Objective probabilistic forecasts of future climate that include parameter
uncertainty can be made by using the Bayesian prediction integral with the
prior set to Jeffreys' Prior. The calculations involved in determining the
prior can then be simplified by making parametric assumptions about the
distribution of the output from the climate model. The most obvious assumption
to make is that the climate model output is normally distributed, in which case
evaluating the prior becomes a question of evaluating gradients in the
parameters of the normal distribution. In previous work we have considered the
special cases of diagonal (but not constant) covariance matrix, and constant
(but not diagonal) covariance matrix. We now derive expressions for the general
multivariate normal distribution, with non-constant non-diagonal covariance
matrix. The algebraic manipulation required is more complex than for the
special cases, and involves some slightly esoteric matrix operations including
taking the expectation of a vector quadratic form and differentiating the
determinants, traces and inverses of matrices