Markov Chain Monte Carlo (MCMC), Gibbs Sampling, Metropolis Algorithms, and Simulated Annealing Bioinformatics Course Supplement

Transcription

1 Markov Chan Monte Carlo MCMC, Gbbs Samplng, Metropols Algorthms, and Smulated Annealng 2001 Bonformatcs Course Supplement SNU Bontellgence Lab Outlne! Markov Chan Monte Carlo MCMC! Metropols-Hastngs Algorthm! Metropols Algorthm! Gbbs Samplng! Smulated Annealng c 2001 SNU Bontellgence Lab 2

2 Introducton! Markov Chan Monte Carlo Monte Carlo ntegraton usng Markov chans Monte Carlo ntegraton draw samples from the requred dstrbuton, and then forms sample averages to approxmate expectatons Markov chan Monte Carlo draws samples by runnng a cleverly constructed Markov chan for a long tme MCMC s usually used for Bayesan nference c 2001 SNU Bontellgence Lab 3 Bayesan Inference 1! Bayesan nference Most applcatons of MCMC are orented D : observed data, θ : model parameters Pθ : pror dstrbuton, PD θ : lkelhood Full probablty model PD, θ =PD θ Pθ Posteror dstrbuton of θ: PD θ! Havng observed D, Bayes theorem s used! Object of all Bayesan nference P θ D = P θ P D θ P θ P D θ dθ c 2001 SNU Bontellgence Lab 4

3 Bayesan Inference 2! Any features of the posteror dstrbuton are legtmate for Bayesan nference The posteror expectaton of a functon fθ:! Dffculty E[ f θ D] = f θ P θ P D θ dθ P θ P D θ dθ Integraton, especally n hgh dmensons s mpossble Numercal evaluaton s also dffcult and naccurate Analytc approxmaton: Laplace approxmaton, Monte Carlo ntegraton MCMC c 2001 SNU Bontellgence Lab 5 Calculatng Expectatons 1! Terms π Task! A vector of k random varables wth dstrbuton π! In Bayesan applcatons, wll comprse model parameters θ! Posteror dstrbuton for Bayesans, e Pθ D =PD θpθ / PD! Lkelhood for frequentsts, e PD θ! Evaluate the expectaton for some functon f E[ f ] = f x π x dx π x dx c 2001 SNU Bontellgence Lab 6

4 Calculatng Expectatons 2! Problem πxdx s unknown Generalty of! takesvaluesnk-dmensonal Eucldean space! Dscrete random varables! Mxture of dscrete and contnuous random varables! k can tself be varable c 2001 SNU Bontellgence Lab 7 Monte Carlo Integraton 1! Drawng samples { t, t=1,, n} from π! Approxmatng 1 E[ f ] n n t= 1 f t! When the samples { t } are ndependent, laws of large numbers ensure that the approxmaton can be made as accurate as desred by ncreasng the sample sze n c 2001 SNU Bontellgence Lab 8

5 Monte Carlo Integraton 2! Problem Drawng { t } ndependently from π s not feasble, snce π can be qute non-standard! { t } need not necessarly be ndependent { t } can be generated by any process whch draws samples throughout the support of π n the correct proportons MCMC! One way of dong ths s through a Markov chan havng π as ts statonary dstrbuton c 2001 SNU Bontellgence Lab 9 Markov Chan 1! Consder a generated sequence { 0, 1, 2, }! t+1 s sampled from a dstrbuton P t+1 t Markov Chan : ths sequence P : transton kernel of the chan Assume that the chan s tme-homogenous c 2001 SNU Bontellgence Lab 10

6 Markov Chan 2! Effect of 0 to t : P t t 0 Subject to regularty condtons, the chan wll gradually forget ts ntal state and P t 0 wlleventually converge to a unque statonary dstrbuton φ As t ncreases, the sampled ponts { t } wll look ncreasngly lke dependent samples from φ c 2001 SNU Bontellgence Lab 11 Markov Chan 3! After a suffcently long burn-n of, say, m teratons, ponts { t : t = m+1,, n} wll be dependent samples approxmately from φ! Use the output from the Markov chan to estmate E[f], where has dstrbuton φ! Ergodc average f 1 = n m n t= m+ 1 f t c 2001 SNU Bontellgence Lab 12

7 The Metropols-Hastngs Algorthm 1! How to construct a Markov chan such that ts statonary dstrbuton φ s precsely our dstrbuton of nterest π?! At each tme t, the next state t+1 s chosen by frst samplng canddate pont Y from a proposal dstrbuton q t! The canddate pont Y s then accepted wth probablty α t, Y π Y q Y α, Y = mn 1, π q Y! If the canddate pont s accepted, the next state becomes t+1 =Y! If the canddate s rejected, the chan does not move, e t+1 = t c 2001 SNU Bontellgence Lab 13 The Metropols-Hastngs Algorthm 2 Intalze 0 ;sett =0 Repeat { Sample a pont Y from q t Sample a Unform0,1 random varable U If U α t, Y set t+1 = Y Otherwse set t+1 = t Increment t } c 2001 SNU Bontellgence Lab 14

8 Implementaton Issues! Canoncal forms of proposal dstrbuton Any proposal dstrbuton wll ultmately delver samples from the target dstrbuton π However, the rate of convergence to the statonary dstrbuton wll depend crucally on the relatonshp between q and π For computatonal effcency, q should be chosen so that t can be easly sampled and evaluated c 2001 SNU Bontellgence Lab 15 Metropols Algorthm Symmetrc proposals qy =q Y whch generates Y condtonally ndependently, gven t π Y α, Y = mn 1, π! Random-walk Metropols: qy =q -Y Scale of a proposal dstrbuton may need to be chosen carefully! A cautous proposal dstrbuton generatng small steps! A bold proposal dstrbuton generatng large steps! Avod both these extremes c 2001 SNU Bontellgence Lab 16

9 c 2001 SNU Bontellgence Lab 17 q =N, 05 q =N, 01 q =N, 100 Statonary dstrbuton N,1 c 2001 SNU Bontellgence Lab 18 Gbbs Samplng 1! Sngle-Component Metropols-Hastngs Dvdng nto components { 1, 2,, h } of possbly dfferng dmenson, and then updatng component one by one! An teraton of the sngle-component Metropols-Hastngs algorthm comprses h updatng steps! The th proposal dstrbuton q, generates a canddate only for the th component of, and may depend on the current values of any of the components of =,, 1, mn,, Y q Y q Y Y π π α

10 Gbbs Samplng 2! Gbbs Samplng A specal case of sngle-component Metropols-Hastngs Most statstcal applcatons of MCMC have used Gbbs samplng q Y, = π Y Acceptance probablty s 1; that s, Gbbs sampler canddates are always accepted c 2001 SNU Bontellgence Lab 19 Smulated Annealng 1! Statstcal Mechancs Smulated annealng SA explots an analogy between the way n whch a metal cools and freezes nto a mnmum energy crystallne structure the annealng process and the search for a mnmum n a more general system! Boltzmann-Gbbs Dstrbuton The probablty of beng n state s at temperature T P s = P x1,, xn = e f s/ kt Z c 2001 SNU Bontellgence Lab 20

11 Smulated Annealng 2! The mplementaton of the SA algorthm Representaton of possble solutons Generator of random changes n solutons Means of evaluatng the problem functons Annealng schedule: anntal temperature and rules for lowerng t as the search progresses Intalze Generate New State s Accept? Yes Update State Lower the Temperature No Termnate? No c 2001 SNU Bontellgence Lab Yes21 Stop Smulated Annealng 3! Advantages SA can reach the Boltzmann-Gbbs equlbrum dstrbuton n a reasonable tme, whle any MCMC method fals n general SA's another advantage over other methods s an ablty to avod becomng trapped at local optmum Whle smulated annealng s usually used n combnaton wth the Metropols algorthm, t s n fact applcable to any MCMC method, and n partcular Gbbs samplng c 2001 SNU Bontellgence Lab 22