# Does a Gaussian mixture model always imply a within-class multivariate normal probability distribution?

If I use a latent profile analysis (Gaussian Mixture Model) to model my observed multivariate probability distribution as a mixture (K-classes) of conditionally-independent normal pdfs, does this model imply that each within-class multivariate pdf is multivariate normal?

Can you show my why, or why not, this is true?

If it is true, can tests for multivariate normallity be used to help evaluate the model?

Except in degenerate cases, it won't be multivariate Normal. The easiest way to see that is in the $\mathbb R^2$ case. The probability density function of a GMM is a sum over Normal densities $p(x) = \sum_i \pi_i \cdot \mathcal N(x|\mu_i,\Sigma_i)$. Since a multivariate Normal distribution is unimodal with its only a single maximum at the mean, the mixture should also be unimodal. However, if $\mu_i\not=\mu_j$ and $\pi_i\not=0$, you get several "bumps" since there is a Gaussian sitting on each mean $\mu_i$. Therefore, the distribution you can't be multivariate Normal.
• Hmm, I try to guess what you mean. The sampling model for a mixture of Normals would be: randomly draw a normal distribution from the $K$ you got according to $\pi_1, ...,\pi_K$. Then draw a sample from that Normal distribution. This means that conditioned on the component your data point is multivariate Normal. Dec 28 '13 at 10:51