## 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

## 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.