Statistical Computing - Monte Carlo Methods

       Stat 535C  - CPSC 535D

Course scheduleTwo 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

• Assignment #1  Pdf     Check here the results you should have obtained
• Assignement #2  Pdf   Dataset (matlab format)  Dataset (txt file)

Handouts

• 05/01/06: Lecture 1 - Introduction   Pdf  Ps  Ps-4pages

• 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

• 12/01/06: No lecture

• 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

• 31/01/06: Lecture 7 -  Introduction to Monte Carlo Pdf  Ps  Ps-4pages

     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.

• 07/02/06: Lecture 9 - Importance Sampling    Pdf
  Ps
  Ps-4pages

      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 [url=http://www.jstor.org/cgi-bin/jstor/printpage/00063444/di992425/99p0485a/0.pdf?backcontext=results&dowhat=Acrobat&config=&userID=8e670872@ubc.ca/01cce44035244fd109897ca0ea&0.pdfhttp://www.jstor.org/cgi-bin/jst ... ext=results&dowhat=Acrobat&config=&userID=8e670872@ubc.ca/01cce44035244fd109897ca0ea&0.pdf]here[/url]
             - 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, [size=-1]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, [size=-1]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 [url=http://www.jstor.org/cgi-bin/jstor/printpage/01621459/di986005/98p0228m/0?frame=noframe&dpi=3&userID=8e670872@ubc.ca/01cc99333c4a7310a18efee15&backcontext=page&backurl=/cgi-bin/jstor/viewitem/01621459/di986005/98p0228m/0%3fframe%3dnoframe%26dpi%3d3%26userID%3d8e670872@ubc.ca/01cc99333c4a7310a18efee15%26config%3d%26PAGE%3d0&action=download&config=jstor]Pdf file here[/url]

• 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

            
ObjectivesTo 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

• Francesca Dominici's course at Johns Hopkins
• Sujit Ghosh's course at NCSU
• Paula Sebastiani's course at UMass
• Lawrence Joseph's course at McGill
• Gary Rosner's course at Rice
• Kate Cowles's course at Iowa
• Brani Vidakovic's course at Georgia Tech
• David Madigan's course at Rutgers

• Bayesian Java applications
• There are R Tutorials here
• The BUGS Project
• R
• Figueiredo's Bayesian Tutorial
• Wasserman's Decision Theory Notes
• Stuart Cole's Bayesian Notes