# Difference between multivariate Gaussian distribution and multivariate Gaussian mixture model

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.

• See, for example, this question for a multivariate Gaussian mixture model. – Dilip Sarwate Sep 3 '15 at 21:30

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.