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Uncertainty propagation for empirical Bayes interval estimates: A fiducial approach

1 December 2017 Comments off

D. R. Bickel, “Confidence distributions applied to propagating uncertainty to inference based on estimating the local false discovery rate: A fiducial continuum from confidence sets to empirical Bayes set estimates as the number of comparisons increases,” Communications in Statistics – Theory and Methods 46, 10788-10799 (2017). Published article | Free access (limited time)2014 preprint

Publication Cover

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.

The generalized fiducial distribution: A kinder, more objective posterior?

1 June 2017 Comments off

MR3561954

Hannig, JanIyer, HariLai, Randy C. S.Lee, Thomas C. M.
Generalized fiducial inference: a review and new results. (English summary)
J. Amer. Statist. Assoc. 111 (2016), no. 515, 1346–1361.
62A01 (62F99 62G05 62J05)
This review article introduces generalized fiducial inference, the flavor of fiducial statistics developed by the authors and their collaborators since the beginning of the millennium. This research program has been driven by a vision of fiducial distributions as posterior distributions untainted by the subjectivity seen in prior distributions.
Other approaches to fiducial inference bring subjectivity more to the forefront. For example, G. N. Wilkinson had highlighted the incoherence of fiducial distributions formulated in a more Fisherian flavor [J. Roy. Statist. Soc. Ser. B 39 (1977), no. 2, 119–171; MR0652326]. More recently, R. J. Bowater [AStA Adv. Stat. Anal. 101 (2017), no. 2, 177–197] endorsed an explicitly subjective interpretation of fiducial probability. For the place of generalized fiducial inference in the context of other fiducial approaches, see [D. L. Sonderegger and J. Hannig, in Contemporary developments in statistical theory, 155–189, Springer Proc. Math. Stat., 68, Springer, Cham, 2014; MR3149921] and the papers it {MR3149921} cites.
In addition to providing an inspiring exposition of generalized fiducial inference, the authors report these new contributions:
  1. A weak-limit definition of a generalized fiducial distribution.
  2. Sufficient conditions for a generalized fiducial distribution to have asymptotic frequentist coverage.
  3. Novel formulas for computing a generalized fiducial distribution and a fiducial probability of a model.

The fiducial probability of a model is applicable to both model selection and model averaging. A seemingly different fiducial method of averaging statistical models was independently proposed by D. R. Bickel [“A note on fiducial model averaging as an alternative to checking Bayesian and frequentist models”, preprint, Fac. Sci. Math. Stat., Univ. Ottawa, 2015].

Reviewed by David R. Bickel

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Empirical Bayes single-comparison procedure

1 July 2016 Comments off

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 version2011 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

7 January 2016 Comments off

Fiducial nonparametrics

24 August 2015 Comments off

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)

This chapter provides both asymptotic and finite-sample properties of a fiducial solution to the problem of free-knot splines with four or more degrees, assuming a known number of knot points. The authors lay a foundation for the solution by proving the asymptotic normality of certain multivariate fiducial estimators. After demonstrating that the fiducial solution meets the sufficient conditions for asymptotic normality, they quantify small-sample performance on the basis of simulations. The authors conclude that fiducial inference provides a promising alternative to Bayesian inference for the free-knot spline problem addressed.
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

10 August 2015 Comments off

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 article2012 preprint | 2011 preprint | Slides

SMA

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

10 May 2015 Comments off