马尔科夫链蒙特卡洛方法.pdf

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Markov Chain Monte Carlo Methods

Introduction

Integration problems in Bayesian inference

Markov Chain Monte Carlo Integration

Markov Chain

The Metropolis-Hastings Algorithm

Metropolis-Hastings Sampler

The Metropolis Sampler

Random Walk Metropolis

The Independence Sampler

Single-component Metropolis Hastings Algorithms

Application: Logistic regression

Lecture 8: Markov Chain Monte Carlo Methods () (ŒŒŒ¯¯¯AAAkkk{{{) Monday 26th October, 2009

Contents 1 Markov Chain Monte Carlo Methods 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Integration problems in Bayesian inference . . . . . . 1.1.2 Markov Chain Monte Carlo Integration . . . . . . . 1.1.3 Markov Chain . . . . . . . . . . . . . . . . . . . . . 1 1 2 4 8 1.2 The Metropolis-Hastings Algorithm . . . . . . . . . . . . . . 13 1.2.1 Metropolis-Hastings Sampler . . . . . . . . . . . . . 14 1.2.2 The Metropolis Sampler . . . . . . . . . . . . . . . . 22 1.2.3 Random Walk Metropolis . . . . . . . . . . . . . . . 23 . . . . . . . . . . . . . . 33 1.2.4 The Independence Sampler 1.3 Single-component Metropolis Hastings Algorithms . . . . . 37 1.4 Application: Logistic regression . . . . . . . . . . . . . . . . 39 Previous Next First Last Back Forward 1

Chapter 1 Markov Chain Monte Carlo Methods 1.1 Introduction MCMC(Markov Chain Monte Carlo) {ne–ºw Metropo- lis et al. (1953)–9 Hastings (1970), –9ƒ«0MCMC;˝. !•0ø«{˜gA^. 5¿3c¡0Monte Carlo{O¨' g(t)dt, A ·rd¨'L«⁄Ø,V˙f (t)eˇ". l¨'OﬂK=z ⁄l8IV˙f (t)¥) ¯. Previous Next First Last Back Forward 1

3MCMC{¥, ˜kÆŒ¯, ƒf (t)†›'. K –$1dŒ¯¿'m ´æ†›', @ol8I'f (t)¥ )¯, ·l†›GŒ¯¥)·». • 0A«ÆøŒ¯{: Metropolis{, Metropolis- Hastings{, –9Gibbs˜{. —ATkﬂ•(rapid mixing)5—l?¿uØﬂ†›'. 1.1.1 Integration problems in Bayesian inference Bayesian¥NıﬂK·MCMC{A^. lBayesian*:5w, .¥*CºŒ·¯C. ˇd, x = (x1, · · · , xn)º ŒθØ'–L« fx,θ(x, θ) = fx|θ(x1, · · · , xn)π(θ). Previous Next First Last Back Forward 2

–O. ('f'1¥Kz~Œ-). ø·~—5, ˇ 3 ØıﬂKe, Kz~ŒØJO. ,¡, 3ØıNﬂK¥d¨'vkw«L«, Œ{ØJO , AO·3p, MCMC{Øda¨'Jł Ø—O{. 1.1.2 Markov Chain Monte Carlo Integration ¨' E[g(θ|x)] = g(θ)fθ|x(θ|x)dθ Monte CarloO m i=1 ¯g = 1 m g(xi), ¥x1, · · · , xml'fθ|x(θ|x)¥˜. x1, · · · , xmÆ, K –Œ˘n“uˆ¡, ¯g´æE[g(θ|x). ·3ﬂK¥, l'fθ|x(θ|x)¥˜·~(J, MCMC {·d8), 1”MC”, Markov Chain, L«l8I' ¥˜, 1”MC”, Monte Carlo, KL«3˜e, A^Monte Previous Next First Last Back Forward 4

Carlo¨'{Ø¨'?1O. MCM{n3ueª4‰n: Theorem 1. {Xt, t ≥ 0},–ˇGmΩŒ…, π†›', π0·—'', K πt → π, t → ∞. øpπtL«Œ…3t>S'. d‰n‘†dŒ…$1¿' m(t = n, nØ), K3n, Xn'π, —''ˆ’. Theorem 2. (Markov chain Law of Large Numbers)eX1, X2, · · · H {†›'πŒ…(¥zXt–·ı), KXn'´æ 'π¯CX, Ø?¿…Œf , Eπ|f (X)|3, n → ∞, K n t=1 ¯fn = 1 n f (Xt) → Eπf (X), a.s. Previous Next First Last Back Forward 5

e¡•OH{ ¯fn. Pf t = f (Xt), γk = cov(f t, f t+k), Kf tσ2 = γ0, k’XŒρk = γk/σ2. l n V arπ( t=1 f t) f t) = 1 n n n−t t=1 1 n k=1 1 n n = nV arπ( ¯fn) = nV ar( τ 2 n t=1 = 1 n V arπ(f t) + 2 cov(f t, f t+k) n−1 n − k n n−1 n − k n ρk]. = σ2 + 2 γk = σ2[1 + 2 k=1 –y† . ˇdH{ ¯fn n → τ 2 = σ2[1 + 2 τ 2 n−1 [1 + 2 σ2 n k=1 V arπ( ¯fn) = k=1 ρk] ∞ k=1 n − k n ρk]. Previous Next First Last Back Forward 6

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