Statistical Computing - Monte Carlo Methods
Stat 535C - CPSC 535D
Course schedule
Two lectures per week (Tuesday and Thursday from 10.00 to 11.30)
in LSK 301.
If you want to arrange a meeting, just send me an email at the
following address arnaud at cs dot ubc dot ca
Announcements
- I have posted the slides of Lecture 21.
Assignements
Handouts
- 10/01/06: Lecture 2 -
Sufficiency, Likelihood and Conditionality
Principles Revised
version 10/01/06 Pdf
Ps Ps-4pages
Additional
reading: Chapter 1 of the Bayesian Choice by C.P. Robert or the nice
handouts h1
h2
by B. Vidakovic
- 17/01/06: Lecture 3 -
Introduction to Bayesian Statistics Revised
version 18/01/06 Pdf
Ps Ps-4pages
Additional
reading: Chapter 1 of the Bayesian Choice by C.P. Robert or h3
by B. Vidakovic
- 19/01/06: Lecture 4 -
More Bayesian Statistics (Examples, Testing
hypothesis, Bayes factors) Revised
version 23/01/06 Pdf
Ps Ps-4pages
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
- 24/01/06: Lecture 5 - And
more Bayesian Statistics (Bayesian
model selection) Revised
version 24/01/06 Pdf
Ps Ps-4pages
Additional reading:
- 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
- P. Brown, T. Fearn and M. Vannucci,
Bayesian wavelet regression on curves with application to a
spectroscopic calibration problem, JASA, 2001 here
- 26/01/06: Lecture 6 - And
more Bayesian Statistics
(From prior information to prior distribution) Pdf Ps
Ps-4pages
Additional reading: Chapter 3 of
Bayesian Choice by C.P. Robert or h5
h6
by B. Vidakovic
Additional reading:
- Section 3.1 and 3.2 of Monte
Carlo Statistical Methods.
- 02/02/06: Lecture 8
- Classical Methods (inverse
transform, accept/reject) Pdf Ps
Ps-4pages
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.
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
- 09/02/06: Lecture 10 -
Introduction to Markov chain Monte
Carlo Pdf Ps Ps-4pages
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
- 21/02/06: Lecture 11 -
Gibbs samplers - Case Studies 1: Variable
Selection and Mixture Models
Pdf Ps Ps-4pages
- There is no
specific additional reading for this lecture.
- 23/02/06: Lecture 12 -
Gibbs samplers - Case Studies 2: Time
Series Models Revised
version 27/02/06
Pdf Ps Ps-4pages
- C. Andrieu
and A. Doucet, Iterative Algorithms for
State Estimation in Jump Markov systems, IEEE Signal Processing, 2001 here
- B. Carlin,
N. Polson and D. Stoffer, A Monte Carlo Approach to Nonnormal and
Nonlinear State-Space Modeling, JASA, 1992 here
- C.
Carter and R. Kohn, On Gibbs Sampling for State-Space Models,
Biometrika, 1994 here
-
S. Chib, Calculating Posterior Distributions and Modal Estimates in
Markov
Mixture Models, J. Econometrics, 1996 here
- E.
Jacquier, N. Polson and P. Rossi, Bayesian Analysis of Stochastic
Volatility Models, J. Bus. Econ. Statist., 1994 here
- 28/02/06: Lecture 13 -
Metropolis-Hastings and
Generalizations Pdf
Ps Ps-4pages
- S. Chib and
E. Greenberg, Understanding the Metropolis-Hastings algorithm, The
American Statistician, 1995 here
- Chapter 7
of Robert & Casella.
- You can
play with the following java
applets.
- 02/03/06: Lecture 14 -
More about the Metropolis-Hastings
Algorithm: mixture, composition, hybrid algorithms Pdf Ps
Ps-4pages
- Chapter 10
and in particular Section 10.3 of Robert & Casella
- 07/03/06: Lecture 15 - MH
algorithm - Case Studies 3: Generalized
Linear Models Revised
version 08/02/06 Pdf Ps Ps-4pages
- Chapter 10
and in particular Section 10.3 of Robert & Casella
- As an
exercise, you could fit the logistic model p. 15 of Robert &
Casella.
- The
bank dataset is here
-
Another less trivial but interesting example is here
(start with the number of sinusoids fixed)
- 09/03/06: Lecture 16
- MH algorithm - Case Studies 4: More Time Series Pdf Ps
Ps-4pages
- L. Held,
Conditional Prior Proposals in Dynamic Models, Scand. J. Statist., 1999
Pdf file here
- M.K. Pitt
& N. Shephard, Likelihood Analysis of Non-Gaussian Measurement Time
Series, Biometrika, 1996 Pdf
file here
- 14/03/06: Lecture 17 -
Transdimensional MCMC algorithms Pdf Ps
Ps-4pages
- 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
- 16/03/06: Lecture 18 -
More Transdimensional MCMC algorithms Pdf Ps
Ps-4pages
-
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
- 21/03/06: Lecture 19 -
Advanced MCMC: Tempering, annealing, slice sampling Pdf Ps
Ps-4pages
- 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
- 23/03/06: Lecture 20 -
Introduction to Sequential Monte Carlo Pdf Ps
Ps-4pages
- Chapter 11 of Robert
& Casella
- A. Doucet, N. De Freitas
and N.J. Gordon, An introduction to Sequential Monte Carlo, Ps file here
- 28/03/06: Lecture 21 -
Sequential Monte Carlo for Filtering Pdf
Ps Ps-4pages
- J. Carpenter, P.
Clifford and P. Fearnhead, An Improved
Particle Filter for Non-linear Problems, Pdf file
here
- A. Doucet, S.J. Godsill
and C. Andrieu, On Sequential Monte Carlo sampling methods for Bayesian
filtering, Stat. Comp., 2000 (reprinted 2005) Pdf
file here
- M.K. Pitt and N.
Shephard, Filtering via Simulation: Auxiliary Particle Filter, JASA,
1999 Pdf
file here
- 30/03/06: Lecture 22 -
More Sequential Monte Carlo: Beyond
standard optimal filtering Pdf Ps Ps-4pages
- C. Andrieu and A.
Doucet, Particle Filtering for Partially Observed Gaussian State-Space
Models, JRSS B, 2002 Pdf
file here
- A. Kong, J.S. Liu
and
W.H. Wong, Sequential Imputations and Bayesian Missing Data Problems,
JASA, 1994
Pdf
file here
- R. Chen and J.S.
Liu, Predictive Updating Methods with Application to Bayesian
Classification, JRSS B, 1996 Pdf
file here
- J.S. Liu and R.
Chen, Sequential Monte Carlo methods for dynamic systems, JASA, 1998 Pdf
file here
- 04/04/06: Lecture 23 -
General Sequential Monte Carlo Pdf Ps Ps-4pages
- P. Del Moral, A.
Doucet and A. Jasra, Sequential Monte Carlo samplers, JRSSB, 2006 Pdf
file here
- P. Del Moral, A.
Doucet and A. Jasra, Sequential Monte Carlo for Bayesian Computation,
Bayesian Statistics, 2006 Pdf
file here (first draft! do not distribute!)
- 06/04/06: Lecture 24 -
General Sequential Monte Carlo Pdf Ps Ps-4pages
Objectives
To provide students an introduction to modern computational methods
used in (Bayesian) statistics. The computational methods
presented here will be illustrated by a large number of complex
statistical models: (dynamic) generalised linear models, mixture
and hidden Markov models, Dirichlet processes, nonlinear regression and
classification models, stochastic volatility models etc.
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
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, a midterm exam
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