I have seen two different approaches to the explanation of the Linear Discriminant Analysis. The following is the description of the rough understanding of the approaches:
1) The first one refers to the within-class and between-class variances, computes their ratio and tries to minimize within-class variance at the same time maximize between-class variances. It is done by computing the eigenvalues of the inverse of the matrix of within-class variance (before projection) multiplied by the between-class variances. Then choosing the eigenvectors that correspond to the biggest eigenvalues in absoulute value.
2) The second one is the probabilistic method. It tries to maximize the probability of detecting the class given the inputs. For this purpose, it uses the Bayesian rule by assuming that the inputs, given the class of the inputs, have a multivariate normal distribution and a common covariance. Then it computes log-likelihood and tries to maximize it over the class.
The question is how are these two methods connected with each other?