Monte Carlo Methods in Statistical Mechanics: Foundations and ...

(A) Irreducibility. For each pair x y 2 S, there exists ann 0 for which p (n)
xy > 0.
(B) Stationarity of . For each y 2 S,
X
x
x pxy = y: (4.1)
A su cient condition for (B), which isoften more convenient toverify, is:
(B 0 ) Detailed balance for . For each pairx y 2 S xpxy = ypyx.
Avery general method for constructing transition matrices satisfying detailed balance
for a given distribution was introduced in 1953 by Metropolis et al. [17], with
a slight extension two decades later by Hastings [18]. The idea is the following: Let
P (0) = fp (0)
xy g be an arbitrary irreducible transition matrix on S. We callP (0) the
proposal matrix weshall use it to generate proposed moves x ! y that will then be
accepted or rejected with probabilities axy and 1; axy, respectively. Ifaproposed move
is rejected, then we make a\null transition" x ! x. Therefore, the transition matrix
P = fpxyg of the full algorithm is
pxy = p (0)
xy axy
pxx = p (0)
xx + X
y6=x
for x 6= y
p (0)
xy (1 ; axy) (4.2)
where of course we must have 0 axy 1 for all x y. Itiseasyto see that P satis es
detailed balance for if and only if
axy
ayx
= y p (0)
yx
x p (0)
xy
for all pairs x 6= y. Butthis is easily arranged: just set
axy = F
y p (0)
yx
x p (0)
xy
where F :[0 +1] ! [0 1] is any function satisfying
F (z)
F (1=z)
The choice suggested by Metropolis et al. is
!
(4.3)
(4.4)
= z for all z: (4.5)
F (z) = min(z1) (4.6)
this is the maximal function satisfying (4.5). Another choice sometimes used is
F (z) = z
1+z
17
: (4.7)