Carlos M. Carvalho

High-Dimensional Sparse Factor Modeling: Applications in Gene Expression Genomics

We describe studies in molecular profiling and biological pathway analysis that use sparse latent factor and regression models for microarray gene expression data. We discuss breast cancer applications and key aspects of the modeling and computational methodology. Our case studies aim to investigate and characterize heterogeneity of structure related to specific oncogenic pathways, as well as links between aggregate patterns in gene expression profiles and clinical biomarkers. Based on the metaphor of statistically derived “factors” as representing biological “subpathway” structure, we explore the decomposition of fitted sparse factor models into pathway subcomponents and investigate how these components overlay multiple aspects of known biological activity. Our methodology is based on sparsity modeling of multivariate regression, ANOVA, and latent factor models, as well as a class of models that combines all components. Hierarchical sparsity priors address questions of dimension reduction and multiple comparisons, as well as scalability of the methodology. The models include practically relevant non-Gaussian/nonparametric components for latent structure, underlying often quite complex non-Gaussianity in multivariate expression patterns. Model search and fitting are addressed through stochastic simulation and evolutionary stochastic search methods that are exemplified in the oncogenic pathway studies. Supplementary supporting material provides more details of the applications, as well as examples of the use of freely available software tools for implementing the methodology.

Particle Learning and Smoothing

Particle learning (PL) provides state filtering, sequential parameter learning and smoothing in a general class of state space models. Our approach extends existing particle methods by incorporating the estimation of static parameters via a fully-adapted filter that utilizes conditional sufficient statistics for parameters and/or states as particles. State smoothing in the presence of parameter uncertainty is also solved as a by-product of PL. In a number of examples, we show that PL outperforms existing particle filtering alternatives and proves to be a competitor to MCMC.

Experiments in Stochastic Computation for High-Dimensional Graphical Models

We discuss the implementation, development and performance of methods of stochastic computation in Gaussian graphical models. We view these methods from the perspective of high-dimensional model search, with a particular interest in the scalability with dimension of Markov chain Monte Carlo (MCMC) and other stochastic search methods. After reviewing the structure and context of undirected Gaussian graphical models and model uncertainty (covariance selection), we discuss prior specifications, including new priors over models, and then explore a number of examples using various methods of stochastic computation. Traditional MCMC methods are the point of departure for this experimentation; we then develop alternative stochastic search ideas and contrast this new approach with MCMC. Our examples range from low (12–20) to moderate (150) dimension, and combine simple synthetic examples with data analysis from gene expression studies. We conclude with comments about the need and potential for new computational methods in far higher dimensions, including constructive approaches to Gaussian graphical modeling and computation.

A Genomic Strategy to Elucidate Modules of Oncogenic Pathway Signaling Networks

Recent studies have emphasized the importance of pathway-specific interpretations for understanding the functional relevance of gene alterations in human cancers. Although signaling activities are often conceptualized as linear events, in reality, they reflect the activity of complex functional networks assembled from modules that each respond to input signals. To acquire a deeper understanding of this network structure, we developed an approach to deconstruct pathways into modules represented by gene expression signatures. Our studies confirm that they represent units of underlying biological activity linked to known biochemical pathway structures. Importantly, we show that these signaling modules provide tools to dissect the complexity of oncogenic states that define disease outcomes as well as response to pathway-specific therapeutics. We propose that this model of pathway structure constitutes a framework to study the processes by which information propogates through cellular networks and to elucidate the relationships of fundamental modules to cellular and clinical phenotypes.

Dynamic matrix-variate graphical models

This paper introduces a novel class of Bayesian models for multivariate time series analysis based on a synthesis of dynamic linear models and graphical models. The synthesis uses sparse graphical modelling ideas to introduce struc- tured, conditional independence relationships in the time-varying, cross-sectional covariance matrices of multiple time series. We dene this new class of models and their theoretical structure involving novel matrix-normal/hyper-inverse Wishart distributions. We then describe the resulting Bayesian methodology and compu- tational strategies for model tting and prediction. This includes novel stochastic evolution theory for time-varying, structured variance matrices, and the full se- quential and conjugate updating, ltering and forecasting analysis. The models are then applied in the context of nancial time series for predictive portfolio analysis. The improvements dened in optimal Bayesian decision analysis in this example context vividly illustrate the practical benets of the parsimony induced via appro- priate graphical model structuring in multivariate dynamic modelling. We discuss theoretical and empirical aspects of the conditional independence structures in such models, issues of model uncertainty and search, and the relevance of this new framework as a key step towards scaling multivariate dynamic Bayesian modelling methodology to time series of increasing dimension and complexity.

Simulation-based sequential analysis of Markov switching stochastic volatility models

We propose a simulation-based algorithm for inference in stochastic volatility models with possible regime switching in which the regime state is governed by a first-order Markov process. Using auxiliary particle filters we developed a strategy to sequentially learn about states and parameters of the model. The methodology is tested against a synthetic time series and validated with a real financial time series: the IBOVESPA stock index (Sao Paulo Stock Exchange).

Feature-Inclusion Stochastic Search for Gaussian Graphical Models

We describe a serial algorithm called feature-inclusion stochastic search, or FINCS, that uses online estimates of edge-inclusion probabilities to guide Bayesian model determination in Gaussian graphical models. FINCS is compared to MCMC, to Metropolis-based search methods, and to the popular lasso; it is found to be superior along a variety of dimensions, leading to better sets of discovered models, greater speed and stability, and reasonable estimates of edge-inclusion probabilities. We illustrate FINCS on an example involving mutual-fund data, where we compare the model-averaged predictive performance of models discovered with FINCS to those discovered by competing methods.

FLEXIBLE COVARIANCE ESTIMATION IN GRAPHICAL GAUSSIAN MODELS

In this paper, we propose a class of Bayes estimators for the covariance matrix of graphical Gaussian models Markov with respect to a decomposable graph G. Working with the W PG family defined by Letac and Massam [Ann. Statist. 35 (2007) 1278-1323] we derive closed-form expressions for Bayes estimators under the entropy and squared-error losses. The W PG family includes the classical inverse of the hyper inverse Wishart but has many more shape parameters, thus allowing for flexibility in differentially shrinking various parts of the covariance matrix. Moreover, using this family avoids recourse to MCMC, often infeasible in high-dimensional problems. We illustrate the performance of our estimators through a collection of numerical examples where we explore frequentist risk properties and the efficacy of graphs in the estimation of high-dimensional covariance structures.

Decoupling Shrinkage and Selection in Bayesian Linear Models: A Posterior Summary Perspective

Selecting a subset of variables for linear models remains an active area of research. This article reviews many of the recent contributions to the Bayesian model selection and shrinkage prior literature. A posterior variable selection summary is proposed, which distills a full posterior distribution over regression coefficients into a sequence of sparse linear predictors.

A Bayesian Analysis Strategy for Cross-Study Translation of Gene Expression Biomarkers

We describe a strategy for the analysis of experimentally derived gene expression signatures and their translation to human observational data. Sparse multivariate regression models are used to identify expression signature gene sets representing downstream biological pathway events following interventions in designed experiments. When translated into in vivo human observational data, analysis using sparse latent factor models can yield multiple quantitative factors characterizing expression patterns that are often more complex than in the controlled, in vitro setting. The estimation of common patterns in expression that reflect all aspects of covariation evident in vivo offers an enhanced, modular view of the complexity of biological associations of signature genes. This can identify substructure in the biological process under experimental investigation and improved biomarkers of clinical outcomes. We illustrate the approach in a detailed study from an oncogene intervention experiment where in vivo factor profiling of an in vitro signature generates biological insights related to underlying pathway activities and chromosomal structure, and leads to refinements of cancer recurrence risk stratification across several cancer studies.