Pilot-based estimation of the squared Euclidean norm of the channel vector of a Rayleigh fading system is considered. Unlike most previous work in the area of estimation of multiple antenna channels, we consider Bayesian estimation where the long-term channel statistics are known a priori. Closed-form expressions of the minimum mean square error (MMSE) estimator and its mean squared error (MSE) are derived for the cases of either an unweighted or a weighted unitary pilot matrix. The problem of finding the optimal pilot weighting, in the sense of minimizing the average MSE, is solved and a simple algorithm is proposed to achieve this power allocation numerically. The numerical evaluation shows that an optimal weighting can significantly improve the estimation quality in spatially correlated environments.