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	Multilevel Hamiltonian Monte Carlo	Multilevel Variational Inference
分野	ベイズ	ベイズ
系統	Bayesian methods	Bayesian methods
提唱年	2010s	2016
提唱者	Beskos, Jasra, Law, Tempone, Zhou (multilevel MCMC); Neal (HMC component)	Ranganath, Altosaar, Tran, Blei (hierarchical VI formalization, 2016); Blei et al. (VI framework, 2017)
種類	Bayesian computational sampler	approximate Bayesian inference
原典	Beskos, A., Jasra, A., Law, K., Tempone, R., & Zhou, Y. (2017). Multilevel sequential Monte Carlo samplers. Stochastic Processes and their Applications, 127(5), 1417–1440.	Blei, D. M., Kucukelbir, A., & McAuliffe, J. D. (2017). Variational inference: A review for statisticians. Journal of the American Statistical Association, 112(518), 859-877.
別名	Multilevel HMC, MLHMC, multilevel HMC sampler, multilevel leapfrog MCMC	hierarchical variational inference, multilevel VI, variational Bayes for multilevel models, MLVI
関連	5	4
概要	Multilevel Hamiltonian Monte Carlo (Multilevel HMC) combines the variance-reduction strategy of multilevel Monte Carlo with the efficient gradient-driven exploration of Hamiltonian Monte Carlo. By running coupled HMC chains at increasing levels of model fidelity or discretisation, it achieves accurate posterior estimates at a computational cost substantially lower than a single fine-level HMC chain.	Multilevel variational inference (MLVI) is a scalable approximate Bayesian method that fits hierarchical (multilevel) models by optimizing a variational approximation to the posterior, rather than drawing MCMC samples. It exploits the grouped structure of multilevel data — individuals nested within groups, groups nested within higher-level units — to derive efficient coordinate-wise updates, making Bayesian inference tractable for large clustered datasets.