Monte Carlo Methods

First Conference on Computational Interdisciplinary Sciences
Instituto Nacional de Pesquisas Espaciais,
August 23rd to 27th 2010
Sao Jose dos Campos, Brazil

Tutorial material (PDF FILE)

Tutorial outline

• Part 1: Monte Carlo Methods
• Part 2: Markov Chain Monte Carlo Methods

R code

• Example 0. Approximating pi.
• Example i. Monte Carlo integration.
• Example ii. Monte Carlo integration via importance sampling.
• Example iii. Rejection method and sampling importance resampling.
• Example iv. Rejection method, SIR and Metropolis-Hastings algorithms.
• Example v. Rejection method, SIR and Metropolis-Hastings algorithms.
• Example vi. Random walk MH and independent MH algorithms.
• Example vii. Simulated annealing.
• Example viii. Gibbs sampler.
• Raoblackwellization.

Other useful links

• MCMC: Stochastic Simulation for Bayesian Inference
• The R project
• Markov Chain Monte Carlo Preprint

Classical Monte Carlo papers

• Metropolis and Ulam (1949) The Monte Carlo method. JASA, 44, 335-341.
• Metropolis, Rosenbluth, Rosenbluth, Teller and Teller (1953) Equation of state calculations by fast computing machines. Journal of Chemical Physics, 21, 2087-1092.
• Hastings (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57, 97-109.
• Peskun (1973) Optimum Monte Carlo sampling using Markov chains. Biometrika, 60, 607-612.
• Besag (1974) Spatial Interaction and the Statistical Analysis of Lattice Systems. JRSS-B, 36, 192-236.
• Kirkpatrick, Gelatt and Vecchi (1983) Optimization by Simulated Annealing. Science, 220 (4598), 671-680.
• Geman and Geman (1984) Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Analysis and Machine Intelligence, 6, 721-741.
• Pearl (1987) Evidential reasoning using stochastic simulation of causal models. Artificial intelligence, 32, 245-257.
• Tanner and Wong (1987) The Calculation of Posterior Distributions by Data Augmentation. JASA, 82, 528-540.
• Geweke (1989) Bayesian Inference in Econometric Models Using Monte Carlo Integration. Econometrica, 57, 1317-1339.
• Gelfand and Smith (1990) Sampling-Based Approaches to Calculating Marginal Densities. JASA, 85, 398-409.
• Casella and George (1992) Explaining the Gibbs Sampler. The American Statistician, 46, 167-174.
• Gilks and Wild (1992) Adaptive Rejection Sampling for Gibbs Sampling. Applied Statistics, 41, 337-348.
• Smith and Gelfand (1992) Bayesian Statistics without Tears: A Sampling-Resampling Perspective. The American Statistician, 46, 84-88.
• Chib and Greenberg (1995) Understanding the Metropolis-Hastings algorithm. The American Statistician, 49, 327-335.