## 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.

## Optimal strength of evidence

D. R. Bickel, “Minimax-optimal strength of statistical evidence for a composite alternative hypothesis,” *International Statistical Review* **81**, 188-206 (2013). 2011 version | Simple explanation (added 2 July 2017)

This publication generalizes the likelihood measure of evidential support for a hypothesis with the help of tools originally developed by information theorists for minimizing the number of letters in a message. The approach is illustrated with an application to proteomics data.

## Confidence levels as degrees of belief

D. R. Bickel, “A frequentist framework of inductive reasoning,” *Sankhya A* **74**, 141-169 (2013). published version | 2009 version | relationship to a working paper | simple explanation (added 17 July 2017)

A confidence measure is a parameter distribution that encodes all confidence intervals for a given data set, model, and pivot. This article establishes some properties of the confidence measure that commend it as a viable alternative to the Bayesian posterior distribution.

Confidence (correct frequentist coverage) and coherence (compliance with Ramsey-type restrictions on rational belief) are both presented as desirable properties. The only distributions on a scalar parameter space that have both properties are confidence measures.