  To include parameter uncertainty into probabilistic climate forecasts one
must first specify a prior. We advocate the use of objective priors, and, in
particular, the Jeffreys' Prior. In previous work we have derived expressions
for the Jeffreys' Prior for the case in which the observations are independent
and normally distributed. These expressions make the calculation of the prior
much simpler than evaluation directly from the definition. In this paper, we
now relax the independence assumption and derive expressions for the Jeffreys'
Prior for the case in which the observations are distributed with a
multivariate normal distribution with constant covariances. Again, these
expressions simplify the calculation of the prior: in this case they reduce it
to the calculation of the differences between the ensemble means of climate
model ensembles based on different parameter settings. These calculations are
simple enough to be applied to even the most complex climate models.
