Homework Assignments

Lecture Notes

1. Introduction (1 week)
   • Standard approaches to statistical inference: Bayesian, frequentist, and beyond

     Example: Y ~ N(mu, 1), where a < mu < b with known a and b.

     References

     • R. A. Fisher (1922). On the Mathematical Foundations of Theoretical Statistics, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, Vol. 222, (1922), pp. 309-368
     • R. Martin and C. Liu (2015). Inferential Models: Reasoning with Uncertainty Chapman and Hall/CRC

   • Computing with data using the R software

     Slides: R fundamentals

   • Machine learning and deep learning: an overlook

2. Advanced Programming and Distributed Computing in R (0 weeks)
   • Some basics in programming with R
     1. Creating an map of the world [of a single R session] by using the functions library(), attach(), search(), as.environment(), environment(), parent.env(), getNamespace(), loadedNamespaces(), and getNamespaceImports(). Exampe 1 [parents of loaded namespaces].; see also this.

     2. Visualizing R in action with the functions eval(), evalq(), local(), with(), quote(), bquote(), and parent.frame() (also see this).
     3. Mastering inter-process communication (IPC) with the functions showConnections(), socketConnection(), socketSelect(), fifo(), writeLines(), readLines(), serialize(), and unserialize().

   • An experimental system of distributed computing [Instructions for installation on skew (prepared by Glen DePalma).]

3. Some Basics (2 weeks)
   • Introduction: Matrix algebra, multivariate normal model, Wishart distribution, and the sweep operator
     • The sweep operator in R

   • Random number generation
     • Uniform random number generators: Random number generation, Mersenne Twister and a linear congruential generator in R
     • Section 1.4 Random Variable Generation in the book entitled Advanced Markov Chain Monte Carlo Methods: Learning from Past Samples

       Example: rou.rnorm

   • A Stochastic Approximation Method [Stochastic Approximation, wikipedia, An example of the Robbins-Monro algorithm]

4. Numerical Differentiation and Integration (1 week)
   • Numerical integeration and Monte Carlo methods

     Alzheimer.R Alzheimer.dat

   • Large sample-based (or approximate) inference and numerical differentiation
     • Computing observed Fisher information matrix [Liu (1998), Jamshidian and Jennrich (2000), He and Liu (2008), ]

5. Gradient-based Optimization Methods (2 weeks)
   • The Method Of Steepest Descent [Wikipedia, Qian (1999), An example: Gradient Descent for a 2-dimensional quadratic function x'Ax ]
   • Conjugate Gradient Methods [A survey by Hager and Zhang (2006), A recent CG algorithm by Andrei (2006)]
   • Quasi-Newton methods (GH: Chapter 2, Liu and Vander Wiel (2007))

   • See also the Nelder-Mead method (or downhill simplex method). It is applied to nonlinear optimization problems for which derivatives may not be known.

6. The EM-type Algorithms (2 weeks)
   • The EM algorithm [Dempster, Laird, and Rubin (1977); Wu (1983)]
   • The ECM and ECME algorithms [Meng and Rubin (1994); Liu and Rubin (1994)]
   • The PX-EM algorithm [Liu, Rubin, and Wu (1998), Lewandowski, Liu, Vander Wiel (2010)]
   • The DECME algorithm [He and Liu (2012)]
   • Incremental EM (Neal and Hinton, 1998)
   • The DEM Algorithm [Srivastava, DePalma, and Liu (2019)]
7. Markov Chain Monte Carlo Methods (3 weeks)
• Markov chain: an overview
• Data Augmentation [Examples: Incomplete Multivariate Normal Data, Multiple linear regression with censored data (Liu and Sun, 2000 Technometrics)]
• The Gibbs sampler [See Gelfand and Smith (1990, JASA)]
• The Metropolis-Hastings algorithm [See Hastings (1970, Biometrika)]
• Stochastic Approximation in Monte Carlo Computation: Faming Liang's presentation

• Stochastic Gradient Markov chain Monte Carlo (Yi-An Ma, Tianqi Chen, and Emily B. Fox, 2015)
• Stochastic Quasi-Newton Markov chain Monte Carlo (Umut Simseklim, Roland Badeau, A. Taylan Cemgi, and Gaël Richard, 2016)
• Stochastic Hamiltonian Markov chain Monte Carlo (Tianqi Chen, Emily B. Fox, and Carlos Guestrin, 2014)

  [See also Srivastava, DePalma, and Liu (2019) and He and Liu (2012) in the context of EM-type algorithms?]

• Unbiased Markov chain Monte Carlo with couplings ( Pierre E. Jacob, John O'Leary, Yves F. Atchadé, 2019; Jeremy Heng and Pierre E. Jacob, 2018)

  [See also Liu and Rubin (1996)?]

8. Selected Topics
• Advanced MCMC methods for efficient computing
• Computing for exact inference with inferential models
• Algorithms in data analysis with machine learning, especially deep learning models.
• Divide and conquer algorithms

Computer Code

Two simple EM examples