Bayesian Biostatistics by E Lesaffre & AB Lawson, Chapter 6 学習ノート
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事前分布を正規分布として取り扱った場合のMH法。
とにかく、プログラムを動かしてみる。
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> # MH SAMPLING a t(3) distribution using a normal distribution > > > nchain <- 1500 > > theta.1 <- rep(0,nchain) > > # proposal density is normal distribution > # two choices: N(3,4) and N(0,2) > > > # first choice: N(3,4) > > > munormal <- 0 > sigmanormal <- 4 > > # Target distribution is t(3) with location mut and scale sigmat > > mut <- 3 > sigmat <- 2 > df <- 3 > > > set.seed(729) > > ichain <- 1 > accept <- 0 > > theta.1[ichain] <- 0 > > while (ichain < nchain) { + theta.1.tilde <- rnorm(1,munormal,sigmanormal) + + tk <- (theta.1[ichain] - mut)/sigmat + ttilde <- (theta.1.tilde - mut)/sigmat + r1 <- dt(ttilde,df)/dt(tk,df) + + nk <- (theta.1[ichain] - munormal)/sigmanormal + ntilde <- (theta.1.tilde - munormal)/sigmanormal + r2 <- dnorm(ntilde,0,1)/dnorm(nk,0,1) + + r <- r1/r2 + + u <- runif(1,0,1) + + if(u < r) {theta.1[ichain+1] <- theta.1.tilde;accept=accept+1} + else {theta.1[ichain+1] <- theta.1[ichain]} + + if(ichain < 15){print(ichain);print(theta.1[ichain]);print(theta.1.tilde);print(theta.1[ichain+1]) } + + ichain <- ichain +1 + + } [1] 1 [1] 0 [1] -3.092103 [1] 0 [1] 2 [1] 0 [1] 3.836091 [1] 3.836091 [1] 3 [1] 3.836091 [1] 6.532608 [1] 6.532608 [1] 4 [1] 6.532608 [1] -4.467422 [1] 6.532608 [1] 5 [1] 6.532608 [1] -2.45395 [1] -2.45395 [1] 6 [1] -2.45395 [1] 4.874305 [1] 4.874305 [1] 7 [1] 4.874305 [1] 4.595936 [1] 4.595936 [1] 8 [1] 4.595936 [1] 0.7565358 [1] 0.7565358 [1] 9 [1] 0.7565358 [1] 5.63561 [1] 5.63561 [1] 10 [1] 5.63561 [1] -5.965046 [1] 5.63561 [1] 11 [1] 5.63561 [1] -3.581635 [1] 5.63561 [1] 12 [1] 5.63561 [1] -3.416178 [1] 5.63561 [1] 13 [1] 5.63561 [1] -3.8426 [1] 5.63561 [1] 14 [1] 5.63561 [1] -0.7893432 [1] -0.7893432 > > rate.accept <- accept/nchain > rate.accept [1] 0.486 > > nstart <- 501 > > # After burn-in part > > > gridt <- seq(-5,5,0.01) > grid <- gridt*sigmat+mut > mgrid <- min(grid) > Mgrid <- max(grid) > griddt <- dt(gridt,df)/sigmat > > par(mar=c(5,4,4,2)) > > par(mfrow=c(1,2)) > par(pty="s") > hist(theta.1[nstart:nchain],xlab="t",ylab="", + nclas=30,prob=T,xlim=c(mgrid,Mgrid),ylim=c(0,0.32), + main="",col="light blue",cex.lab=1.5,cex=1.2,cex.axis=1.3) > lines(grid,griddt,lwd=3) > text(-5.8,0.31,"(a)",cex=1.4) > > nchain <- 1500 > > theta.1 <- rep(0,nchain) > > > # second choice: N(0,2) > > > munormal <- 0 > sigmanormal <- 2 > > # Target distribution is t(3) with location mut and scale sigmat > > mut <- 3 > sigmat <- 2 > df <- 3 > > > set.seed(729) > > ichain <- 1 > accept <- 0 > > theta.1[ichain] <- 0 > > while (ichain < nchain) { + theta.1.tilde <- rnorm(1,munormal,sigmanormal) + + tk <- (theta.1[ichain] - mut)/sigmat + ttilde <- (theta.1.tilde - mut)/sigmat + r1 <- dt(ttilde,df)/dt(tk,df) + + nk <- (theta.1[ichain] - munormal)/sigmanormal + ntilde <- (theta.1.tilde - munormal)/sigmanormal + r2 <- dnorm(ntilde,0,1)/dnorm(nk,0,1) + + r <- r1/r2 + + u <- runif(1,0,1) + + if(u < r) {theta.1[ichain+1] <- theta.1.tilde;accept=accept+1} + else {theta.1[ichain+1] <- theta.1[ichain]} + + if(ichain < 15){print(ichain);print(theta.1[ichain]);print(theta.1.tilde);print(theta.1[ichain+1]) } + + ichain <- ichain +1 + + } [1] 1 [1] 0 [1] -1.546051 [1] -1.546051 [1] 2 [1] -1.546051 [1] 1.918046 [1] 1.918046 [1] 3 [1] 1.918046 [1] 3.266304 [1] 3.266304 [1] 4 [1] 3.266304 [1] -2.233711 [1] 3.266304 [1] 5 [1] 3.266304 [1] -1.226975 [1] -1.226975 [1] 6 [1] -1.226975 [1] 2.437153 [1] 2.437153 [1] 7 [1] 2.437153 [1] 2.297968 [1] 2.297968 [1] 8 [1] 2.297968 [1] 0.3782679 [1] 0.3782679 [1] 9 [1] 0.3782679 [1] 2.817805 [1] 2.817805 [1] 10 [1] 2.817805 [1] -2.982523 [1] 2.817805 [1] 11 [1] 2.817805 [1] -1.790817 [1] 2.817805 [1] 12 [1] 2.817805 [1] -1.708089 [1] 2.817805 [1] 13 [1] 2.817805 [1] -1.9213 [1] 2.817805 [1] 14 [1] 2.817805 [1] -0.3946716 [1] -0.3946716 > > rate.accept <- accept/nchain > rate.accept [1] 0.368 > > > > > hist(theta.1[nstart:nchain],xlab="t",ylab="", + nclas=30,prob=T,xlim=c(mgrid,Mgrid),ylim=c(0,0.32), + main="",col="light blue",cex.lab=1.5,cex=1.2,cex.axis=1.3) > lines(grid,griddt,lwd=3) > text(-5.8,0.31,"(b)",cex=1.4) > > summary(theta.1[nstart:nchain]) Min. 1st Qu. Median Mean 3rd Qu. Max. -6.047 1.421 2.563 2.278 3.504 5.722 |
