# GMM weighting change during marginalization

Suppose one has a multivariate Gaussian Mixture Model:

$$\text{pdf}(\vec{x}) = \sum_{i=1}^N w_i \mathcal{N}(\mu^{(i)}, \Sigma^{(i)})$$

Suppose $\vec{x} = \{\vec{a},\vec{b}\}$ and we marginalize out $\vec{b}$. For a single Gaussian component we would get:

$$\mathcal{N}\left(\mu_a \mid \Sigma_a \right)$$

where $\mu_a$ is the corresponding part of $\mu = [\mu_a, \mu_b]'$ and $\Sigma = \begin{bmatrix}\Sigma_{a} & \Sigma_{ab} \\ \Sigma_{ba} & \Sigma_{b} \end{bmatrix}$.

For the marginalized mixture, does one need to update the weights?:

$$\text{pdf}(\vec{a}) = \sum_{i=1}^N w'_i \mathcal{N}(\mu_a^{(i)}, \Sigma^{(i)}_a)$$

In other words, is $w_i' = w_i$ for all of the components?