Can the gaussian mixture model combined in clustering? Suppose I have a data with two clusters. Suppose further that I cluster the data using, for example, K-means. Then, can I fit a mixture model to each cluster? That is, can I fit a gaussian mixture model to each cluster?
Sometimes the data is known to have 2 classes or clusters, for example, disease data. It contains information of infected and non-infected people. However, researcher still use k-means to cluster the data!! I really do not understand why they do this? Another example, sport data which contains female and male information. So, why we need to cluster then again?
 A: (1) "Suppose I have a data with two clusters. Suppose further that I cluster the data using, for example, K-means. Then, can I fit a mixture model to each cluster? That is, can I fit a gaussian mixture model to each cluster?" You can, but in most cases this will not be very good. It is true that many real clusters are not Gaussian, and it is possible to model them with a (for example) unimodal mixture of Gaussians, see, e.g.,
C. Hennig (2010) Methods for merging Gaussian mixture components. Advances in Data Analysis and Classification 4, 3-34.  https://link.springer.com/article/10.1007/s11634-010-0058-3 (There is some more work by others on this problem.)
However, in that case, $K$-means with $K$ being the number of true clusters (if known) is not a good starting point, as there is no guarantee that the $K$-means clusters align well with the true ones, and there are good reasons to expect that often they won't. There is some work that starts from $K$-means with very large $K$ and then merges clusters though, see
Melnykov, V., Michael, S. Clustering Large Datasets by Merging K-Means Solutions. J Classif 37, 97–123 (2020). https://doi.org/10.1007/s00357-019-09314-8
(2)  "Sometimes the data is known to have 2 classes or clusters, for example, disease data. It contains information of infected and non-infected people. However, researcher still use k-means to cluster the data!!" There is often more than one meaningful clustering in the data. So even if one grouping is "given", it may be of interest to find others. For example, athletes could be clustered according to their relative performance profile in different sports, and male/female may not have much to do with that, even if this information is in the data.
