Weekly syllabus
| week | topics |
|---|---|
| 1 | Pseudo random number generators, Mapping methods for direct sampling, Rejection sampling, Multivariate normal sampling via Cholesky factorization, Monte Carlo estimation and error bars. |
| 2 | Markov chain Monte Carlo (MCMC): Definition of Markov chains and invariant distributions, The Perron Frobenius theorem, Detailed balance and Metropolis sampling, Partial resampling. |
| 3 | Reducing the static variance: Control variates, Systematic sampling (stratafied sampling, Latin hypercube), Rao Blackwellization, Importance sampling for variance reduction, Other applications of importance sampling. |
| 4 | Testing MCMC codes for correctness, histograms and other unit tests. Error bars for MCMC, the auto-covariance function and the auto-correlation time. |
| 5 | Improved MCMC samplers. Adaptive samplers. Affine invariant ensemble samplers. Metropolized dynamics and Hamiltonian samplers. Methods that use derivative information. |
| 6 | Optimizing within Monte Carlo. Monte Carlo sensitivity analysis, gradient estimation. Robbins Munro -- stochastic gradient descent. Sample average approximation (SAA). Affine invariant descent methods, Robbins and Lai. |
| 7 | Rare event simulation and large deviation theory. Some examples -- Cramer Rao problem, exit time problems, the critical path. |
| 8 | Rare events via importance sampling. |
| 9 | Umbrella sampling, simulated tempering, parallel tempering, the multi-histogram method. |
| 10 | Rare event sampling without large deviations, the bifurcation method. Sampling with inequality constraints. |
| 11 | Model selection, estimating the evidence integral, nested sampling. |
| 12 | Analysis of MCMC methods, I. The role of the spectral gap with and without detailed balance. Linear Gaussian examples. |
| 13 | Analysis of MCMC methods, II, using Poincare and Sobolev type inequalities. |
| 14 | Analysis of MCMC methods, III, the work of the Lovasz school. |