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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?

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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.

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The weights are the "responsibilities".

They are normalized to 1, which is motivated from the assumption that all the data has to be explained by the model, then using the law of total probability. So in that sense they are not just weights - they are the probabilities of the point being part of the cluster.

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