ST3456: Modern Statistical Methods II

## Module Code: ST3456 (descriptor)Module Title: Modern Statistical Methods II (Simulation methods)Cohorts: Maths Week range: 03-08, 10-14Semester: S1 (Michaelmas) Total hours: 33 (28 classes + 5 labs)Timetable: - Monday, h. 12-14, room LB01- Tuesday, h. 10-11, room LB 1.07 (or ICT1 lab when labs are scheduled)Textbook: Lecture notes will be uploaded to this webpageOther references:Ross, "Simulation"Rubinstein & Kroese, "Simulation and Monte Carlo Method"Robert & Casella, "Monte Carlo statistical methods"Robert & Casella, "Introducing Monte Carlo methods with R"Devroye, "Non-uniform random variate generation"Links:www.random.org for true random number generation (link to Box-Muller method for normal random generation)Group project (password protected)instructions (pdf), deadline: 07/01/2019

• 10/09/18: Section 1: motivating examples (pdf)
• 11/09/18: Section 2.1: inverse transform method (pdf)
• 17/09/18: Section 2.2: transformation methods (pdf)
• 18/09/18: Section 2.3: acceptance-rejection method (pdf)
• 01/10/18: Section 3, 3.1: controlling the variance; common and antithetic variables (pdf)
• 08/10/18: Section 3.2, 3.3.1: control variables; conditional expectation (pdf)
• 15/10/18: Section 3.3.2: variance reduction by conditioning (pdf)
• 15/10/18: Lecture notes first 6 weeks: Sections 1, 2 and 3 (pdf)
• 12/11/18: Sections 4.1 and 4.2: essential of Markov chains; Metropolis-Hastings for discrete state-space (pdf)
• 19/11/18: Sections 4.3 and 4.4: Metropolis-Hastings for continuous state-space; independent M-H; random walk M-H (pdf)
• 26/11/18: Lecture notes, final version (pdf) (last updated 29/11/18, corrections appear in red and are listed here)

• 24/09/18: problem set 1 (pdf), solutions (pdf)
• 12/11/18: problem set 2 (pdf), solutions (pdf)

• 10/09/18: Class 1 (pdf)
• 12/11/18: Example: random walk with limiting Poisson distribution (ppt)
• 19/11/18: Example: independent M-H vs Acceptance-Rejection; Example: random walk M-H and role of the parameter sigma (ppt)
• 26/11/18: Example: Gibbs sampling vs Metropolis-Hastings; Example: diagnostic tools for MCMC (ppt)

• 25/09/18: Lab 1 - instructions (pdf), R script (R) and solutions (R)
• 02/10/18: Lab 2 - instructions (pdf), R script (R) and solutions (R)
• 09/10/18: Lab 3 - instructions (pdf), R script (R) and solutions (R)
• 16/10/18: Lab 4 - instructions (pdf), R script (R) and solutions (R)
• 06/11/18: Lab 5 - instructions (pdf), R script (R) and solutions (R)
• 13/11/18: Lab 6 - instructions (pdf), R script (R) and solutions (R)
• 20/11/18: Lab 7 - instructions (pdf), R script (R) and solutions (R)

Syllabus (lecture by lecture):

• 10/09/18 (2 h): motivating examples: Monte Carlo integration, Bayesian Inference, analysis of leukaemia data.
• 11/09/18 (1 h): inverse transform method (with proof), ITM for exponential random variables (example 2.5), max and min of IID random variables (first method).
• 17/09/18 (2 h): max and min of IID random variables (ITM method), ITM for discrete random variable, Transformation methods, Box-Muller method.
Assignment: see example 2.7 (ITM for Gamma(n, lambda)).
• 18/09/18 (1 h): acceptance-rejection (AR) method, with proof.
• 24/09/18 (2 h): geometric distribution for the number of needed iterations in AR, AR for densities known up to a constant, example: optimal choice of exponential envelope for gamma target density.
• 25/09/18 (1 h): lab 1 - ITM for discrete random variables (Binomial and Poisson).
• 01/10/18 (2 h): Monte Carlo estimation of expected value, mean squared error (MSE) and relative error, principle of variance reduction, estimating probability of rare event via crude Monte Carlo (example 3.1), antithetic variables (example 3.2), common variables (example 3.3).
• 02/10/18 (1 h): lab 2 - ITM (Exponential) and AR (Gamma).
• 08/10/18 (2 h): control variables, conditional expectation, conditional variance formula.
• 09/10/18 (1 h): lab 3 - Simulating Normals via AR and Box-Muller.
• 15/10/18 (2 h): variance reduction by conditioning, Example: comparing methods in estimating pi.
• 16/10/18 (1 h): lab 4 - Integral estimation via simulation
• 30/10/18 (1 h): importance sampling method, example: estimating the probability of the rare event {Z>4.5}, with Z standard normal.
• 05/11/18 (2 h): exponentially tilted distributions for importance sampling, exponentially tilted Bernoulli distribution, example: estimating the probability of a rare event with tilted proposal.
• 06/11/18 (1 h): lab 5 - Importance sampling with exponentially tilted proposal.
• 12/11/18 (2 h): Essentials of Markov chains, Metropolis-Hastings algorithm for discrete state-space, Example: Poisson simulation via M-H
• 13/11/18 (1 h): lab 6 - Metropolis-Hastings for discrete state-space
• 19/11/18 (2 h): Metropolis-Hastings for continuous state space, independent M-H, independent M-H vs acceptance rejection, random-walk M-H, acceptance rate.
• 13/11/18 (1 h): lab 7 - Metropolis-Hastings for continuous state-space
• 26/11/18 (2 h): Two-stage Gibbs sampling with systematic scan, two-stage Gibbs sampling with random scan, multistage Gibbs sampling, connection between Gibbs sampling and Metropolis-Hastings, Diagnostic tools for MCMC (very briefly).
• 27/11/18 (1 h): lab - work on group project
• 28/11/18 (1 h): tutorial - overview of exam of May 2017, Problem 4 in problem set 2 on Gibbs sampling.

Videos:
• 18/09/18: acceptance-rejection for bounded density on compact support:

#### AR_movie.mov

• 18/09/18: acceptance-rejection for truncated Normal(0,1) with Exponential(1) envelope:

#### AR_movie_HALF.mov

• 05/11/18: Markov chain with Poisson limiting distribution and initial state x0=1:

#### Poisson_x01_histogram_traceplot.mov

• 05/11/18: Markov chain with Poisson limiting distribution and initial state x0=22:

#### Poisson_x22_histogram_traceplot.mov

• 19/11/18: Random walk Metropolis-Hastings for a bivariate and bimodal distribution, with standard deviation for the marginal normal proposal sigma=0.1:

#### MHsigma01.mov

• 19/11/18: Random walk Metropolis-Hastings for a bivariate and bimodal distribution, with standard deviation for the marginal normal proposal sigma=1:

#### MHsigma1.mov

• 19/11/18: Random walk Metropolis-Hastings for a bivariate and bimodal distribution, with standard deviation for the marginal normal proposal sigma=10:

#### MHsigma10.mov

• 26/11/18: Gibbs sampling for a bivariate and bimodal distribution: