Sammenlign metoder

	Laplace Approximation	Forventningsudbredelse (EP)	Markov Chain Monte Carlo (MCMC)
Fagområde	Bayesiansk	Bayesiansk	Bayesiansk
Familie	Bayesian methods	Bayesian methods	Bayesian methods
Oprindelsesår	1986	2001	—
Ophavsperson	Pierre-Simon Laplace (1774); Bayesian formalisation: Tierney & Kadane (1986)	Thomas P. Minka	—
Type	Analytical posterior approximation	Approximate inference algorithm	Posterior sampling algorithm
Oprindelig kilde	Tierney, L. & Kadane, J. B. (1986). Accurate approximations for posterior moments and marginal densities. Journal of the American Statistical Association, 81(393), 82–86.	Minka, T. P. (2001). Expectation propagation for approximate Bayesian inference. In Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence (UAI-01), pp. 362–369. Morgan Kaufmann.	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
Aliasser	Laplace's method, saddle-point approximation (Bayesian), second-order Gaussian approximation, LA	EP, expectation propagation, EP algorithm, assumed-density filtering generalisation	markov chain monte carlo, MCMC sampling, MCMC (Markov Zinciri Monte Carlo)
Relaterede	3	3	3
Resumé	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).	Expectation Propagation (EP) is a deterministic message-passing algorithm for approximate posterior inference in Bayesian models, introduced by Thomas P. Minka at UAI 2001. It iteratively refines a set of local approximate factors — each drawn from the exponential family — so that their product closely matches the true intractable posterior, achieving higher accuracy than mean-field variational inference on many probabilistic machine learning tasks.	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.