Inference methods to constraint posterior moments?

Suppose I want to learn a posterior that stays close to a safe value, say

E_q(z)[||z-z_safe||^2] = r

for some soft trust radius r. This isn’t quite Bayesian because I’m constraining the learned posterior q(z). What is the state of the art in performing inference in these semi-Bayesian settings?

no idea but i think this is usually called “posterior regularization” so i’d search for that keyword e.g. here

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