[Bayes] qgamma & rgamma: Central Credible Interval

gamma分布的density的奇怪特性，如下：

---

Poisson的Gamma先验

 h(x) 的置信区间 的 获取

> n =
> sumx=
>
> alpha=
> beta=0.01
> pmean=(alpha+sumx)/(beta+n)
> L=qgamma(0.025, alpha+sumx, beta+n)　　// 获得cdf的边界
> U=qgamma(0.975, alpha+sumx, beta+n)
>
> cat("Posterior mean: ", pmean, " (", L, ",",U,")")
Posterior mean:  382.8796  ( 378.1376 , 387.6506 )
// 95% 置信区间的边界值 还有期望。

Monte Carlo sampling:

N= # or 500 or 5000
L=U=rep(NA,length=)
for (i in :) {
dat=sort(rgamma(N,alpha+sumx,beta+n))
L[i]=dat[0.025*N]
U[i]=dat[0.975*N]
}

// 获得某分位点的大量样本

widthL=max(L)-min(L)
widthU=max(U)-min(U)
par(mfrow=c(,))
hist(L,probability=T,xlab="Lower 95% CI bound")
points(qgamma(0.025,alpha+sumx,beta+n),,pch=,col=)
hist(U,probability=T,xlab="Upper 95% CI bound")
points(qgamma(0.975,alpha+sumx,beta+n),,pch=,col=)
cat("L interval variability (range):",widthL,"\n")
cat("U interval variability (range):",widthU,"\n")

Sampling估计的分位点，看来与True value差不多呢。

> mean(L)
[] 378.0747
> mean(U)
[] 387.5853

问题来了，N要多大才能保证要求的分位点估计精度：Sol 要 according to Central Limit Thearem.

 p(y) 的预测

alpha=; beta=0.01
sumx=; n=65

theta=rgamma(,alpha+sumx,beta+n)　　// 后验sita
y=rpois(,theta)　　// poisson分布sampling，带入后验sita的f(y|sita),5000个相应的预测值
hist(y,probability=T,ylab="Density",main="Posterior predictive distribution")
// sampling法得到直方图。
// 相当吻合！
// 精确函数得到散点图。
xx=:
pr=dnbinom(xx,sumx+alpha,-/(beta+n+))
#lines(xx,pr,col=) # The (incorrect) continuous version - ok as an approx.
#The (correct) discrete version:
for (i in :length(xx)) {
　　lines(c(xx[i],xx[i+]),rep(pr[i],),col=,lwd=)
}