## Uncertainty propagation for empirical Bayes interval estimates: A fiducial approach

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

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?

**MR3561954**

Generalized fiducial inference: a review and new results. (English summary)

*J. Amer. Statist. Assoc.*111 (2016), no. 515, 1346–1361.

62A01 (62F99 62G05 62J05)

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.

- A weak-limit definition of a generalized fiducial distribution.
- Sufficient conditions for a generalized fiducial distribution to have asymptotic frequentist coverage.
- 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

## 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.

## 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