The probability density of a point x with respect to a multivariate normal distribution having a mean ? and covariance matrix ? is given by the equa- tion

Using the sample mean x and covariance matrix S as estimates of the mean


? and covariance matrix ?, respectively, show that the log(prob(x)) is equal


to the Mahalanobis distance between a data point x and the sample mean x


plus a constant that does not depend on x.





If we use the sample mean and covariance as estimates of ? and ?, respec-


tively, then











The constant and the constant factor do not affect the ordering of this quan-


tity, only their magnitude. Thus, if we want to base a distance on this





quantity, we can keep only the variable part, which is the Mahalanobis dis-


tance.

Computer Science & Information Technology

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