  Ratios of normalizing constants for two distributions are needed in both
Bayesian statistics, where they are used to compare models, and in statistical
physics, where they correspond to differences in free energy. Two approaches
have long been used to estimate ratios of normalizing constants. The `simple
importance sampling' (SIS) or `free energy perturbation' method uses a sample
drawn from just one of the two distributions. The `bridge sampling' or
`acceptance ratio' estimate can be viewed as the ratio of two SIS estimates
involving a bridge distribution. For both methods, difficult problems must be
handled by introducing a sequence of intermediate distributions linking the two
distributions of interest, with the final ratio of normalizing constants being
estimated by the product of estimates of ratios for adjacent distributions in
this sequence. Recently, work by Jarzynski, and independently by Neal, has
shown how one can view such a product of estimates, each based on simple
importance sampling using a single point, as an SIS estimate on an extended
state space. This `Annealed Importance Sampling' (AIS) method produces an
exactly unbiased estimate for the ratio of normalizing constants even when the
Markov transitions used do not reach equilibrium. In this paper, I show how a
corresponding `Linked Importance Sampling' (LIS) method can be constructed in
which the estimates for individual ratios are similar to bridge sampling
estimates. I show empirically that for some problems, LIS estimates are much
more accurate than AIS estimates found using the same computation time,
although for other problems the two methods have similar performance. Linked
sampling methods similar to LIS are useful for other purposes as well.
