21 IIIHP, 2008 5

III

10

1

1

257

1 1

1.1 . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 . . . . . 3

2 4

2.1 . . . . . . . . . . . . . . . . . . . . 4

2.2 . . . . . . . . . . . . . . . . . . . . . . 6

2.3 . . . . . . . . . . . . . . . . . . . . . . . . 8

3 9

3.1 . . . . . . . . 9

3.2 MH . . . . . . . . . . . . . . . . . . . . 12

3.3 MH . . . . . . . . . . . . . . . . 12

4 14

4.1 . . . . . . . . . . . . . . 14

4.2 MH . . 15

4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 18

5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6 21

6.1 . . . . . . . . . . . . . . . . . . 22

6.2 . . . . . . . . . . . . . . . 24

258

1

Markov chain Monte Carlo methodMCMC

MCMC

1

1.1

x (x)1x h(x)

(1) I = E [h(x)] =X

h(x)(x)dx

x X RdE[] (x)

(x)

(x)

2(x) (x(1), . . . ,x(n))(1)

(2) IM =1n

ni=1

h(x(i))

1 2sampling

Devroye(1986)Ripley (1987)Gentle (2003)

259

n IM I nI

IM

Monte Carlo integration

(x)

(x)

q(x)3

(1)

I =X

h(x)(x)q(x)

q(x)dx = Eq

[h(x)

(x)q(x)

](x(1), . . . ,x(n)) q(x)I

(3) IIS =1n

ni=1

h(x(i))w(x(i))

w(x(i)) = (x(i))/q(x(i))x(i)

(3)

importance sampling

q(x)n

IIS I

q(x)q(x) |h(x)|(x)

Rubinstein (1981)Geweke (1989)

Evans and Swartz (2000)4

(x)

(x) = p(x)/X p(x)dx p(x)

5(1)

I =X

h(x)p(x)

X p(x)dxdx =

X h(x)p(x)dx

X p(x)dx

3q(x)support (x)4 5 1 p(x)

p(x)/R

X p(x)dxR

X p(x)dx

x

260

q(x)

(4) I =

X

h(x)p(x)q(x)

q(x)dxX

p(x)q(x)

q(x)dx=

Eq

[h(x)

p(x)q(x)

]Eq

[p(x)q(x)

](4)

IIS =1n

ni=1 h(x

(i))w(x(i))1n

ni=1 w(x(i))

=n

i=1

h(x(i))w(x(i))

w(x(i)) = p(x(i))/q(x(i))w(x(i)) = w(x(i))/n

j=1 w(x(j))

n

i=1 w(x(i)) = 1w(x(i))

IM IIS IIISn

IIS I Geweke (1989)

IIS IISRobert

and Casella (2004)

1.2

1

x (x)

(x) q(x)

x

1990

MCMCMCMCMH

261

2MCMC

6

3MH 4

5

6MCMC

2

MCMC

Markov chain

7

2.1

t x(t) (x(0),x(1), . . .)

x(t) (t = 0, 1, . . .)X

state space

x(t) 1X = {1, . . . , k}

(x(0),x(1), . . .) i, j X t 0

Pr(x(t+1) = j|x(0) = i0,x(1) = i1, . . . ,x(t1) = it1,x(t) = i

)= Pr(x(t+1) = j|x(t) = i)(5)

(x(0),x(1), . . .)ik X

kx(t+1)6MCMCLiu (2001)Robert and Casella (2004)Gamer-

man and Lopes (2006) (2001) (2003) (2005) (2005)

7Karlin and Taylo (1975) Ross (1995)

262

t {x(0) = i0,x(1) = i1, . . . ,x(t1) = it1,x(t) = i}

t + 1

t

Markov property

(5) p(i, j) = Pr(x(t+1) = j|x(t) = i)

transition probabilityp(i, j) (i, j)

k k

T =

p(1, 1) p(1, 2) p(1, k)

p(2, 1) p(2, 2) p(2, k)...

.... . .

...

p(k, 1) p(k, 2) p(k, k)

transition matrix

p(i, j)

i j

p(i, j) 0k

j=1 p(i, j) = 1

2.1. X = {1, 2, 3}

T =

1/2 1/3 1/6

3/4 0 1/4

0 1 0

1

1/2 1/3 2 1/6

3 2 1 3

3 2

x(0) (0)

(0) = ((0)1 , (0)2 , . . . ,

(0)k ) =

(Pr(x(0) = 1), Pr(x(0) = 2), . . . , Pr(x(0) = k)

)(1), (2), . . .x(1),x(2), . . .

(t) = ((t)1 , (t)2 , . . . ,

(t)k ) =

(Pr(x(t) = 1), Pr(x(t) = 2), . . . , Pr(x(t) = k)

)

263

x(1) (1)

(1)j = Pr(x

(1) = j) =k

i=1

Pr(x(0) = i,x(1) = j)

=k

i=1

Pr(x(0) = i) Pr(x(1) = j|x(0) = i) =k

i=1

(0)i p(i, j)

(1) = (0)T x(2) (2) =

(1)T = (0)T2 (t) = (0)Tt

2.2

(0)

T

(1) x(0) (0)

(2) t = 0, 1, . . .x(t+1) (T)x(t)

(T)x(t)T x(t)

(x(0),x(1), . . .) x(t) (t)

MCMC(t)

(1)

irreducibility(2) aperiodicity(3) invariant

distribution

T

i, j X (Tn)ij > 0 n

(Tn)ij Tn (i, j)

264

2.2. 2.1

T =

0.6 0.4 0

0.3 0.7 0

0.2 0.2 0.6

3

1 2 3

i X {n 1 : (Tn)ii > 0}

iperiod

1

2.3.

T =

0 0.5 0 0.5

0.5 0 0.5 0

0 0.5 0 0.5

0.5 0 0.5 0

{n 1 : (Tn)ii > 0} = {2, 4, 6, . . .} (i =

1, . . . , 4) 2

T

= (1, . . . , k)

(1) i 0 (i X )k

i=1 i = 1 (2) = T

T

2.4. 2.1 = (1/2, 1/3, 1/6)3

i=1 i = 1

= T T

(t)

Haggstrom (2002)

265

(). (x(0),x(1), . . .)

TT

(0)t 12k

i=1 |(t)i i| 0

8

2.3

(x(0),x(1), . . .)

m(x(m+1),x(m+2), . . .)

detailed balance condition

(6) ip(i, j) = jp(j, i) (i, j X )

reversible

9MCMC

8

9(6) i Pk

i=1 ip(i, j) =Pk

i=1 jp(j, i) = j

266

10

Pr(x(t+1) A|x(t) = x) =

AT (x,y)dy (x X , A X )

T (x,y) T (x,y)

transition kernelT (x,y)

(y) =X

(x)T (x,y)dx

(x)

(x)T (x,y) = (y)T (y,x)

3

MCMC

MetropolisHastingsMH

Metropolis et al. (1953) Hastings (1970)

11MH

3.1

(x) (x)

target distributionMH

proposal distribution

12

q(y|x)10Nummelin (1984) Meyn and

Tweedie (1993)11Dongarra and Sullivan (2000) 20 10

1MH12candidate generating distribution

267

MH

acceptance probability

(x)

MH

(1) x(0)

(2) t = 0, 1, . . .

(i) y q(y|x(t))

(ii) u U(0, 1)13

x(t+1) =

y u (x(t),y)

x(t)

(x,y) = min{

1, (y)q(x|y)(x)q(y|x)}

(x,y)

MH

x(t+1) = x(t)

MH

MH q(y|x)

MH

3.1. (x)x(t) =

x

y = x + , N (0, 2I)

14random walk chain

q(y|x) = q(x|y) (x,y) =

13U(a, b) (a, b)14N (,)

268

min {1, (y)/(x)} Metropolis et al. (1953)

q(y|x) = q(x|y)MH

U(, )

t T(0, 2I)15

step size

1

16

3.2.

y = x +2

2 log (x)

x+ , N (0, 2I)

Langevin chainRoberts and Rosenthal (1998)Christensen et al. (2001)

Christensen and Waagepetersen (2002) x

log (x)/x 0

x

log (x)/x = 0

3.3. y x

q(y|x) = q(y)independent chain

(x,y) = min{

1, (y)/q(y)(x)/q(x)}

(x)/q(x)y x

y y

q(y) (x)

15T(,) t1 t t(, 2)

16Roberts et al. (1997)Roberts and Rosenthal(2001)

269

(x)

{ log (x)/xx}1 t

Chib and Greenberg (1995)

3.2 MH

MH (x(0),x(1), . . .)

MH

x y

(7) T (x,y) = q(y|x)(x,y) + r(x)x(y)

2

r(x) =X q(y|x) {1 (x,y)} dyx(y) = I(x = y)

17

(7) q(y|x)(x,y)(x) = q(x|y)(y,x)(y)

r(x)x(y)(x) = r(y)y(x)(y)

(x)T (x,y) = q(y|x)(x,y)(x) + r(x)x(y)(x)

= q(x|y)(y,x)(y) + r(y)y(x)(y) = (y)T (y,x)

MH

(x)

Roberts and Smith (1994)Tierney (1994)MH

MH (x(0),x(1), . . .)

m (x(m+1),x(m+2), . . .) (x)

m

burn-in period

3.3 MH

MH

MH17I()

270

(x)2MH

T1(x,y)T2(x,y)

w T1(x,y)MH

1w T2(x,y)MH

w

1 w

(8) T (x,y) = wT1(x,y) + (1 w)T2(x,y)

mixture of tran-

sition kernelsTi(x,y) (i = 1, 2)

(x)(8) T (x,y) (x)

MH

MH

MH

T1(x,x)MH

x x T2(x,y)

MH x yx

y

(9) T (x,y) =X

T1(x,x)T2(x,y)dx

(9)cycle of transition kernels

(x)X

(x)T (x,y)dx =X

X

(x)T1(x,x)T2(x,y)dxdx

=X

(x)T2(x,y)dx = (y)

(x) T (x,y)18182 MH

Tierney (1994)

271

MCMC

4

MH

Gibbs samplingGeman and Geman

(1984)Gelfand

and Smith (1990)

4.1

x k x = (x1, . . . ,xk)

xi (xi|xi)19

xi = (x1, . . . ,xi1,xi+1, . . . ,xk) (xi|xi)

(full conditional distribution)

(1) x(0) = (x(0)1 , . . . ,x(0)k )

(2) t = 0, 1, . . .

(i) x(t+1)1 (x1|x(t)2 , . . . ,x

(t)k )

(ii) x(t+1)2 (x2|x(t+1)1 ,x

(t)3 , . . . ,x

(t)k )

...

(k) x(t+1)k (xk|x(t+1)1 , . . . ,x

(t+1)k1 )

MH

(xi|xi)

xi

xiLiu et al. (1995)19 5

272

4.1. (x1, x2) = nCx1xx1+12 (1 x2)nx1+1

x1 {0, 1, . . . , n}x2 [0, 1]x1 x2

(x1|x2) nCx1xx12 (1 x2)

nx1 , (x2|x1) xx1+12 (1 x2)nx1+1

x1|x2 Bi(n, x2)x2|x1

Be(x1 + , n x1 + )20

4.2 MH

k

k = 2

MH

MHmultipleblock MH

algorithm

x = (x1,x2) y = (y1,y2)(x1,x2) (y1,x2)

(y1,y2)MH(x1,x2)

(y1,x2) (x1|x2)MH

q1(y1|x1,x2)

1(x1,y1) = min{

1,(y1|x2)q1(x1|y1,x2)(x1|x2)q1(y1|x1,x2)

}MH T1(x1,y1|x2)

(y1,x2) (y1,y2)

(x2|x1) q2(y2|y1,x2)

2(x2,y2) = min{

1,(y2|y1)q2(x2|y1,y2)(x2|y1)q2(y2|y1,x2)

}MHT2(x2,y2|y1)

20Bi(n, p) (x) px(1 p)nx Be(a, b) (x) xa1(1 x)b1

273

MHx y

T (x,y) = T1(x1,y1|x2)T2(x2,y2|y1)

T1(x1,y1|x2) (x1|x2)X

(x)T (x,y)dx =

(x1,x2)T1(x1,y1|x2)T2(x2,y2|y1)dx1dx2

= [

(x1|x2)T1(x1,y1|x2)dx1]

(x2)T2(x2,y2|y1)dx2

=

(y1|x2)(x2)T2(x2,y2|y1)dx2

(y1|x2) =

(y1)(x2|y1)/(x2) X

(x)T (x,y)dx =

(y1)(x2|y1)T2(x2,y2|y1)dx2

= (y1)(y2|y1) = (y)

MH

(x)

MH

MH

q1(y1|x1,x2) = (y1|x2)

1(x1,y1) = min{

1,(y1|x2)(x1|x2)(x1|x2)(y1|x2)

}= 1

q2(y2|y1,x2) = (y2|y1) 2(x2,y2) = 1

1

MH2121 (x)

274

(x)22

MH

MH

MHMetropolis within

GibbsMuller (1991)

4.3

(x) z Z

(x, z) (x, z)

Z (x, z)dz = (x)

MCMC x (x)

data augmentation methodTanner and

Wong (1987)

Chib (1992)Albert and

Chib (1993)

(x|z) (z|x)

x

zmissing data(z|x)

Rubin (1987)imputation

xx

EMDempster et al. (1970)

(z|x)E(x|z)

MLiu (2002)

x z (x) p(x)

l(x)(x) p(x)l(x)

22Chan (1993) Roberts and Smith (1994)

275

z(x, z)

(10) (x, z) I[z < l(x)]p(x)

x

slice sampling

(10)(z|x) U(0, l(x))z

(x|z) {x : l(x) > z}

p(x)

Damien et al. (1999)

Besag and Green (1993)Higdon (1998)Damien and Walker (2001)

Neal (2003)

5

5.1

MCMC

(x(0),x(1), . . .)

MCMC

Cowles and Carlin (1996)Mengersen et al. (1999)

Heidelberger and Welch (1983)

Gelman and Rubin (1992)Geweke (1992)Raftery and Lewis (1992)

276

23

MCMC

multiple chain1 1

single chain

5.2

MCMC

(x(0),x(1), . . . ,x(m+n))E[h(x)]

(11) I =1n

ni=1

h(x(m+i))

m

n IE[h(x)]Tierney

(1994)I

Var(I) =2

n

1 + 2n1j=1

(1 j

n

)j

2n1 + 2

j=1

j

2 = Var[h(x)]j h(x(t))

h(x(t+j))

j > 0MCMC

23R CODA

277

MCMC 1

j

(11)

MCMC

1 + 2

j=1 j

inefficiency factor24

I 2/n

5.3

mixingMH

MH

1

Liu (1994)

x = (x1,x2,x3)(x1,x2)

(x1,x2)

(x1|x2) (x2|x1)x3 (x3|x1,x2)

24L 1+2

PLj=1 j

278

Gelfand et al. (1995)

Roberts and Sahu (1997)

EM

Liu and Wu (1999)Meng and van Dyk (1999)van Dyk and Meng

(2001)EMMCMC

Neal (1996)

Liu and Sabatti (2000)

Liu (2003)

parallel temperingGeyer (1991) simulated tempring

Geyer and Thompson (1995)

multiple try Liu et al. (2000)multiple point

Qin and Liu (2001)

xxxMCMC

Moral et al. (2006)

6

MCMC

279

25

MCMC

6.1

Breslow and Clayton (1993)

21

26 i

ni yiBreslow and Clayton (1993)

(12) yi Bi(ni, pi), logpi

1 pi= xi + bi, bi N (0, 2)

xi bi 2

N (0,B0)2 IG (n0/2, s0/2) 27

2bi (i = 1, . . . 21)

2

2|, {bi} IG

(21 + n0

2,

21i=1 b

2i + s0

2

)

(|2, {bi}) 21i=1

pyii (1 pi)niyi exp

{1

2( 0)B10 ( 0)

}MH

(13) yi = xi + bi + i, i N (0, 2i )25 Berger (1980) 26Crowder (1978)27IG(a, b) (x) (1/x)a+1 exp(b/x)

280

McCullagh and Nelder (1989)yi = xi + bi +

(yi nipi)/{nipi(1 pi)}2i = 1/{nipi(1 pi)}MH

(13)

N (, V)

V1 =21i=1

xixi2i

+ B10 , = V

{21i=1

xi(yi bi)2i

+ B10 0

} t

t T(, V)

bi

(bi|, 2) pyii (1 pi)niyi exp

( b

2i

22

) t(bi, v2i )MH

v2i = 22i /(2 + 2i )

bi = v2i (yi xi)/2i

Breslow and Clayton (1993)

(12) 1

0 = 0B0 = 100In0 = 1s0 = 0.01

= 2028 1

29 90%

1

Breslow and Clayton (1993)

30

(12)

yi Bi(ni, pi), logpi

1 pi= i, i N (xi, 2)

28 2930 5000

10000

281

MH

0 2500 5000 7500 10000 12500 15000

1.0

0.5

0.0 0

0 5 10 15 20 25

0.5

1.0ACF0

0 2500 5000 7500 10000 12500 15000

1

0

1 1

0 5 10 15 20 25

0.5

1.0ACF1

0 2500 5000 7500 10000 12500 15000

1

22

0 5 10 15 20 25

0.5

1.0ACF2

0 2500 5000 7500 10000 12500 15000

1

13

0 5 10 15 20 25

0.5

1.0ACF3

0 2500 5000 7500 10000 12500 15000

1

0

10

0 5 10 15 20 25

0.5

1.0ACF0

0 2500 5000 7500 10000 12500 15000

1

0

1 1

0 5 10 15 20 25

0.5

1.0ACF1

0 2500 5000 7500 10000 12500 15000

1

2

32

0 5 10 15 20 25

0.5

1.0ACF2

0 2500 5000 7500 10000 12500 15000

2

1

0

13

0 5 10 15 20 25

0.5

1.0ACF3

1:

1:

-0.546 0.177 -0.542 0.190 0.091 0.293 0.146 0.308 1.333 0.250 1.339 0.270 -0.803 0.399 -0.825 0.430 0.242 0.125 0.313 0.121

MH|2, {i}

N (, V)31V1 =21

i=1 xixi/

2+B10

= V(21

i=1 xii/2 +B10 0) 5

1

6.2

Leroux and Puterman (1992)

5

t yt

31i 2

282

yt Po(st) (t = 1, . . . , T )

Po(st) st st {1, 2}

Leroux and Puterman (1992)

T =

p11 p12p21 p22

32T 0 = (01, 02)

s1 0st

({i}, {pii}, {st})

i

Ga(ai, bi)

i|{pii}, {st} Ga

ai + tSi

yt, bi + ni

(i = 1, 2)33Si = {t : st = i}ni Si

pii Be(i, i)pii

pii|{i}, {st} Be(i + nii, i + nij) (i = 1, 2)

nij i j st

st

(st = i|{i}, {pii}, {st}t =t)

f(yt|i)0ipi,st+1 t = 1

f(yt|i)pst1,ipi,st+1 1 < t < T

f(yt|i)pst1,i t = T

f(y|)

st 132Leroux and Puterman (1992){st}33 Ga(a, b)(x) xa1 exp(bx)

283

Chib (1996) {st}

2

2{st} 1

i

{st}

1

0 2000 4000 6000 8000 10000

0.1

0.2

0.3

0.41

0 10 20 30 40 50

0.25

0.50

0.75

1.00ACF1

0 2000 4000 6000 8000 10000

2

4

6 2

0 10 20 30 40 50

0.25

0.50

0.75

1.00ACF2

0 2000 4000 6000 8000 10000

0.1

0.2

0.3

0.4 1

0 10 20 30 40 50

0.25

0.50

0.75

1.00ACF1

0 2000 4000 6000 8000 10000

2

4

62

0 10 20 30 40 50

0.25

0.50

0.75

1.00ACF2

2:

2

IF34

2: 1

IF IF1 0.220 0.050 14.346 0.220 0.048 19.289

2 2.282 0.769 13.445 2.273 0.744 24.257

p11 0.968 0.024 16.195 0.968 0.023 20.295

p22 0.672 0.154 2.6689 0.671 0.148 4.7609

34Chib (1996) 5000 10000 50

284

[1] (2005). , II, 3106,

[2] (2001). , 31, 305344.

[3] (2003).MCMC

[4] (2005)

[5] Albert, J. and Chib, S. (1993). Bayesian analysis of binary and polychoto-

mous response data, Journal of the American Statistical Association 88,

669679.

[6] Berger, J.O. (1980). Statistical Decision Theory and Bayesian Analysis (2nd

ed.). Springer, New York.

[7] Besag, J. and Green, P. (1993). Spatial statistics and Bayesian computa-

tion, Journal of the Royal Statistics Society B55, 2537.

[8] Breslow, N.E. and Clayton, D.G. (1993). Approximate inference in gener-

alized linear mixed models, Journal of the American Statistical Association

88, 925.

[9] Chan, K.S. (1993). Asymptotic behavior of the Gibbs sampler, Journal of

the American Statistical Association 88, 320326.

[10] Chib, S. (1992). Bayes regression for the tobit censored regression model,

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[11] Chib, S. (1996). Calculating posterior distributions and modal estimates in

Markov mixture models, Journal of Econometrics 75, 7997.

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