在逻辑回归中，我们将二元响应\（Y_i \）回归到协变量\（X_i \）上。下面的代码使用Metropolis采样来探索\（\ beta_1 \）和\（\ beta_2 \）的后验YiYi到协变量XiXi。
定义expit和分对数链接函数
logit<-function(x){log(x/(1-x))} 此函数计算\（（\ beta_1，\ beta_2）\）的联合后验。它返回后验的对数以获得数值稳定性。(β1,β2)(β1,β2)。它返回后验的对数以获得数值稳定性。 log_post<-function(Y,X,beta){ prob1 <- expit(beta[1] + beta[2]*X) like+prior}
这是MCMC的主要功能.can.sd是候选标准偏差。
Bayes.logistic<-function(y,X, n.samples=10000, can.sd=0.1){ keep.beta <- matrix(0,n.samples,2) keep.beta[1,] <- beta acc <- att <- rep(0,2) for(i in 2:n.samples){ for(j in 1:2){ att[j] <- att[j] + 1 # Draw candidate: canbeta <- beta canbeta[j] <- rnorm(1,beta[j],can.sd) canlp <- log_post(Y,X,canbeta) # Compute acceptance ratio: R <- exp(canlp-curlp) U <- runif(1) if(U<R){ acc[j] <- acc[j]+1 } } keep.beta[i,]<-beta } # Return the posterior samples of beta and # the Metropolis acceptance rates list(beta=keep.beta,acc.rate=acc/att)}
生成一些模拟数据
set.seed(2008) n <- 100 X <- rnorm(n) true.p <- expit(true.beta[1]+true.beta[2]*X) Y <- rbinom(n,1,true.p)
拟合模型
burn <- 10000 n.samples <- 50000 fit <- Bayes.logistic(Y,X,n.samples=n.samples,can.sd=0.5) tock <- proc.time()[3] tock-tick ## elapsed ## 3.72
结果
abline(true.beta[1],0,lwd=2,col=2)
abline(true.beta[2],0,lwd=2,col=2)
hist(fit$beta[,1],main="Intercept",xlab=expression(beta[1]),breaks=50)
hist(fit$beta[,2],main="Slope",xlab=expression(beta[2]),breaks=50) abline(v=true.beta[2],lwd=2,col=2)
print("Posterior mean/sd") ## [1] "Posterior mean/sd" print(round(apply(fit$beta[burn:n.samples,],2,mean),3)) ## [1] -0.076 0.798 print(round(apply(fit$beta[burn:n.samples,],2,sd),3)) ## [1] 0.214 0.268 print(summary(glm(Y~X,family="binomial"))) ## ## Call: ## glm(formula = Y ~ X, family = "binomial") ## ## Deviance Residuals: ## Min 1Q Median 3Q Max ## -1.6990 -1.1039 -0.6138 1.0955 1.8275 ## ## Coefficients: ## Estimate Std. Error z value Pr(>|z|) ## (Intercept) -0.07393 0.21034 -0.352 0.72521 ## X 0.76807 0.26370 2.913 0.00358 ** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## (Dispersion parameter for binomial family taken to be 1) ## ## Null deviance: 138.47 on 99 degrees of freedom ## Residual deviance: 128.57 on 98 degrees of freedom ## AIC: 132.57 ## ## Number of Fisher Scoring iterations: 4
