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57
Convergence Assessment for Reversible Jump MCMC Simulations
, 1998
"... In this paper we introduce the problem of assessing convergence of reversible jump MCMC algorithms on the basis of simulation output. We discuss the various direct approaches which could be employed, together with their associated drawbacks. Using the example of fitting a graphical Gaussian model vi ..."
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In this paper we introduce the problem of assessing convergence of reversible jump MCMC algorithms on the basis of simulation output. We discuss the various direct approaches which could be employed, together with their associated drawbacks. Using the example of fitting a graphical Gaussian model via RJMCMC, we show how the simulation output for models which can be parameterised so that parameters of primary interest retain a coherent interpretation throughout the simulation, can be used to assess convergence. In the context of this example, we extend the work of Gelman and Rubin (1992) and Brooks and Gelman (1998), to provide convergence assessment procedures for graphical model determination problems, but which may be applied to any form of model choice problem and, indeed, MCMC simulations more generally.
Learning Bayes net structure from sparse data sets
, 2001
"... There are essentially two kinds of approaches for learning the structure of Bayesian Networks (BNs) from data. The first approach tries to find a graph which satis es all the constraints implied by the empirical conditional independencies measured in the data [PV91, SGS00a, Shi00]. The second approa ..."
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Cited by 14 (2 self)
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There are essentially two kinds of approaches for learning the structure of Bayesian Networks (BNs) from data. The first approach tries to find a graph which satis es all the constraints implied by the empirical conditional independencies measured in the data [PV91, SGS00a, Shi00]. The second approach searches through the space of models (either DAGs or PDAGs), and uses some scoring metric (typically Bayesian or some approximation, such as BIC/MDL) to evaluate the models [CH92, Hec95, Hec98, Kra98], typically returning the highest scoring model found. Our main interest is in learning BN structure from gene expression data [FLNP00, HGJY01, MM99, SGS00b]. In domains such as this, where the ratio of the number of observations to the number of variables is low (i.e., when we have sparse data), selecting a threshold for the conditional independence (CI) tests can be tricky, and repeated use of such tests can lead to inconsistencies [DD99]. Bayesian s...
Bayesian Variable Selection Using the Gibbs Sampler
, 2000
"... Specification of the linear predictor for a generalised linear model requires determining which variables to include. We consider Bayesian strategies for performing this variable selection. In particular we focus on approaches based on the Gibbs sampler. Such approaches may be implemented using the ..."
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Cited by 13 (2 self)
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Specification of the linear predictor for a generalised linear model requires determining which variables to include. We consider Bayesian strategies for performing this variable selection. In particular we focus on approaches based on the Gibbs sampler. Such approaches may be implemented using the publically available software BUGS. We illustrate the methods using a simple example. BUGS code is provided in an appendix. 1 Introduction In a Bayesian analysis of a generalised linear model, model uncertainty may be incorporated coherently by specifying prior probabilities for plausible models and calculating posterior probabilities using f(mjy) = f(m)f(yjm) P m2M f(m)f(y jm) ; m 2 M (1.1) where m denotes the model, M is the set of all models under consideration, f (m) is the prior probability of model m and f (yjm; fi m ) the likelihood of the data y under model m. The observed data y contribute to the posterior model probabilities through f(yjm), the marginal likelihood calculated...
Bayesian Variable and Link Determination for Generalised Linear Models
, 2000
"... this paper, we describe full Bayesian inference for generalised linear models where uncertainty ..."
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this paper, we describe full Bayesian inference for generalised linear models where uncertainty
Penalized Likelihood for Sparse Contingency Tables with an Application to FullLength cDNA Libraries
"... Background: The joint analysis of several categorical variables is a common task in many areas of biology, and is becoming central to systems biology investigations whose goal is to identify potentially complex interaction among variables belonging to a network. Interactions of arbitrary complexity ..."
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Background: The joint analysis of several categorical variables is a common task in many areas of biology, and is becoming central to systems biology investigations whose goal is to identify potentially complex interaction among variables belonging to a network. Interactions of arbitrary complexity are traditionally modeled in statistics by loglinear models. It is challenging to extend these to the high dimensional and potentially sparse data arising in computational biology. An important example, which provides the motivation for this article, is the analysis of socalled fulllength cDNA libraries of alternatively spliced genes, where we investigate relationships among the presence of various exons in transcript species. Results: We develop methods to perform model selection and parameter estimation in loglinear models for the analysis of sparse contingency tables, to study the interaction of two or more factors. Maximum Likelihood estimation of loglinear model coefficients is not appropriate because of the presence of zeros in the table’s cells, and new methods are required. We propose a computationally efficient ℓ1 penalization approach extending the Lasso algorithm to this context, and compare it to other procedures in a simulation study. We then illustrate these algorithms on contingency tables arising from fulllength cDNA libraries. Conclusions: We propose regularization methods that can be used successfully to detect complex interaction
Gibbs Variable Selection using BUGS
 Artificial Intelligence
, 1999
"... In this paper we discuss and present in detail the implementation of Gibbs variable selection as defined by Dellaportas et al. (2000, 2002) using the BUGS software (Spiegelhalter et al., 1996a,b,c). The specification of the likelihood, prior and pseudoprior distributions of the parameters as well a ..."
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In this paper we discuss and present in detail the implementation of Gibbs variable selection as defined by Dellaportas et al. (2000, 2002) using the BUGS software (Spiegelhalter et al., 1996a,b,c). The specification of the likelihood, prior and pseudoprior distributions of the parameters as well as the prior term and model probabilities are described in detail. Guidance is also provided for the calculation of the posterior probabilities within BUGS environment when the number of models is limited. We illustrate the application of this methodology in a variety of problems including linear regression, loglinear and binomial response models.
Efficient Model Determination for Discrete Graphical Models
, 2000
"... We present a novel methodology for bayesian model determination in discrete decomposable graphical models. We assign, for each given graph, a Hyper Dirichlet distribution on the matrix of cell probabilities. To ensure compatibility across models such prior distributions are obtained by marginalis ..."
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We present a novel methodology for bayesian model determination in discrete decomposable graphical models. We assign, for each given graph, a Hyper Dirichlet distribution on the matrix of cell probabilities. To ensure compatibility across models such prior distributions are obtained by marginalisation from the prior conditional on the complete graph. This leads to a prior distribution automatically satisfying the hyperconsistency criterion. Our contribution is twofold. On one hand we improve an existing methodology, the MC 3 algorithm by Madigan and York (1995). On the other hand we introduce an original methodology based on the use of the Reversible jump sampler by Green (1995) and Giudici and Green (1999). Legal movement, that is leading to a decomposable graph, are identied making use of the junction tree representation of the considered graph. Keywords: Bayesian model selection; Contingency table; Dirichlet distribution; Hyper Markov distribution; Junction tree; Marko...
Bayesian covariance matrix estimation using a mixture of decomposable graphical models. Unpublished manuscript
, 2005
"... Summary. Estimating a covariance matrix efficiently and discovering its structure are important statistical problems with applications in many fields. This article takes a Bayesian approach to estimate the covariance matrix of Gaussian data. We use ideas from Gaussian graphical models and model sele ..."
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Cited by 6 (2 self)
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Summary. Estimating a covariance matrix efficiently and discovering its structure are important statistical problems with applications in many fields. This article takes a Bayesian approach to estimate the covariance matrix of Gaussian data. We use ideas from Gaussian graphical models and model selection to construct a prior for the covariance matrix that is a mixture over all decomposable graphs, where a graph means the configuration of nonzero offdiagonal elements in the inverse of the covariance matrix. Our prior for the covariance matrix is such that the probability of each graph size is specified by the user and graphs of equal size are assigned equal probability. Most previous approaches assume that all graphs are equally probable. We give empirical results that show the prior that assigns equal probability over graph sizes outperforms the prior that assigns equal probability over all graphs, both in identifying the correct decomposable graph and in more efficiently estimating the covariance matrix. The advantage is greatest when the number of observations is small relative to the dimension of the covariance matrix. Our method requires the number of decomposable graphs for each graph size. We show how to estimate these numbers using simulation and that the simulation results agree with analytic results when such results are known. We also show how