I am trying to do some exercises of this book: http://www.inference.phy.cam.ac.uk/itprnn/book.pdf
Currently, I am stuck at question 22.7 (on page 318) which is the following:
Make a version of the K-means algorithm that models the data as a mixture of K arbitrary Gaussians, i.e., Gaussians that are not constrained to be axis-aligned.
As far as I understand, this means that I need to model $K$ multivariate Gaussians and determine the means and convariance matrices of these Gaussians. Is that correct?
If that is the case, I think it would be useful to find the log likelihood of the Gaussian mixture and derive the expression with respect to $\mu^{(k)}$ (the mean of the $k^{th}$ Gaussian) and with respect to $\Sigma^{(k)}$ (the covariance matrix of the $k^{th}$ Gaussian). Then, I would find the second derivatives and with those, I could find the Newton-Rhapson update rules. Am I correct? Or is there an easier way?