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无模型框架下线性回归的子模型一致界AI 解读

For the last two decades, high-dimensional data and methods have proliferated throughout the literature. Yet, the classical technique of linear regression has not lost its usefulness in applications. In fact, many high-dimensional estimation techniques can be seen as variable selection that leads to a smaller set of variables (a “submodel”) where classical linear regression applies. We analyze linear regression estimators resulting from model selection by proving estimation…

费城犯罪趋势：结合粒子优化的贝叶斯聚类AI 解读

Accurate estimation of the change in crime over time is a critical first step toward better understanding of public safety in large urban environments. Bayesian hierarchical modeling is a natural way to study spatial variation in urban crime dynamics at the neighborhood level, since it facilitates principled "sharing of information" between spatially adjacent neighborhoods. Typically, however, cities contain many physical and social boundaries that may manifest as spatial…

查尔斯·斯坦与不变性：从 Hunt–Stein 定理谈起AI 解读

When statistical decision theory was emerging as a promising new paradigm, Charles Stein was to play a major role in the development of minimax theory for invariant statistical problems. In some of his earliest work with Gil Hunt, he set out to prove that, in problems where invariant procedures have constant risk, any best invariant test would be minimax among all tests. Although finding it not quite true in general, this led to the legendary Hunt–Stein theorem, which…

Modern data-driven approaches to modeling make extensive use of covariate/model selection. Such selection incurs a cost: it invalidates classical statistical inference. A conservative remedy to the problem was proposed by Berk et al. (Ann. Statist. 41 (2013) 802–837) and further extended by Bachoc, Preinerstorfer and Steinberger (2016). These proposals, labeled “PoSI methods,” provide valid inference after arbitrary model selection. They are computationally NP-hard and have…

尖峰与厚尾LASSOAI 解读

Despite the wide adoption of spike-and-slab methodology for Bayesian variable selection, its potential for penalized likelihood estimation has largely been overlooked. In this article, we bridge this gap by cross-fertilizing these two paradigms with the Spike-and-Slab LASSO procedure for variable selection and parameter estimation in linear regression. We introduce a new class of self-adaptive penalty functions that arise from a fully Bayes spike-and-slab formulation,…

通过自动稀疏旋转实现快速贝叶斯因子分析AI 解读

Rotational post hoc transformations have traditionally played a key role in enhancing the interpretability of factor analysis. Regularization methods also serve to achieve this goal by prioritizing sparse loading matrices. In this work, we bridge these two paradigms with a unifying Bayesian framework. Our approach deploys intermediate factor rotations throughout the learning process, greatly enhancing the effectiveness of sparsity inducing priors. These automatic rotations…

EMVS：基于 EM 算法的贝叶斯变量选择AI 解读

Despite rapid developments in stochastic search algorithms, the practicality of Bayesian variable selection methods has continued to pose challenges. High-dimensional data are now routinely analyzed, typically with many more covariates than observations. To broaden the applicability of Bayesian variable selection for such high-dimensional linear regression contexts, we propose EMVS, a deterministic alternative to stochastic search based on an EM algorithm which exploits a…

采用广义g先验的完全贝叶斯因子AI 解读

For the normal linear model variable selection problem, we propose selection criteria based on a fully Bayes formulation with a generalization of Zellner’s g-prior which allows for p > n. A special case of the prior formulation is seen to yield tractable closed forms for marginal densities and Bayes factors which reveal new model evaluation characteristics of potential interest.

预测密度估计的容许性AI 解读

Let X|μ∼Np(μ, vxI) and Y|μ∼Np(μ, vyI) be independent p-dimensional multivariate normal vectors with common unknown mean μ. Based on observing X=x, we consider the problem of estimating the true predictive density p(y|μ) of Y under expected Kullback–Leibler loss. Our focus here is the characterization of admissible procedures for this problem. We show that the class of all generalized Bayes rules is a complete class, and that the easily interpretable conditions of Brown and…

多元回归的预测密度估计AI 解读

Suppose we observe X ∼ Nm(Aβ,σ2I) and would like to estimate the predictive density p(y|β) of a future Y ∼ Nn(Bβ,σ2I). Evaluating predictive estimates by Kullback–Leibler loss, we develop and evaluate Bayes procedures for this problem. We obtain general sufficient conditions for minimaxity and dominance of the “noninformative” uniform prior Bayes procedure. We extend these results to situations where only a subset of the predictors in A is thought to be potentially…

摘要暂缺

Kullback–Leibler 损失下改进的极小极大预测密度AI 解读

Let X|μ∼Np(μ,vxI) and Y|μ∼Np(μ,vyI) be independent p-dimensional multivariate normal vectors with common unknown mean μ. Based on only observing X=x, we consider the problem of obtaining a predictive density p̂(y|x) for Y that is close to p(y|μ) as measured by expected Kullback–Leibler loss. A natural procedure for this problem is the (formal) Bayes predictive density p̂U(y|x) under the uniform prior πU(μ)≡1, which is best invariant and minimax. We show that any Bayes…

变量选择问题AI 解读

The problem of variable selection is one of the most pervasive model selection problems in statistical applications. Often referred to as the problem of subset selection, it arises when one wants to model the relationship between a variable of interest and a subset of potential explanatory variables or predictors, but there is uncertainty about which subset to use. This vignette reviews some of the key developments that have led to the wide variety of approaches for this…

灵活的小波经验贝叶斯估计AI 解读

Summary Wavelet shrinkage estimation is an increasingly popular method for signal denoising and compression. Although Bayes estimators can provide excellent mean-squared error (MSE) properties, the 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…

多元回归的风险膨胀准则AI 解读

A new criterion is proposed for the evaluation of variable selection procedures in multiple regression. This criterion, which we call the risk inflation, is based on an adjustment to the risk. Essentially, the risk inflation is the maximum increase in risk due to selecting rather than knowing the "correct" predictors. A new variable selection procedure is obtained which, in the case of orthogonal predictors, substantially improves on AIC, $C_p$ and BIC and is close to…

通过Gibbs采样进行变量选择AI 解读

A crucial problem in building a multiple regression model is the selection of predictors to include. The main thrust of this article is to propose and develop a procedure that uses probabilistic considerations for selecting promising subsets. This procedure entails embedding the regression setup in a hierarchical normal mixture model where latent variables are used to identify subset choices. In this framework the promising subsets of predictors can be identified as those…

未知变点之前的参数估计AI 解读

Consider the problem of estimating $\mu$, based on the observation of $Y_0, Y_1, \ldots, Y_n$, where it is assumed only that $Y_0, Y_1, \ldots, Y_\kappa \operatorname{iid} N(\mu, \sigma^2)$ for some unknown $\kappa$. Unlike the traditional change-point problem, the focus here is not on estimating $\kappa$, which is now a nuisance parameter. When it is known that $\kappa = k$, the sample mean $\bar{Y}_k = \sum^k_0Y_i/(k + 1)$, provides, in addition to wonderful efficiency…

摘要暂缺

利润导向损失函数下的估计AI 解读

Consider the problem of estimating a price-sensitivity parameter in a demand model. Depending on the context in which the estimate will be used, traditional squared-error loss may be inappropriate. The authors consider the situation in which the estimate will be used by a manufacturer to set the price. Effectively, the manufacturer's goal of profit maximization induces a loss function that turns out to be asymmetric. Estimates that perform well with respect to such loss…

利用 Gibbs 抽样进行捕获—再捕获估计AI 解读

Capture-recapture models are widely used in the estimation of population sizes. Based on data augmentation considerations, we show how Gibbs sampling can be applied to calculate Bayes estimates in this setting. As a result, formulations which were previously avoided because of analytical and numerical intractability can now be easily considered for practical application. We illustrate this potential by using Gibbs sampling to calculate Bayes estimates for a hierarchical…

价格与促销弹性的收缩估计：似乎不相关方程AI 解读

Consider the problem where a retailer or manufacturer wants to estimate product price and promotional elasticities based on supermarket scanner data. Classical linear modeling suffers from the following aggregation dilemma. Price and promotional elasticities appear to vary considerably among chains and brands so that one overall model is too restrictive. Alternatively, the use of a different model for each chain and brand leads to noisy and often nonsensical estimates of…

极小极大收缩估计量的组合AI 解读

When one estimates a multivariate normal mean, the use of Stein estimation entails both the grouping of coordinates and the selection of a set of targets toward which to shrink each group. In this article we propose new minimax multiple shrinkage estimators that allow for multiple specifications of these aspects. We provide examples that are evaluated on real and simulated data, including an estimator that adaptively resolves the issue of combining possibly related…

极小极大多重收缩估计AI 解读

For the canonical problem of estimating a multivariate normal mean under squared-error-loss, this article addresses the problem of selecting a minimax shrinkage estimator when vague or conflicting prior information suggests that more than one estimator from a broad class might be effective. For this situation a new class of alternative estimators, called multiple shrinkage estimators, is proposed. These estimators use the data to emulate the behavior and risk properties of…

集体诉讼中的损害金额估计AI 解读

In a class action litigation, actual damages are not known exactly and must be estimated. Various estimators are proposed and assessed by using a model that identifies possible sources of error. Estimators that have been used in practice are shown to be seriously biased. An empirical Bayes estimator and an empirical minimal mean squared error estimator are found to be more satisfactory methods for estimating damages.