A frequentist framework of inductive reasoning

@article{Bickel2012AFF,
  title={A frequentist framework of inductive reasoning},
  author={David R. Bickel},
  journal={arXiv: Statistics Theory},
  year={2012},
  volume={74},
  pages={141-169}
}
  • D. Bickel
  • Published 17 February 2006
  • Mathematics
  • arXiv: Statistics Theory
A betting game establishes a sense in which confidence measures, confidence distributions in the form of probability measures, are the only reliable inferential probability distributions. In addition, because confidence measures are Kolmogorov probability distributions, they are as coherent as Bayesian posterior distributions in their avoidance of sure loss under the usual Dutch-book betting game. 
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References

SHOWING 1-10 OF 87 REFERENCES
Coherent Frequentism: A Decision Theory Based on Confidence Sets
By representing fair betting odds according to one or more pairs of confidence set estimators, dual parameter distributions called confidence posteriors secure the coherence of actions without any
Topics on the Foundations of Robust Bayesian Analysis
TLDR
This chapter surveys whether a similar situation holds for robust Bayesian analysis, overviewing foundational results leading to standard computations in robust Bayesian analysis.
The Uncertain Reasoner's Companion: A Mathematical Perspective
Introduction 1. Motivation 2. Belief as probability 3. Justifying belief as probability 4. Dempster-Shafer belief 5. Truth-functional belief 6. Inference processes 7. Principles of uncertain
Conditional Confidence Statements and Confidence Estimators
Abstract Procedures are given for assessing the conclusiveness of a decision in terms of a (chance) conditional confidence coefficient Γ that has frequentist interpretability analogous to that of a
Prior Inferences for Posterior Judgements
We consider the minimal assumptions of temporal consistency that will allow us to make meaningful prior statements about our posterior judgements. These consistency conditions suggest a natural
Logic of Statistical Inference
TLDR
This chapter discusses the long run frequencies, statistical tests, and theories of testing that led to Bayes' theory, as well as the subjective theory.
The Role of Likelihood in Interval Estimation
TLDR
It is common practice to treat coverage probability as the only requirement for interval estimation, but neither is sufficient, and this violates the law of likelihood.
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