We propose to change the default P-value threshold for statistical significance from 0.05 to 0.005 for claims of new discoveries. T he lack of reproducibility of scientific studies has caused growing concern over the credibility of claims of new discoveries based on 'statistically significant' findings. There has been much progress toward documenting and addressing several causes of this lack of reproducibility (for example, multiple testing, P-hacking, publication bias and under-powered studies). However, we believe that a leading cause of non-reproducibility has not yet been adequately addressed: statistical standards of evidence for claiming new discoveries in many fields of science are simply too low. Associating statistically significant findings with P < 0.05 results in a high rate of false positives even in the absence of other experimental, procedural and reporting problems.For fields where the threshold for defining statistical significance for new discoveries is P < 0.05, we propose a change to P < 0.005. This simple step would immediately improve the reproducibility of scientific research in many fields. Results that would currently be called significant but do not meet the new threshold should instead be called suggestive. While statisticians have known the relative weakness of using P ≈ 0.05 as a threshold for discovery and the proposal to lower it to 0.005 is not new 1,2 , a critical mass of researchers now endorse this change.We restrict our recommendation to claims of discovery of new effects. We do not address the appropriate threshold for confirmatory or contradictory replications of existing claims. We also do not advocate changes to discovery thresholds in fields that have already adopted more stringent standards (for example, genomics and high-energy physics research; see the 'Potential objections' section below).We also restrict our recommendation to studies that conduct null hypothesis significance tests. We have diverse views about how best to improve reproducibility, and many of us believe that other ways of summarizing the data, such as Bayes factors or other posterior summaries based on clearly articulated model assumptions, are preferable to P values. However, changing the P value threshold is simple, aligns with the training undertaken by many researchers, and might quickly achieve broad acceptance.

Zellner's g-prior remains a popular conventional prior for use in Bayesian variable selection, despite several undesirable consistency issues. In this paper, we study mixtures of g-priors as an alternative to default g-priors that resolve many of the problems with the original formulation, while maintaining the computational tractability that has made the g-prior so popular. We present theoretical properties of the mixture g-priors and provide real and simulated examples to compare the mixture formulation with fixed g-priors, Empirical Bayes approaches and other default procedures.

For the problem of model choice in linear regression, we introduce a Bayesian adaptive sampling algorithm (BAS), that samples models without replacement from the space of models. For problems that permit enumeration of all models, BAS is guaranteed to enumerate the model space in 2 p iterations where p is the number of potential variables under consideration. For larger problems where sampling is required, we provide conditions under which BAS provides perfect samples without replacement. When the sampling probabilities in the algorithm are the marginal variable inclusion probabilities, BAS may be viewed as sampling models "near" the median probability model of Barbieri and Berger. As marginal inclusion probabilities are not known in advance, we discuss several strategies to estimate adaptively the marginal inclusion probabilities within BAS. We illustrate the performance of the algorithm using simulated and real data and show that BAS can outperform Markov chain Monte Carlo methods. The algorithm is implemented in the R package BAS available at CRAN. This article has supplementary material online.

This paper discusses Bayesian methods for multiple shrinkage estimation in wavelets. Wavelets are used in applications for data denoising, via shrinkage of the coe cients towards zero, and for data compression, by shrinkage and setting small coe cients to zero. We approach wavelet shrinkage by using Bayesian hierarchical models, assigning a positive prior probability to the wavelet coe cients being zero. The resulting estimator for the wavelet coe cients is a multiple shrinkage estimator that exhibits a wide variety of nonlinear shrinkage patterns. We discuss fast computational implementations, with a focus on easy-to-compute analytic approximations as well as importance sampling and Markov chain Monte Carlo methods. Multiple shrinkage estimators prove to have excellent mean squared error performance in reconstructing standard test functions. We demonstrate this in simulated test examples, comparing various implementations of multiple shrinkage to commonly used shrinkage rules. Finally, we illustrate our approach with an application to the so-called \glint" data.

The evolution of Bayesian approaches for model uncertainty over the past decade has been remarkable. Catalyzed by advances in methods and technology for posterior computation, the scope of these methods has widened substantially. Major thrusts of these developments have included new methods for semiautomatic prior specification and posterior exploration. To illustrate key aspects of this evolution, the highlights of some of these developments are described.

SummaryWavelet shrinkage estimation is an increasingly popular method for signal denoising and compression.Although Bayes estimators can provide excellent mean squared error (MSE) properties, selection of an effective prior is a difficult task. To address this problem, we propose Empirical Bayes (EB) prior selection methods for various error distributions including the normal and the heavier tailed Student t distributions.Under such EB prior distributions, we obtain threshold shrinkage estimators based on model selection, and multiple shrinkage estimators based on model averaging. These EB estimators are seen to be computationally competitive with standard classical thresholding methods, and to be robust to outliers in both the data and wavelet domains. Simulated and real examples are used to illustrate the flexibility and improved MSE performance of these methods in a wide variety of settings.

There are two generations of Gibbs sampling methods for semiparametric models involving the Dirichlet process. The first generation suffered from a severe drawback: the locations of the clusters, or groups of parameters, could essentially become fixed, moving only rarely. Two strategies that have been proposed to create the second generation of Gibbs samplers are integration and appending a second stage to the Gibbs sampler wherein the cluster locations are moved. We show that these same strategies are easily implemented for the sequential importance sampler, and that the first strategy dramatically improves results. As in the case of Gibbs sampling, these strategies are applicable to a much wider class of models. They are shown to provide more uniform importance sampling weights and lead to additional Rao-Blackwellization of estimators. RESUMEDeux gtntrations d'tchantillonneurs de Gibbs ont it6 popularistes pour les modtles semiparamttriques oh intervient le processus de Dirichlet. La premitre gtntration Ctait lourdement handicap6.e du fait que la localisation des grappes, ou groupes de paramttres, pouvait devenir quasi-stationnaire. La d e u x i h e gtnCration dkchantillonneurs a comblt cette lacune en incorporant deux strattgies: I'inttgration et I'ajout d'une Ctape de relocalisation des grappes. Les auteurs montrent ici que ces stratkgies sont tgalement applicables a la mtthode d'tchantillonnage stquentiel pondtrt et que I'inttgration est particulitrement efficace dans ce contexte. Comme dans le cas de I'tchantillonnage de Gibbs, les deux strategies s'appliquent en fait ? I une classe beaucoup plus grande de modtles. Comme les auteurs le font valoir, elles produisent des poids d'khantillonnage plus uniformes et permettent une Rao-Blackwellisation additionnelle des estimateurs.

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