## Confidence-based decision theory

D. R. Bickel, “Coherent frequentism: A decision theory based on confidence sets,” *Communications in Statistics – Theory and Methods* **41**, 1478-1496 (2012). Full article (open access) | 2009 version | Simple explanation (link added 27 June 2018)

To combine the self-consistency of Bayesian statistics with the objectivity of frequentist statistics, this paper formulates a framework of inference for developing novel statistical methods. The framework is based on a confidence posterior, a parameter probability distribution that does not require any prior distribution. While the Bayesian posterior is defined in terms of a conditional distribution given the observed data, the confidence posterior is instead defined such that the probability that the parameter value lies in any fixed subset of parameter space, given the observed data, is equal to the coverage rate of the corresponding confidence interval. Inferences based on the confidence posterior are reliable in the sense that the certainty level of a composite hypothesis is a weakly consistent estimate of the 0-1 indicator of hypothesis truth. At the same time, the confidence posterior is as non-contradictory as the Bayesian posterior since both satisfy the same coherence axioms. Using the theory of coherent upper and lower probabilities, the confidence posterior is generalized for situations in which no approximate or exact confidence set is available. Examples of hypothesis testing and estimation illustrate the range of applications of the proposed framework.

Additional summaries appear in the abstract and in Section 1.3 of the paper.

## Observed confidence levels for microarrays, etc.

D. R. Bickel, “Estimating the null distribution to adjust observed confidence levels for genome-scale screening,” *Biometrics* **67**, 363-370 (2011). Abstract and article | French abstract | Supplementary material | Simple explanation

This paper describes the first application of observed confidence levels to data of high-dimensional biology. The proposed method for multiple comparisons can take advantage of the estimated null distribution without any prior distribution. The new method is applied to microarray data to illustrate its advantages.