Syllabus: Monte Carlo Methods
Description: Computationally intensive statistical methodologies and their theoretical foundations. Design, implementation and analysis of correct, scalable inference software.
For Term 2 of 2017-2018, the course will focus on Monte Carlo methods, from foundations to recent advances. If time permits, topics at the interface with optimization will be covered.
Relevant topics:
- Motivations and foundations of Monte Carlo inference (pseudo randomness, simple exact sampling, simple approximate sampling via importance sampling, basic asymptotic theory, correctness, debugging and performance evaluation of randomized algorithms, control variates, limitations and failure cases of simple methods);
- Scalable inference methods for sequences and their generalizations (motivation: hidden and state space models, exact recurrences such as Kalman filters and dynamic programming, Sequential Monte Carlo algorithms, evidence estimation, SMC samplers and change of measure, correctness, debugging and performance evaluation, failure cases);
- Scalable inference in high-dimensional models (motivation taken from random effect models, spatial statistics, genetics, Metropolis Hastings framework and its theoretical analysis, auxiliary variable methods, correctness, debugging and performance evaluation, failure cases);
- Inference in intractable models (motivation, pseudo-marginal methods, Approximate Bayesian Computation);
- Selected advanced topics, such as non-reversible methods, piecewise-deterministic Markov processes, topics in probabilistic programming, topics in variational inference, other advanced MCMC/SMC methods.
Assessment will consist of implementation, analysis and comparison of various computationally intensive statistical methodologies.
References:
There is no single textbook that I would say covers all the ground unfortunately. I will provide notes which can be your primary source. If you would like to look at some reference textbooks, consider:
- Resources on Monte Carlo:
- Monte Carlo theory, methods and examples (2013). Art Owen.
- Available from the author's webpage..
- Excellent treatment of Importance Sampling in particular.
- Monte Carlo strategies in scientific computing (2001). Jun S. Liu.
- Available through SpringerLink behind UBC paywalls or VPN.
- Good reference for the basics but missing important new developments from the past 15 years.
- Handbook of Monte Carlo Methods (2011). Dirk P. Kroese, Thomas Taimre, and Zdravko I. Botev.
- Seems to cover some slightly more recent algorithms but still missing lots of developments that happened in this decade.
- Monte Carlo theory, methods and examples (2013). Art Owen.
- Resources on Bayesian inference (one of the many motivations for Monte Carlo).
- Lecture notes available for my course on Bayesian statistics (2015). Notes from a previous iteration of the course (2014) that focussed more on Bayesian non-parametric statistics.
- The Bayesian Choice (2007). Christian Robert.
- Great coverage of the theoretical foundations of Bayesian inference,
- and also of Monte Carlo approximations needed, in particular, when computing Bayes factors.
- Bayesian Data Analysis (2013). Gelman et al.
- Another popular reference on Bayesian inference.
- Resources on probabilistic programming
- Blang, which efficiently supports both discrete, combinatorial and continuous random variables; especially useful for hard non-convex problems.
- Frameworks focussing on autodiff+HMC, e.g. for continuous random variable and target not too multimodal: Stan, Edward, pyMC3, pyro (some support discrete variable at various degrees of efficiency).
- Languages focussing on maximum expressivity: Anglican, Birch.
- Many more.
Evaluation
- Assignment (they will involve coding) 50%
- Includes some exercise and problems based on participation
- Final research project (teams encouraged) 50%
Final project:
The course project involves independent work on a topic of your choice, with the constraint that you should make use of some of the theory covered in class, or extension of these techniques. There are three main types of projects: application, methodology, and theory, as described in class. Combinations of these is also encouraged. Extending the exercises is a good way to start thinking about project ideas.
- Teams are encouraged, in which case you should outline the final report who did what.
- Format: A latex document (and code if there is an empirical aspect), submitted electronically using the usual method.
- Grading: I will base the grade on the same factors one would usually consider in a paper reviewing process (but taking into account the fact that the time frame is shorter than the typical time to write a research paper). Is the goal clearly defined? Is it well motivated? Is the approach sound? Creative? Is the paper well-written? Are there interesting connections made to the existing literature?
Office hours, TA, piazza, etc: See Contact
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