I am a Japanese student studying machine learning and stuff like that. When I have studied Mean Field Approximation HERE, I got question.
In that post, equation (14) is as follows: $$ L[q_1,...,q_m] - \sum_{j=1}^{m}\lambda_j\int_{z_j}q_j(z_j)dz_j = 0 $$
But I think it should be like: $$ L[q_1,...,q_m] - \sum_{j=1}^{m}\lambda_j\left[\int_{z_j}q_j(z_j)dz_j - 1 \right] $$ Because we construct the Lagrangian with careful the following constraint: $$ \int_{z_j}q_j(z_j)dz_j = 1 ~~~~~\textrm{for all }j $$ It means that each $q_j$ is probabilistic density function.
Please help me.
Strictly speaking, you're right. If you want to construct the Lagrangian such that you can recover the constraints by setting the derivatives with respect to the Lagrangian multipliers to zero, you need to do it the way you say. However, since the constraints are already known, there's usually no need to recover them from the Lagrangian, and the Lagrangian is usually only used to derive the equations of motion by differentiating with respect to the $q_i$ (and $\dot q_i$). Since a constant in a constraint doesn't make a difference in the derivatives with respect to $q_i$, constants in constraints are often dropped in the Lagrangian for simplicity.