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I would like to know the difference between a multivariate Gaussian distribution and multivariate Gaussian mixture model. Can someone provide an intuitive and/or detailed explanation?

Thanks.

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  • $\begingroup$ See, for example, this question for a multivariate Gaussian mixture model. $\endgroup$ – Dilip Sarwate Sep 3 '15 at 21:30
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A (non-degenerate) multivariate Gaussian density function has a specific form $$f(\mathbf x) = \frac{1}{(2\pi)^{n/2}\sqrt{\det(\Sigma)}} \exp\left(-\frac{1}{2}\left(\mathbf x - \mathbf m\right)\Sigma^{-1} (\mathbf x - \mathbf m)^T \right).\tag{1}$$

A multivariate Gaussian mixture model is a weighted sum of densities such as $(1)$:

$$f(\mathbf x) = \sum_{i=1}^N p_i\cdot\frac{1}{(2\pi)^{n/2}\sqrt{\det(\Sigma_i)}} \exp\left(-\frac{1}{2}\left(\mathbf x - \mathbf m_i\right)\Sigma^{-1}_i (\mathbf x - \mathbf m_i)^T \right)$$ where the $p_i$ are positive numbers such that $\sum_{i=1}^N p_i = 1$. Effectively, we have $N$ mutually exclusive events $A_i$ such that the conditional density given that $A_i$ occurred is $$\frac{1}{(2\pi)^{n/2}\sqrt{\det(\Sigma_i)}} \exp\left(-\frac{1}{2}\left(\mathbf x - \mathbf m_i\right)\Sigma^{-1}_i (\mathbf x - \mathbf m_i)^T \right)$$ and $(2)$ is just the law-of-total-probability expression for the unconditional density.

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  • $\begingroup$ I would appreciate it if you can have a look at my other question stats.stackexchange.com/questions/171307/… $\endgroup$ – notArefill Sep 6 '15 at 10:38
  • $\begingroup$ This can be done completely generally by using distribution or characteristic functions instead of densities. That allows for one or more of the Gaussians being mixed to be degenerate. $\endgroup$ – Mark L. Stone Feb 18 '17 at 23:40

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