The Accept-Reject method is a classical sampling method which allows one to sample from a distribution which is difficult or impossible to simulate by an inverse transformation. Instead, draws are taken from an instrumental density and accepted with a carefully chosen probability. The resulting draw is a draw from the target density.
Accept-Reject Algorithm
The objective is to sample from a target density , where
,
is the unnormalized target density, and
the potentially unknown normalizing constant. Suppose that we can sample from another density
and that there exists a constant
such that
for all
. To obtain a draw from
:
- Draw a candidate
from
and
from
, the uniform distribution on the interval
.
- If
, return
- Otherwise, return to 1.
The expected number of iterations required to accept a draw is . To ensure efficiency, the optimal choice of
is:
References
-
Chib, S. and E. Greenberg (1995). Understanding the Metropolis Hastings Algorithm. American Statistical Journal 49, 327–335.
-
Robert, C.P., and G. Casella (2004). Monte Carlo Statistical Methods, Second Edition. New York: Springer.