For a Gaussian discriminant classifier, where you model P(y = 1) (using a binomial) and P(x|y = 0) and P(x|y = 1). The latter have distinct means 0...

For a Gaussian discriminant classifier, where you model P(y = 1) (using a binomial) and P(x|y = 0) and P(x|y = 1). The latter have distinct means µ0 and µ1, and a shared covariance matrix Σ .

If we classify an example for which we know inputs x1, . . . xn−1, but the value of xn is missing.  The value of xn is filled by its class-conditional means, E(xn|y = 0) and E(xn|y = 1). Using the log-odds ratio, can someone provide a mathematical justification