The overall mean and variance of a Gaussian mixture of i components can be calculated from the means and variances of the individual components multiplied by their weights:

M = sumi(wi mi) and V = sumi(vi + mi2) - M2

where M, V = mean and variance of mixture

sumi(...) = sum over all i

wi, mi, vi = weight, mean and variance of ith component

sumi(wi) = 1

Is it then legitimate to use a single Guassian defined by M and V in place of the original mixture? I am only interested in descriptive stats here rather than inferential statistics. Are there any implications for uncertainty calculations?

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