This article discusses the connection between the matrix models and algebraic geometry. In particular, it considers three specific applications of matrix models to algebraic geometry, namely: the Kontsevich matrix model that describes intersection indices on moduli spaces of curves with marked points; the Hermitian matrix model free energy at the leading expansion order as the prepotential of the Seiberg-Witten-Whitham-Krichever hierarchy; and the other orders of free energy and resolvent expansions as symplectic invariants and possibly amplitudes of open/closed strings. The article first describes the moduli space of algebraic curves and its parameterization via the Jenkins-Strebel differentials before analysing the relation between the so-called formal matrix models (solutions of the loop equation) and algebraic hierarchies of Dijkgraaf-Witten-Whitham-Krichever type. It also presents the WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) equations, along with higher expansion terms and symplectic invariants.
Hedibert Lopes and Nicholas Polson
This article discusses the use of Bayesian multiscale spatio-temporal models for the analysis of economic data. It demonstrates the utility of a general modelling approach for multiscale analysis of spatio-temporal processes with areal data observations in an economic study of agricultural production in the Brazilian state of Espìrito Santo during the period 1990–2005. The article first describes multiscale factorizations for spatial processes before presenting an exploratory multiscale data analysis and explaining the motivation for multiscale spatio-temporal models. It then examines the temporal evolution of the underlying latent multiscale coefficients and goes on to introduce a Bayesian analysis based on the multiscale decomposition of the likelihood function along with Markov chain Monte Carlo (MCMC) methods. The results from agricultural production analysis show that the spatio-temporal framework can effectively analyse massive economics data sets.
James S. Clark, Dave Bell, Michael Dietze, Michelle Hersh, Ines Ibanez, Shannon LaDeau, Sean McMahon, Jessica Metcalf, Emily Moran, Luke Pangle, and Mike Wolosin
This article focuses on the use of Bayesian methods in assessing the probability of rare climate events, and more specifically the potential collapse of the meridional overturning circulation (MOC) in the Atlantic Ocean. It first provides an overview of climate models and their use to perform climate simulations, drawing attention to uncertainty in climate simulators and the role of data in climate prediction, before describing an experiment that simulates the evolution of the MOC through the twenty-first century. MOC collapse is predicted by the GENIE-1 (Grid Enabled Integrated Earth system model) for some values of the model inputs, and Bayesian emulation is used for collapse probability analysis. Data comprising a sparse time series of five measurements of the MOC from 1957 to 2004 are analysed. The results demonstrate the utility of Bayesian analysis in dealing with uncertainty in complex models, and in particular in quantifying the risk of extreme outcomes.
Antonia Tulino and Sergio Verdu
This article examines asymptotic singular value distributions in information theory, with particular emphasis on some of the main applications of random matrices to the capacity of communication channels. Results on the spectrum of random matrices have been adopted in information theory. Furthermore, information theorists, motivated by certain channel models, have obtained a number of new results in random matrix theory (RMT). Most of those results are related to the asymptotic distribution of the (square of) the singular values of certain random matrices that model data communication channels. The article first provides an overview of three transforms that are useful in expressing the asymptotic spectrum results — Stieltjes transform, η-transform, and Shannon transform — before discussing the main results on the limit of the empirical distributions of the eigenvalues of various random matrices of interest in information theory.
Antonio Pievatolo and Fabrizio Ruggeri
This article discusses the results of a Bayes linear uncertainty analysis for oil reservoirs based on multiscale computer experiments. Using the Gullfaks oil and gas reservoir located in the North Sea as a case study, the article demonstrates the applicability of Bayes linear methods to address highly complex problems for which the full Bayesian analysis may be computationally intractable. A reservoir simulation model, run at two different levels of complexity, is used, and a simulator of a hydrocarbon reservoir represents properties of the reservoir on a three-dimensional grid. The article also describes a general formulation for the approach to uncertainty analysis for complex physical systems given a computer model for that system. Finally, it presents the results of simulations and forecasting for the Gullfaks reservoir.
Jonathan A. Cumming and Michael Goldstein
This article discusses the results of a study in Bayesian analysis and decision making in the maintenance and reliability of nuclear power plants. It demonstrates the use of Bayesian parametric and semiparametric methodology to analyse the failure times of components that belong to an auxiliary feedwater system in a nuclear power plant at the South Texas Project (STP) Electric Generation Station. The parametric models produce estimates of the hazard functions that are compared to the output from a mixture of Polya trees model. The statistical output is used as the most critical input in a stochastic optimization model which finds the optimal replacement time for a system that randomly fails over a finite horizon. The article first introduces the model for maintenance and reliability analysis before presenting the optimization results. It also examines the nuclear power plant data to be used in the Bayesian models.
Dani Gamerman, Tufi M. Soares, and Flávio Gonçalves
This article discusses the use of a Bayesian model that incorporates differential item functioning (DIF) in analysing whether cultural differences may affect the performance of students from different countries in the various test items which make up the OECD’s Programme for International Student Assessment (PISA) test of mathematics ability. The PISA tests in mathematics and other subjects are used to compare the educational attainment of fifteen-year old students in different countries. The article first provides a background on PISA, DIF and item response theory (IRT) before describing a hierarchical three-parameter logistic model for the probability of a correct response on an individual item to determine the extent of DIF remaining in the mathematics test of 2003. The results of Bayesian analysis illustrate the importance of appropriately accounting for all sources of heterogeneity present in educational testing and highlight the advantages of the Bayesian paradigm when applied to large-scale educational assessment.
Bayesian approaches to aspects of the Vioxx trials: Non-ignorable dropout and sequential meta-analysis
Jerry Cheng and David Madigan
This article discusses Bayesian approaches to aspects of the Vioxx trials study, with a focus on non-ignorable dropout and sequential meta-analysis. It first provides a background on Vioxx, a COX-2 selective, non-steroidal anti-inflammatory drug (NSAID) approved by the FDA in May 1999 for the relief of the signs and symptoms of osteoarthritis, the management of acute pain in adults, and for the treatment of menstrual symptoms. However, Vioxx was found to cause an array of cardiovascular side-effects such as myocardial infarction, stroke, and unstable angina. As a result, Vioxx was withdrawn from the market. The article describes an approach to sequential meta-analysis in the context of Vioxx before considering dropouts in the key APPROVe study. It also presents a Bayesian approach to handling dropout and showcases the utility of Bayesian analysis in addressing multiple, challenging statistical issues and questions arising from clinical trials.
Bayesian causal inference: Approaches to estimating the effect of treating hospital type on cancer survival in Sweden using principal stratification
Donald Rubin, Xiaoqin Wang, Li Yin, and Elizabeth Zell
This article discusses the use of Bayesian causal inference, and more specifically the posterior predictive approach of Rubin’s causal model (RCM) and methods of principal stratification, in estimating the effects of ‘treating hospital type’ on cancer survival. Using the Karolinska Institute in Stockholm, Sweden, as a case study, the article investigates which type of hospital (large patient volume vs. small volume) is superior for treating certain serious conditions. The study examines which factors may reasonably be considered ignorable in the context of covariates available, as well as non-compliance complications due to transfers between hospital types for treatment. The article first provides an overview of the general Bayesian approach to causal inference, primarily with ignorable treatment assignment, before introducing the proposed approach and motivating it using simple method-of-moments summary statistics. Finally, the results of simulation using Markov chain Monte Carlo (MCMC) methods are presented.
Peter Green, Kanti Mardia, Vysaul Nyirongo, and Yann Ruffieux
This article describes Bayesian modelling for matching and alignment of biomolecules. One particular task where statistical modelling and inference can be useful in scientific understanding of protein structure is that of matching and alignment of two or more proteins. In this regard, statistical shape analysis potentially has something to offer in solving biomolecule matching and alignment problems. The article discusses the use of Bayesian methods for shape analysis to assist with understanding the three-dimensional structure of protein molecules, with a focus on the problem of matching instances of the same structure in the CoMFA (Comparative Molecular Field Analysis) database of steroid molecules. It introduces a Bayesian hierarchical model for pairwise matching and for alignment of multiple configurations before concluding with an overview of some advantages of the Bayesian approach to problems in protein bioinformatics, along with modelling and computation issues, alternative approaches, and directions for future research.