Archive
Empirical Bayes single-comparison procedure
D. R. Bickel, “Small-scale inference: Empirical Bayes and confidence methods for as few as a single comparison,” International Statistical Review 82, 457-476 (2014). Published version | 2011 preprint | Simple explanation (link added 21 June 2017)
Parametric empirical Bayes methods of estimating the local false discovery rate by maximum likelihood apply not only to the large-scale settings for which they were developed, but, with a simple modification, also to small numbers of comparisons. In fact, data for a single comparison are sufficient under broad conditions, as seen from applications to measurements of the abundance levels of 20 proteins and from simulation studies with confidence-based inference as the competitor.
Coherent inference after checking a prior
D. R. Bickel, “Bayesian revision of a prior given prior-data conflict, expert opinion, or a similar insight: A large-deviation approach,” Working Paper, University of Ottawa, deposited in uO Research at http://hdl.handle.net/10393/34089/ (2015). 2015 preprint
Fiducial nonparametrics
Sonderegger, Derek L.; Hannig, Jan
Fiducial theory for free-knot splines. Contemporary developments in statistical theory, 155–189,
Springer Proc. Math. Stat., 68, Springer, Cham, 2014.
62F12 (62F10 62F99 65D07)
The research reported reflects the recent surge in developments of Fisher’s fiducial argument [S. Nadarajah, S. Bityukov and N. Krasnikov, Stat. Methodol. 22 (2015), 23–46; MR3261595]. The work of this chapter is carried out within the framework of generalized fiducial inference [J. Hannig, Statist. Sinica 19 (2009), no. 2, 491–544; MR2514173 (2010h:62071)], which is built on the functional-model formulation of fiducial statistics [A. P. Dawid, M. Stone and M. Stone, Ann. Statist. 10 (1982), no. 4, 1054–1074; MR0673643 (83m:62008)] rather than on the broadly equivalent confidence-based tradition beginning with [G. N. Wilkinson, J. Roy. Statist. Soc. Ser. B 39 (1977), no. 2, 119–171; MR0652326 (58 #31491)] and generalized by [E. E. M. van Berkum, H. N. Linssen and D. Overdijk, J. Statist. Plann. Inference 49 (1996), no. 3, 305–317; MR1381161 (97k:62007)].
{For the entire collection see MR3149911.}
Reviewed by David R. Bickel
This review first appeared at “Fiducial theory for free-knot splines” (Mathematical Reviews) and is used with permission from the American Mathematical Society.
Maximum entropy over a set of posteriors
D. R. Bickel, “Blending Bayesian and frequentist methods according to the precision of prior information with applications to hypothesis testing,” Statistical Methods & Applications 24, 523-546 (2015). Published article | 2012 preprint | 2011 preprint | Slides | Simple explanation
This framework of statistical inference facilitates the development of new methodology to bridge the gap between the frequentist and Bayesian theories. As an example, a simple and practical method for combining p-values with a set of possible posterior probabilities is provided.
In this general approach, Bayesian inference is used when the prior distribution is known, frequentist inference is used when nothing is known about the prior, and both types of inference are blended according to game theory when the prior is known to be a member of some set. (The robust Bayes framework represents knowledge about a prior in terms of a set of possible priors.) If the benchmark posterior that corresponds to frequentist inference lies within the set of Bayesian posteriors derived from the set of priors, then the benchmark posterior is used for inference. Otherwise, the posterior within that set that minimizes the cross entropy to the benchmark posterior is used for inference.
Fiducial model averages from model checks
D. R. Bickel, “A note on fiducial model averaging as an alternative to checking Bayesian and frequentist models,” Working Paper, University of Ottawa, deposited in uO Research at http://hdl.handle.net/10393/32313 (2015). 2015 preprint
Erratum: “Simple estimators of false discovery rates given as few as one or two p-values without strong parametric assumptions”
Main entry: Small dimensional empirical Bayes inference
Small-scale empirical Bayes & fiducial estimators
M. Padilla and D. R. Bickel, “Empirical Bayes and fiducial effect-size estimation for small numbers of tests,” Working Paper, University of Ottawa, deposited in uO Research at http://hdl.handle.net/10393/32151 (2015). 2015 preprint
Fiducial error propagation for empirical Bayes set estimates
D. R. Bickel, “A fiducial continuum from confidence sets to empirical Bayes set estimates as the number of comparisons increases,” Working Paper, University of Ottawa, deposited in uO Research at http://hdl.handle.net/10393/31898 (2014). 2014 preprint
Two problems confronting the eclectic approach to statistics result from its lack of a unifying theoretical foundation. First, there is typically no continuity between a p-value reported as a level of evidence for a hypothesis in the absence of the information needed to estimate a relevant prior on one hand and an estimated posterior probability of a hypothesis reported in the presence of such information on the other hand. Second, the empirical Bayes methods recommended do not propagate the uncertainty due to estimating the prior.
The latter problem is addressed by applying a coherent form of fiducial inference to hierarchical models, yielding empirical Bayes set estimates that reflect uncertainty in estimating the prior. Plugging in the maximum likelihood estimator, while not propagating that uncertainty, provides continuity from single comparisons to large numbers of comparisons.
Coherent fiducial distributions
D. R. Bickel and M. Padilla, “A prior-free framework of coherent inference and its derivation of simple shrinkage estimators,” Journal of Statistical Planning and Inference 145, 204–221 (2014). 2012 version
Small dimensional empirical Bayes inference
D. R. Bickel, “Simple estimators of false discovery rates given as few as one or two p-values without strong parametric assumptions,” Statistical Applications in Genetics and Molecular Biology 12, 529–543 (2013). 2011 version | erratum
To address multiple comparison problems in small-to-high-dimensional biology, this paper introduces estimators of the local false discovery rate (LFDR), reports their main properties, and illustrates their use with proteomics data. The new estimators have the following advantages:
- proven asymptotic conservatism;
- simplicity of calculation without the tuning of smoothing parameters;
- no strong parametric assumptions;
- applicability to very small numbers of hypotheses as well as to very large numbers of hypotheses.
The link to the erratum was added 31 March 2015.
David Bickel, statistician 
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