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Karl J. Friston - One of the best experts on this subject based on the ideXlab platform.

  • Sample size and the fallacies of Classical Inference.
    NeuroImage, 2013
    Co-Authors: Karl J. Friston
    Abstract:

    Abstract I would like to thank Michael Ingre, Martin Lindquist and their co-authors for their thoughtful responses to my ironic Comments and Controversies piece. I was of two minds about whether to accept the invitation to reply — largely because I was convinced by most of their observations. I concluded that I should say this explicitly, taking the opportunity to consolidate points of consensus and highlight outstanding issues.

  • Classical and Bayesian Inference
    2003
    Co-Authors: Karl J. Friston, William D Penny
    Abstract:

    Since its inception, statistical parametric mapping (SPM) has proved useful for characterizing neuroimaging data sequences. However, SPM is limited because it is based on Classical Inference procedures. This chapter introduces a more general framework, which places SPM in a broader context and points to alternative ways of characterizing and making Inferences about regionally specific effects in neuroimaging. In particular, procedures used in conventional data analysis in terms of hierarchical linear models are formulated and the connection between Classical Inference and empirical Bayesian Inference is established through covariance component estimation. This estimation is based on the EM algorithm. This chapter also emphasizes on the applications of the theory to a range of important issues in neuroimaging. These issues include estimating nonsphericity or variance components in fMRI time series that can arise from serial correlations within subject or that are induced by multi-subject studies. It also includes, Bayesian models for imaging data, in which effects at one voxel are constrained by responses in others. © 2004 Elsevier Inc. All rights reserved.

  • Classical and bayesian Inference in neuroimaging theory
    NeuroImage, 2002
    Co-Authors: Karl J. Friston, William D Penny, Christophe Phillips, Stefan J Kiebel, Geoffrey E Hinton, John Ashburner
    Abstract:

    This paper reviews hierarchical observation models, used in functional neuroimaging, in a Bayesian light. It emphasizes the common ground shared by Classical and Bayesian methods to show that conventional analyses of neuroimaging data can be usefully extended within an empirical Bayesian framework. In particular we formulate the procedures used in conventional data analysis in terms of hierarchical linear models and establish a connection between Classical Inference and parametric empirical Bayes (PEB) through covariance component estimation. This estimation is based on an expectation maximization or EM algorithm. The key point is that hierarchical models not only provide for appropriate Inference at the highest level but that one can revisit lower levels suitably equipped to make Bayesian Inferences. Bayesian Inferences eschew many of the difficulties encountered with Classical Inference and characterize brain responses in a way that is more directly predicated on what one is interested in. The motivation for Bayesian approaches is reviewed and the theoretical background is presented in a way that relates to conventional methods, in particular restricted maximum likelihood (ReML). This paper is a technical and theoretical prelude to subsequent papers that deal with applications of the theory to a range of important issues in neuroimaging. These issues include; (i) Estimating nonsphericity or variance components in fMRI time-series that can arise from serial correlations within subject, or are induced by multisubject (i.e., hierarchical) studies. (ii) Spatiotemporal Bayesian models for imaging data, in which voxels-specific effects are constrained by responses in other voxels. (iii) Bayesian estimation of nonlinear models of hemodynamic responses and (iv) principled ways of mixing structural and functional priors in EEG source reconstruction. Although diverse, all these estimation problems are accommodated by the PEB framework described in this paper.

David H. Annis - One of the best experts on this subject based on the ideXlab platform.

Thierry Magnac - One of the best experts on this subject based on the ideXlab platform.

  • Set identification, moment restrictions, and Inference
    Annual Review of Economics, 2017
    Co-Authors: Christian Bontemps, Thierry Magnac
    Abstract:

    For the past 10 years, the topic of set identification has been much studied in the econometric literature. Classical Inference methods have been generalized to the case in which moment inequalities and equalities define a set instead of a point. We review several instances of partial identification by focusing on examples in which the underlying economic restrictions are expressed as linear moments. This setting illustrates the fact that convex analysis helps not only for characterizing the identified set but also for Inference. From this perspective, we review Inference methods using convex analysis or inversion of tests and detail how geometric characterizations can be useful.

  • Set Identification, Moment Restrictions, and Inference
    2017
    Co-Authors: Christian Bontemps, Thierry Magnac
    Abstract:

    For the last ten years, the topic of set identification has been much studied in the econometric literature. Classical Inference methods have been generalized to the case in which moment inequalities and equalities define a set instead of a point. We review several instances of partial identification by focusing on examples in which the underlying economic restrictions are expressed as linear moments. This setting illustrates the fact that convex analysis helps not only in characterizing the identified set but also for Inference. In this perspective, we review Inference methods using convex analysis or inversion of tests and detail how geometric characterizations can be useful.

Andreas Buja - One of the best experts on this subject based on the ideXlab platform.

  • Models as Approximations - A Conspiracy of Random Regressors and Model Deviations Against Classical Inference in Regression
    Statistical Science, 2015
    Co-Authors: Andreas Buja, Richard A. Berk, Lawrence D. Brown, Edward I. George, Emil Pitkin, Mikhail Traskin, Linda Zhao, K. Zhang
    Abstract:

    Abstract. More than thirty years ago Halbert White inaugurated a “modelrobust” form of statistical Inference based on the “sandwich estimator” of standard error. It is asymptotically correct even under “model misspecification,” that is, when models are approximations rather than generative truths. It is well-known to be “heteroskedasticity-consistent”, but it is less well-known to be “nonlinearity-consistent” as well. Nonlinearity, however, raises fundamental issues: When fitted models are approximations, conditioning on the regressor is no longer permitted because the ancillarity argument that justifies it breaks down. Two effects occur: (1) parameters become dependent on the regressor distribution; (2) the sampling variability of parameter estimates no longer derives from the conditional distribution of the response alone. Additional sampling variability arises when the nonlinearity conspires with the randomness of the regressors to generate a 1/ √ N contribution to standard errors. Asymptotically, standard errors from “model-trusting” fixedregressor theories can deviate from those of “model-robust” randomregressor theories by arbitrary magnitudes. In the case of linear models, a test will be proposed for comparing the two types of standard errors.

  • The Conspiracy of Random Predictors and Model Violations against Classical Inference in Regression
    2014
    Co-Authors: Andreas Buja, Richard A. Berk, Lawrence D. Brown, Edward I. George, Emil Pitkin, Mikhail Traskin, K. Zhang, Linda Zhao
    Abstract:

    xed, White permits models to be \misspecied" and predictors to be random. Careful reading of his theory shows that it is a synergistic eect | a \conspiracy" | of nonlinearity and randomness of the predictors that has the deepest consequences for statistical Inference. It will be seen that the synonym \heteroskedasticity-consistent estimator" for the sandwich estimator is misleading because nonlinearity is a more consequential form of model deviation than heteroskedasticity, and both forms are handled asymptotically correctly by the sandwich estimator. The same analysis shows that a valid alternative to the sandwich estimator is given by the \pairs bootstrap" for which we establish a direct connection to the sandwich estimator. We continue with an asymptotic comparison of the sandwich estimator and the standard error estimator from Classical linear models theory. The comparison shows that when standard errors from linear models theory deviate from their sandwich analogs, they are usually too liberal, but occasionally they can be too conservative as well. We conclude by answering questions that would occur to statisticians acculturated to the assumption of model correctness and conditionality on the predictors: (1) Why should we be interested in Inference for models that are not correct? (2) What are the arguments for conditioning on predictors, and why might they not be valid? In this review we limit ourselves to linear least squares regression as the demonstration object, but the qualitative insights hold for all forms of regression.

  • Models as Approximations: How Random Predictors and Model Violations Invalidate Classical Inference in Regression
    arXiv: Methodology, 2014
    Co-Authors: Andreas Buja, Richard A. Berk, Lawrence D. Brown, Edward I. George, Emil Pitkin, Mikhail Traskin, K. Zhan, Linda Zhao
    Abstract:

    We review and interpret the early insights of Halbert White who over thirty years ago inaugurated a form of statistical Inference for regression models that is asymptotically correct even under "model misspecification," that is, under the assumption that models are approximations rather than generative truths. This form of Inference, which is pervasive in econometrics, relies on the "sandwich estimator" of standard error. Whereas linear models theory in statistics assumes models to be true and predictors to be fixed, White's theory permits models to be approximate and predictors to be random. Careful reading of his work shows that the deepest consequences for statistical Inference arise from a synergy --- a "conspiracy" --- of nonlinearity and randomness of the predictors which invalidates the ancillarity argument that justifies conditioning on the predictors when they are random. Unlike the standard error of linear models theory, the sandwich estimator provides asymptotically correct Inference in the presence of both nonlinearity and heteroskedasticity. An asymptotic comparison of the two types of standard error shows that discrepancies between them can be of arbitrary magnitude. If there exist discrepancies, standard errors from linear models theory are usually too liberal even though occasionally they can be too conservative as well. A valid alternative to the sandwich estimator is provided by the "pairs bootstrap"; in fact, the sandwich estimator can be shown to be a limiting case of the pairs bootstrap. We conclude by giving meaning to regression slopes when the linear model is an approximation rather than a truth. --- In this review we limit ourselves to linear least squares regression, but many qualitative insights hold for most forms of regression.

Christian Bontemps - One of the best experts on this subject based on the ideXlab platform.

  • Set identification, moment restrictions, and Inference
    Annual Review of Economics, 2017
    Co-Authors: Christian Bontemps, Thierry Magnac
    Abstract:

    For the past 10 years, the topic of set identification has been much studied in the econometric literature. Classical Inference methods have been generalized to the case in which moment inequalities and equalities define a set instead of a point. We review several instances of partial identification by focusing on examples in which the underlying economic restrictions are expressed as linear moments. This setting illustrates the fact that convex analysis helps not only for characterizing the identified set but also for Inference. From this perspective, we review Inference methods using convex analysis or inversion of tests and detail how geometric characterizations can be useful.

  • Set Identification, Moment Restrictions, and Inference
    2017
    Co-Authors: Christian Bontemps, Thierry Magnac
    Abstract:

    For the last ten years, the topic of set identification has been much studied in the econometric literature. Classical Inference methods have been generalized to the case in which moment inequalities and equalities define a set instead of a point. We review several instances of partial identification by focusing on examples in which the underlying economic restrictions are expressed as linear moments. This setting illustrates the fact that convex analysis helps not only in characterizing the identified set but also for Inference. In this perspective, we review Inference methods using convex analysis or inversion of tests and detail how geometric characterizations can be useful.