# Calculate number of standard deviations separating two multivariate Gaussians?

Given a set of multivariate Gaussian distributions (from fitting a Gaussian mixture model) I would like to be able to calculate the likelihood that a data point drawn from one Gaussian will improperly be classified as belonging to one of the other Gaussians.

Specifically, I have this data (ignore the inset).

If you're interested, this is figure 5 from this paper. The ellipses are drawn 3 and 4 standard deviations around each cluster. Given a model of the cluster covariance matrix as a function of the centroid, I'd like to calculate a set of centroids such that the probability of misidentifying a data point from that cluster is less than some specified probability. Put another way, I'd like to move these clusters together until the 4 standard deviation contours touch.

It seems like the Bhattacharyya distance measures what I want, or is very close to it, but I'm not sure how to interpret that distance in terms of a likelihood of a point from one distribution being misidentified as a different one.

• Interesting. Is a given point classified based on the nearest centroid, or the multivariate Gaussian with largest density for the point? – Sean Easter Jun 5 '14 at 23:17
• I would classify the points by the multivariate Gaussian they are most likely to belong to (which I guess is the one with highest density for that point?) – Kurt Jun 6 '14 at 17:26