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.