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	Approssimazione di Laplace	Catena di Markov Monte Carlo (MCMC)
Campo	Bayesiano	Bayesiano
Famiglia	Bayesian methods	Bayesian methods
Anno di origine	1986	—
Ideatore	Pierre-Simon Laplace (1774); Bayesian formalisation: Tierney & Kadane (1986)	—
Tipo	Analytical posterior approximation	Posterior sampling algorithm
Fonte seminale	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
Alias	Laplace's method, saddle-point approximation (Bayesian), second-order Gaussian approximation, LA	markov chain monte carlo, MCMC sampling, MCMC (Markov Zinciri Monte Carlo)
Correlati	3	3
Sintesi	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.