Examples of simulations in different fields (computer science, statistics, statistical mechanics, operations research, financial mathematics). Generation of uniform random variables. Generation of random variables with arbitrary distributions (quantile transform, accept-reject, importance sampling), simulation of Gaussian processes and diffusions. The precision of simulations, methods for variance reduction. Introduction to Markov chains and Markov chain Monte Carlo (Metropolis-Hastings, Gibbs sampler, Hamiltonian Monte Carlo, reversible jump MCMC).
August 30th, 2016:
Beginning of lecture: Tuesday, 20/09/2016. (Exercises start on 27/09/2016 at 16:15).
September 19th, 2016:
Lecture notes and slides for the first lecture are online.
September 23rd, 2016:
R-code for the example shown in class and slides for next lecture are online.
September 29th, 2016:
New versions of slides and lecture notes are now online.
October 10th, 2016:
Slides for lecture 4, series 2 and solution to series 1 are now online.
October 14th, 2016:
Slides for lecture 5 online.
October 19th, 2016:
Series 3 online.
October 24th, 2016:
Slides for lecture 6 online.
October 26th, 2016:
R-code for lecture 5 and series 4 online.
November 2nd, 2016:
Updated version of slides for lectures 2 and 3, and of the organisation sheet.
November 28th, 2016:
Series 6 online and new version of the organisation sheet.
November 29th, 2016:
Slides for lecture 11 online. New version of the script (reversible jump MCMC updated).
- Lecture Notes (subject to change during the semester)
- Slides used in the course will be available in due time. Exercises as well as solutions will also be provided.
- More information available in Organization sheet
|1||20/09||L-L-L||Introduction, distribution of estimators: trimmed mean, Bootstrap, Simulation in Bayesian statistics|
|2||27/09||L-L-E||Simulation in Statistical Mechanics and Operations Research|
|3||04/10||L-L-L||Simulation in Financial Mathematics, other applications, accuracy of MC methods|
|4||11/10||L-L-E||Generating uniform random variables|
|5||18/10||L-L-L||Quantile transformation, rejection sampling, relations between distributions, permutations|
|6||25/10||L-L-E||Importance sampling , simulation of stochastic differential equations|
|7||01/11||L-L-E||Variance reduction: antithetic variables, control variates, importance sampling|
|8||08/11||L-L-L||Quasi Monte-Carlo, introduction MCMC, basics of Markov chains|
|10||22/11||L-L-L||Independence sampler, random walk Metropolis, componentwise modification, Gibbs sampler|
|11||29/11||L-L-E||Hamiltonian MCMC, Metropolis-Hastings for variable dimension models|
|12||06/12||L-L-L||Metropolis-Hastings for variable dimension models (cont.), reversible jump MCMC|
|13||13/12||L-L-L||Accuracy of MCMC approximations|
|14||20/12||L-L-L||reserve / buffer|
Exercises will be held roughly bi-weekly, but on an irregular schedule. The statistical software package R is recommended for solving the exercises. The exercises will take place at the specified date from 16.15-17.00 in the same place as the lectures.
Series and solutions
Submitting solutions to the exercise is not compulsory except for some phd students. You can hand in your solution during the class or by email until the designated date and will receive some feedback in due time.
|27/09||Distribution of estimators||Series 1||04/10|
|11/10||Bayes and Ising||Series 2||18/10|
|25/10||Generation of random variables||Series 3||01/11|
|01/11||Importance sampling||Series 4||08/11|
|15/11||Control variates and Antithetic variables||Series 5||22/11|
|29/11||MCMC: Gibbs sampler, random walk Metropolis algorithm||Series 6||06/12|
|13/12||Hamiltonian MC||Series 7 (schools.csv)||20/12|
- G. S. Fishman, A First Course in Monte Carlo. Thomson Brooks/Cole, 2006.
- S. M. Ross. Simulation. Academic Press, 2012 (5th edition).
Books on a similar level as the course
- Ch. Robert and G. Casella. Introducing Monte Carlo Methods with R. Springer Science & Business Media, 2009.
- Ch. Robert, G. Casella. Monte Carlo Statistical Methods. Springer 2004 (2nd edition).
- S. Asmussen, P. W. Glynn, Stochastic Simulation, Algorithms and Analysis. Springer, 2007.
- P. Glasserman, Monte Carlo Methods in Financial Engineering. Springer 2004.
- B. D. Ripley. Stochastic Simulation. Wiley, 1987.
- W. R. Gilks, S. Richardson, D. J. Spiegelhalter. Markov Chain Monte Carlo in Practice. Chapman & Hall, 1996.
- S. Brooks, A. Gelman, G. Jones, and X.L. Meng, eds. Handbook of Markov Chain Monte Carlo. CRC press, 2011.