## Model fusion & multiple testing in the likelihood paradigm

D. R. Bickel, “Model fusion and multiple testing in the likelihood paradigm: Shrinkage and evidence supporting a point null hypothesis,” Working Paper, University of Ottawa, deposited in uO Research at http://hdl.handle.net/10393/31897 (2014). 2014 preprint | Supplement (link added 10 February 2015)

Errata for Theorem 4:

- The weights of evidence should not be conditional.
- Some of the equal signs should be “is a member of” signs.

## 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.

## Self-consistent frequentism without fiducialism

D. R. Bickel, “Self-consistent confidence sets and tests of composite hypotheses applicable to restricted parameters,” Working Paper, University of Ottawa, deposited in uO Research at http://www.ruor.uottawa.ca/handle/10393/31530 (2014). 2014 preprint

## Confidence + coherence = fiducial shrinkage

D. R. Bickel, “A prior-free framework of coherent inference and its derivation of simple shrinkage estimators,” Working Paper, University of Ottawa, deposited in uO Research at http://hdl.handle.net/10393/23093 (2012). 2012 preprint

This paper proposes a new method of shrinking point and interval estimates on the basis of fiducial inference. Since problems with the interpretation of fiducial probability have prevented its widespread use, this manuscript first places fiducial inference within a general framework that has Bayesian and frequentist foundations.

## Estimating probabilities of enrichment

Z. Yang, Z. Li, and D. R. Bickel, “Empirical Bayes estimation of posterior probabilities of enrichment,” Technical Report, Ottawa Institute of Systems Biology, Technical Report, Ottawa Institute of Systems Biology, arXiv:1201.0153 (2011). Full preprint | 2010 seed

This paper adapts novel empirical Bayes methods for the problem of detecting enrichment in the form of differential representation of genes associated with a biological category with respect to a list of genes identified as differentially expressed. A microarray case study illustrates the methods using Gene Ontology (GO) terms, and a simulation study compares their performance. We report that which enrichment methods work best depends strongly on how many GO terms or other biological categories are of interest.