Many outstanding answers here detail the fundamentals of linear discriminant analysis. These include descriptions of its use in dimensionality reduction, an explanation of classification using Bayes' rule, and a description of within- and between-class scatter matrices.
Common methods for measuring importance of discriminants to classification as a whole (eigenvalue and correlation-based) are included in the first answer. These seem to tackle the problem, "How important is discriminant d for separating clusters in general?" I am searching for something slightly different:
How important is discriminant d, given the value of an input point, x, that we would like to classify?
What I've tried so far: iteratively removing discriminants and calculating class probabilities at x using Bayes' rule. Stopping when statistical difference between iteration i and i+1 exceeds a threshold. I would prefer, however, a non-iterative solution that is more similar to the direct calculation of discriminant coefficients.
