Improve docs for variational inference

Summary:

I would like to use the ADVI implemented in Stan but can't find helpful docs.

Description:

My model needs to fit a distribution based on KL divergence. This is possible with variational inference and as far as I understand, Stan has support for VI with it's ADVI functionality.

However, the user guide and reference manual contain only very superficial information on how VI is implemented internally. They don't provide instructions on how to use it.

It would be very helpful for me to see a working toy example. Let's say I have a Poisson distribution with gamma=10, and would like to represent it with a Gaussian which minimizes KL divergence.

Additional Information:

I've asked an equivalent question on SO: https://stackoverflow.com/questions/59276428/stan-how-to-use-variational-inference-to-fit-a-distribution

Current Version:

v2.21.0

Start by reviewing the Stan user guide and reference manual sections covering ADVI, then compare them with the linked Stack Overflow question. Add a working toy example using a Poisson distribution with gamma=10 and a Gaussian approximation minimizing KL divergence, with instructions sufficient for users to run it.