[Bayes] runif: Inversion Sampling

runifum

Inversion Sampling 看样子就是个路人甲。

Ref: [Bayes] Hist & line: Reject Sampling and Importance Sampling

> func=function(n) {
+     return(-0.5+sqrt(0.25+2*runif(n)))
+ }
// 反函数的x的均匀sampling值 => y 就是原函数的x，刚好作为hist的输入参数
> hist(func(5000),probability=T, xlab=expression(theta), ylab="Density", main="Samples from f(x)")

// 以下是真实值，用线表示
> xx=seq(0, 1, length=100)
> lines(xx, xx+0.5, col=2)

 

---

 

func=function(n) {
　　u=runif(n)
　　return(c(0.5*log(2*u[u<0.5]),-0.5*log(2*(1-u[u>0.5]))) )
}
hist(func(5000),probability=T,xlab=expression(theta),ylab="Density",main="Samples from f(x)",nclass=31,ylim=c(0,1))

xx=seq(-5,5,length=100)
lines(xx,exp(-sign(xx)*2*xx),col=2)

 

-sign(xx) 算是一个表达技巧，将原本两个半图，合并在了一起，如下：

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简化技巧

Y = −λ log(1 − U ).
Note that if U is uniformly distributed on [0, 1], then 1 − U is also uniformly distributed on [0, 1] so that we could simply compute
Y = −λ log U