i am trying to understand the derivation of the mean field equations - and my text books shows the following formulas (as part of a larger derivation).
$$L(q_j)=\sum_{x}\prod_{i}q_i(x_i)[\log p(x)-\sum_{k}\log q_k(x_k)]$$
$$ = \sum_{x_j}\sum_{x_{-j}}q_j(x_j)\prod_{i \neq j}q_i(x_i)[\log p(x)-\sum_{k}\log q_k(x_k)]$$
I have trouble understanding, in this specific case, why we can split the summation signal into $$\sum_{x_j}\sum_{x_{-j}}q_j(x_j)$$
In general - when confronted with similar issues in the future around tracing / understanding the individual algabraic steps of a derivation / proof, i would love to hear if there are tips or tricks on how to approach these types of challenges. Especially when it is harder to "insert numbers" to show equality between different steps of a derivation (as in this example) - i find it hard to tackle these types of questions myself. Advice much appreciated.