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37
Schwarz, Wallace, and Rissanen: Intertwining Themes in Theories of Model Selection
, 2000
"... Investigators interested in model order estimation have tended to divide themselves into widely separated camps; this survey of the contributions of Schwarz, Wallace, Rissanen, and their coworkers attempts to build bridges between the various viewpoints, illuminating connections which may have pr ..."
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Investigators interested in model order estimation have tended to divide themselves into widely separated camps; this survey of the contributions of Schwarz, Wallace, Rissanen, and their coworkers attempts to build bridges between the various viewpoints, illuminating connections which may have previously gone unnoticed and clarifying misconceptions which seem to have propagated in the applied literature. Our tour begins with Schwarz's approximation of Bayesian integrals via Laplace's method. We then introduce the concepts underlying Rissanen 's minimum description length principle via a Bayesian scenario with a known prior; this provides the groundwork for understanding his more complex nonBayesian MDL which employs a "universal" encoding of the integers. Rissanen's method of parameter truncation is contrasted with that employed in various versions of Wallace's minimum message length criteria.
ON THE POSTERIOR DISTRIBUTION OF THE NUMBER OF COMPONENTS IN A FINITE MIXTURE
"... The posterior distribution of the number of components k in a finite mixture satisfies a set of inequality constraints. The result holds irrespective of the parametric form of the mixture components and under assumptions on the prior distribution weaker than those routinely made in the literature on ..."
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The posterior distribution of the number of components k in a finite mixture satisfies a set of inequality constraints. The result holds irrespective of the parametric form of the mixture components and under assumptions on the prior distribution weaker than those routinely made in the literature on Bayesian analysis of finite mixtures. The inequality constraints can be used to perform an “internal ” consistency check of MCMC estimates of the posterior distribution of k and to provide improved estimates which are required to satisfy the constraints. Bounds on the posterior probability of k components are derived using the constraints. Implications on prior distribution specification and on the adequacy of the posterior distribution of k as a tool for selecting an adequate number of components in the mixture are also explored. 1. Introduction. Finite
Performance of Bayesian Model Selection Criteria for Gaussian Mixture Models
, 2009
"... Bayesian methods are widely used for selecting the number of components in a mixture models, in part because frequentist methods have difficulty in addressing this problem in general. Here we compare some of the Bayesianly motivated or justifiable methods for choosing the number of components in a o ..."
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Bayesian methods are widely used for selecting the number of components in a mixture models, in part because frequentist methods have difficulty in addressing this problem in general. Here we compare some of the Bayesianly motivated or justifiable methods for choosing the number of components in a onedimensional Gaussian mixture model: posterior probabilities for a wellknown proper prior, BIC, ICL, DIC and AIC. We also introduce a new explicit unitinformation prior for mixture models, analogous to the prior to which BIC corresponds in regular statistical models. We base the comparison on a simulation study, designed to reflect published estimates of mixture model parameters from the scientific literature across a range of disciplines. We found that BIC clearly outperformed the five other
Detecting Mines in Minefields with Linear Characteristics
, 1999
"... We consider the problem of detecting minefields using aerial images. A first stage of image processing has reduced the image to a set of points, each one representing a possible mine. Our task is to decide which ones are actual mines. We assume that the minefield consists of approximately parallel r ..."
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We consider the problem of detecting minefields using aerial images. A first stage of image processing has reduced the image to a set of points, each one representing a possible mine. Our task is to decide which ones are actual mines. We assume that the minefield consists of approximately parallel rows of mines laid out according to a probability distribution that encourages evenly spaced, linear patterns. The noise points are assumed to be distributed as a Poisson process. We construct a Markov chain Monte Carlo algorithm to estimate the model and obtain posterior probabilities for each point being a mine. The algorithm performs well on
Stable Limit Laws for Harmonic Mean Estimators of Marginal Likelihoods
, 2010
"... The task of calculating marginal likelihoods arises in a wide array of statistical inference problems, including the evaluation of Bayes factors for model selection and hypothesis testing. Although Markov chain Monte Carlo methods have simplified many posterior calculations needed for practical Baye ..."
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The task of calculating marginal likelihoods arises in a wide array of statistical inference problems, including the evaluation of Bayes factors for model selection and hypothesis testing. Although Markov chain Monte Carlo methods have simplified many posterior calculations needed for practical Bayesian analysis, the evaluation of marginal likelihoods remains difficult. We consider the behavior of the wellknown harmonic mean estimator (Newton and Raftery, 1994) of the marginal likelihood, which converges almostsurely but may have infinite variance and so may not obey a central limit theorem. We give examples illustrating the convergence in distribution of the harmonic mean estimator to a onesided stable law with characteristic exponent 1 < α < 2. While the harmonic mean estimator does converge almost surely, we show that it does so at rate n −ǫ where ǫ = 1 − α −1 is often as small as 0.10 or 0.01. In such a case, the reduction of Monte Carlo sampling error by a factor of two requires increasing the Monte Carlo sample size by a factor of 2 1/ǫ, or in excess of 2.5 ·10 30 when ǫ = 0.01. We explore the possibility of estimating the parameters of the limiting stable distribution to provide accelerated convergence.
αStable Limit Laws for Harmonic Mean Estimators of Marginal Likelihoods
, 2010
"... The task of calculating marginal likelihoods arises in a wide array of statistical inference problems, including the evaluation of Bayes factors for model selection and hypothesis testing. Although Markov chain Monte Carlo methods have simplified many posterior calculations needed for practical Baye ..."
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The task of calculating marginal likelihoods arises in a wide array of statistical inference problems, including the evaluation of Bayes factors for model selection and hypothesis testing. Although Markov chain Monte Carlo methods have simplified many posterior calculations needed for practical Bayesian analysis, the evaluation of marginal likelihoods remains difficult. We consider the behavior of the wellknown harmonic mean estimator (Newton and Raftery, 1994) of the marginal likelihood, which converges almostsurely but may have infinite variance and so may not obey a central limit theorem.
Delivery: an open–source model–based Bayesian seismic inversion program
"... We introduce a new opensource toolkit for modelbased Bayesian seismic inversion called Delivery. The prior model in Delivery is a trace–local layer stack, with rock physics information taken from log analysis and layer times initialised from picks. We allow for uncertainty in both the fluid type a ..."
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We introduce a new opensource toolkit for modelbased Bayesian seismic inversion called Delivery. The prior model in Delivery is a trace–local layer stack, with rock physics information taken from log analysis and layer times initialised from picks. We allow for uncertainty in both the fluid type and saturation in reservoir layers: variation in seismic responses due to fluid effects are taken into account via Gassman’s equation. Multiple stacks are supported, so the software implicitly performs a full AVO inversion using approximate Zoeppritz equations. The likelihood function is formed from a convolutional model with specified wavelet(s) and noise level(s). Uncertainties and irresolvabilities in the inverted models are captured by the generation of multiple stochastic models from the Bayesian posterior, all of which acceptably match the seismic data, log data, and rough initial picks of the horizons. Postinversion analysis of the inverted stochastic models then facilitates the answering of commercially useful questions, e.g. the probability of hydrocarbons, the expected reservoir volume and its uncertainty, and the distribution of net sand. Delivery is written in java, and thus platform independent, but the SU data backbone makes the inversion particularly suited to Unix/Linux environments and cluster systems.
Semiparametric Bayesian . . .
, 2004
"... We describe procedures for Bayesian estimation and testing in cross sectional, panel data and nonlinear smooth coefficient models. The smooth coefficient model is a generalization of the partially linear or additive model wherein coefficients on linear explanatory variables are treated as unknown fu ..."
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We describe procedures for Bayesian estimation and testing in cross sectional, panel data and nonlinear smooth coefficient models. The smooth coefficient model is a generalization of the partially linear or additive model wherein coefficients on linear explanatory variables are treated as unknown functions of an observable covariate. In the approach we describe, points on the regression lines are regarded as unknown parameters and priors are placed on differences between adjacent points to introduce the potential for smoothing the curves. The algorithms we describe are quite simple to implement for example, estimation, testing and smoothing parameter selection can be carried out analytically in the crosssectional smooth coefficient model. We apply our methods using data from the National Longitudinal Survey of Youth (NLSY). Using the NLSY data we first explore the relationship between ability and log wages and flexibly model how returns to schooling vary with measured cognitive ability. We also examine model of female labor supply and use this example to illustrate how the described techniques can been applied in nonlinear settings.
Modeling and Analysis of SAGE Libraries
, 2005
"... Professor Andrej Rotter and Dr. Magdalena Popesco for introducing us to SAGE and kindly sharing their data with us. The supplementary material and the programs implementing the algorithms may be accessed at ..."
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Professor Andrej Rotter and Dr. Magdalena Popesco for introducing us to SAGE and kindly sharing their data with us. The supplementary material and the programs implementing the algorithms may be accessed at
Approximating the marginal likelihood using copula
, 810
"... Model selection is an important activity in modern data analysis and the conventional Bayesian approach to this problem involves calculation of marginal likelihoods for different models, together with diagnostics which examine specific aspects of model fit. Calculating the marginal likelihood is a d ..."
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Model selection is an important activity in modern data analysis and the conventional Bayesian approach to this problem involves calculation of marginal likelihoods for different models, together with diagnostics which examine specific aspects of model fit. Calculating the marginal likelihood is a difficult computational problem. Our article proposes some extensions of the Laplace approximation for this task that are related to copula models and which are easy to apply. Variations which can be used both with and without simulation from the posterior distribution are considered, as well as use of the approximations with bridge sampling and in random effects models with a large number of latent variables. The use of a tcopula to obtain higher accuracy when multivariate dependence is not well captured by a Gaussian copula is also discussed.