When:	Tues 11:40am-(break)-2:30pm		Where:	025 Old Chemistry Building
	(start Aug 28th : end Nov 27th)
Prof:	Mike West, Statistical Science		TA:	Dawn Woodard, PhD, Statistical Science
Office Hours:	By appt.			Mon: 9-10 & 14:15-15:15

Prerequisites: STA 215, 244 and STA 290, or (very) close equivalents. This is a second level statistics graduate course that relies on the background in core mathematical statistics, Bayesian and non-Bayesian inference, applied modelling in statistics and computation in these first level graduate courses. All students must also have expertise and facility in computing and applied statistics at the level of STA 290. Use of Matlab will be required and routine in in-class examples, homeworks and exams.

This course is concerned with the theory and structure of selected classes of statistical and stochastic process models that are common in many applications, the theory, methods and uses of stochastic simulation in statistics, multivariate distribution theory and manipulation of models and distributions in statistical analysis, and computational exploration and implementation. Core goals include those of (a) developing facility for manipulating multivariate distributions in such models - involving mastery of linear algebra and multivariate calculus for probabilistic and statistical models; (b) developing the theory and implementation of key methods of simulation of multivariate distributions, including Markov chain Monte Carlo methods; (c) familiarisation with a range of multilinear model frameworks including structured multivariate normal models in a graphical model context, linear data and model decompositions, Markovian time series and dynamic models (linear systems - state space models); and (d) aspects of theory, structure and exploration of graphical models: in Gibbs sampling, and in Gaussian graphical models in multivariate analysis.

Computation: Students will be expected to develop a broad range of computational algorithms and exploratory modelling exercises with minimal supervision using Matlab.

Coverage/Topics: Please note that (a) this is not a week-by-week outline - contact with individual topics will be interspersed throughout; and (b) the list will evolve as semester progresses - some topics will not be covered (due to time constraints and interests of students) and other may be added. Examples: The course will be leavened throughout with technical examples and with real problems involving data from areas including finance, genetics, neuroscience, genomics and climatology. Aspects of multivariate distribution theory Multivariate normal, and associated linear algebra and multivariate calculus Families of normal and Wishart distributions, singular cases Multinomial and Dirichlet distributions Dirichlet processes and fitting models to histograms Mixture distributions: convolutions, discrete mixtures as models, and mixtures arising in statistical model analyses, imputation Mixture modelling in problems of density estimation and clustering in 1,2 and more dimensions Convolutions, composition, transforms and generating functions Data and model decompositions, and tools of multivariate analysis Eigenmethods, SVD, PCA, linear systems (difference equations, stochastic versions) eigentheory Simulating multivariate distributions Direct methods, composition, convolution Importance sampling, rejection MCMC theory and methods Gibbs sampling Metropolis Hastings Markov chains on continuous, multivariate state-spaces Latent variables, missing data, data augmentation Simulation in regression models, Bayesian analysis in regression, heavy-tailed models/robust regression Aspects of Markov chain theory Stationarity, convergence, reversibility Discrete and continuous state spaces, intuition behind convergence theory Gaussian Markov processes: AR models Linear algebra and linear systems theory related to AR models Theory of AR models, stationary processes and variants Simulation and relation to Markov chain methods Aspects of model fitting and inference Examples of hidden Markov models Linear filtering and related ideas Simulation in DLMs/HMMs Stochastic volatility models - Examples of HMMs, Markov processes, mixture models, MCMC Aspects of graphical and other structured multivariate normal models: Directed and undirected graphs, elementary graph theory ideas - multivariate distributions over graphs Gaussian graphical models: normal theory association, precision, covariance selection; links to linear regressions Potential later topics, if time permits and depending on student interest: More advanced topics in Hidden Markov modelling Semi- and non-Markov models, time-varying parameter (non-stationary) AR processes Linear latent factor models: introduction to structured latent factor analysis, and Relationships between graphical models and factor models