I have trained a model in GPflow and ultimately I would like to take this posterior distribution and use it as the prior in a new instance. I reviewed the docs and couldn't see anything. I did see the following link which sounded very good, but it appears to be a placeholder and is empty.
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There is a difference between "using the posterior as a new prior" and "adding more data to the model". The former is a bit more of a research question, see for example https://github.com/thangbui/streaming_sparse_gp (GPflow 1.0) and https://github.com/pmorenoz/RecyclableGP (PyTorch+GPy). The latter is easy to do (for GPflow's VGP model, see the implementation in the Trieste Bayesian Optimization library built on top of GPflow).

STJ
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