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The method of Lagrange multipliers finds critical points (including maxima and minima) of a differentiable function subject to differentiable constraints.
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How to maximize the ELBO in coordinate ascent variational inference
From (22) to (23), it involves functional derivatives. The same question can be found here.
From (23) to (24), it just set (23) equals to 0 and get the corresponding $q_j(z_j)$. The constant in the ex …