Multivariate Gaussian distribution and classification

Question

This is a general question. I understand the theory behind using multivariate Gaussian distributions, however one question still bugs me and I have not been able to get a "layman's" answer to it.

Given a new data point $X$, and a data set comprising various data points that can be classified into $N$ multivariate Gaussians. How is the point classified?

What I am asking is, given the $\{x_1,..,x_n\}$ dimensions that make up the data vector to be classified, how is the decision made whether that point is part of one Gaussian distribution vs. another?

Answer 1

Assuming you have N multivariate Gaussian PDFs $f_i(X)$, you could compare the PDF values like in maximum likelihood: $f_i(X)$ vs. $f_j(X)$, then pick the one with the highest value.