Hello! You must be searching for what people are referring to as “Recursive Bayes”. This will also lead you to this discussion. It’s definitely worth also mentioning the work from the github organization biaslab RxInfer.jl, which I think it’s not in the first discussion.
WIth traditional MCMC you will have to resample all your model data with every new measurement you receive. All different techniques I think them more or less as an efficient approximation of that.
If you are interested in a discount factor, i.e. the parameters of interests are evolving with time, you definitely need to move away from traditional MCMC. Sequential Monte Carlo (SMC), for example, targets this. (there is this tool SMC.jl although I haven’t tried that out)
With respect to time series, people have been using a lot Kalman filters. An alternative, if you find that restricting, are Gaussian processes which is a very powerful tool. Both are considered Bayesian and provide nice uncertainty estimations. However the modeling decisions are fixed to (MV-)normal, but that shouldn’t scare you for the second case too much because they manage to get very flexible.