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	Hierarchical Markov Chain Monte Carlo	Inferenza Variazionale
Campo	Bayesiano	Bayesiano
Famiglia	Bayesian methods	Bayesian methods
Anno di origine	1990	1999
Ideatore	Gelfand & Smith (1990), building on Geman & Geman (1984)	Jordan, Ghahramani, Jaakkola & Saul
Tipo	Bayesian computational sampler	Approximate Bayesian inference
Fonte seminale	Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A. & Rubin, D. B. (2013). Bayesian Data Analysis (3rd ed.). CRC Press. ISBN: 978-1439840955	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	hierarchical MCMC, MCMC for multilevel models, Bayesian hierarchical MCMC, multilevel MCMC sampling	VI, variational Bayes, VB, mean-field variational inference
Correlati	6	4
Sintesi	Hierarchical Markov chain Monte Carlo applies MCMC sampling to hierarchical Bayesian models, jointly drawing from the posterior over both observation-level parameters and the hyperparameters that govern them. This allows principled uncertainty propagation across all levels of a multilevel structure, from individuals to groups to population, using algorithms such as Gibbs sampling, Metropolis-Hastings, or Hamiltonian Monte Carlo.	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.