I was trying to follow the proof given in the solutions of question b in fifth point of the Problem about Mixture of Linear regression. It is argued that the losses at the two points \(w^\star, \pi^\star\) and \(\hat{w}, \hat{\pi}\) are the same, hence the model is not identifiable. But what guarantees that the points are really different, i.e. that the components of \(w^\star\) and \(\pi^\star\) are not all the same?
Thank you to anyone who can help me figure this out.
Problem 3.5.b in Problem Set 6
I was trying to follow the proof given in the solutions of question b in fifth point of the Problem about Mixture of Linear regression. It is argued that the losses at the two points \(w^\star, \pi^\star\) and \(\hat{w}, \hat{\pi}\) are the same, hence the model is not identifiable. But what guarantees that the points are really different, i.e. that the components of \(w^\star\) and \(\pi^\star\) are not all the same?
Thank you to anyone who can help me figure this out.
Hi,
Thanks for your question. Yes, we need to assume that the components $w_1^\star$ and $w_2^\star$ are not equal.
In general, if $w_1^\star = w_2^\star$, then these two components are actually the same component and could be reduced to just one component.
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