Hi @drezap! The first question has been an interest of mine for awhile, but I’m not aware of an absolutely robust way to do this with Stan besides simply refitting the model with all the data.
Here’s a thread with more discussion and links: Updating model based on new data via PSIS
The method outlined has worked well for me in the past and can limit the number of refits in a context where you have people uploading data all the time and hoping for fresh estimation. There is literature on Sequential MCMC, but I haven’t dived deeply into it since I fixed my immediate issues with PSIS. If anyone has plans to add a sequential sampler algorithm that works with existing Stan models, I would like to digitally hug that person.