Not a strict proof, but think of it this way: The liklihood of getting a particular value of x has 2 parts. 1st x has to be generated from h, the liklihood of this happening is h(x), 2nd the point has to be accepted with conditional probability f(x)/(c*h(x)). If we multiply we get h(x) * f(x)/ ( c* h(x) ) and the 2 h(x)'s cancel out leaving the liklihood of getting x as f(x)/c. The /c just indicates that approximately 1-1/c points will be rejected and thrown out and the final normalized distribution is f(x), which was the goal.
Hope this helps, -- Gregory (Greg) L. Snow Ph.D. Statistical Data Center Intermountain Healthcare [EMAIL PROTECTED] (801) 408-8111 > -----Original Message----- > From: [EMAIL PROTECTED] > [mailto:[EMAIL PROTECTED] On Behalf Of Leeds, > Mark (IED) > Sent: Friday, July 13, 2007 2:45 PM > To: [email protected] > Subject: [R] Question about acception rejection sampling - > NOT R question > > 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. > -------------------------------------------------------- > > This is not an offer (or solicitation of an offer) to > buy/se...{{dropped}} > > ______________________________________________ > [email protected] mailing list > 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. > ______________________________________________ [email protected] mailing list 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.
