Traditional analyses of the curve fitting problem maintain that the data do not indicate what form the fitted curve should take. Rather, this issue is said to be settled by prior probabilities, by simplicity, or by a background theory. In this paper, we describe a result due to Akaike [1973], which shows how the data can underwrite an inference concerning the curve's form based on an estimate of how predictively accurate it will be. We argue that this approach throws light (...) on the theoretical virtues of parsimoniousness, unification, and non ad hocness, on the dispute about Bayesianism, and on empiricism and scientific realism. * Both of us gratefully acknowledge support from the Graduate School at the University of Wisconsin-Madison, and NSF grant DIR-8822278 (M.F.) and NSF grant SBE-9212294 (E.S.). Special thanks go to A. W. F. Edwards.William Harper. Martin Leckey. Brian Skyrms, and especially Peter Turney for helpful comments on an earlier draft. (shrink)

We show how the Full Bayesian Significance Test (FBST) can be used as a model selection criterion. The FBST was presented by Pereira and Stern as a coherent Bayesian significance test. Key Words: Bayesian test; Evidence; Global optimization; Information; Model selection; Numerical integration; Posterior density; Precise hypothesis; Regularization. AMS: 62A15; 62F15; 62H15.

The history of statistics is filled with many controversies, in which the prime focus has been the difference in the “interpretation of probability” between Fre- quentist and Bayesian theories. Many philosophical arguments have been elabo- rated to examine the problems of both theories based on this dichotomized view of statistics, including the well-known stopping-rule problem and the catch-all hy- pothesis problem. However, there are also several “hybrid” approaches in theory, practice, and philosophical analysis. This poses many fundamental questions. This paper (...) reviews three cases and argues that the interpretation problem of probabil- ity is insufficient to begin a philosophical analysis of the current issues in the field of statistics. A novel viewpoint is proposed to examine the relationship between the stopping-rule problem and the catch-all hypothesis problem. (shrink)

Abstract: The Pull Bayesian Significance Test (FBST) for precise hy- potheses is applied to a Multivariate Normal Structure (MNS) model. In the FBST we compute the evidence against the precise hypothesis. This evi- dence is the probability of the Highest Relative Surprise Set (HRSS) tangent to the sub-manifold (of the parameter space) that defines the null hypothesis. The MNS model we present appears when testing equivalence conditions for genetic expression measurements, using micro-array technology.

The unit root problem plays a central role in empirical applications in the time series econometric literature. However, significance tests developed under the frequentist tradition present various conceptual problems that jeopardize the power of these tests, especially for small samples. Bayesian alternatives, although having interesting interpretations and being precisely defined, experience problems due to the fact that that the hypothesis of interest in this case is sharp or precise. The Bayesian significance test used in this article, for the unit root (...) hypothesis, is based solely on the posterior density function, without the need of imposing positive probabilities to sets of zero Lebesgue measure. Furthermore, it is conducted under strict observance of the likelihood principle. It was designed mainly for testing sharp null hypotheses and it is called FBST for Full Bayesian Significance Test. (shrink)

Statistical tests that detect and measure deviation from the Hardy-Weinberg equilibrium (HWE) have been devised but are limited when testing for deviation at multiallelic DNA loci is attempted. Here we present the full Bayesian significance test (FBST) for the HWE. This test depends neither on asymptotic results nor on the number of possible alleles for the particular locus being evaluated. The FBST is based on the computation of an evidence index in favor of the HWE hypothesis. A great deal of (...) forensic inference based on DNA evidence assumes that the HWE is valid for the genetic loci being used. We applied the FBST to genotypes obtained at several multiallelic short tandem repeat loci during routine parentage testing; the locus Penta E exemplifies those clearly in HWE while others such as D10S1214 and D19S253 do not appear to show this. (shrink)

The Full Bayesian Significance Test (FBST) for precise hypotheses is presented, with some applications relevant to reliability theory. The FBST is an alternative to significance tests or, equivalently, to p-ualue.s. In the FBST we compute the evidence of the precise hypothesis. This evidence is the probability of the complement of a credible set "tangent" to the sub-manifold (of the para,rreter space) that defines the null hypothesis. We use the FBST in an application requiring a quality control of used components, based (...) on remaining life statistics. (shrink)

The full Bayesian signi/cance test (FBST) for precise hypotheses is presented, with some illustrative applications. In the FBST we compute the evidence against the precise hypothesis. We discuss some of the theoretical properties of the FBST, and provide an invariant formulation for coordinate transformations, provided a reference density has been established. This evidence is the probability of the highest relative surprise set, “tangential” to the sub-manifold (of the parameter space) that defines the null hypothesis.

The Full Bayesian Significance Test, FBST, is extensively reviewed. Its test statistic, a genuine Bayesian measure of evidence, is discussed in detail. Its behavior in some problems of statistical inference like testing for independence in contingency tables is discussed.

In statistics, there are two main paradigms: classical and Bayesian statistics. The purpose of this article is to investigate the extent to which classicists and Bayesians can agree. My conclusion is that, in certain situations, they cannot. The upshot is that, if we assume that the classicist is not allowed to have a higher degree of belief in a null hypothesis after he has rejected it than before, then he has to either have trivial or incoherent credences to begin with (...) or fail to update his credences by conditionalization. (shrink)

Science depends on judgments of the bearing of evidence on theory. Scientists must judge whether an observation or the result of an experiment supports, disconfirms, or is simply irrelevant to a given hypothesis. Similarly, scientists may judge that, given all the available evidence, a hypothesis ought to be accepted as correct or nearly so, rejected as false, or neither. Occasionally, these evidential judgments can be made on deductive grounds. If an experimental result strictly contradicts a hypothesis, then the truth of (...) the data deductively entails the falsity of the hypothesis. In the great majority of cases, however, the connection between evidence and hypothesis is non-demonstrative, or inductive. In particular, this is so whenever a general hypothesis is inferred to be correct on the basis of the available data, since the truth of the data will not deductively entail the truth of the hypothesis. It always remains possible that the hypothesis is false even though the data are correct. (shrink)

No scientific conclusion follows automatically from a statistically non-significant result, yet people routinely use non-significant results to guide conclusions about the status of theories (or the effectiveness of practices). To know whether a non-significant result counts against a theory, or if it just indicates data insensitivity, researchers must use one of: power, intervals (such as confidence or credibility intervals), or else an indicator of the relative evidence for one theory over another, such as a Bayes factor. I argue Bayes factors (...) allow theory to be linked to data in a way that overcomes the weaknesses of the other approaches. Specifically, Bayes factors use the data themselves to determine their sensitivity in distinguishing theories (unlike power), and they make use of those aspects of a theory’s predictions that are often easiest to specify (unlike power and intervals, which require specifying the minimal interesting value in order to address theory). Bayes factors provide a coherent approach to determining whether non-significant results support a null hypothesis over a theory, or whether the data are just insensitive. They allow accepting and rejecting the null hypothesis to be put on an equal footing. Concrete examples are provided to indicate the range of application of a simple online Bayes calculator, which reveal both the strengths and weaknesses of Bayes factors. (shrink)

The Jeffreys–Lindley paradox displays how the use of a \ value ) in a frequentist hypothesis test can lead to an inference that is radically different from that of a Bayesian hypothesis test in the form advocated by Harold Jeffreys in the 1930s and common today. The setting is the test of a well-specified null hypothesis versus a composite alternative. The \ value, as well as the ratio of the likelihood under the null hypothesis to the maximized likelihood under the (...) alternative, can strongly disfavor the null hypothesis, while the Bayesian posterior probability for the null hypothesis can be arbitrarily large. The academic statistics literature contains many impassioned comments on this paradox, yet there is no consensus either on its relevance to scientific communication or on its correct resolution. The paradox is quite relevant to frontier research in high energy physics. This paper is an attempt to explain the situation to both physicists and statisticians, in the hope that further progress can be made. (shrink)

The interpretation of tests of a point null hypothesis against an unspecified alternative is a classical and yet unresolved issue in statistical methodology. This paper approaches the problem from the perspective of Lindley's Paradox: the divergence of Bayesian and frequentist inference in hypothesis tests with large sample size. I contend that the standard approaches in both frameworks fail to resolve the paradox. As an alternative, I suggest the Bayesian Reference Criterion: it targets the predictive performance of the null hypothesis in (...) future experiments; it provides a proper decision-theoretic model for testing a point null hypothesis and it convincingly accounts for Lindley's Paradox. (shrink)

Frequentism and Bayesianism represent very different approaches to hypothesis testing, and this presents a skeptical challenge for Bayesians. Given that most empirical research uses frequentist methods, why (if at all) should we rely on it? While it is well known that there are conditions under which Bayesian and frequentist methods agree, without some reason to think these conditions are typically met, the Bayesian hasn’t shown why we are usually safe in relying on results reported by significance testers. In this article, (...) I provide arguments that such conditions will usually be met; the Bayesian can maintain her theoretical disagreement with the frequentist while holding that her error is mostly harmless in practice. (shrink)

Bayesian methods are ubiquitous in contemporary observational cosmology. They enter into three main tasks: cross-checking datasets for consistency; fixing constraints on cosmological parameters; and model selection. This article explores some epistemic limits of using Bayesian methods. The first limit concerns the degree of informativeness of the Bayesian priors and an ensuing methodological tension between task and task. The second limit concerns the choice of wide flat priors and related tension between parameter estimation and model selection. The Dark Energy Survey and (...) its recent Year 1 results illustrate both these limits concerning the use of Bayesianism. (shrink)

Published research findings are sometimes refuted by subsequent evidence, says Ioannidis, with ensuing confusion and disappointment.

The three main approaches in statistical inference—classical statistics, Bayesian and likelihood—are in current use in phylogeny research. The three approaches are discussed and compared, with particular emphasis on theoretical properties illustrated by simple thought-experiments. The methods are problematic on axiomatic grounds, extra-mathematical grounds relating to the use of a prior or practical grounds. This essay aims to increase understanding of these limits among those with an interest in phylogeny.

Using a sample of Chinese listed firms for the period of 2004–2010, this study examines the impact of religion on corporate philanthropic giving. Based on hand-collected data of religion and corporate philanthropic giving, we provide strong and robust evidence that religion is significantly positively associated with Chinese listed firms’ philanthropic giving. This finding is consistent with the view that religiosity has remarkable effects on individual thinking and behavior, and can serve as social norms to influence corporate philanthropy. Moreover, religion and (...) corporate philanthropic giving have a significantly weaker (less pronounced) positive association for state-owned enterprises than for non-state-owned enterprises. The results are robust to a variety of sensitivity tests. Our results highlight religious influence on corporate philanthropic giving in contemporary China, an old traditional country with a typical communist economy. (shrink)

The aim of the article is to propose a way to determine to what extent a hypothesis H is confirmed if it has successfully passed a classical significance test. Bayesians have already raised many serious objections against significance testing, but in doing so they have always had to rely on epistemic probabilities and a further Bayesian analysis, which are rejected by classical statisticians. Therefore, I will suggest a purely frequentist evaluation procedure for significance tests that should also be accepted by (...) a classical statistician. This procedure likewise indicates some additional problems of significance tests. In some situations, such tests offer only a weak incremental support of a hypothesis, although an absolute confirmation is necessary, and they overestimate positive results for small effects, since the confirmation of H is often rather marginal in these cases. In specific cases, for example, in cases of ESP-hypotheses, such as precognition, this phenomenon leads too easily to a significant confirmation and can be regarded as a form of the probabilistic falsification fallacy. (shrink)

Benjamin et al. Nature Human Behavior 2, 6–10 proposed decreasing the significance level by an order of magnitude to improve the replicability of psychology. This modest, practical proposal has been widely criticized, and its prospects remain unclear. This article defends this proposal against these criticisms and highlights its virtues.

Bayesian confirmation theory is a leading theory to decide the confirmation/refutation of a hypothesis based on probability calculus. While it may be much discussed in philosophy of science, is it actually practiced in terms of hypothesis testing by scientists? Since the assignment of some of the probabilities in the theory is open to debate and the risk of making the wrong decision is unknown, many scientists do not use the theory in hypothesis testing. Instead, they use alternative statistical tests that (...) can measure the risk or the reliability in decision making, circumventing some of the theoretical problems in practice. Therefore, the theory is not very popular in hypothesis testing among scientists at present. However, there are some proponents of Bayesian hypothesis testing, and software packages are made available to accelerate utilization by scientists. Time will tell whether Bayesian confirmation theory can become both a leading theory and a widely practiced method. In addition, this theory can be used to model the belief of scientists when testing hypotheses. (shrink)

A frequentist confidence interval can be constructed by inverting a hypothesis test, such that the interval contains only parameter values that would not have been rejected by the test. We show how a similar definition can be employed to construct a Bayesian support interval. Consistent with Carnap’s theory of corroboration, the support interval contains only parameter values that receive at least some minimum amount of support from the data. The support interval is not subject to Lindley’s paradox and provides an (...) evidence-based perspective on inference that differs from the belief-based perspective that forms the basis of the standard Bayesian credible interval. (shrink)

A frequentist confidence interval can be constructed by inverting a hypothesis test, such that the interval contains only parameter values that would not have been rejected by the test. We show how a similar definition can be employed to construct a Bayesian support interval. Consistent with Carnap’s theory of corroboration, the support interval contains only parameter values that receive at least some minimum amount of support from the data. The support interval is not subject to Lindley’s paradox and provides an (...) evidence-based perspective on inference that differs from the belief-based perspective that forms the basis of the standard Bayesian credible interval. (shrink)

Many people regard utility theory as the only rigorous foundation for subjective probability, and even de Finetti thought the betting approach supplemented by Dutch Book arguments only good as an approximation to a utility-theoretic account. I think that there are good reasons to doubt this judgment, and I propose an alternative, in which the probability axioms are consistency constraints on distributions of fair betting quotients. The idea itself is hardly new: it is in de Finetti and also Ramsey. What is (...) new is that it is shown that probabilistic consistency and consequence can be defined in a way formally analogous to the way these notions are defined in deductive (propositional) logic. The result is a free-standing logic which does not pretend to be a theory of rationality and is therefore immune to, among other charges, that of “logical omniscience”. (shrink)

A pure significance test would check the agreement of a statistical model with the observed data even when no alternative model was available. The paper proposes the use of a modified p -value to make such a test. The model will be rejected if something surprising is observed. It is shown that the relation between this measure of surprise and the surprise indices of Weaver and Good is similar to the relationship between a p -value, a corresponding odds-ratio, and a (...) logit or log-odds statistic. The s -value is always larger than the corresponding p -value, and is not uniformly distributed. Difficulties with the whole approach are discussed. (shrink)

The Jeffreys–Lindley paradox exposes a rift between Bayesian and frequentist hypothesis testing that strikes at the heart of statistical inference. Contrary to what most current literature suggests, the paradox was central to the Bayesian testing methodology developed by Sir Harold Jeffreys in the late 1930s. Jeffreys showed that the evidence for a point-null hypothesis $${\mathcal {H}}_0$$ H 0 scales with $$\sqrt{n}$$ n and repeatedly argued that it would, therefore, be mistaken to set a threshold for rejecting $${\mathcal {H}}_0$$ H 0 (...) at a constant multiple of the standard error. Here, we summarize Jeffreys’s early work on the paradox and clarify his reasons for including the $$\sqrt{n}$$ n term. The prior distribution is seen to play a crucial role; by implicitly correcting for selection, small parameter values are identified as relatively surprising under $${\mathcal {H}}_1$$ H 1. We highlight the general nature of the paradox by presenting both a fully frequentist and a fully Bayesian version. We also demonstrate that the paradox does not depend on assigning prior mass to a point hypothesis, as is commonly believed. (shrink)