I thought Mahalanobis Distance was
which has both mean and covariance in the calculation.
The Mahalanobis distance can tell you how similar a measurement is to a set of other measurements with associate mean variance. Now in my case, the measurement is a probability distribution (normal), so I know what it's mean is and I know what its variance is. The Mahalanobis distance would just use the mean of the measurement, but it would be useful if I could incorporate its variance into it as well.
I poked around and found the Hellinger distance on wikipedia, but it isn't described how it can be implemented for multivariate data, where measurements are vectors and you're dealing with covariance matrices. Could someone please point me to a resource that explains it?