THE EGS5 CODE SYSTEM

The result of this algorithm is that ˆx will have the density function f given by
n∑
f(x) = α i f i (x) g i (x) . (2.22)
i=1
Thus, if we have a function f which can be put in this form for which the f i can be sampled
easily and the g i evaluated relatively easily, then we have a good method of sampling from f. It
can be shown that the mean number of tries to accept a value is ∑ α i . If all the g i = 1, we have
the pure “composition” method; and if n = 1, we have the pure “rejection” method. For short we
will call this the “mixed” method. We shall sometimes use the mixed method for some of the f i
also. In these cases we will use the notation (for example)
and so on.
f 2 (x) =
n 2 ∑
i=1
α 2j f 2j (x) g 2j (x) . (2.23)
Finally, we consider the problem of sampling from a joint density function f(x 1 , x 2 , . . . , x n ).
Define the marginal density functions
∫
g m (x 1 , x 2 , . . . , x m ) = f(x 1 , x 2 , . . . , x n ) dx m+1 dx m+2 . . . dx n
∫
= g m+1 (x 1 , x 2 , . . . , x m , x m+1 ) dx m+1 .
We see that g n = f. Now consider h m given by
We see that
(2.24)
h m (x m |x 1 , x 2 , . . . , x m−1 ) ≡ g m (x 1 , x 2 , . . . , x m )/g m−1 (x 1 , x 2 , . . . , x m−1 ) . (2.25)
∫
h m (x m |x 1 , x 2 , . . . , x m−1 ) dx m = 1 (2.26)
from the definition of the g’s. We see that h m is the conditional density function for x m given the
specified values for x 1 , x 2 , . . . , x m−1 . It can be easily seen that f can be factored into a product
of the h’s; namely,
f(x 1 , x 2 , . . . , x n ) = h 1 (x 1 ) h 2 (x 2 |x 1 ) h 3 (x 3 |x 1 , x 2 ) . . . (2.27)
. . . h n (x n |x 1 , x 2 , . . . , x n−1 ) .
The procedure then is to get a sample value x 1 using density function h 1 . Then use this value x 1 to
determine a density function h 2 (x 2 |x 1 ) from which to sample x 2 . Similarly, the previously sampled
x 1 , x 2 , . . . , x m determine the density function h m+1 (x m+1 |x 1 , x 2 , . . . , x m ) for x m+1 . This scheme
is continued until all x i have been sampled. Of course, if the ˆx i are independent random variables,
the conditional densities will just be the marginal densities and the variables can be sampled in
any order.
There are other methods analogous to the one-dimensional sampling methods which can be
used for sampling joint distributions. The reader is referred to the references cited above for more
details.
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