Coherent Frequentism: A Decision Theory Based on Confidence Sets

Bickel, David R.

doi:10.1080/03610926.2010.543302

Cited by 18 publications

(29 citation statements)

References 58 publications

(52 reference statements)

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“…It follows that p (x) = S (0; x) is a p-value for testing the hypothesis that θ * = 0 and that [S −1 (α; X) , S −1 (β; X)] is a (β − α) 100% con dence interval for θ * given any α ∈ [0, 1] and β ∈ [α, 1]. Thus, whether a signi cance function is found from p-values over a set of simple null hypotheses or instead from a set of nested con dence intervals, it contains the information needed to derive either (Schweder and Hjort, 2002;Singh et al, 2007;Bickel, 2012aBickel, , 2011a. Letθ * denote the random variable of the probability measureP * that has S (•; x) as its distribution function.…”

Section: A Con Dence Benchmark Posterior 321 Con Dence Posterior Thmentioning

confidence: 99%

“…The term con dence posterior (Bickel, 2012a(Bickel, , 2011a) is preferred here over the usual term con dence distribution (Schweder and Hjort, 2002) to emphasize its use as an alternative to Bayesian posterior distributions. …”

Section: A Con Dence Benchmark Posterior 321 Con Dence Posterior Thmentioning

confidence: 99%

“…The resulting distribution is known as a reduction (Bickel, 2012a) of the set. Perhaps the best known inference process for a discrete parameter set Θ is that of the m ximum entropy prin iple, which would select a member ofṖ such that it has higher entropy than any other member of the set (see Remark 2).…”

Section: Bayesian Posteriorsmentioning

confidence: 99%

“…(In the rst case, nested con dence intervals with inexact coverage rates generate a setP * of multiple con dence posteriors rather than the single con dence posterior that is generated by exact con dence intervals (Bickel, 2012a).) Suppose there exists a functionτ :P * → Θ such thatP , the probability distribution ofτ P * , is de ned on (Θ, A).P is called the en hm rk posterior @distri utionA, andθ =τ P * is the inferenti l t rget ofP * .…”

Section: Benchmark Posteriorsmentioning

confidence: 99%

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Blending Bayesian and frequentist methods according to the precision of prior information with applications to hypothesis testing

Bickel

2015

Stat Methods Appl

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The following zero-sum game between nature and a statistician blends Bayesian methods with frequentist methods such as p-values and con dence intervals. Nature chooses a posterior distribution consistent with a set of possible priors. At the same time, the statistician selects a parameter distribution for inference with the goal of maximizing the minimum Kullback-Leibler information gained over a con dence distribution or other benchmark distribution. In cases of hypothesis testing, the statistician reports a posterior probability of the hypothesis that is informed by both Bayesian and frequentist methodology, each weighted according how well the prior is known.As is generally acknowledged, the Bayesian approach is ideal given knowledge of a prior distribution that can be interpreted in terms of relative frequencies. On the other hand, frequentist methods such as con dence intervals and p-values have the advantage that they perform well without knowledge of such a distribution of the parameters. However, neither the Bayesian approach nor the frequentist approach is entirely satisfactory in situations involving partial knowledge of the prior distribution, the proposed procedure reduces to a Bayesian method given complete knowledge of the prior, to a frequentist method given complete ignorance about the prior, and to a blend between the two methods given partial knowledge of the prior. The blended approach resembles the Bayesian method rather than the frequentist method to the precise extent that the prior is known.The proposed framework o ers a simple solution to the enduring problem of testing a point null hypothesis. The blended probability that the null hypothesis is true is equal to the p-value or a lower bound of an unknown Bayesian posterior probability, whichever is greater. Thus, given total ignorance represented by a lower bound of 0, the p-value is used instead of any Bayesian posterior probability. At the opposite extreme of a known prior, the p-value is ignored. In the intermediate case, the possible Bayesian posterior probability that is closest to the p-value is used for inference. Thus, both the Bayesian method and the frequentist method in uence the inferences made.Similarly, blended inference may help resolve ongoing controversies in testing multiple hypotheses. Whereas the adjusted p-value is often considered the multiple comparison procedure (MCP) of choice for small numbers of hypotheses, large numbers of p-values enable accurate estimation of the local false discovery rate, a physical posterior probability of hypothesis truth. Each blended posterior probability reduces to either the adjusted p-value or the LFDR estimate by e ectively determining on a hypothesis-by-hypothesis basis whether the LFDR can be estimated with su cient accuracy. This blended MCP is applied to both a microarray data set and a more conventional biostatistics data set to illustrate its generality.2

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Section: A Con Dence Benchmark Posterior 321 Con Dence Posterior Thmentioning

confidence: 99%

Section: A Con Dence Benchmark Posterior 321 Con Dence Posterior Thmentioning

confidence: 99%

Section: Bayesian Posteriorsmentioning

confidence: 99%

Section: Benchmark Posteriorsmentioning

confidence: 99%

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Blending Bayesian and frequentist methods according to the precision of prior information with applications to hypothesis testing

Bickel

2015

Stat Methods Appl

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“…In the former case, σ x is the Borel eld over Θ. Polansky (2007), Singh et al (2007), and Bickel (2011bBickel ( , 2012a present alternative denitions of condence distributions of vector basic parameters. The denition used here is a slight generalization of the condence posterior found in Bickel (2012b, 2.3).…”

Section: Introductionmentioning

confidence: 99%

A prior-free framework of coherent inference and its derivation of simple shrinkage estimators

Bickel

Padilla

2014

Journal of Statistical Planning and Inference

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The reasoning behind uses of condence intervals and p-values in scientic practice may be made coherent by modeling the inferring statistician or scientist as an idealized intelligent agent. With other things equal, such an agent regards a hypothesis coinciding with a condence interval of a higher condence level as more certain than a hypothesis coinciding with a condence interval of a lower condence level. The agent uses dierent methods of condence intervals conditional on what information is available. The coherence requirement means all levels of certainty of hypotheses about the 1 parameter agree with the same distribution of certainty over parameter space. The result is a unique and coherent ducial distribution that encodes the post-data certainty levels of the agent.While many coherent ducial distributions coincide with condence distributions or Bayesian posterior distributions, there is a general class of coherent ducial distributions that equates the two-sided p-value with the probability that the null hypothesis is true.The use of that class leads to point estimators and interval estimators that can be derived neither from the dominant frequentist theory nor from Bayesian theories that rule out data-dependent priors. These simple estimators shrink toward the parameter value of the null hypothesis without relying on asymptotics or on prior distributions.

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Evaluation of operational system structures based on reliability data

Polansky

Ryu

2021

Quality & Reliability Eng

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Systems of components have a structure that plays an important role in determining how the reliability of the individual components relates to the reliability of the system. The system reliability can be computed from component reliabilities using results from basic probability theory in the simplest case with all of the components assumed to act independently of one another. However, in the case of dependence, such calculations can be much more involved. When reliability data have been independently collected on both the system and each component in the system, it can be difficult to model any possible dependence between components. Established methods use the known structure of a system, along with these data, to assess whether the reliability of the individual components are mutually independent. In this paper, we expand this methodology to include an assessment of the type of dependence that may exist between the components. This is based on finding the system structure that would most likely produce the observed reliability data, under independence. In the frequentist setting, the likelihood approach is used to find these structures and an observed confidence measure is used to assess the strength of the statistical evidence in favor of each possible structure. In the Bayesian setting, posterior probabilities along with Bayes factors are used. An example demonstrates how these methods can be used in an applied setting.

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Coherent Frequentism: A Decision Theory Based on Confidence Sets

Cited by 18 publications

References 58 publications

Blending Bayesian and frequentist methods according to the precision of prior information with applications to hypothesis testing

Blending Bayesian and frequentist methods according to the precision of prior information with applications to hypothesis testing

A prior-free framework of coherent inference and its derivation of simple shrinkage estimators

Evaluation of operational system structures based on reliability data

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