เปรียบเทียบวิธี

	การประมาณค่าแบบลาปลาซ	การแพร่กระจายความคาดหวัง (EP)	Markov Chain Monte Carlo (MCMC)
สาขาวิชา	เบย์	เบย์	เบย์
ตระกูล	Bayesian methods	Bayesian methods	Bayesian methods
ปีกำเนิด	1986	2001	—
ผู้ริเริ่ม	Pierre-Simon Laplace (1774); Bayesian formalisation: Tierney & Kadane (1986)	Thomas P. Minka	—
ประเภท	Analytical posterior approximation	Approximate inference algorithm	Posterior sampling algorithm
แหล่งต้นตำรับ	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
ชื่อเรียกอื่น	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)
ที่เกี่ยวข้อง	3	3	3
สรุป	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.