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	Inferenza Bayesiana Multilivello	Inferenza Variazionale
Campo	Bayesiano	Bayesiano
Famiglia	Bayesian methods	Bayesian methods
Anno di origine	1980s–2000s	1999
Ideatore	Gelman, Hill, Raudenbush, Bryk	Jordan, Ghahramani, Jaakkola & Saul
Tipo	Bayesian hierarchical model	Approximate Bayesian inference
Fonte seminale	Gelman, A., & Hill, J. (2007). Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press. ISBN: 978-0521686891	Jordan, M. I., Ghahramani, Z., Jaakkola, T. S., & Saul, L. K. (1999). An introduction to variational methods for graphical models. Machine Learning, 37(2), 183–233.
Alias	Bayesian multilevel model, Bayesian hierarchical model, Bayesian mixed-effects model, Bayesian random-effects model	VI, variational Bayes, VB, mean-field variational inference
Correlati	6	4
Sintesi	Multilevel Bayesian inference combines Bayesian probability with hierarchical data structures, treating group-level parameters as drawn from a common population distribution. It simultaneously estimates unit-level effects and the hyperparameters governing their variation, propagating full uncertainty through every level of the hierarchy via posterior sampling.	Variational inference (VI) is a family of techniques that turn Bayesian posterior computation into an optimisation problem. Instead of drawing samples from the exact posterior — as Markov chain Monte Carlo does — VI posits a simpler, tractable family of distributions and finds the member of that family closest to the true posterior by maximising the evidence lower bound (ELBO). Introduced in its modern graphical-model form by Jordan, Ghahramani, Jaakkola and Saul (1999) and given a comprehensive statistical treatment by Blei, Kucukelbir and McAuliffe (2017), VI is now the standard scalable inference engine in probabilistic machine learning.