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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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    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)? – Danica Mar 27 '16 at 15:25
  • I have two GMM samples. I want to compare whether they are different or not. –  Mar 27 '16 at 18:21
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    What do you mean by "GMM samples"? – Tim Mar 27 '16 at 18:53
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    -1 and vote to close as unclear for reasons listed above. – amoeba Mar 27 '16 at 18:57
  • I think this makes perfect sense. I have two GMMs and I want to determine whether they are from different populations or not –  Mar 27 '16 at 18:58
  • You have two samples or two models? Those are not the same. – Tim Mar 27 '16 at 19:02
  • Two samples and I fit a GMM for each of those samples; hence I have 2 GMMs. –  Mar 27 '16 at 19:02
  • What would you like to compare in this scenario? You have two different models fitted to two different datasets... – Tim Mar 27 '16 at 19:19
  • Whether they are statistically different! –  Mar 27 '16 at 19:19
  • Possible duplicate of [Zero inflate models vs generalized mixture model](http://stats.stackexchange.com/questions/78973/zero-inflate-models-vs-generalized-mixture-model) – gung - Reinstate Monica Mar 27 '16 at 19:45
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    Whether *what* is statistically different? The data (what aspect of it?), the models? – Tim Mar 27 '16 at 21:00
  • Yes the models, what else? –  Mar 27 '16 at 21:05
  • @zero but those are two different models, computed on different data. Example: you want to compare model that assumes normal distribution with parameters $\mu$ and $\sigma$ for human height to another model that assumes binomial distribution for number of heads in $n$ coin tosses parametrized by $p$ -- what and how would you like to compare in here? It is like comparing taste of pizza to speed of a sports car... – Tim Mar 28 '16 at 18:21
  • This is just a two sample test where each of the samples is bimodal in nature. –  Mar 29 '16 at 15:09

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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

Uri Goren
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