬äàòÇ 50 / 55

70. Št HMC{Š¨zå”žð…cr´òÖéò¯e—_tw”–|@q žA^•˜— NUTS{HMCzÕcPÐèà–¾Ûž|ß”Bcr[˜— Mamba.jlXJuliazöJ0v´òÖè–z¶ö×W 51 / 55

71. ˆæ Bishop, C. M. (2006). Pattern recognition and machine learning. New York: springer. (pJÙ (2012) ´òÖéò¯] Ð¾–òéå tA†»? A, pp.237-273. ½ch) Hoffman, M. D., Gelman, A. (2014). The No-U-Turn Sampler : Adaptively Setting Path Lengths in Hamiltonian Monte Carlo. Journal of Machine Learning Research, 15, 1351–1381. MacKay, D. J. C. (2003). Information Theory, Inference, and Learning Algorithms. Cambridge University Press. Neal, R. M. (2011). MCMC Using Hamiltonian Dynamics. In Handbook of Markov Chain Monte Carlo, pp.113-162. Chapman Hall/CRC. iJ;. (2008). Ýê²Ô×7áòÅªêì] ÒÞ„? 52 / 55

72. VŠ] 53 / 55

73. Juliaz¼–¸²–ÈtÍ²–Èz£àv·Í function build_tree(L::Function, ∇L::Function, θ::Vector{Float64}, r::Vector{Float64}, u::Float64, v::Int if j == 0 θ′, r′ = leapfrog(∇L, θ, r, v * ϵ) C′ = u ≤ exp(L(θ′) - r′ ⋅ r′ / 2) ? Set([(θ′, r′)]) : Set([]) s′ = int(L(θ′) - r′ ⋅ r′ / 2 log(u) - Δmax) return θ′, r′, θ′, r′, C′, s′ else θ⁻, r⁻, θ⁺, r⁺, C′, s′ = build_tree(L, ∇L, θ, r, u, v, j - 1, ϵ) if v == -1 θ⁻, r⁻, _, _, C″, s″ = build_tree(L, ∇L, θ⁻, r⁻, u, v, j - 1, ϵ) else _, _, θ⁺, r⁺, C″, s″ = build_tree(L, ∇L, θ⁺, r⁺, u, v, j - 1, ϵ) end s′ = s′ * s″ * ((θ⁺ - θ⁻) ⋅ r⁻ ≥ 0) * ((θ⁺ - θ⁻) ⋅ r⁺ ≥ 0) C′ = C′ ∪ C″ return θ⁻, r⁻, θ⁺, r⁺, C′, s′ end end nuts.jl 54 / 55

74. ÎÞêÇÊ¡òzÓ/[zËÚ E) EU ) EYJ ) EQJ

75. YJ EU QJ EU #J ) ) ) ) #J + QJ YJ QJ YJ 55 / 55

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