Some Examples on Gibbs Sampling and Metropolis-Hastings methods

Transcription

1 Soe Exaples o Gibbs Saplig ad Metropolis-Hastigs ethods S420/620 Itroductio to Statistical Theory, Fall 2012

2 Gibbs Sapler Saple a ultidiesioal probability distributio fro coditioal desities. Suppose d = 2, θ = (θ 1, θ 2 ). Set a iitial poit, θ (0) = (θ (0) 1, θ(0) 2 ). 1 Geerate a value θ (1) 2 fro π(θ 2 θ 1 = θ (0) 1, X = x). 2 Geerate a value θ (1) 1 fro π(θ 1 θ 2 = θ (1) 2, X = x). Two steps give a ew value θ (1) = (θ (0) 1, θ(0) 2 ) After several iteratios: θ (1), θ (2), θ (3),..., Saples (after traiig period) correspod to π(θ 1, θ 2 x). S420/620 Itroductio to Statistical Theory, Fall 2012

3 Exaple: Bioial-Beta Oe observatio, X Bioial(, θ) ad π(θ) = Beta(a, b) Bayes theore, π(θ x) θ x (1 θ) x θ a 1 (1 θ) b 1 The joit desity for (x, θ), ( ) f (x; θ)π(θ) = θ x x Γ(a + b) (1 θ) x Γ(a)Γ(b) θa 1 (1 θ) b 1 What is the argial distributio of X? Its a Bioial-Beta distributio, ( ) Γ(a + b) Γ(a + x)γ(b + x) f (x) =, x = 0, 1, 2,..., x Γ(a)Γ(b) Γ(a + b + ) S420/620 Itroductio to Statistical Theory, Fall 2012

4 Via Gibbs saplig. Set iitial value (x (0), θ (0) ). For i = 1, 2,... M 1 Saple x (i) f (x θ (i 1) ) = Bioial(, θ (i 1) ) 2 Saple θ (i) π(θ x (i) ) = Beta(a + x (i), b + x (i) ) 3 Repeat (1) ad (2) ay ties. The process produces saples of (x, θ). Approxiate f (x) with the values x (i) For exaple, f (x) = P(X = x) (# of X (i) = x)/m We ay also, (1) Saple θ (i) π(θ x (i 1) ) (2) Saple x (i) f (x θ (i) ). S420/620 Itroductio to Statistical Theory, Fall 2012

5 # Exaple 1:Beta-Bioial siulatio # Prior paraeters a=2; b=4; =16 # prepare objects to save iteratios it =1500 x= rep(na,it); th=rep(na,it) # set iitial value x[1]=1; th[1]=0.5 # Perfor Gibbs iteratios for (i i 2:it) { x[i] =rbio(1,size=,prob=th[i-1]) th[i] = rbeta(1,a+x[i],b+-x[i]) } S420/620 Itroductio to Statistical Theory, Fall 2012

6 Trace plots for siulatios of x ad θ x iteratio th iteratio S420/620 Itroductio to Statistical Theory, Fall 2012

7 Histogra x values (f(x)) Desity x S420/620 Itroductio to Statistical Theory, Fall 2012

8 Histogra θ values (π(θ) ) Desity theta S420/620 Itroductio to Statistical Theory, Fall 2012

9 I d-diesios, we iteratively use π i (θ i θ i, x) where θ i = (θ 1, θ 2,..., θ i1, θ i+1,..., θ d ). Gibbs saplig is a exaple of Markov Chai Mote Carlo (MCMC) ethods. A ew poit θ (s+1) depeds o θ (s) Mote Carlo: pseudo-rado values. The liit or statioary distributio is π(θ x). Gibbs saplig assues direct siulatio fro full coditioals. If full coditioal siulatio is ot possible, the we could use Metropolis-Hastigs. S420/620 Itroductio to Statistical Theory, Fall 2012

10 Exaple: Bioial-Beta-Poisso Treat as ukow with π() = Poisso(λ), (λ kow). X Bioial(, θ); π(θ) = Beta(a, b) The joit distributio for (X, θ, ) is: ( ) ) θ x x Γ(a + b) (1 θ) x Γ(a)Γ(b) θa 1 (1 θ) (e b 1 λ λ! x = 0, 1,..., ; 0 < θ < 1; = 0, 1, 2,.... Agai, iterested o f (x) but ipossible to fid i aalytic-closed for. Alteratively (get rid of costats), f (θ, x, ) ( x )θ a+x 1 (1 θ) b+ x 1 λ! S420/620 Itroductio to Statistical Theory, Fall 2012

11 For Gibbs Saplig, fid full coditioals: f (x θ, ) ( x) θ x (1 θ) x Bioial(, θ), π(θ x, ) θ a+x 1 (1 θ) b+ x 1 Beta(a + x, b + x), ) λ π( θ, x) ( x! (1 θ) x [λ(1 θ)] x ( x)! ; = x, x + 1, x + 2,..., If we set, z = x, the z Poisso(λ(1 θ)) Set (x (0), θ (0), (0) ). For i = 1, 2, 3,..., Saple x (i) Bioial( (i 1), θ (i 1) ). Saple θ (i) Beta(a + x (i), b + (i 1) x (i) ). Saple (i) = x (i) + z, z Poisso(λ(1 θ (i) )) Repeat util covergece is reached. S420/620 Itroductio to Statistical Theory, Fall 2012

12 Trace plot ad histogra of values iteratio Histogra of Desity S420/620 Itroductio to Statistical Theory, Fall 2012

13 Histogra of x values Desity x S420/620 Itroductio to Statistical Theory, Fall 2012

14 Exaple with Bivariate Noral Distributio x = (x 1, x 2 ), µ = (µ 1, µ 2 ) ad Σ is a 2 2 covariace atrix with diagoal etries σ1 2, σ2 2 ad off-diagoals σ 1,2. The pdf is ( f (x µ, Σ) Σ 1/2 exp 1 ) 2 (x µ)t Σ 1 (x µ) Margial distributios: x 1 N(µ 1, σ 2 1 ) ad x 2 N(µ 2, σ 2 2 ). For Gibbs saplig, we eed f (x 1 x 2 ) ad f (x 2 x 1 ). I fact f (x 1 x 2 ) = N(µ 1 + (σ 1,2 /σ 2 2 )(x 2 µ 2 ), σ 2 1 (σ 1,2/σ 2 ) 2 ) f (x 2 x 1 ) = N(µ 2 + (σ 1,2 /σ 2 1 )(x 1 µ 1 ), σ 2 2 (σ 1,2/σ 1 ) 2 ) Exaple with µ = (2, 1), σ 2 1 = σ2 2 = 1 ad σ 1,2 = 0.7. S420/620 Itroductio to Statistical Theory, Fall 2012

15 Trace plots for siulatios of x 1 ad x 2 xs[, 1] Idex xs[, 2] Idex S420/620 Itroductio to Statistical Theory, Fall 2012

16 Histogras for siulatios of x 1 ad x 2 Histogra of xs[-(1:100), 1] Desity x1 Histogra of xs[-(1:100), 2] Desity x2 S420/620 Itroductio to Statistical Theory, Fall 2012

17 British coal iig disasters data o. of disasters tie Histogra of y Desity y S420/620 Itroductio to Statistical Theory, Fall 2012

18 Exaple: Poisso process with a Chage Poit Hierarchical odel proposed i Carli, Gelfad ad Sith (1992) (also see Sectio 3.10). First stage: X i Poisso(µ), i = 1, 2,..., k. X i Poisso(λ), i = k + 1, k + 2,..., so X i represets the observatios ad the paraeter of iterest are (µ, λ, k). Data cosists of coal-iig disasters i the U.K. Secod stage: Idepedet priors o (µ, λ, k). k discrete uifor o {1, 2,..., }, =saple size. µ Ga(a 1, b 1 ) ad λ Ga(a 2, b 2 ) a 1, b 1, a 2, b 2 are fixed. S420/620 Itroductio to Statistical Theory, Fall 2012

19 The joit posterior distributio for (µ, λ, k) has the for: π(µ, λ, k X) f (X µ, λ, k)π(µ)π(λ)π(k) The likelihood fuctio is: f (X θ, λ, k) = = k f (x i µ, k) f (x i λ, k) i=1 k µ x i e µ i=1 x i! i=k+1 i=k+1 λ x i e λ x i! Therefore, π(µ, λ, k X) k µ x i e µ i=1 x i! i=k+1 λ x i e λ x i! ( ) (µ a1 1 e µb 1 )(λ a2 1 e λb 2 1 ) S420/620 Itroductio to Statistical Theory, Fall 2012

20 Or equivaletly, π(µ, λ, k X) µ a 1+ k i=1 x i 1 e µ(k+b 1) λ a 2+ i=k+1 x i 1 e λ( k+b 2) Full coditioal distributios, 1 π(µ λ, k, X) µ a 1+ k i=1 x i 1 e µ(k+b1) = Ga(a 1 + k i=1 x i, k + b 1 ) 2 π(λ µ, k, X) λ a 2+ i=k+1 x i 1 e λ( k+b2) = Ga(a 2 + i=k+1 x i, k + b 2 ) 3 π(k µ, λ, X) µ k i=1 x i λ i=k+1 x i e k(µ λ) L(X µ, λ, k) So, p(k µ, λ, X) = L(X µ,λ,k) k=1, k = 1, 2,..., L(X µ,λ,k) Soe results with the British Coal iig disaster data follow. S420/620 Itroductio to Statistical Theory, Fall 2012

21 k Series draws[, i] Histogra of draws[, i] theta ACF Desity iteratio Lag theta Series draws[, i] Histogra of draws[, i] labda ACF Desity iteratio Lag labda Series draws[, i] Histogra of draws[, i] ACF Desity iteratio Lag k S420/620 Itroductio to Statistical Theory, Fall 2012

22 Metropolis-Hastigs Algorith (cotiuous case) Saple (idirectly) a value X fro a pdf f (x). I Bayesia probles f (x) is the posterior distributio. Need a trial or proposal desity q(x, y) 0. 1 Set a iitial value X (0). For = 1, 2,..., 2 Give X () = x, geerate Y = y, fro q(x, y). 3 Copute, { α = i 1, } f (y)q(x, y) f (x)q(x, y) 4 U U(0, 1). If U α, the X (+1) = y. Otherwise, X (+1) = x 5 Chage to + 1 ad go back to 2. S420/620 Itroductio to Statistical Theory, Fall 2012

23 Coo to ru algorith say 1000 iteratios ad delete the first, say 100 (bur-i). Uder a syetric proposal (Metropolis), q(x, y) = q(y, x) ad { α = i 1, f (y) } f (x) Rado walk proposal, h is a scalig paraeter. y = x + z; z N(0, h 2 ). h is specified so that 40% 60% acceptace rate is achieved. S420/620 Itroductio to Statistical Theory, Fall 2012

24 Exaple with Bivariate Noral Distributio θ = (θ 1, θ 2 ), µ = (µ 1, µ 2 ) ad S is a 2 2 covariace atrix with diagoal etries s 1,1, s 2,2 ad off-diagoals s 1,2 = s 2,1. The pdf is ( π(θ) S 1/2 exp 1 ) 2 (θ µ)t S 1 (θ µ) Margial distributios: θ 1 N(µ 1, s 1,1 ) ad θ 2 N(µ 2, s 2,2 ). Exaple with µ = (2, 1), s 1,1 = s 2,2 = 1 ad s 1,2 = 0.7. S420/620 Itroductio to Statistical Theory, Fall 2012

25 Trial or cadidate desity q(θ, θ ), also a bivariate oral so θ = θ + z; z N(0, Σ) with Σ a diagoal atrix with diagoal etries 0.6, 0.4. z 1 N(0, 0.6) ad z 2 N(0, 0.4). Rado walk proposal so q(θ, θ ) = q(θ, θ). The Metropolis ratio is: { } α(θ, θ ) = i 1, π(θ ) = π(θ) ( ) exp 1 = i 1, 2 (θ µ) t S 1 (θ µ) exp ( 1 2 (θ µ)t S 1 (θ µ) ) S420/620 Itroductio to Statistical Theory, Fall 2012

26 Fro a iitial poit θ (0). For i = 1, 2,..., θ = θ (i 1) + z The copute the acceptace probability, α(θ (i 1), θ ). Make θ (i) = θ if U < α(θ (i 1), θ ). Otherwise, θ (i) = θ (i 1). Fro θ (0) = ( 1, 3), we ll look at: Tie series plots of θ = (θ 1, θ 2 ) values. Histogras of argial saples for θ 1 ad θ 2. Bivariate plots of saples. S420/620 Itroductio to Statistical Theory, Fall 2012

27 theta iteratio theta iteratio S420/620 Itroductio to Statistical Theory, Fall 2012

28 Margial for Theta theta1 Margial for Theta theta2 S420/620 Itroductio to Statistical Theory, Fall 2012

29 2 0-2 theta siulatios fr o the Bivariate Noral theta theta siulatios fro the M-H algorith theta 1 S420/620 Itroductio to Statistical Theory, Fall