Book•
Bayes and Empirical Bayes Methods for Data Analysis
15 May 1996-
TL;DR: Approaches for Statistical Inference: The Bayes Approach, Model Criticism and Selection, and Performance of Bayes Procedures.
Abstract: Approaches for Statistical Inference. The Bayes Approach. The Empirical Bayes Approach. Performance of Bayes Procedures. Bayesian Computation. Model Criticism and Selection. Special Methods and Models. Case Studies. Appendices.
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TL;DR: In this paper, the authors consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined and derive a measure pD for the effective number in a model as the difference between the posterior mean of the deviances and the deviance at the posterior means of the parameters of interest, which is related to other information criteria and has an approximate decision theoretic justification.
Abstract: Summary. We consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined. Using an information theoretic argument we derive a measure pD for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest. In general pD approximately corresponds to the trace of the product of Fisher's information and the posterior covariance, which in normal models is the trace of the ‘hat’ matrix projecting observations onto fitted values. Its properties in exponential families are explored. The posterior mean deviance is suggested as a Bayesian measure of fit or adequacy, and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages. Adding pD to the posterior mean deviance gives a deviance information criterion for comparing models, which is related to other information criteria and has an approximate decision theoretic justification. The procedure is illustrated in some examples, and comparisons are drawn with alternative Bayesian and classical proposals. Throughout it is emphasized that the quantities required are trivial to compute in a Markov chain Monte Carlo analysis.
11,691 citations
Book•
24 Aug 2012
TL;DR: This textbook offers a comprehensive and self-contained introduction to the field of machine learning, based on a unified, probabilistic approach, and is suitable for upper-level undergraduates with an introductory-level college math background and beginning graduate students.
Abstract: Today's Web-enabled deluge of electronic data calls for automated methods of data analysis. Machine learning provides these, developing methods that can automatically detect patterns in data and then use the uncovered patterns to predict future data. This textbook offers a comprehensive and self-contained introduction to the field of machine learning, based on a unified, probabilistic approach. The coverage combines breadth and depth, offering necessary background material on such topics as probability, optimization, and linear algebra as well as discussion of recent developments in the field, including conditional random fields, L1 regularization, and deep learning. The book is written in an informal, accessible style, complete with pseudo-code for the most important algorithms. All topics are copiously illustrated with color images and worked examples drawn from such application domains as biology, text processing, computer vision, and robotics. Rather than providing a cookbook of different heuristic methods, the book stresses a principled model-based approach, often using the language of graphical models to specify models in a concise and intuitive way. Almost all the models described have been implemented in a MATLAB software package--PMTK (probabilistic modeling toolkit)--that is freely available online. The book is suitable for upper-level undergraduates with an introductory-level college math background and beginning graduate students.
8,059 citations
Cites methods from "Bayes and Empirical Bayes Methods f..."
..., (Carlin and Louis 1996; Efron 2010)), so it is widely used by non-Bayesians....
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TL;DR: This work generalizes the method proposed by Gelman and Rubin (1992a) for monitoring the convergence of iterative simulations by comparing between and within variances of multiple chains, in order to obtain a family of tests for convergence.
Abstract: We generalize the method proposed by Gelman and Rubin (1992a) for monitoring the convergence of iterative simulations by comparing between and within variances of multiple chains, in order to obtain a family of tests for convergence. We review methods of inference from simulations in order to develop convergence-monitoring summaries that are relevant for the purposes for which the simulations are used. We recommend applying a battery of tests for mixing based on the comparison of inferences from individual sequences and from the mixture of sequences. Finally, we discuss multivariate analogues, for assessing convergence of several parameters simultaneously.
5,493 citations
Cites background from "Bayes and Empirical Bayes Methods f..."
...Key Words: Convergence diagnosis; Inference; Markov chain Monte Carlo....
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TL;DR: For example, this paper showed that using the adjusted Wald test with null rather than estimated standard error yields coverage probabilities close to nominal confidence levels, even for very small sample sizes, and that the 95% score interval has similar behavior as the adjusted-Wald interval obtained after adding two "successes" and two "failures" to the sample.
Abstract: For interval estimation of a proportion, coverage probabilities tend to be too large for “exact” confidence intervals based on inverting the binomial test and too small for the interval based on inverting the Wald large-sample normal test (i.e., sample proportion ± z-score × estimated standard error). Wilson's suggestion of inverting the related score test with null rather than estimated standard error yields coverage probabilities close to nominal confidence levels, even for very small sample sizes. The 95% score interval has similar behavior as the adjusted Wald interval obtained after adding two “successes” and two “failures” to the sample. In elementary courses, with the score and adjusted Wald methods it is unnecessary to provide students with awkward sample size guidelines.
3,276 citations
References
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TL;DR: In this article, Bayes and empirical Bayes methods for data analysis are presented for Data Analysis. But, they do not consider the use of data augmentation in data analysis.
Abstract: (1997). Bayes and Empirical Bayes Methods for Data Analysis. Technometrics: Vol. 39, No. 3, pp. 337-337.
795 citations
"Bayes and Empirical Bayes Methods f..." refers methods in this paper
...2nd ed (0) Statistics g6102: statistical modeling for data Bayesian Data Analysis (second edition), by Gelman, * B.P. Carlin and T.A. Louis (2000) Bayes and Empirical Bayes Methods for Data Analysis, Bayes and empirical bayes methods for data Book information and reviews for ISBN:1584881704,Bayes…...
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...2nd ed (0) Statistics g6102: statistical modeling for data Bayesian Data Analysis (second edition), by Gelman, * B.P. Carlin and T.A. Louis (2000) Bayes and Empirical Bayes Methods for Data Analysis, Bayes and empirical bayes methods for data Book information and reviews for ISBN:1584881704,Bayes And Empirical Bayes Methods For Data Analysis, Second Edition by Bradley P. Carlin....
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