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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?

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    $\begingroup$ This sounds contradictory: each cluster should correspond to one Gaussian component, the mixture being appropriate for the entire dataset. $\endgroup$
    – Xi'an
    Commented Jun 28, 2022 at 7:58
  • $\begingroup$ @Xi'an: When running k-means with $k=2$ fixed, depending on the data, the two clusters may well be heterogeneous and non-normal or even corresponding to a Gaussian mixture each. Although I agree that it would in that case be better to fit a mixture to the whole dataset. Note though that there's data with non-normal clusters, and fitting a Gaussian mixture to them will often fit a cluster with several Gaussians. "Each cluster should correspond to a Gaussian component" is often wishful thinking, see link.springer.com/article/10.1007/s11634-010-0058-3 $\endgroup$ Commented Jun 28, 2022 at 13:12
  • $\begingroup$ @Alice: "Then, can I fit a mixture model to each cluster? That is, can I fit a gaussian mixture model to each cluster?" What do you want to achieve with this? $\endgroup$ Commented Jun 28, 2022 at 13:14
  • $\begingroup$ "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. $\endgroup$ Commented Jun 28, 2022 at 13:16
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    $\begingroup$ OK, I made an answer out of the key points. $\endgroup$ Commented Jun 28, 2022 at 14:01

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(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.

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