Confidence distributions: A review

Nadarajah, Saralees; Bityukov, Sergey; Krasnikov, N.

doi:10.1016/j.stamet.2014.07.002

Cited by 23 publications

(13 citation statements)

References 110 publications

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“…The long-discredited fiducial argument of Fisher has returned as various theories of confidence distributions (e.g., Schweder and Hjort, 2002;Singh et al, 2005;Polansky, 2007;Singh et al, 2007;Tian et al, 2011;Bityukov et al, 2011;Kim and Lindsay, 2011;Taraldsen and Lindqvist, 2018) and related priorless posterior distributions of the parameter of interest (e.g., Hannig et al, 2006;Hannig, 2009;Xiong and Mu, 2009;Gibson et al, 2011;Wang et al, 2012;Zhao et al, 2012;Balch, 2012;Martin and Liu, 2013;Bickel and Padilla, 2014;Bowater, 2017). Efron (2010), Nadarajah et al (2015), and Schweder and Hjort (2016) provide informative expositions.…”

Section: Empirical Bayes Methods Confidence Methods and Their Evidementioning

confidence: 99%

Confidence distributions and empirical Bayes posterior distributions unified as distributions of evidential support

Bickel

2020

Communications in Statistics - Theory and Methods

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While empirical Bayes methods thrive in the presence of the hundreds or thousands of simultaneous hypothesis tests in genomics and other large-scale applications, significance tests and confidence intervals are considered more appropriate for small numbers of simultaneously tested hypotheses. Indeed, for fewer hypotheses or, more generally, fewer populations, there is more uncertainty in empirical Bayes methods of estimating the prior distribution. Confidence distributions have been used to propagate the uncertainty in the prior to empirical Bayes inference about a parameter, but only by combining a Bayesian posterior distribution with a confidence distribution, a probability distribution that encodes significance tests and confidence intervals. Combining distributions of both types has also been used to combine empirical Bayes methods and confidence intervals for estimating a parameter of interest. To clarify the foundational status of such combinations, the concept of an evidential model is proposed. In the framework of evidential models, both Bayesian posterior distributions and confidence distributions are degenerate special cases of evidential support distributions. Evidential support distributions, by quantifying the sufficiency of the data as evidence, leverage the strengths of Bayesian posterior distributions and confidence distributions for cases in which each type performs well and for cases benefiting from the combination of both. Evidential support distributions also address problems of bioequivalence, bounded parameters, and the lack of a unique confidence distribution.

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Section: Empirical Bayes Methods Confidence Methods and Their Evidementioning

confidence: 99%

Confidence distributions and empirical Bayes posterior distributions unified as distributions of evidential support

Bickel

2020

Communications in Statistics - Theory and Methods

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“…Although the fiducial argument for equation ( 5) when the estimand Pr (ϑ ∈ C (γ; y) | P = p) is a subjective probability has been discredited, equation ( 5) has support from the broadly applicable approximation of one-sided p values to posterior probabilities (Appendix A). If C (γ; Y ) is exact in the sense that formula (4) holds with equality, then the function Pr is a probability distribution known as a confidence distribution (Schweder and Hjort, 2002;Singh et al, 2005;Nadarajah et al, 2015;Schweder and Hjort, 2016;Bickel, 2019b).…”

Section: Uncalibrated Confidence Intervals and P Valuesmentioning

confidence: 99%

Interval estimation, point estimation, and null hypothesis significance testing calibrated by an estimated posterior probability of the null hypothesis

Bickel

2021

Communications in Statistics - Theory and Methods

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Much of the blame for failed attempts to replicate reports of scientific findings has been placed on ubiquitous and persistent misinterpretations of the p value. An increasingly popular solution is to transform a two-sided p value to a lower bound on a Bayes factor. Another solution is to interpret a one-sided p value as an approximate posterior probability.Combining the two solutions results in confidence intervals that are calibrated by an estimate of the posterior probability that the null hypothesis is true. The combination also provides a point estimate that is covered by the calibrated confidence interval at every level of confidence.Finally, the combination of solutions generates a two-sided p value that is calibrated by the estimate of the posterior probability of the null hypothesis. In the special case of a 50% prior probability of the null hypothesis and a simple lower bound on the Bayes factor, the calibrated two-sided p value is about (1 -abs(2.7 p ln p)) p + 2 abs(2.7 p ln p) for small p. The calibrations of confidence intervals, point estimates, and p values are proposed in an empirical Bayes framework without requiring multiple comparisons.

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“…This allows to construct and use confidence distribution of parameter λ under estimation of parameter via measurement of the random variable x. More details about confidence distributions can be found in reviews [14,15].…”

Section: Confidence Densitymentioning

confidence: 99%