## Fiducial model averaging of Bayesian models and of frequentist models

D. R. Bickel, “A note on fiducial model averaging as an alternative to checking Bayesian and frequentist models,” *Communications in Statistics – Theory and Methods* **47**, 3125-3137 (2018). Full article | 2015 preprint

## “A Litany of Problems With p-values”

Bayesian, likelihoodist, and frequentist views appear in the comments in Statistical Thinking: A Litany of Problems With p-values.

## Lower the statistical significance threshold to 0.005—or 0.001?

D. R. Bickel, “Sharpen statistical significance: Evidence thresholds and Bayes factors sharpened into Occam’s razors,” Working Paper, University of Ottawa, <hal-01851322>** **https://hal.archives-ouvertes.fr/hal-01851322 (2018). 2018 preprint

## Pre-data insights update priors via Bayes’s theorem

D. R. Bickel, “Bayesian revision of a prior given prior-data conflict, expert opinion, or a similar insight: A large-deviation approach,” *Statistics* **52**, 552-570 (2018). Full text | 2015 preprint | Simple explanation

## How to adjust statistical inferences for the simplicity of distributions

D. R. Bickel, “Confidence intervals, significance values, maximum likelihood estimates, etc. sharpened into Occam’s razors,” Working Paper, University of Ottawa, <hal-01799519>** **https://hal.archives-ouvertes.fr/hal-01799519 (2018). 2018 preprint | Slides

## 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