(2.7) |
This result gives us guidance about how to choose not just the features, but also . In short, in order to make the projected PDF as good as possible an approximation to , choose and so that is approximately sufficient statistic for the likelihood ratio test between and . But, the sufficiency condition is required for optimality, but is not necessary for 2.6 to be a valid PDF. Here we can see the importance of the theorem which provides a means of creating PDF approximations on the high-dimensional input data space without dimensionality penalty using low-dimensional feature PDFs. It also provides a way to optimize the approximation by controlling both the reference hypothesis as well as the features themselves. This is the remarkable property of Theorem 1 - that the resulting function remains a PDF whether or not the features are sufficient statistics. Since sufficiency means optimality of the classifier, approximate sufficiency mean PDF approximation and approximate optimality.
Baggenstoss 2017-05-19