Feature request: parallel vb sampling from posterior

Summary:

I was wondering how hard would be to sample in parallel from the approximate posterior from variational bayes.

Description:

In my algorithm I explore the tails of many distributions to arbitrary precision to detect outliers, giving the user the possibility to set the desired % of false positives. https://github.com/stemangiola/ppcSeq

Generally the inference is really quick and consistent with HMC, but then the algorithm seats for long minutes sampling from the posterior. Since it is an approximate posterior would be easy to spread that sampling through multiple cores? That would be really great.

Reproducible Steps:

Please report steps to reproduce the issue. If it's not possible to reproduce, please include a description of how you discovered the issue.

If you have a reproducible example, please include it.

Current Output:

The current output. Knowing what is the current behavior is useful.

Expected Output:

Describe what you expect the output to be. Knowing the correct behavior is also very useful.

Additional Information:

Provide any additional information here.

Current Version:

v2.20.0

The issue names no files, tests, or entry points. Start by locating Stan's variational Bayes posterior-sampling path and assessing the requested multi-core behavior against the linked ppcSeq use case. Done would require an agreed scope, a reproducible workload, and validation of parallel sampling.