Using MCMC based model in closed loop

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See @junpenglao’s relevant commentary here:

Updating multivariate priors

Last time the similar question comes up there is no good solution: Updating Priors with Posterior for Vector-Valued Distributions, Updating priors for Gaussian Mixture Model I think one thing worth to explore is to use some kind of approximation (MvNormal, for example). It would be an extension of: Performance speedup for updating posterior with new data and code here: https://github.com/junpenglao/Planet_Sakaar_Data_Science/blob/master/PyMC3QnA/Update%20prior%20with%20Interpolation.ipynb (see …

or here:

The limits of using the traceplot to update priors?

It was recently debated on Twitter, that updating prior (beside in most cases un-interesting conjugate models) under Bayesian framework is actually not trivial. Personally, approximating the posterior with some heavy tail distribution (say a t distribution) is what I would do. Otherwise, Bayesian filter (Kalman filter, particle filter) sounds like a promising framework that I would love to explore more.

The fundamental issue is that MCMC doesn’t produce posterior distributions, but instead samples from posterior distributions. These can’t be directly “integrated out” as priors for new data – but approximate posteriors can be fit to the last model’s posterior samples, and used as the next model’s priors.