Archive
Confidence-based decision theory
D. R. Bickel, “Coherent frequentism: A decision theory based on confidence sets,” Communications in Statistics – Theory and Methods 41, 1478-1496 (2012). Full article (open access) | 2009 version | Simple explanation (link added 27 June 2018)
To combine the self-consistency of Bayesian statistics with the objectivity of frequentist statistics, this paper formulates a framework of inference for developing novel statistical methods. The framework is based on a confidence posterior, a parameter probability distribution that does not require any prior distribution. While the Bayesian posterior is defined in terms of a conditional distribution given the observed data, the confidence posterior is instead defined such that the probability that the parameter value lies in any fixed subset of parameter space, given the observed data, is equal to the coverage rate of the corresponding confidence interval. Inferences based on the confidence posterior are reliable in the sense that the certainty level of a composite hypothesis is a weakly consistent estimate of the 0-1 indicator of hypothesis truth. At the same time, the confidence posterior is as non-contradictory as the Bayesian posterior since both satisfy the same coherence axioms. Using the theory of coherent upper and lower probabilities, the confidence posterior is generalized for situations in which no approximate or exact confidence set is available. Examples of hypothesis testing and estimation illustrate the range of applications of the proposed framework.
Additional summaries appear in the abstract and in Section 1.3 of the paper.
How to use priors with caution
D. R. Bickel, “Controlling the degree of caution in statistical inference with the Bayesian and frequentist approaches as opposite extremes,” Electronic Journal of Statistics 6, 686-709 (2012). Full text (open access) | 2011 preprint
This paper reports a novel probability-interval framework for combining strengths of frequentist and Bayesian methods on the basis of game-theoretic first principles. It enables data analysis on the basis of the posterior distribution that is a blend between a set of plausible Bayesian posterior distributions and a parameter distribution that represents an alternative method of data analysis. This paper’s framework of statistical inference is intended to facilitate the development of new methods to bridge the gap between the frequentist and Bayesian approaches. Four concrete examples illustrate how such intermediate methods can leverage strengths of the two extreme approaches.
Effect-size estimates from hypothesis probabilities
D. R. Bickel, “Empirical Bayes interval estimates that are conditionally equal to unadjusted confidence intervals or to default prior credibility intervals,” Statistical Applications in Genetics and Molecular Biology 11 (3), art. 7 (2012). Full article | 2010 preprint
The method contributed in this paper adjusts confidence intervals in multiple-comparison problems according to the estimated local false discovery rate. This shrinkage method performs substantially better than standard confidence intervals under the independence of the data across comparisons. A special case of the confidence intervals is the posterior median, which provides an improved method of ranking biological features such as genes, proteins, or genetic variants. The resulting ranks of features lead to better prioritization of which features to investigate further.
Minimax strength of statistical evidence
D. R. Bickel, “A predictive approach to measuring the strength of statistical evidence for single and multiple comparisons,” Canadian Journal of Statistics 39, 610–631 (2011). Full text | Revised preprint | 2010 draft
This paper introduces a novel approach to the multiple comparisons problem by generalizing a promising method of model selection developed by information theorists. The first two sections present that method and its main advantages over conventional approaches without burdening statisticians with unfamiliar terms from coding theory. A quantitative proteomics case study facilitates application of the new method to the analysis of data sets involving multiple biological features. The theorems describe its operating characteristics.
The cited medium-scale paper presented previous minimum description length (MDL) methods. Unlike those methods, the new MDL methods of the current paper are based on a conflation of the normalized maximum likelihood (NML) with the weighted likelihood (WL). The previous MDL methods are used in the CJS article for comparison with its NML/WL methods.
Observed confidence levels for microarrays, etc.
D. R. Bickel, “Estimating the null distribution to adjust observed confidence levels for genome-scale screening,” Biometrics 67, 363-370 (2011). Abstract and article | French abstract | Supplementary material | Simple explanation
This paper describes the first application of observed confidence levels to data of high-dimensional biology. The proposed method for multiple comparisons can take advantage of the estimated null distribution without any prior distribution. The new method is applied to microarray data to illustrate its advantages.
Shrinkage estimation of expression fold change
Z. Montazeri*, C. M. Yanofsky*, and D. R. Bickel, “Shrinkage estimation of effect sizes as an alternative to hypothesis testing followed by estimation in high-dimensional biology: Applications to differential gene expression,” Statistical Applications in Genetics and Molecular Biology 9 (1) 23 (2010). Article | Software
* the first two authors contributed equally
Significance v. fold change
C. M. Yanofsky and D. R. Bickel, “Validation of differential gene expression algorithms: Application comparing fold change estimation to hypothesis testing,” BMC Bioinformatics 11, 63 (2010). Article
Gene network reconstruction from transcriptional dynamics
D. R. Bickel, Z. Montazeri, P.-C. Hsieh, M. Beatty, S. J. Lawit, and N. J. Bate, “Gene network reconstruction from transcriptional dynamics under kinetic model uncertainty: A case for the second derivative,” Bioinformatics 25, 772-779 (2009).
Application: cis- & trans-effects on gene expression
M. Guo, S. Yang, M. Rupe, B. Hu, D. R. Bickel, L. Arthur, and O. Smith, “Genome-wide allele-specific expression analysis using Massively Parallel Signature Sequencing (MPSS) reveals cis- and trans-effects on gene expression in maize hybrid meristem tissue,” Plant Molecular Biology 66, 551-563 (2008).
David Bickel, statistician 
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