Why do we use the mean-field approximation for variational Bayes? I often see the mean-field approximation for Variational Bayes.  I understand the independence assumption: what I don't understand is why we make that assumption.  How does it help us?
 A: To flesh the comment I write this answer. 

Why do we use the mean-field approximation for variational Bayes?

Firstly we employ the variational Bayes to obtain the posterior distribution. But the posterior distribution $posterior = p(\theta|y) = \frac{likelihood * prior}{evidence} = \frac{p(y|\theta)*p(\theta)}{p(y)}$ is intractable for the evidence(also called marginal likelihood $\int p(y|\theta)d(\theta)$) is intractable(you cannot integrate over all possible $\theta$). Then to compute the posterior, we need to make the intractable problem tractable. The normal deterministic method is to approximate the marginal likelihood using a simpler distribution $q(\theta)$ which can be factorized to an integration of a sum of terms in the log joint $q(\theta)$ which is called mean field approximation. Absolutely this cannot be the only method you should use, since there are two other variational inference approaches: expectation propagation (Minka et al.) and variational 2nd-order Taylor approximation (Xing). 

why we make that assumption. How does it help us?

Let $q$ be a member of a parametric family of density functions over a certain space(there are no dependencies between latent variables or a generalized version of this), then we can deform the marginal likelihood as: 
$$\eqalign{
&\log p(y) &= \log p(y) \int q(\theta) d(\theta) = \int q(\theta) \log p(y) d\theta \\
&           &= \int q(\theta) \log \{\frac{p(y, \theta)/q(\theta)}{p(\theta|y)/q(\theta)}\}d\theta\\
&           &= \int q(\theta) log \{\frac{p(y, \theta)}{q(\theta)}\}d\theta + \int q(\theta) \log \{\frac{q(\theta)}{p(\theta|y)}\}d\theta  \hspace{2cm}  (1)\\
&           &\ge \int q(\theta) \log \{\frac{p(y,\theta)}{q(\theta)}\}\hspace{7.3cm}  (2)
}$$
Let's denote the two terms in (1) respectively: 
$\underline p(y;q) = exp\int q(\theta) log \{\frac{p(y, \theta)}{q(\theta)}\}d\theta$, which is the evidence lower bound, exp of (2)
$p_{KL} = exp\int q(\theta) \log \{\frac{q(\theta)}{p(\theta|y)}\}d\theta$, which is the exp of Kullback–Leibler divergence between $q$ and $p(·|y)$
Since the lower bound is more tractable than the evidence, we try to minimize the $p_{KL}$(which >= 0) by maximizing the lower bound. The lager the lower bound $\underline p(y;q)$ is, the more similar the $q(\theta)$ is to $p(\theta|y)$, and more accurately we get $\log p(y)$ and hence $p(y)$.
According to the mean field variational approximation, $q(\theta)$ factorizes into $\prod_{i=1}^M q_i(\theta_i)$, for some partition ${\theta_1,..., \theta_M}$ of $\theta$. We refer to each $q_i(\theta_i)$ as a local variational approximation. 
If we keep $\mathcal{L}=\int q(\theta) \log \{\frac{q(\theta)}{p(\theta|y)}\}d\theta$, then $\mathcal{L}=\int \prod_{i=1}^M q_i(\theta_i)\{\log p(y, \theta)-\sum_{i=1}^M \log q_i(\theta_i)\}d\theta_1 \ldots d\theta_M$. 
The do the coordinate ascend for each latent variable and keep all others fixed to compute $argmax_{q(\theta_i)}\mathcal{L}$. 
May this be of help.  
references:
1) https://www.cs.cmu.edu/~epxing/Class/10708-17/notes-17/10708-scribe-lecture13.pdf
2) http://www.maths.usyd.edu.au/u/jormerod/JTOpapers/Ormerod10.pdf
A: Please read my recent paper titled "Gibbs sampler and coordinate ascent variational inference: A set-theoretical review". Visit link https://www.tandfonline.com/doi/full/10.1080/03610926.2021.1921214
My short answer is, by putting such a marginal independence via mean field variational assumption, we can use practical algorithm called coordinate ascent variational inference.
