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The maximum likelihood estimate of a vector, given noisy observations of linear combinations of the vector's components, is a function of the covariance matrices of the noise. Often the matrices are not exactly known, and consequently the maximum likelihood estimate will be in error. An algorithm is developed for computing the covariances of the errors in the maximum likelihood estimate due to uncertainties in the noise covariance matrices. It is assumed that the uncertainties are small and can be described statistically.