suppose I have two parameters: $(\mu_{1},\Sigma_{1},\alpha_{1},\mu_{2},\Sigma_{2})$ and $(\mu_{1},\Sigma_{1},\alpha_{1},\mu_{2},\Sigma_{2})$ and suppose I mapped them to PDF using the gaussian mixture model with one component, let the pdfs are $P_{1}$ and $P_{2}$ so is it possible that if both the parameters are different but $P_{1}=P_{2}$? can someone give any example if it is possible! I have seen some results about it but there they took the components as 1d gaussian and order the variance but here we have 2d gaussian components.
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Because one characterization of multivariate Gaussians is that the distribution of any linear combination of the variables is (univariate) Gaussian, the answer for the univariate case applies to the more general multivariate case. – whuber Dec 08 '23 at 17:23
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I don't really understand the question. The "two parameters" ("vectors") you wrote down there look the same to me. I'm not sure what you mean by "mapping" a two-component mixture to a mixture with one component(or rather "to pdf using...". Also it can't hurt to explain notation even though I can guess here what it means. But you don't want to rely on reader's guesswork. I suspect the result you are alluding to is the identifiability of Gaussian mixtures by the way. See Yakowitz and Spragins, Annals of Mathematical Statistics 1968. – Christian Hennig Dec 09 '23 at 17:08
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I mean suppose the parameters that I have taken are different then does that mean the GMM concerning them is different? – Andyale Dec 09 '23 at 17:15