## LFDR.MLE-package function | R Documentation

Suite of R functions for the estimation of the local false discovery rate (LFDR) using Type II maximum likelihood estimation (MLE):

## Inference to the best explanation of the evidence

The *p* value and Bayesian methods have well known drawbacks when it comes to measuring the strength of the evidence supporting one hypothesis over another. To overcome those drawbacks, this paper proposes an alternative method of quantifying how much support a hypothesis has from evidence consisting of data.

D. R. Bickel, “The strength of statistical evidence for composite hypotheses: Inference to the best explanation,” *Statistica Sinica* **22**, 1147-1198 (2012). Full article | 2010 version

The special law of likelihood has many advantages over more commonly used approaches to measuring the strength of statistical evidence. However, it only can measure the support of a hypothesis that corresponds to a single distribution. The proposed general law of likelihood also can measure the extent to which the data support a hypothesis that corresponds to multiple distributions. That is accomplished by formalizing inference to the best explanation.

## How to make decisions using somewhat reliable posterior distributions

D. R. Bickel, “Departing from Bayesian inference toward minimaxity to the extent that the posterior distribution is unreliable,” Working Paper, University of Ottawa, <hal-01673783>** **https://hal.archives-ouvertes.fr/hal-01673783 (2017). 2017 preprint

## 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.

## “Can You Change Your Bayesian Prior?”

Sometimes. A subjective Bayesian encountering completely unexpected data changes the prior:In the philosophy literature, that has been compared to changing the premises of a deductive argument. It has been argued that just as one may revise a premise without abandoning deductive logic as a tool, one may revise a prior without abandoning Bayesian updating as a tool.

## 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