I have learnt about Variational Inference at this website.
In my understanding, it is a method that sampling random variables from an easy distribution $q$ which is most similar to an intractable distribution $p$.
In order to do that, i must firstly find the easy distribution $q$. To find $q$, my objective is optimize this:
$$\min \limits_q J(q) = \min \limits_q \sum\limits_x q(x)\log\dfrac{q(x)}{p(x)} \text{ with } q(x) = \prod\limits_{i=1}^nq_i(x_i)$$
But I don't know what exactly to do because the $p$ is unknown. Can I give some assumption about $p$ ?
For example, give a dataset of data points coming from an unknown distribution $p$:
$$\mathcal{D} = \{1, 2, 4\}$$
Can I assump that $p$ is Uniform Distribution with $p(1) = \dfrac{1}{3}$, $p(2) = \dfrac{1}{3}$, $p(4) = \dfrac{1}{3}$ ? Then if I assump that $q \sim \mathcal{N}(\mu, \sigma)$ so my objective is just find $\mu$ and $\sigma$ to minimize $J(q)$, like this:
$$\mu^*, \sigma^* = \arg \min \limits_{\mu, \sigma} \sum \limits_{x \in \mathcal{D}} \mathcal{N}(x;\mu, \sigma)\log\mathcal{N}(x;\mu, \sigma)$$
Can anyone tell me that if i'm correct or not ?