  This article considers the sequential Monte Carlo (SMC) approximation of
ratios of normalizing constants associated to posterior distributions which in
principle rely on continuum models. Therefore, the Monte Carlo estimation error
and the discrete approximation error must be balanced. A multilevel strategy is
utilized to substantially reduce the cost to obtain a given error level in the
approximation as compared to standard estimators. Two estimators are considered
and relative variance bounds are given. The theoretical results are numerically
illustrated for the example of identifying a parametrized permeability in an
elliptic equation given point-wise observations of the pressure.
