To add a word of caution:
I’d only work with online learing like that if I really had to. If you can just rerun the whole model, even if it takes a bit more time, just do that. Especially if you keep updating posteriors in a long chain, I’m not sure the final result would be close to the actual posterior, even if all individual approximations are kind of ok. With each step you keep introducing an error, and it is not obvious how those errors interact. Small individual errors might add up to something gigantic. Or at least I don’t know of any results that would tell me that they don’t. 