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1 MCMC

2 I. MCMC MH MCMC 2

3 II

4 I. MCMC MH 5. 4

5 1. MCMC 5

6 2. A P (A) : P (A)=0.02 A B A B Pr B A) Pr B A c Pr B A)=0.8, Pr B A c =0.1 6

7 B A 7

8 8

9 A, : B x, x ( ) 9

10 10

11 ( ) 11

12 1. N( 0, 02 ) ( 0, 02 ) 12

13 , 13

14 2. 2 IG (n 0 /2, n 0 S 0 /2) 14

15 ,, 2 IG (n 1 /2, n 1 S 1 /2) 15

16 . 2 N( 0, 2 /m 0 ). 2 IG (n 0 /2, n 0 S 0 /2) 16

17 , 17

18 18

19 19

20 3.. (, 2 x) ( ) 1. (0), 2(0). 2. (1), 2(1). 20

21 3. ( 3. i=1,2, (i+1), 2(i+1). 4. (i), 2(i) (i=n,n+1, N ) (, 2 x). 21

22 3. ( WinBUGS model{ #, 2 ) = - 2 for (i in 1:N) { x[i] ~ dnorm(mu,tau) # x ~N(, 2 ) } # (, 2 ) tau~dgamma(1.0e-3,1.0e-3) # 2 ~IG(0.001,0.001) mu ~ dnorm(0.0,1.0e-3) # ~N(0,1000) var <- 1/tau } 22

23 3. ( X 1,..,X 30 ~N(10,25) ) list(x=c(13.08, 2.44, 13.82, 14.14, 15.63, 18.99, 4.37, 11.41, 13.53, 9.43, 12.69, 5.30, 17.73, 4.31, 11.42, -2.72, 14.15, 9.56, 18.29, 12.09, 11.16, 8.93, 6.14, 4.72, 7.99, 7.80, 11.35, 11.76, 13.62, 14.47),N=30), list(mu=10 tau=1/25) 1000 (burn-in period),

24 3. ( mu iteration var iteration 24

25 3. ( mu sample: var sample:

26 3. ( MC error %

27 3. ( X X new (i) ~ N( (i), 2(i) ) xnew sample:

28 3. ( ( 1,, m x). 1. (0) =( (0) 1,, (0) m ). 2. i=0,1,2,. a. (i+1) 1 ~ ( 1 (i) 2,, (i) m x) b. (i+1) 2 ~ ( 2 (i+1) 1, (i), 3, (i) m x) c. d. (i+1) m ~ ( m (i+1) 2,, (i+1) m-1 x) 3. i>n ) (i). (i) (i=1,2, ) (. MH 28

29 4.MH MH( target distribution proposal distribution) MH 29

30 4.MH ( ( 1,, m x). 1. (0) =( 1 (0),, m (0) ). 2. i=0,1,2,. 1. (i) =( 1 (i),, m (i) ) Q. Q ( ) q( (i), x) (q( (i), x) ). 2. ( x),, (i+1) =. (i+1) = (i). 3. i>n ) (i). 30

31 4.MH ( MH,.,. (detailed balance equation).. 31

32 4.MH ( 1. MH. (random walk) = (i) +z, z~f(z)., q( (i), x)=f( - (i) ). z 0 t,.,.,. (i) (i=1,2, ),. 32

33 4.MH ( 2. (independence) MH. q( (i), x)=f( x ) q (i). w q.. 33

34 4.MH (, * N(m,V), log ( x) * 2.. V, v_0. 34

35 4.MH ( 1 U 1, 1) 2 IG(n 0 /2,n 0 S 0 /2) 35

36 4.MH ( 2,y IG(n 1 /2,n 1 S 1 /2) 36

37 4.MH ( TN (-1,1) (, 2 ) 37

38 4.MH ( ~TN (-1,1) (, 2 ),. 38

39 4.MH ( 1. (0), 2(0) ). 2. i=0,1,2,. 2(i+1) (i),y IG(n 1 /2,n 1 S 1 /2) 2(i+1),y~TN (-1,1) (, 2 ), (i+1) =, (i+1) = (i). 3. i>n ) (i), 2(i) ). 39

40 4.MH ( Ox. ( ). for(i=0;i<10000;++i){ // 2 a=0.5*(0.001+nobs+1); b=0.5*(0.001+(1-phi^2)*y[0]+sumsqrc(y[1:nobs-1]-phi*y[0:nobs-2])); var_e=1/rangamma(1,1,a,b); // mu_phi=(y[0:nobs-2]'*y[1:nobs-1])/sumsqrc(y[1:nobs-2]); var_phi=var_e/sumsqrc(y[1:nobs-2]); // 2(i+1),y~TN (-1,1) (, 2 ) do{phi_n=mu_phi+rann(1,1)*sqrt(var_phi);}while(fabs(phi_n)>=1); //. if(ranu(1,1)<= (0.5*sqrt(1-phi_n^2))/(0.5*sqrt(1-phi^2))){phi=phi_n;} } 40

41 4.MH ( ( =0.9, 2 =1.0 y 0,,y 1000 ) 96.3% %

42 4.MH ( 1 Sample ACF PHI Sample Path 0.95 PHI Posterior Density 30 PHI

43 4.MH ( 1 Sample ACF SIGMA Sample Path 1.2 SIGMA Posterior Density 10.0 SIGMA

44 4.MH ( WinBUGS ( WinBUGS Slice sampler ) # Model y 0,, y model{ # y, 2 ) = - 2 tau0 <-tau*(1-phi*phi) y[1]~dnorm(0,tau0) # y 0 ~N(0, 2 /(1-2 ) ) for( i in 2 : N ) { ymu[i] <-phi*y[i-1] # y t ~N( y t-1, 2 ) y[i] ~ dnorm(ymu[i],tau)} # (, 2 ) phi ~ dunif(-1,1) # ~U(-1,1) tau~dgamma(1.0e-3,1.0e-3) # 2 ~IG(0.001,0.001) var <- 1/tau } 44

45 4.MH ( =0.9, 2 =1.0 ) list(n=1000,y=c( , )) Fold, list(phi=0 tau=1) 1000 (burn-in period),

46 4.MH ( phi iteration var iteration 46

47 4.MH ( phi sample: var sample:

48 4.MH ( MC error %

49 .,, 49

50 . 3.. mu lag var lag

51 . mu iteration var iteration

52 . Geweke (1992) (10%) (50%) g( ). Gelfand and Smith (1990), Gelman and Rubin (1992), Liu and Liu (1993), Raftery and Lewis (1992) Heidelberger and Welch (1983) Mykland et al. (1995) 52

53 . Cowles and Carlin (1996) 13. Robert and Casella (1999) Mengersen, Robert and Guihenneuc-Jouyaux(1999). S-Plus R.. CODA(Convergence Diagnostic and Output Analysis Software) BOA (Bayesian Output Analysis) Geweke, Gelman and Rubin, Raftery and Lewis, Heidelberger and Welch,,,,, 95% 53

54 . BOA Geweke(1992) GEWEKE CONVERGENCE DIAGNOSTIC: ============================== Fraction in first window = 0.1 Fraction in last window = 0.5 Chain: sample mu var Z-Score p-value

55 . Gibbs Sampler etc. 55

56 6.MCMC WinBUGS. BayesX TSP STATA. CODA, BOA (R Splus. ) Ox, Matlab, GAUSS, Ox C, Fortran 56

57 II

58 1. y i, i=1,,n x i =(x i1,,x ik ) : =( 1,, K ) : i N(0, 2 ). 58

59 1. 59

60 1. 60

61 1. (Y) (X)

62 1. WinBUGS model{ #, 2 ) = - 2 for(i in 1:n){ #n mu[i] <-beta[1]+beta[2]*x[i] # i = x i y[i]~dnorm(mu[i],tau) # y i ~N( i, 2 ) } # (, 2 ) for(i in 1:K){ #K =2 beta[i]~dnorm(0,1.0e-6)} # i ~N(0,10 6 ) tau~dgamma(1.0e-6,1.0e-6) sigma2 <-1/tau # 2 ~IG(10-6, 10-6 ) } 62

63 1. MC error %

64 1. beta[1] sample: beta[2] sample: sigma2 sample: E E E E+5 64

65 1. y n+1, (, 2 ) y n+1, 2, x n+1 ~N(x n+1, 2 ). 65

66 1., i ~N(0, 2 )., t. 0 i 0. 66

67 1. 67

68 1. 68

69 2. y i,. 0, y i*, i=1,,n y i y i* >0,. x i =(x i1,,x ik ) : =( 1,, K ) : i N(0, 2 ). 69

70 2. 70

71 2. 71

72 2. 1 (y) 1. (x1), 2. (x2), 3. (x3), 4. (x4). 0 72

73 Ox 2. for(cn=-cburn;cn<crepeat;++cn){ mb1=invert((1/dsigma2)*mx *mx+invert(mb0)); # vb1=mb1*((1/dsigma2)*mx'*vystar+invert(mb0)*vb0); vb=vb1+choleski(mb1)*rann(cm,1); for(i=0;i<cnobs;++i){ #y* if(vy[i]==0){ do{vystar[i]=mx[i][]*vb+rann(1,1)*sqrt(dsigma2); }while(vystar[i] >=0); } else{vystar[i]=vy[i];}} 2 da=0.5*(0.002+cnobs);db=0.5*(0.002+(vystar-mx*vb)'*(vystar-mx*vb)); dsigma2=1/rangamma(1,1,da,db); } 73

74 % E E E E

75 2. 75

76 3. y i 0 1. : (y i =1), (y i =0). (y i =1), (y i =0) 76

77 3. F( ) 0 1 F F y i * y*, 1. 77

78 3. 78

79 3. 79

80 3. :,. 95 Pindyck and Rubinfeld (1998),. Y: 1, 0 X_1: X_2: 1, 0 X_3: 1, 0 X_4: ( X_5: ( ) 80

81 3. Ox for(cn=-cburn;cn<crepeat;++cn){ # ~N(b 1,B 1 ) vb1=mb1*(mx'*vystar+invert(mb0)*vb0); vb=vb1+choleski(mb1)*rann(cm,1); # y* for(i=0;i<cnobs;++i){ if(vy[i]==0){do{vystar[i]=mx[i][]*vb+rann(1,1); }while(vystar[i] >=0);} else{do{vystar[i]=mx[i][]*vb+rann(1,1); }while(vystar[i] <0);} } } 81

82 3. 82

83 3. 83

84 %

85 4. 85

86 4. ( ) MH N(,V).,. 86

87 4. ( ), ( m,v),.. 87

88 4. ( ) WinBUGS model { for (i in 1:n) { y[i] ~ dbern(p[i]) logit(p[i]) <- b[1] + b[2]*x[i,1]+ b[3]*x[i,2] + b[4]*x[i,3] + b[5]*x[i,4] } #prior for (j in 1:K) {b[j] ~ dnorm(0,0.001)} } ( WinBUGS ) 88

89 3. Ox for(cn=-cburn;cn<crepeat;++cn){ vb_o=vb;flk(vb_o,&dl_o,0,0); # dh_o=dlhat+vl1hat *(vb_o-vbhat) # +0.5*(vb_o-vbhat)'*ml2hat*(vb_o-vbhat); vb_n=vb1+mb1root*rann(cm,1); # flk(vb_n,&dl_n,0,0); # dh_n=dlhat+vl1hat *(vb_n-vbhat) # +0.5*(vb_n-vbhat)'*ml2hat*(vb_n-vbhat); dfrac=exp(dl_n+dh_o-dl_o-dh_n); # if(ranu(1,1)<=dfrac){vb=vb_n;} } MH 72.75% 89

90 4. ( ) 90

91 %

92 5. Seemingly Unrelated Model y i mx1, i=1,,n t m X i = mxk =( 1,, K ) : i N(0, ). 92

93 5.SUR 93

94 5.SUR 94

95 5.SUR :General Electric Westinghouse Gross investment (Y): Value of the Firm (V) Stock of plant and equipment(k) y i = V i 3 K i i Grunfeld Investment Data, 100 yearly observations

96 5.SUR ( ) WinBUGS model { for (t in 1:T) { y[t,1:2] ~ dmnorm(mu[t,],omega[, ]); mu[t,1] <- beta[1,1]+beta[1,2]*v[t,1]+beta[1,3]*k[t,1] mu[t,2] <- beta[2,1]+beta[2,2]*v[t,2]+beta[2,3]*k[t,2] } # priors on regression coefficients for (i in 1:M) {for (j in 1:3) {beta[i,j] ~ dnorm(0,0.001)}} Omega[1 : M, 1 : M] ~ dwish(r[, ], 2) # cross-correlation matrix of dimension M=2 Sigma[1:M,1:M] <- inverse(omega[, ]) } 96

97 5.SUR ( ) beta[1,1] sample: beta[1,2] sample: beta[1,3] sample: rho sample:

98 5.SUR ( ) beta[2,1] sample: beta[2,2] sample: beta[2,3] sample:

99 5.SUR SUR MC rror % E E E

100 Chen, Shao and Ibrahim(2000) Monte Carlo Methods in Bayesian Computation. Springer. Gamerman, D. (1997) Markov Chain Monte Carlo. Chapman & Hall Gelman, Carlin, Stern and Rubin(1995) Bayesian Data Analysis. Chapman & Hall Ibrahim, Chen and Shinha (2001) Bayesian Survival Analysis. Springer. Koop, G. (2003) Bayesian Econometrics. Wiley., J.S. (2001) Monte Carlo Strategies in Scientific Computing. Springer. 100

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