Võrdle meetodeid

	Laplace'i lähendus	Markovi ahel-Monte Carlo (MCMC)	Variational Inference
Valdkond	Bayesi meetodid	Bayesi meetodid	Bayesi meetodid
Perekond	Bayesian methods	Bayesian methods	Bayesian methods
Tekkeaasta	1986	—	1999
Looja	Pierre-Simon Laplace (1774); Bayesian formalisation: Tierney & Kadane (1986)	—	Jordan, Ghahramani, Jaakkola & Saul
Tüüp	Analytical posterior approximation	Posterior sampling algorithm	Approximate Bayesian inference
Algallikas	Tierney, L. & Kadane, J. B. (1986). Accurate approximations for posterior moments and marginal densities. Journal of the American Statistical Association, 81(393), 82–86.	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.
Rööpnimetused	Laplace's method, saddle-point approximation (Bayesian), second-order Gaussian approximation, LA	markov chain monte carlo, MCMC sampling, MCMC (Markov Zinciri Monte Carlo)	VI, variational Bayes, VB, mean-field variational inference
Seotud	3	3	4
Kokkuvõte	The Laplace approximation is a classical analytic technique that replaces an intractable posterior distribution with a multivariate Gaussian centred at the posterior mode, using the curvature of the log-posterior at that mode to set the covariance. Formalised for Bayesian statistics by Tierney and Kadane (1986) in their landmark Journal of the American Statistical Association paper, it provides a fast, deterministic alternative to Markov chain Monte Carlo and forms the mathematical core of Integrated Nested Laplace Approximations (INLA).	Markov Chain Monte Carlo (MCMC) is a family of computational algorithms for sampling from complex probability distributions, most commonly the posterior distributions that arise in Bayesian inference. Rather than computing posteriors analytically — which is rarely possible for realistic models — MCMC constructs a Markov chain whose stationary distribution is the target posterior and draws dependent samples from it, enabling full probabilistic inference for virtually any model.	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.