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This might be a trivial question

I have a probabilistic model, say a Gaussian mixture model of known parameters, and with that I have a set of defined sample points.

I would like to know how likely it is that the sample points were generated from (i.e. that they match) that model.

What would be the theoretical solution to this problem?

The issue is that simply using the pdf of a GMM does not seem to make sense because:

  1. It is a continuous distribution while the points are infinitesimal samples
  2. If all samples lie exactly at the mean of the sharpest Gaussian component, their probability according to the pdf would be higher than having them spread among all the Gaussian components according to the components' weights, although the latter case would match the underlying model better.
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If I understand your question correctly, you are looking for goodness-of-fit tests. Check out this link for a basic description: https://onlinecourses.science.psu.edu/stat414/node/259

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