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Say that I have two Gaussian Mixture Models. How would I determine whether they are statistically different from one another?

EDIT:

I'm thinking about doing pairwise tests for each Gaussian?

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    $\begingroup$ What exactly do you mean by this? Do you have two data sets, each with a best-fit GMM, and you want to test whether the generating GMMs are different? Are you just given two sets of GMM parameters and you want to determine if they're "different enough" (in which case we'll need much more info)? $\endgroup$
    – Danica
    Commented Mar 27, 2016 at 15:25
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    $\begingroup$ What do you mean by "GMM samples"? $\endgroup$
    – Tim
    Commented Mar 27, 2016 at 18:53
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    $\begingroup$ -1 and vote to close as unclear for reasons listed above. $\endgroup$
    – amoeba
    Commented Mar 27, 2016 at 18:57
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    $\begingroup$ I think this makes perfect sense. I have two GMMs and I want to determine whether they are from different populations or not $\endgroup$
    – user46925
    Commented Mar 27, 2016 at 18:58
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    $\begingroup$ Whether they are statistically different! $\endgroup$
    – user46925
    Commented Mar 27, 2016 at 19:19

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I would suggest using a Bayesian setting,

For example, assuming you have 2 Gaussians in each model, The posterior distribution on the two models is a good way to measure how well each model describe the sample.

Bayes Factor can be used to measure difference/similarity between the two models

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  • $\begingroup$ Nice, but I'd like to stick to the Frequentist school. $\endgroup$
    – user46925
    Commented Mar 27, 2016 at 18:48