[Bayes] runif: Inversion Sampling

runifum

Inversion Sampling 看樣子就是個路人甲。

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

> func=function(n) { +     return(-0.5+sqrt(0.25+2*runif(n))) + } // 反函數的x的均勻sampling值 => y 就是原函數的x，恰好做爲hist的輸入參數
> hist(func(5000),probability=T, xlab=expression(theta), ylab="Density", main="Samples from f(x)") 
// 如下是真實值，用線表示
> xx=seq(0, 1, length=100) > lines(xx, xx+0.5, col=2)

 

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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) 算是一個表達技巧，將本來兩個半圖，合併在了一塊兒，以下：

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簡化技巧

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