Statistical Computing - Monte Carlo Methods
CPSC 535D
Course schedule
Two lectures per week (Tuesday and Thursday from 3.30 to 5.00)
in ICCS 238.
If you want to arrange a meeting, just send me an email at the
following address: arnaud at cs dot ubc dot ca
Announcements
- NO LECTURE ON
THURSDAY 5th APRIL
- Project Presentation on Thursday 12th April: 15 minutes per
student.
- Third assignement posted.
Projects
Handouts
- Lecture 1 (Tuesday 16th
January): Introduction ps
Additional
reading: Chapter 1 of the Bayesian Choice by C.P. Robert or the nice
handouts h1
h2
by B. Vidakovic
- Lecture 2 (Thursday 18th
January): Review of Basic Principles of Statistics ps
- Lecture 3 (Tuesday 23rd January): Basics of
Bayesian Statistics ps
Additional reading: Chapter 1 of the Bayesian Choice by C.P. Robert or h3
by B. Vidakovic
- Lecture 4 (Thursday 25th January): More
Bayesian Statistics (Example, Testing hypothesis, Bayes factors) ps
Additional reading:
- h6
by B. Vidakovic
- R. Kass and A. Raftery, Bayes
Factors, JASA, 1995 paper
- R. Kass, Bayes Factors in
Practice, The Statistician, 1992 here
- M. Lavine and M.J. Schervish,
Bayes Factors: What they are and what they are not, The American
Statistician, 1999 here
- Lecture 5 (Tuesday 30th January): Bayesian model selection ps
- Chapter 7 of the Bayesian Choice by
C.P. Robert
- J. Hoeting, D. Madigan, A. Raftery and
C. Volinsky, Bayesian model averaging: A tutorial, Statistical Science,
1999 here
- A. Raftery, D. Madigan and J. Hoeting,
Bayesian model averaging for linear regression models, JASA, 1997 here
- Lecture 6 (Thursday 1st February):
Introduction to Monte Carlo ps
Additional reading:
- Section 3.1 and 3.2 of Monte
Carlo Statistical Methods.
- Lecture 7 (Tuesday 6th
February): Classical Methods (inverse
transform, accept/reject) pdf
Additional reading:
- Chapter 2 of Monte
Carlo Statistical Methods.
- Scale mixture of
Gaussians, JRSS B, 1974 here:
very useful representation of non-Gaussian distributions as
infinite mixture of Gaussians
- W. Gilks and P. Wild, Adaptive
rejection sampling for Gibbs sampling, Applied Statistics, 1992 here
- B.D. Flury,
Rejection sampling made easy, SIAM Review, 1990 here
More advanced
- A. Peterson
and R. Kronmal, On mixture methods for the computer generation of
random variables, The American Statistician, 1982 here
- J. Halton,
Reject the rejection technique, J. Scientific Computing, 1992. (by the
way please don't reject it)
- A. Beskos and G.
Roberts, Exact simulation of diffusions, Annals of Applied Proba, 2005.
here
Check
Proposition 1 and its proof for a very clever and useful remark about
rejection sampling.
-
Lecture 8 (Thursday 8th February): Importance
Sampling pdf
Additional
reading:
- Chapter 3 of
Monte Carlo Statistical Methods.
-
Y. Chen, Another
look at rejection sampling through importance sampling, Stat. Proba.
Lett., 2005 here
- J. Geweke,
Bayesian inference in econometric models using Monte Carlo integration,
Econometrica, 1989 here
- H. Van Dijk, J.
Hop, A. Louter, An Algorithm for the Computation of Posterior Moments
and Densities Using Simple Importance Sampling, The Statistician, 1987 here
Optional reading
- A. Owen and
Y. Zhou, Safe and effective importance sampling, JASA, 2000 here
- Lecture 9 (Tuesday 13th
February): More Importance Sampling, Sampling Importance Resampling,
Sequential Importance Sampling pdf
- Chapter 11 of Robert
& Casella
- A.F.M. Smith, A.E.
Gelfand, Bayesian Statistics without Tears: A Sampling Importance
Resampling Perspective, The American Statistician, 1992 Pdf
file here
- A. Doucet, N. De Freitas
and N.J. Gordon, An introduction to Sequential Monte Carlo, SMC in
Practice, 2001 Ps file here
- A. Kong, J.S. Liu
and
W.H. Wong, Sequential Imputations and Bayesian Missing Data Problems,
JASA, 1994
Pdf
file here
- J.S. Liu and R.
Chen, Sequential Monte Carlo methods for dynamic systems, JASA, 1998 Pdf
file here
- Lecture 10 (Thursday
15th February) More Sequential Importance Sampling, Sequential
Monte Carlo extended version of slides9 pdf
- Lecture 11 (Tuesday 27th
February): We will fnish SMC methods and will start Gibbs sampling pdf
Matlab code to generate fractal
image code
Additional reading:
- D.
Mackay, Introduction to Monte Carlo methods, here
- R. Neal,
Probabilistic Inference Using Markov Chain Monte Carlo Methods,
Technical report, 1993 here
- C.
Andrieu, A. Doucet, N. De Freitas and M. Jordan, Markov chain Monte
Carlo for Machine Learning, Machine Learning, 2003 here
- S. Brooks,
Markov chain Monte Carlo Methods and Its Application, The Statistician,
1998 here
- G. Casella and
E.I. George, Explaining the Gibbs sampler. The American Statistician,
1992 here
- S. Chib and
E. Greenberg, Understanding the Metropolis-Hastings algorithm, The
American Statistician, 1995 here
- Lecture 12 (Thursday
1st March): More Gibbs sampling
extended version of previous slides here
- Lecture 13 (Tuesday 6th
March): More Gibbs sampling for Hidden
Markov models here
- Lecture 14 (Tuesday 13th
March): Metropolis-Hastings algorithm here
- Lecture 15 (Thursday
15th March): More Metropolis-Hastings here
- Lecture 16 (Tuesday 20th
March): We will finish Metropolis-Hastings and start Transdimensional
algorithms here
-
Chapter 11
of Robert & Casella
- P.J. Green,
Transdimensional Markov chain Monte Carlo, Highly Structured Stochastic
Systems, OUP, 2003 Pdf
file here
- S. Sisson, Trans-dimensional Markov chains: A decade of progress and
future perspectives., JASA, 2005 Pdf
file here
- Lecture 17 (Thursday
22nd
March): More transdimensional
algorithms here
- Lecture 18 (Tuesday
27th
March): Computing normalizing constants here
- Gelman and Meng, Simulating normalizing
constants: from importance
sampling to bridge sampling to path sampling, Statistical
Science, 1998. here
- Neal, Annealed importance sampling, Stat.
Computing, 2001. here
- Lecture 19 (Thursday
29th March): Advanced methods here
- Chapter 8
of Robert & Casella
- C. Andrieu,
L. Breyer & A. Doucet, Convergence of Simulated Annealing using
Foster-Lyapunov
Criteria, Journal Applied Probability, 2001. Pdf
file here
- Paul
Damien, Jon Wakefield, Stephen Walker, Gibbs Sampling for Bayesian
Non-Conjugate and Hierarchical Models by Using Auxiliary Variables,
JRSS B, 1999 Pdf
file here
- R. Neal,
Sampling from Multimodal Distributions using Tempered Transitions,
Statistics and Computing, 1996 Pdf file
here
- C. Geyer & E. Thompson, Annealing Markov Chain Monte Carlo with
Applications to Ancestral Inference, JASA, 1995 Pdf
file here
- Lecture 20 (Tuesday 3rd
April): Dirichlet processes here
- M.I. Jordan
Tutorial on Nonparametric Bayes, NIPS 2005 Ps file
here
- R. Neal MCMC
for
Dirichlet Process Mixture Models, JCGS, 2000. Pdf
file here
- Lecture 21 (Tuesday 10th
April): Introduction to Markov chains theory here
Assignements
Volatility data
Probabilistic Nearest Neighbour
paper here
Course contents
- Introduction to Bayesian
Statistics.
- Probability as measure of uncertainty.
- Posterior distribution as compromise between data and prior
information.
- Prior distributions: conjugacy and noninformative priors.
- Bayes factors.
- Large sample inference.
- Introduction to Monte Carlo Methods
- Limitations of deterministic numerical methods.
- Monte Carlo integration and Non-Uniform random variable
generation (inverse method, accept/reject)
- Importance sampling.
- Variance reduction techniques (Rao-Blackwellisation, antithetic
variables).
- Markov Chain Monte Carlo
Methods - Basics
- Introduction to general state-space Markov chain theory.
- Metropolis-Hastings algorithm.
- Gibbs sampler.
- Hybrid algorithms.
- Case studies: Capture-Recapture experiments, Regression and
Variable selection, Generalised linear models, Models for Robust
inference
- Case studies: Mixture models and Hidden Markov models,
Nonparametric Bayes, Markov random fields.
- Markov Chain Monte Carlo
Methods - Advanced Topics
- Variable dimension algorithms (Reversible jump MCMC).
- Simulated tempering.
- Monte Carlo optimization (MCEM, simulated annealing).
- Perfect simulation.
- Case studies: Nonlinear Regression and Variable selection,
Mixture models, Hidden Markov models, Bayes CART.
- Sequential Monte Carlo Methods
& Particle Filtering Methods
- Dynamic generalized linear models, hidden Markov models,
nonlinear non-Gaussian state-space models.
- Sequential importance sampling and resampling.
- Filtering/smoothing and parameter estimation.
- Sequential Monte Carlo for static problems and extensions.
- Case studies: Switching State-Space models, Stochastic
Volatility models, Contingency tables, Linkage analysis.
Textbook
- Christian P. Robert and George Casella, Monte Carlo Statistical
Methods, Springer, 2nd edition (you don't need to buy it but it
is a good reference).
We will also use
- Jean-Michel Marin and Christian P. Robert, Bayesian Core: A
Practical Approach to Computational Bayesian Statistics, Springer, to
appear.
- Denis G.T. Denison, Chris C. Holmes, Bani K. Mallick and Adrian
F.M. Smith, Bayesian Methods for Nonlinear Classification and
Regression, Wiley.
- Arnaud Doucet, Nando De Freitas and Neil J. Gordon (eds),
Sequential Monte Carlo in Practice, Springer.
- Andrew Gelman, John B. Carlin, Hal Stern and Donald B. Rubin,
Bayesian Data Analysis, Chapman&Hall/CRC, 2nd edition.
- Christian P. Robert, The Bayesian Choice, Springer, 2nd edition.
Grading
This will be based on several assignments and a
final project (exact weighting yet to be decided). The computational
part of the
assignments will be done
using the R
statistical language or Matlab.
If you don't know what these are, I urge you to familiarize yourself
with them.
Note that R is open source and can be downloaded
for free.
Some interesting links - other Bayesian
computational courses