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56
Prior Information and Uncertainty in Inverse Problems
, 2001
"... Solving any inverse problem requires understanding the uncertainties in the data to know what it means to fit the data. We also need methods to incorporate dataindependent prior information to eliminate unreasonable models that fit the data. Both of these issues involve subtle choices that may ..."
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Cited by 29 (5 self)
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Solving any inverse problem requires understanding the uncertainties in the data to know what it means to fit the data. We also need methods to incorporate dataindependent prior information to eliminate unreasonable models that fit the data. Both of these issues involve subtle choices that may significantly influence the results of inverse calculations. The specification of prior information is especially controversial. How does one quantify information? What does it mean to know something about a parameter a priori? In this tutorial we discuss Bayesian and frequentist methodologies that can be used to incorporate information into inverse calculations. In particular we show that apparently conservative Bayesian choices, such as representing interval constraints by uniform probabilities (as is commonly done when using genetic algorithms, for example) may lead to artificially small uncertainties. We also describe tools from statistical decision theory that can be used to...
Learning to be Bayesian without supervision
 in Adv. Neural Information Processing Systems (NIPS*06
, 2007
"... Bayesian estimators are defined in terms of the posterior distribution. Typically, this is written as the product of the likelihood function and a prior probability density, both of which are assumed to be known. But in many situations, the prior density is not known, and is difficult to learn from ..."
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Cited by 25 (6 self)
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Bayesian estimators are defined in terms of the posterior distribution. Typically, this is written as the product of the likelihood function and a prior probability density, both of which are assumed to be known. But in many situations, the prior density is not known, and is difficult to learn from data since one does not have access to uncorrupted samples of the variable being estimated. We show that for a wide variety of observation models, the Bayes least squares (BLS) estimator may be formulated without explicit reference to the prior. Specifically, we derive a direct expression for the estimator, and a related expression for the mean squared estimation error, both in terms of the density of the observed measurements. Each of these priorfree formulations allows us to approximate the estimator given a sufficient amount of observed data. We use the first form to develop practical nonparametric approximations of BLS estimators for several different observation processes, and the second form to develop a parametric family of estimators for use in the additive Gaussian noise case. We examine the empirical performance of these estimators as a function of the amount of observed data. 1
Accounting for Phylogenetic Uncertainty in Biogeography: A Bayesian Approach to DispersalVicariance Analysis of the Thrushes (Aves: Turdus)
"... Abstract. — The phylogeny of the thrushes (Aves: Turdus) has been difficult to reconstruct due to short internal branches and lack of node support for certain parts of the tree. Reconstructing the biogeographic history of this group is further complicated by the fact that current implementations of ..."
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Cited by 23 (0 self)
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Abstract. — The phylogeny of the thrushes (Aves: Turdus) has been difficult to reconstruct due to short internal branches and lack of node support for certain parts of the tree. Reconstructing the biogeographic history of this group is further complicated by the fact that current implementations of biogeographic methods, such as dispersalvicariance analysis (DIVA; Ronquist, 1997), require a fully resolved tree. Here, we apply a Bayesian approach to dispersalvicariance analysis that accounts for phylogenetic uncertainty and allows a more accurate analysis of the biogeographic history of lineages. Specifically, ancestral area reconstructions can be presented as marginal distributions, thus displaying the underlying topological uncertainty. Moreover, if there are multiple optimal solutions for a single node on a certain tree, integrating over the posterior distribution of trees often reveals a preference for a narrower set of solutions. We find that despite the uncertainty in tree topology, ancestral area reconstructions indicate that the Turdus clade originated in the eastern Palearctic during the Late Miocene. This was followed by an early dispersal to Africa from where a worldwide radiation took place. The uncertainty in tree topology and short branch lengths seems to indicate that this radiation took place within a limited time span during the Late Pliocene. The
Least Squares Estimation Without Priors or Supervision
, 2011
"... Selection of an optimal estimator typically relies on either supervised training samples (pairs of measurements and their associated true values) or a prior probability model for the true values. Here, we consider the problem of obtaining a least squares estimator given a measurement process with kn ..."
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Cited by 5 (1 self)
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Selection of an optimal estimator typically relies on either supervised training samples (pairs of measurements and their associated true values) or a prior probability model for the true values. Here, we consider the problem of obtaining a least squares estimator given a measurement process with known statistics (i.e., a likelihood function) and a set of unsupervised measurements, each arising from a corresponding true value drawn randomly from an unknown distribution. We develop a general expression for a nonparametric empirical Bayes least squares (NEBLS) estimator, which expresses the optimal least squares estimator in terms of the measurement density, with no explicit reference to the unknown (prior) density. We study the conditions under which such estimators exist and derive specific forms for a variety of different measurement processes. We further show that each of these NEBLS estimators may be used to express the mean squared estimation error as an expectation over the measurement density alone, thus generalizing Stein’s unbiased
Using Loss Functions for DIF Detection: An Empirical Bayes Approach
"... We investigated a DIF flaggbzg method based on loss functions. The approach builds on earlier research that involved the development of an empirical Bayes (EB) enhancement to MantelHaenszel (MH) DIF analysis. The posterior distribution of DIF parameters was estimated nd used to obtain the posterio ..."
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Cited by 5 (2 self)
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We investigated a DIF flaggbzg method based on loss functions. The approach builds on earlier research that involved the development of an empirical Bayes (EB) enhancement to MantelHaenszel (MH) DIF analysis. The posterior distribution of DIF parameters was estimated nd used to obtain the posterior expected loss for the proposed approach ndfor competing classification rules. Under reasonable assumptions about the relative seriousness of Type I and Type II errors, the lossfunctionbased DIF etection rule was found to perform better than the comnzonly used "A, " "B, " and "C " DIF classification system, especially in small samples. The results of a ManteIHaenszel (MH; 1959) analysis of differential item functioning (DIF) typically include an index of the magnitude of DIF, along with an estimated standard error (see Holland & Thayer, 1988). Decisions about whether to discard items or flag them for review are typically based on the statistical significance of the MH chisquare or the magnitude of the MH odds ratio estimate. An approach to DIF classification that incorporates both these criteria is the system developed by Educational Testing Service (ETS) for categorizing DIF as negligible ("A"), slight to moderate ("B"), or moderate to severe ("C"). In this study, we explore an alternative DIF detection method based on loss functions. A decision is made to identify an item as a potential DIF item if the expected loss associated with failing to flag the item is greater than that associated with flagging the item. The work was a spinoff of earlier research (Zwick, Thayer, & Lewis, 1997; in press) in which we used a Bayesian variant of the ETS DIF classification system to estimate the probabilities that the true DIF for
Empirical Bayes modeling, computation, and accuracy
"... This article is intended as an expositional overview of empirical Bayes modeling methodology, presented in a simplified framework that reduces technical difficulties. The two principal empirical Bayes approaches, called fmodeling and gmodeling here, are described and compared. A series of computat ..."
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Cited by 4 (0 self)
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This article is intended as an expositional overview of empirical Bayes modeling methodology, presented in a simplified framework that reduces technical difficulties. The two principal empirical Bayes approaches, called fmodeling and gmodeling here, are described and compared. A series of computational formulas are developed to assess the frequentist accuracy of empirical Bayes applications. Several examples, both artificial and genuine, show the strengths and limitations of the two methodologies.
Empirical Bayes least squares estimation without an explicit prior (Tech
 York University
, 2007
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Empirical Bayes Methods for Smoothing Data and for Simultaneous Estimation of Many Parameters
"... A recent successful development is found in a series of innovative, new statistical methods for smoothing data that are based on the empirical Bayes method. This paper emphasizes their practical usefulness in medical sciences and their theoretically close relationship with the problem of simultaneou ..."
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A recent successful development is found in a series of innovative, new statistical methods for smoothing data that are based on the empirical Bayes method. This paper emphasizes their practical usefulness in medical sciences and their theoretically close relationship with the problem of simultaneous estimation of parameters, depending on strata. The paper also presents two examples of analyzing epidemiological data obtained in Japan using the smoothing methods to illustrate their favorable performance.
The Status of Evaluating Accuracy of Regional Forecasts.’’ Review of Regional Studies 33(July 2003a):85–103. ———.‘‘Structural Regional Factors that Determine Absolute and Relative Accuracy of U.S
"... Recent reviews of the regional growth and change literature conclude that the strand of research attempting to identify determinants of growth in general terms has been reasonably successful, but the strand attempting to provide public policy direction has been far less successful. A critical distin ..."
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Recent reviews of the regional growth and change literature conclude that the strand of research attempting to identify determinants of growth in general terms has been reasonably successful, but the strand attempting to provide public policy direction has been far less successful. A critical distinction between these two strands is that the former studies the past while the latter makes a forecast. One of the reasons we may not have been particularly effective in guiding public policy is that we haven’t addressed the question, “Is the accuracy of our regional forecasting record acceptable? ” This paper reviews the literature on evaluating regional forecast accuracy, discusses the question of “When is a regional forecast ‘accurate’? ” and outlines how accuracy analysis can be used to improve forecast precision. A concluding section suggests possible avenues for future research.
Reverse engineering gene networks using genomic timecourse data
, 2010
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.