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

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## Abstract

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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... Theorem 1. For any utility function satisfying equation (16), the Bacon actions from the prior probableness distribution and the likelihood function on that are specified by equations (10) and (11) for all φ ∈ are identical to the Bayes actions from p (•), where is assumed to be a finite set such that p (φ) > 0 and p (φ| y) > 0 for all φ ∈ . Those actions are given by a (π | y) = a (p | y) = arg sup φ∈ p (φ) f (y |φ) u (φ) = arg sup φ∈ p (φ| y) u (φ) (17) in the notation of Sections 2.2-2.3, ...
... For any utility function satisfying equation (18), the Bacon actions from the prior probableness distribution and the likelihood function on that are specified by equations (10) and (11) for all φ ∈ are identical to the limiting Bayes actions from p (•), which is assumed to satisfy p (φ) > 0 and p (φ| y) > 0 for all φ ∈ . Those actions are given by equation (17) with a 0 (p | y) in place of a (p | y). ...
... Suppose P y , a p-value based on the sample y, is available but f (y | H 1 ) / f (y | H 0 ) is not. Then f (y | H 1 ) / f (y | H 0 ) can be estimated by an upper bound that depends on P y when P y is sufficiently small [10,9]. Many options for such upper bounds are reviewed by Held and Ott [29]. ...
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A Bayesian model has two parts. The first part is a family of sampling distributions that could have generated the data. The second part of a Bayesian model is a prior distribution over the sampling distributions. Both the diagnostics used to check the model and the process of updating a failed model are widely thought to violate the standard foundations of Bayesianism. That is largely because models are checked before specifying the space of all candidate replacement models, which textbook presentations of Bayesian model averaging would require. However, that is not required under a broad class of utility functions that apply when approximate model truth is an important consideration, perhaps among other important considerations. From that class, a simple criterion for model checking emerges and suggests a coherent approach to updating Bayesian models found inadequate. The criterion only requires the specification of the prior distribution up to ratios of prior densities of the models considered until the time of the check. That criterion, while justified by Bayesian decision theory, may also be derived under possibility theory from a decision-theoretic framework that generalizes the likelihood interpretation of possibility functions.
... These tests included, family-wise error control using the Holm's correction (Holm), False Discover Rate control using the Benjamini-Hochberg correction (BH), permutation test using the maximum statistic across the tests (Perm_max) as well as permutation test for individual test (Perm). We employed two p-value calibration methods, a first proposed by Selke and colleagues [Sellke et al., 2001] (pcalSBB, pcal function from the pcal R package) and a second proposed by Bickel [Bickel, 2021b] (pcalBickel, implemented as pcalBickel = ...
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For the one-sided hypothesis testing problem it is shown that it is possible to reconcile Bayesian evidence against H0, expressed in terms of the posterior probability that H0 is true, with frequentist evidence against H0, expressed in terms of the p value. In fact, for many classes of prior distributions it is shown that the infimum of the Bayesian posterior probability of H0 is equal to the p value; in other cases the infimum is less than the p value. The results are in contrast to recent work of Berger and Sellke (1987) in the two-sided (point null) case, where it was found that the p value is much smaller than the Bayesian infimum. Some comments on the point null problem are also given.
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We describe a range of routine statistical problems in which marginal posterior distributions derived from improper prior measures are found to have an unBayesian property—one that could not occur if proper prior measures were employed. This paradoxical possibility is shown to have several facets that can be successfully analysed in the framework of a general group structure. The results cast a shadow on the uncritical use of improper prior measures. A separate examination of a particular application of Fraser's structural theory shows that it is intrinsically paradoxical under marginalization.
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Let X be a random variable which for simplicity we shall assume to have discrete values x and which has a probability distribution depending in a known way on an unknown real parameter A, $$p\left( {x|\lambda } \right) = Pr[X = x|\Lambda = \lambda ],$$ (1) A itself being a random variable with a priori distribution function $$G\left( \lambda \right) = \operatorname{P} r[\Lambda {\text{ }}\underline \leqslant {\text{ }}\lambda ].$$ (2)
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Subjectivism has become the dominant philosophical foundation for Bayesian inference. Yet in practice, most Bayesian analyses are performed with so-called “noninformative” priors, that is, priors constructed by some formal rule. We review the plethora of techniques for constructing such priors and discuss some of the practical and philosophical issues that arise when they are used. We give special emphasis to Jeffreys's rules and discuss the evolution of his viewpoint about the interpretation of priors, away from unique representation of ignorance toward the notion that they should be chosen by convention. We conclude that the problems raised by the research on priors chosen by formal rules are serious and may not be dismissed lightly: When sample sizes are small (relative to the number of parameters being estimated), it is dangerous to put faith in any “default” solution; but when asymptotics take over, Jeffreys's rules and their variants remain reasonable choices. We also provide an annotated bibliography.
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This is the first volume of a two-volume work on Probability and Induction. Because the writer holds that probability logic is identical with inductive logic, this work is devoted to philosophical problems concerning the nature of probability and inductive reasoning. The author iejects a statistical frequency basis for probability in favor of a logical relation between two statements or propositions. Probability "is the degree of confirmation of a hypothesis (or conclusion) on the basis of some given evidence (or premises)." Furthermore, all principles and theorems of inductive logic are analytic, and the entire system is to be constructed by means of symbolic logic and semantic methods. This means that the author confines himself to the formalistic procedures of word and symbol systems. The resulting sentence or language structures are presumed to separate off logic from all subjectivist or psychological elements. Despite the abstractionism, the claim is made that if an inductive probability system of logic can be constructed it will have its practical application in mathematical statistics, and in various sciences. 16-page bibliography. (PsycINFO Database Record (c) 2012 APA, all rights reserved)