# What is the interpretation of the weights in the GMM?

GMM is $$p(x|\theta) = w_1 \mathcal{N}(x|\mu_1,\,\sigma_1^{2})\ + w_2 \mathcal{N}(x|\mu_2,\,\sigma_2^{2}) + w_3 \mathcal{N}(x|\mu_3,\,\sigma_3^{2})\,$$
What is the interpretation of the weights here?
Do they mean "How confident the model is about a particular gaussian(ex, $$\mathcal{N}(x|\mu_2,\,\sigma_2^{2})\,$$ )"?
And Why is the model confident about specific gaussian?

Here is one quite intuitive interpretation. These weights are non-negative and sum up to $$1$$. So, it's like saying we have a RV that comes from $$\mathcal{N_1}$$ with probability $$w_1$$, from $$\mathcal{N_2}$$ with probability $$w_2$$, and from $$\mathcal{N_3}$$ with probability $$w_3$$. By the way, this way of thinking is not a special property of GMMs, but several kinds of mixtures.