Some definitions are introduced and exemplified that may help to relate Bayesian statistics to frequentist statistics. The idea is interesting. More work is required.
Practical implications would be:
- Opens up the possibility to use MCMC algorithms sampling parameters given data, e.g., Stan or WinBUGS, for frequentist hypothesis testing.
Conceptual implications would be:
- Formally relate results from Bayesian statistics to those from frequentist statistics.
- Define approaches and situations, in which frequentist and Bayesian approaches give identical results, or to explain differences obtained with different approaches.
- Kass RE, Wasserman L (1996) The Selection of Prior Distributions by Formal Rules. JASA, 91 (435), 1343-1370
- Berger, James. "The case for objective Bayesian analysis." Bayesian Analysis 1.3 (2006): 385-402.
- Bernardo, José M. "Intrinsic credible regions: An objective Bayesian approach to interval estimation." Test 14.2 (2005): 317-384.
- Efron, Bradley. "A 250-year argument: Belief, behavior, and the bootstrap." Bulletin of the American Mathematical Society 50.1 (2013): 129-146.
- Berger, James O, Brunero Liseo, and Robert L Wolpert. "Integrated likelihood methods for eliminating nuisance parameters." Statistical Science 14.1 (1999): 1-28.
- Severini, Thomas A. "Integrated likelihood functions for non-Bayesian inference." Biometrika 94.3 (2007): 529-542.
- Berger, James O, José M Bernardo, and Dongchu Sun. "The formal definition of reference priors." The Annals of Statistics (2009): 905-938.
- Bjørnstad, Jan F. "On the generalization of the likelihood function and the likelihood principle." Journal of the American Statistical Association 91.434 (1996): 791-806.