[R] Question about acception rejection sampling - NOT R question
Leeds, Mark (IED)
Mark.Leeds at morganstanley.com
Fri Jul 13 22:45:29 CEST 2007
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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