  Statistical Data Assimilation (SDA) is the transfer of information from field
or laboratory observations to a user selected model of the dynamical system
producing those observations. The data is noisy and the model has errors; the
information transfer addresses properties of the conditional probability
distribution of the states of the model conditioned on the observations. The
quantities of interest in SDA are the conditional expected values of functions
of the model state, and these require the approximate evaluation of high
dimensional integrals. We introduce a conditional probability distribution and
use the Laplace method with annealing to identify the maxima of the conditional
probability distribution. The annealing method slowly increases the precision
term of the model as it enters the Laplace method. In this paper, we extend the
idea of precision annealing (PA) to Monte Carlo calculations of conditional
expected values using Metropolis-Hastings methods.
