该研究比较尾概率的贝叶斯与频率学派估计，借助 Jensen 不等式和 Taylor 近似给出前者偏高的充分条件，并将原因归结为分布函数的凸性。
摘要：
本文研究了尾概率的贝叶斯估计量始终高于频率学派估计量的原因。通过利用 Jensen 不等式并考察 Taylor 级数近似，建立了该现象的充分条件；这两种方法都指向分布函数的凸性。
英文摘要：
In this paper, we investigate the reasons that the Bayesian estimator of the tail probability is always higher than the frequentist estimator. Sufficient conditions for this phenomenon are established both by using Jensen's Inequality and by looking at Taylor series approximations, both of which point to the convexity of the distribution function.
This paper explores the intrinsic connections between the Bayesian false discovery rate (FDR) control procedures and their counterpart of frequentist procedures. We attempt to offer a unified view of FDR control within and beyond the setting of testing exchangeable hypotheses. Under the standard two-groups model and the Oracle condition, we show that the Bayesian and the frequentist methods can achieve asymptotically equivalent FDR control at arbitrary levels. Built on this result, we further illustrate that rigorous post-fitting model diagnosis is necessary and effective to ensure robust FDR …
The prediction interval has been increasingly used in meta-analyses as a useful measure for assessing the magnitude of treatment effect and between-studies heterogeneity. In calculations of the prediction interval, although the Higgins-Thompson-Spiegelhalter method is used most often in practice, it might not have adequate coverage probability for the true treatment effect of a future study under realistic situations. An effective alternative candidate is the Bayesian prediction interval, which has also been widely used in general prediction problems. However, these prediction intervals are co…
Bayesian approaches for handling covariate measurement error are well established, and yet arguably are still relatively little used by researchers. For some this is likely due to unfamiliarity or disagreement with the Bayesian inferential paradigm. For others a contributory factor is the inability of standard statistical packages to perform such Bayesian analyses. In this paper we first give an overview of the Bayesian approach to handling covariate measurement error, and contrast it with regression calibration (RC), arguably the most commonly adopted approach. We then argue why the Bayesian …
This comment emphasizes the importance of model checking and model fitting when making inferences about finite population quantities. It also suggests the value of using unit level models when making inferences for small subpopulations, that is, "small area" analyses [arXiv:1108.2356].
We propose modified frequentist definition for the determination of confidence intervals for the case of Poisson statistics. Namely, we require that 1-β' \geq \sum_{n=o}^{n_{obs}+k} P(n|λ) \geq α'. We show that this definition is equivalent to the Bayesian method with prior π(λ) \sim λ^{k}. We also propose modified frequentist definition for the case of nonzero background.
According to Hansen, Madow and Tepping [J. Amer. Statist. Assoc. 78 (1983) 776--793], "Probability sampling designs and randomization inference are widely accepted as the standard approach in sample surveys." In this article, reasons are advanced for the wide use of this design-based approach, particularly by federal agencies and other survey organizations conducting complex large scale surveys on topics related to public policy. Impact of Bayesian methods in survey sampling is also discussed in two different directions: nonparametric calibrated Bayesian inferences from large samples and hiera…
Using a collection of simulated an real benchmarks, we compare Bayesian and frequentist regularization approaches under a low informative constraint when the number of variables is almost equal to the number of observations on simulated and real datasets. This comparison includes new global noninformative approaches for Bayesian variable selection built on Zellner's g-priors that are similar to Liang et al. (2008). The interest of those calibration-free proposals is discussed. The numerical experiments we present highlight the appeal of Bayesian regularization methods, when compared with non-B…
This note is an extended review of the book Error and Inference, edited by Deborah Mayo and Aris Spanos, about their frequentist and philosophical perspective on testing of hypothesis and on the criticisms of alternatives like the Bayesian approach.
We investigate the relation between frequentist and Bayesian approaches. Namely, we find the "frequentist" Bayes prior π_{f}(λ,x_{obs}) = -\frac{\int_{-\infty}^{x_{obs}}\frac{\partial f(x,λ)}{\partial λ}dx}{f(x_{obs},λ)} (here f(x,λ) is the probability density) for which the results of frequentist and Bayes approaches to the determination of confidence intervals coincide. In many cases (but not always) the "frequentist" prior which reproduces frequentist results coincides with the Jeffreys prior.
We propose information criteria that measure the prediction risk of a predictive density based on the Bayesian marginal likelihood from a frequentist point of view. We derive criteria for selecting variables in linear regression models, assuming a prior distribution of the regression coefficients. Then, we discuss the relationship between the proposed criteria and related criteria. There are three advantages of our method. First, this is a compromise between the frequentist and Bayesian standpoints because it evaluates the frequentist's risk of the Bayesian model. Thus, it is less influenced b…
We review and contrast frequentist and Bayesian definitions of tolerance regions. We give conditions under which for large samples a Bayesian region also has frequentist validity, and study the latter for smaller samples in a simulation study. We discuss a computational strategy for computing a Bayesian two-sided tolerance interval for a Gaussian future variable, and apply this to the case of possibly unbalanced linear mixed models. We illustrate the method on a quality control experiment from the pharmaceutical industry.
We introduce new estimation methods for a sub-class of the Gaussian scale mixture models for wavelet trees by Wainwright, Simoncelli & Willsky that rely on modern results for composite likelihoods and approximate Bayesian inference. Our methodology is illustrated for denoising and edge detection problems in two-dimensional images.
Multivariable Mendelian randomization with weak instruments: a comparison of Bayesian and frequentist methods
Sydney School of Public Health, University of Sydney（悉尼大学公共卫生学院）；MRC Biostatistics Unit, University of Cambridge（剑桥大学医学研究委员会生物统计单元）；Cardiovascular Epidemiology Unit, University of Cambridge（剑桥大学心血管流行病学单元）
Despite their importance in supporting experimental conclusions, standard statistical tests are often inadequate for research areas, like the life sciences, where the typical sample size is small and the test assumptions difficult to verify. In such conditions, standard tests tend to be overly conservative, and fail thus to detect significant effects in the data. Here we define a novel statistical test for the two-sample problem. Several characteristics make it an attractive alternative to classical two-sample tests: 1) It is based on Bayesian model selection, and thus takes into account uncer…
Analogues of the frequentist chi-square and F tests are proposed for testing goodness-of-fit and consistency for Bayesian models. Simple examples exhibit these tests' detection of inconsistency between consecutive experiments with identical parameters, when the first experiment provides the prior for the second. In a related analysis, a quantitative measure is derived for judging the degree of tension between two different experiments with partially overlapping parameter vectors.
A new computation method of frequentist $p$-values and Bayesian posterior probabilities based on the bootstrap probability is discussed for the multivariate normal model with unknown expectation parameter vector. The null hypothesis is represented as an arbitrary-shaped region. We introduce new parametric models for the scaling-law of bootstrap probability so that the multiscale bootstrap method, which was designed for one-sided test, can also computes confidence measures of two-sided test, extending applicability to a wider class of hypotheses. Parameter estimation is improved by the two-step…
Bayesian, frequentist and fiducial (BFF) inferences are much more congruous than they have been perceived historically in the scientific community (cf., Reid and Cox 2015; Kass 2011; Efron 1998). Most practitioners are probably more familiar with the two dominant statistical inferential paradigms, Bayesian inference and frequentist inference. The third, lesser known fiducial inference paradigm was pioneered by R.A. Fisher in an attempt to define an inversion procedure for inference as an alternative to Bayes' theorem. Although each paradigm has its own strengths and limitations subject to thei…
The problem of testing mutually exclusive hypotheses with dependent test statistics is considered. Bayesian and frequentist approaches to multiplicity control are studied and compared to help gain understanding as to the effect of test statistic dependence on each approach. The Bayesian approach is shown to have excellent frequentist properties and is argued to be the most effective way of obtaining frequentist multiplicity control, without sacrificing power, when there is considerable test statistic dependence.
Nonresponse weighting adjustment using the response propensity score is a popular tool for handling unit nonresponse. Statistical inference after the nonresponse weighting adjustment is complicated because the effect of estimating the propensity model parameter needs to be incorporated. In this paper, we propose an approximate Bayesian approach to handle unit nonresponse with parametric model assumptions on the response probability, but without model assumptions for the outcome variable. The proposed Bayesian method is calibrated to the frequentist inference in that the credible region obtaine…
The following zero-sum game between nature and a statistician blends Bayesian methods with frequentist methods such as p-values and confidence 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 confidence distribution or other benchmark distribution. An application to testing a simple null hypothesis leads the statistician to report a posterior probability of the hypothesis that is informed…
Following an extensive simulation study comparing the operating characteristics of three different procedures used for establishing equivalence (the frequentist "TOST", the Bayesian "HDI-ROPE", and the Bayes factor interval null procedure), Linde et al. (2021) conclude with the recommendation that "researchers rely more on the Bayes factor interval null approach for quantifying evidence for equivalence." We redo the simulation study of Linde et al. (2021) in its entirety but with the different procedures calibrated to have the same predetermined maximum type 1 error rate. Our results suggest t…
Consider a linear regression model with regression parameter beta and normally distributed errors. Suppose that the parameter of interest is theta = a^T beta where a is a specified vector. Define the parameter tau = c^T beta - t where c and t are specified and a and c are linearly independent. Also suppose that we have uncertain prior information that tau = 0. Kabaila and Giri, 2009, JSPI, describe a new frequentist 1-alpha confidence interval for theta that utilizes this uncertain prior information. We compare this confidence interval with Bayesian 1-alpha equi-tailed and shortest credible in…
While gravitational waves have not yet been measured directly, data analysis from detection experiments commonly includes an upper limit statement. Such upper limits may be derived via a frequentist or Bayesian approach; the theoretical implications are very different, and on the technical side, one notable difference is that one case requires maximization of the likelihood function over parameter space, while the other requires integration. Using a simple example (detection of a sinusoidal signal in white Gaussian noise), we investigate the differences in performance and interpretation, and t…
Many recently developed Bayesian methods have focused on sparse signal detection. However, much less work has been done addressing the natural follow-up question: how to make valid inferences for the magnitude of those signals after selection. Ordinary Bayesian credible intervals suffer from selection bias, owing to the fact that the target of inference is chosen adaptively. Existing Bayesian approaches for correcting this bias produce credible intervals with poor frequentist properties, while existing frequentist approaches require sacrificing the benefits of shrinkage typical in Bayesian met…
University of Tübingen（图宾根大学）；MPI for Intelligent Systems, Tübingen（图宾根智能系统研究所）
Despite their compelling theoretical properties, Bayesian neural networks (BNNs) tend to perform worse than frequentist methods in classification-based uncertainty quantification (UQ) tasks such as out-of-distribution (OOD) detection. In this paper, based on empirical findings in prior works, we hypothesize that this issue is because even recent Bayesian methods have never considered OOD data in their training processes, even though this "OOD training" technique is an integral part of state-of-the-art frequentist UQ methods. To validate this, we treat OOD data as a first-class citizen in BNN t…
The first-order autoregressive process, AR (1), has been widely used and implemented in time series analysis. Different estimation methods have been employed in order to estimate the autoregressive parameter. This article focuses on subjective Bayesian estimation as opposed to objective Bayesian estimation and frequentist procedures. The truncated normal distribution is considered as a prior, to impose stationarity. The posterior distribution as well as the Bayes estimator are derived. A comparative study between the newly derived estimator and other existing estimation methods (frequentist) i…
There are two main opposing schools of statistical reasoning, Frequentist and Bayesian approaches. Until recent days, the frequentist or classical approach has dominated the scientific research, but Bayesianism has reappeared with a strong impulse that is starting to change the situation. Recently the controversy about the primacy of one of the two approaches seems to be unfinished at a philosophical level, but scientific practices are giving an increasingly important position to the Bayesian approach. This paper eludes philosophical debate to focus on the pragmatic point of view of scientists…
Frequentist coverage of $(1-α)$-highest posterior density (HPD) credible sets is studied in a signal plus noise model under a large class of noise distributions. We consider a specific class of spike-and-slab prior distributions. Different regimes are identified and we derive closed form expressions for the $(1-α)$-HPD on each of these regimes. Similar to the earlier work by Marchand and Strawderman, it is shown that under suitable conditions, the frequentist coverage can drop to $1-3α/2.$
In many application domains, such as computational biology, the goal of graphical model structure learning is to uncover discrete relationships between entities. For example, in our problem of interest concerning HIV vaccine design, we want to infer which HIV peptides interact with which immune system molecules (HLA molecules). For problems of this nature, we are interested in determining the number of nonspurious arcs in a learned graphical model. We describe both a Bayesian and frequentist approach to this problem. In the Bayesian approach, we use the posterior distribution over model struct…
The present paper investigates theoretical performance of various Bayesian wavelet shrinkage rules in a nonparametric regression model with i.i.d. errors which are not necessarily normally distributed. The main purpose is comparison of various Bayesian models in terms of their frequentist asymptotic optimality in Sobolev and Besov spaces. We establish a relationship between hyperparameters, verify that the majority of Bayesian models studied so far achieve theoretical optimality, state which Bayesian models cannot achieve optimal convergence rate and explain why it happens.
