If two classes $w_1$ and $w_2$ have normal distribution with known parameters ($M_1$, $M_2$ as their means and $\Sigma_1$,$\Sigma_2$ are their covariances) how we can calculate error of the Bayes classifier for them theorically?

Also suppose the variables are in N-dimensional space.

Note: A copy of this question is also available at http://math.stackexchange.com/q/11891/4051 that is still unanswered.