Stochastic Computation

The inaugural SAMSI program on Stochastic Computation focused on synthesis and developmental research in four inter-related areas in which the problems of searching for and fitting mathematical models to observational data are characteristic of core challenges facing modern mathematical and computational sciences: problems in which realistic applied models are mathematically complex; involve many, and often very, very many uncertain parameters to be estimated; in which the form and specification of the mathematical model itself is uncertain so that scientists need tools for searching over vast spaces of candidate models; and in which data is, relative to model and parameter dimensions, relatively sparse. 

The major themes of the StoCom research - stochastic computation - concerns methods, with accompanying probability theory and computational algorithms, for performing calculations to search effectively over large spaces of models and potentially very high-dimensional parameters within models. Some key innovations and results of StoCom include the following. In the area of large-scale variable selection and regression model search -- perhaps the canonical problem facing modern mathematical modelling in scientific investigation in every area, exemplified by problems such as environmental risk assessment, for example -- StoCom introduced novel methods for rapid exploration of huge model spaces with efficient simulation algorithms that can substantially improve regression model search when faced with many candidate predictors, and also substantially improve prediction performance as a result. In problems of discrete data analysis using contingency tables subject to problems of high-dimension and missing data -- such as arise in applications in areas as diverse as government data-base confidentiality and security, and in human population genetics, for example -- StoCom research produced significant advances in computational methods, including quite novel and effective algorithms, as well as defining interconnections between earlier poorly connected branches of core mathematical/algebraic research and more applied statistical research. In the area of large-scale graphical models -- a core theoretical framework for studying and evaluating complex patterns of association among very many variables, such as arise in applications in areas as diverse as genome-scale gene expression studies in modern computational biology, and in large-scale market research studies, for example -- StoCom led to substantial advances in the capacity to explore very high-dimensional spaces of graphs, extending theory and computational tools to develop radically more effective than earlier available. This component also explored and generated studies of the approach in exploratory analysis of large-scale genomic data via cluster computing, with resulting software tools provided to the research community. In the fourth area, StoCom brought 
together statistical and mathematical researchers in financial modelling to define new approaches to stochastic computation in some of the most challenging financial volatility and pricing models, and defined new research directions from this collaboration in studies of volatility at multiple scales, among other topics.

Specific details of participants, areas of focus, papers and software produced and that represent the full program are given below within each of the four focus areas.

Links to key manuscripts, software etc from the Contingency Tables group:

• A. Dobra, C. Tebaldi and M. West
  Bayesian Inference In Incomplete Multi-Way Tables
• Mark Huber, Yuguo Chen, Ian Dinwoodie, Adrian Dobra and Mike Nicholas
  Monte Carlo algorithms for Hardy-Weinberg Proportions

• Yuguo Chen, Ian Dinwoodie, Adrian Dobra and Mark Huber
  Lattice points, contingency tables, and sampling

• Ian Dinwoodie and Brenda MacGibbon
  Exact Analysis of a Paired Sibling Study

• Ian Dinwoodie
  Estimation of parameters in a network reliability model with spatial dependence

• Ian Dinwoodie, Laura Felicia Matusevich and Ed Mosteig
  Transform methods for the hypergeometric distribution    

• Beatrix Jones, Carlos Carvalho, Adrian Dobra, Chris Hans, Chris Carter & Mike West
  Experiments in Stochastic Computation for High-Dimensional Graphical Models
• Adrian Dobra, Chris Hans, Beatrix Jones, Joseph R Nevins and Mike West
  Sparse Graphical Models for Exploring Gene Expression Data
• Frederick Wong, Christopher K. Carter and Robert Kohn
  Efficient Estimation of Covariance Selection Models (Biometrika 90, 809-830)

• Cripps, E., Carter, C. K., and Kohn, R. (2003)

  Variable selection and covariance selection in multivariate regression models

• GGM - implementations stochastic computation (Metropolis Hastings MCMC and shotgun stochastic search) for model exploration and selection in Gaussian Graphical Models (B. Jones, A. Dobra, C. Hans, C. Carvalho & M. West)

• Merlise Clyde and Edward I. George
  Model Uncertainty

Links to key manuscripts, software etc from the Financial Models group:

• German Molina, Chuan-Hsiang Han and Jean-Pierre Fouque
  MCMC Estimation of Multiscale Stochastic Volatility Models