[R] Question about acception rejection sampling - NOT R question

From: Leeds, Mark (IED) <Mark.Leeds_at_morganstanley.com>
Date: Fri, 13 Jul 2007 16:45:29 -0400

This is not related to R but I was hoping that someone could help me. I am reading the "Understanding the Metropolis Hastings Algorithm" paper from the American Statistician by Chip and Greenberg, 1995, Vol 49, No 4. Right at the beginning they explain the algorithm for basic acceptance rejection sampling in which you want to simulate a density from f(x) but it's not easy and you are able to generate from another density called h(x). The assumption is that there exists some c such that f(x) <= c(h(x) for all x

They clearly explain the steps as follows :

1. generate Z from h(x).
2. generate u from a Uniform(0,1)
3. if u is less than or equal to f(Z)/c(h(Z) then return Z as the generated RV; otherwise reject it and try again.

I think that, since f(Z)/c(h(z) is U(0,1), then u has the distrbution as f(Z)/c(h(Z).  

But, I don't understand why the generated and accepted Z's have the same density as f(x) ?

Does someone know where there is a proof of this or if it's reasonably to explain, please feel free to explain it. They authors definitely believe it's too trivial because they don't. The reason I ask is because, if I don't understand this then I definitely won't understand the rest of the paper because it gets much more complicated. I willing to track down the proof but I don't know where to look. Thanks.

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https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. Received on Fri 13 Jul 2007 - 20:53:40 GMT