Yuzo Maruyama, William E. Strawderman
We derive minimax generalized Bayes estimators of regression coefficients in the general linear model with spherically symmetric errors under invariant quadratic loss for the case of unknown scale. The class of estimators generalizes the class considered in Maruyama and Strawderman (2005) to include non-monotone shrinkage functions.
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Minimax Simple Bayes Estimators of a Normal Variance
本文提出了一种简洁的贝叶斯估计器，解决了正态方差在熵损失下的最小最大性问题，提供了闭式结果。
The paper deals with the non-parametric estimation in the regression with the multiplicative noise. Using the local polynomial fitting and the bayesian approach, we construct the minimax on isotropic Hölder class estimator. Next applying Lepski's method, we propose the estimator which is optimally adaptive over the collection of isotropic Hölder classes. To prove the optimality of the proposed procedure we establish, in particular, the exponential inequality for the deviation of locally bayesian estimator from the parameter to be estimated. These theoretical results are illustrated by simulati…
In this article, we consider two forms of shrinkage estimators of the mean $θ$ of a multivariate normal distribution $X\sim N_{p}\left(θ, σ^{2}I_{p}\right)$ where $σ^{2}$ is unknown. We take the prior law $θ\sim N_{p}\left(\upsilon, τ^{2}I_{p}\right)$ and we constuct a Modified Bayes estimator $δ_{B}^{\ast}$ and an Empirical Modified Bayes estimator $δ_{EB}^{\ast}$. We are interested in studying the minimaxity and the limits of risks ratios of these estimators, to the maximum likelihood estimator $X$, when $n$ and $p$ tend to infinity.
We obtain the optimal Bayesian minimax rate for the unconstrained large covariance matrix of multivariate normal sample with mean zero, when both the sample size, n, and the dimension, p, of the covariance matrix tend to infinity. Traditionally the posterior convergence rate is used to compare the frequentist asymptotic performance of priors, but defining the optimality with it is elusive. We propose a new decision theoretic framework for prior selection and define Bayesian minimax rate. Under the proposed framework, we obtain the optimal Bayesian minimax rate for the spectral norm for all rat…
In this paper, we consider the estimation of a mean vector of a multivariate normal population where the mean vector is suspected to be nearly equal to mean vectors of $k-1$ other populations. As an alternative to the preliminary test estimator based on the test statistic for testing hypothesis of equal means, we derive empirical and hierarchical Bayes estimators which shrink the sample mean vector toward a pooled mean estimator given under the hypothesis. The minimaxity of those Bayesian estimators are shown, and their performances are investigated by simulation.
One-step ahead prediction for the multinomial model is considered. The performance of a predictive density is evaluated by the average Kullback-Leibler divergence from the true density to the predictive density. Asymptotic approximations of risk functions of Bayesian predictive densities based on Dirichlet priors are obtained. It is shown that a Bayesian predictive density based on a specific Dirichlet prior is asymptotically minimax. The asymptotically minimax prior is different from known objective priors such as the Jeffreys prior or the uniform prior.
This paper studies a Bayesian approach to non-asymptotic minimax adaptation in nonparametric estimation. Estimating an input function on the basis of output functions in a Gaussian white-noise model is discussed. The input function is assumed to be in a Sobolev ellipsoid with an unknown smoothness and an unknown radius. Our purpose in this paper is to present a Bayesian approach attaining minimaxity up to a universal constant without any knowledge regarding the smoothness and the radius. Our Bayesian approach provides not only a rate-exact minimax adaptive estimator in large sample asymptotics…
We give a sufficient condition for admissibility of generalized Bayes estimators of the location vector of spherically symmetric distribution under squared error loss. Compared to the known results for the multivariate normal case, our sufficient condition is very tight and is close to being a necessary condition. In particular we establish the admissibility of generalized Bayes estimators with respect to the harmonic prior and priors with slightly heavier tail than the harmonic prior. We use the theory of regularly varying functions to construct a sequence of smooth proper priors approaching …
This paper deals with the problem of estimating predictive densities of a matrix-variate normal distribution with known covariance matrix. Our main aim is to establish some Bayesian predictive densities related to matricial shrinkage estimators of the normal mean matrix. The Kullback-Leibler loss is used for evaluating decision-theoretical optimality of predictive densities. It is shown that a proper hierarchical prior yields an admissible and minimax predictive density. Also, superharmonicity of prior densities is paid attention to for finding out a minimax predictive density with good numeri…
Nearly minimax empirical Bayesian prediction of independent Poisson observables
本研究提出一类经验贝叶斯预测分布，用于独立泊松观测量，在K-L损失下优于Jeffreys先验贝叶斯预测，且风险小于极小极大下界的1.04倍。
In many statistical problems, stochastic signals can be represented as a sequence of noisy wavelet coefficients. In this paper, we develop general empirical Bayes methods for the estimation of true signal. Our estimators approximate certain oracle separable rules and achieve adaptation to ideal risks and exact minimax risks in broad collections of classes of signals. In particular, our estimators are uniformly adaptive to the minimum risk of separable estimators and the exact minimax risks simultaneously in Besov balls of all smoothness and shape indices, and they are uniformly superefficient …
We study asymptotically minimax predictive distributions in an infinite sequence model. First, we discuss the connection between the prediction in the infinite sequence model and the prediction in the function model. Second, we construct an asymptotically minimax predictive distribution when the parameter space is a known ellipsoid. We show that the Bayesian predictive distribution based on the Gaussian prior distribution is asymptotically minimax in the ellipsoid. Third, we construct an asymptotically minimax predictive distribution for any Sobolev ellipsoid. We show that the Bayesian predict…
In the present paper, we derive lower bounds for the risk of the nonparametric empirical Bayes estimators. In order to attain the optimal convergence rate, we propose generalization of the linear empirical Bayes estimation method which takes advantage of the flexibility of the wavelet techniques. We present an empirical Bayes estimator as a wavelet series expansion and estimate coefficients by minimizing the prior risk of the estimator. As a result, estimation of wavelet coefficients requires solution of a well-posed low-dimensional sparse system of linear equations. The dimension of the syste…
Bayesimax Theory: Selecting Priors by Minimizing Total Information
本文提出 Bayesimax 理论，通过极小极大先验披露博弈选择先验，以最小化总信息量（条件熵最大化），并给出算法与理论性质。
Adversarial Meta-Learning of Gamma-Minimax Estimators That Leverage Prior Knowledge
本文针对先验知识模糊时无法用单一先验表达的问题，提出为一般模型定义Gamma-minimax估计器，并设计对抗元学习算法在广义矩约束下计算，附收敛性保证，在熵估计和生物多样性预测中验证。
This paper reviews minimax best equivariant estimation in these invariant estimation problems: a location parameter, a scale parameter and a (Wishart) covariance matrix. We briefly review development of the best equivariant estimator as a generalized Bayes estimator relative to right invariant Haar measure in each case. Then we prove minimaxity of the best equivariant procedure by giving a least favorable prior sequence based on non-truncated Gaussian distributions. The results in this paper are all known, but we bring a fresh and somewhat unified approach by using, in contrast to most proofs …
In this article, we investigate certain asymptotic optimality properties of a very broad class of one-group continuous shrinkage priors for simultaneous estimation and testing of a sparse normal mean vector. Asymptotic optimality of Bayes estimates and posterior concentration properties corresponding to the general class of one-group priors under consideration are studied where the data is assumed to be generated according to a multivariate normal distribution with a fixed unknown mean vector. Under the assumption that the number of non-zero means is known, we show that Bayes estimators arisin…
Minimaxity under the half-Cauchy prior
针对二次损失下 p 元正态均值估计问题，本文利用区间算术从理论上证明了半 Cauchy 先验对应的 Bayes 估计量具有极小极大性。
For normal canonical models, and more generally a vast array of general spherically symmetric location-scale models with a residual vector, we consider estimating the (univariate) location parameter when it is lower bounded. We provide conditions for estimators to dominate the benchmark minimax MRE estimator, and thus be minimax under scale invariant loss. These minimax estimators include the generalized Bayes estimator with respect to the truncation of the common non-informative prior onto the restricted parameter space for normal models under general convex symmetric loss, as well as non-nor…
Matrix norm shrinkage estimators and priors
该研究针对Frobenius损失下的正态均值矩阵，构造了推广James-Stein等经典估计量的极小极大估计量，提出推广Stein先验的超调和先验，经模拟验证其性能良好。
This paper is concerned with the simultaneous estimation of $k$ population means when one suspects that the $k$ means are nearly equal. As an alternative to the preliminary test estimator based on the test statistics for testing hypothesis of equal means, we derive Bayesian and minimax estimators which shrink individual sample means toward a pooled mean estimator given under the hypothesis. It is shown that both the preliminary test estimator and the Bayesian minimax shrinkage estimators are further improved by shrinking the pooled mean estimator. The performance of the proposed shrinkage esti…
The art of BART: Minimax optimality over nonhomogeneous smoothness in high dimension
本文在非齐次各向异性光滑性和可能非连续的条件下，提出分片异质各向异性Hölder函数类并推导高维极小极大下界，证明贝叶斯森林结合Dirichlet子集选择先验可自适应未知光滑性、非连续性与稀疏性，并在数值上优于随机森林和深度神经网络。
In a remarkable series of papers beginning in 1956, Charles Stein set the stage for the future development of minimax shrinkage estimators of a multivariate normal mean under quadratic loss. More recently, parallel developments have seen the emergence of minimax shrinkage estimators of multivariate normal predictive densities under Kullback--Leibler risk. We here describe these parallels emphasizing the focus on Bayes procedures and the derivation of the superharmonic conditions for minimaxity as well as further developments of new minimax shrinkage predictive density estimators including mult…
Minimaxity under half-Cauchy type priors
针对半 Cauchy 先验下正态均值向量估计的极小极大性问题，本文推广到 U 形先验类，给出充分条件、后验采样算法及数值验证。
It is generally believed that ensemble approaches, which combine multiple algorithms or models, can outperform any single algorithm at machine learning tasks, such as prediction. In this paper, we propose Bayesian convex and linear aggregation approaches motivated by regression applications. We show that the proposed approach is minimax optimal when the true data-generating model is a convex or linear combination of models in the list. Moreover, the method can adapt to sparsity structure in which certain models should receive zero weights, and the method is tuning parameter free unlike competi…
We study admissibility of a subclass of generalized Bayes estimators of a multivariate normal vector when the variance is unknown, under scaled quadratic loss. Minimaxity is also established for certain of these estimators.
In this paper we deal with the regression problem in a random design setting. We investigate asymptotic optimality under minimax point of view of various Bayesian rules based on warped wavelets and show that they nearly attain optimal minimax rates of convergence over the Besov smoothness class considered. Warped wavelets have been introduced recently, they offer very good computable and easy-to-implement properties while being well adapted to the statistical problem at hand. We particularly put emphasis on Bayesian rules leaning on small and large variance Gaussian priors and discuss their si…
Group testing is an active area of current research and has important applications in medicine, biotechnology, genetics, and product testing. There have been recent advances in design and estimation, but the simple Dorfman procedure introduced by R. Dorfman in 1943 is widely used in practice. In many practical situations the exact value of the probability p of being affected is unknown. We present both minimax and Bayesian solutions for the group size problem when p is unknown. For unbounded p we show that the minimax solution for group size is 8, while using a Bayesian strategy with Jeffreys …
Bayesian and minimax estimators of loss
该研究针对损失估计的序贯与同步两种情形，推导了多种概率模型下的贝叶斯解与极小极大估计量，明确极小极大结果依赖于第二阶段估计量为常数的贝叶斯解性质。
In practice, it is desired to have estimates that are invariant under reparameterization. The invariance property of the estimators helps to formulate a unified solution to the underlying estimation problem. In robust Bayesian analysis, a frequent criticism is that the optimal estimators are not invariant under smooth reparameterizations. This paper considers the problem of posterior regret gamma-minimax (PRGM) estimation of the natural parameter of the exponential family of distributions under intrinsic loss functions. We show that under the class of Jeffrey's Conjugate Prior (JCP) distributi…
While the Bayesian decision-theoretic framework offers an elegant solution to the problem of decision making under uncertainty, one question is how to appropriately select the prior distribution. One idea is to employ a worst-case prior. However, this is not as easy to specify in sequential decision making as in simple statistical estimation problems. This paper studies (sometimes approximate) minimax-Bayes solutions for various reinforcement learning problems to gain insights into the properties of the corresponding priors and policies. We find that while the worst-case prior depends on the s…
Bayesian methods suffer from the problem of how to specify prior beliefs. One interesting idea is to consider worst-case priors. This requires solving a stochastic zero-sum game. In this paper, we extend well-known results from bandit theory in order to discover minimax-Bayes policies and discuss when they are practical.
We prove a new minimax theorem connecting the worst-case Bayesian regret and minimax regret under partial monitoring with no assumptions on the space of signals or decisions of the adversary. We then generalise the information-theoretic tools of Russo and Van Roy (2016) for proving Bayesian regret bounds and combine them with the minimax theorem to derive minimax regret bounds for various partial monitoring settings. The highlight is a clean analysis of `non-degenerate easy' and `hard' finite partial monitoring, with new regret bounds that are independent of arbitrarily large game-dependent co…
Simulation from the truncated multivariate normal distribution in high dimensions is a recurrent problem in statistical computing, and is typically only feasible using approximate MCMC sampling. In this article we propose a minimax tilting method for exact iid simulation from the truncated multivariate normal distribution. The new methodology provides both a method for simulation and an efficient estimator to hitherto intractable Gaussian integrals. We prove that the estimator possesses a rare vanishing relative error asymptotic property. Numerical experiments suggest that the proposed scheme …
We construct an adaptive wavelet estimator that attains minimax near-optimal rates in a wide range of Besov balls. The convergence rates are affected only by the weakest dependence amongst the channels, and take into account both noise sources.
In this paper, we are interested in the study of beta kernel estimators from an asymptotic minimax point of view. It is well known that beta kernel estimators are, on the contrary of classical kernel estimators, "free of boundary effect" and thus are very useful in practice. The goal of this paper is to prove that there is a price to pay: for very regular functions or for certain losses, these estimators are not minimax. Nevertheless they are minimax for classical regularities such as regularity of order two or less than two, supposed commonly in the practice and for some classical losses.
In this paper, we observe a sparse mean vector through Gaussian noise and we aim at estimating some additive functional of the mean in the minimax sense. More precisely, we generalize the results of (Collier et al., 2017, 2019) to a very large class of functionals. The optimal minimax rate is shown to depend on the polynomial approximation rate of the marginal functional, and optimal estimators achieving this rate are built.
Minimax Quasi-Bayesian estimation in sparse canonical correlation analysis via a Rayleigh quotient function
针对稀疏 CCA 中速率最优估计计算昂贵的问题，提出基于 Rayleigh 商拟似然与 spike-and-slab 先验的 MCMC 拟贝叶斯方法，达到 minimax 速率并在实验及 Covid-19 数据分析中表现优越。
In this paper, we introduce a new estimator for the emission densities of a nonparametric hidden Markov model. It is adaptive and minimax with respect to each state's regularity--as opposed to globally minimax estimators, which adapt to the worst regularity among the emission densities. Our method is based on Goldenshluger and Lepski's methodology. It is computationally efficient and only requires a family of preliminary estimators, without any restriction on the type of estimators considered. We present two such estimators that allow to reach minimax rates up to a logarithmic term: a spectral…
In this paper, a lower bound is determined in the minimax sense for change point estimators of the first derivative of a regression function in the fractional white noise model. Similar minimax results presented previously in the area focus on change points in the derivatives of a regression function in the white noise model or consider estimation of the regression function in the presence of correlated errors.
For the problem of nonparametric estimation of signal in Gaussian noise we point out the strong asymptotically minimax estimators on maxisets for linear estimators (see \cite{ker93,rio}). It turns out that the order of rates of convergence of Pinsker estimator on this maxisets is worse than the order of rates of convergence for the class of linear estimators considered on this maxisets. We show that balls in Sobolev spaces are maxisets for Pinsker estimators.
The $L\_2$-minimax risk in Sobolev classes of densities with non-integer smoothness index is shown to have an analog form to that in integer Sobolev classes. To this end, the notion of Sobolev classes is generalized to fractional derivatives of order $β\in\mathbb R^+$. A minimax kernel density estimator for such a classes is found. Although there exists no corresponding proof in the literature so far, the result of this article was used implicitly in numerous papers. A certain necessity that this gap had to be filled, can thus not be denied.
Constructing Bayes Minimax Estimators through Integral Transformations
本文研究二次损失下多元正态均值的贝叶斯极小极大估计，提出利用 I-变换与 Laplace 变换并配合球形先验来构造估计量，并通过示例验证其有效性。
This article discusses estimation of a multivariate normal mean based on heteroscedastic observations. Under heteroscedasticity, estimators shrinking more on the coordinates with larger variances, seem desirable. Although they are not necessarily minimax in the ordinary sense, we show that such James-Stein type estimators can be ensemble minimax, minimax with respect to the ensemble risk, related to empirical Bayes perspective of Efron and Morris.
Many statistical practices involve choosing between a full model and reduced models where some coefficients are reduced to zero. Data were used to select a model with estimated coefficients. Is it possible to do so and still come up with an estimator always better than the traditional estimator based on the full model? The James-Stein estimator is such an estimator, having a property called minimaxity. However, the estimator considers only one reduced model, namely the origin. Hence it reduces no coefficient estimator to zero or every coefficient estimator to zero. In many applications includi…
Consider the problem of estimating a multivariate normal mean with a known variance matrix, which is not necessarily proportional to the identity matrix. The coordinates are shrunk directly in proportion to their variances in Efron and Morris' (J. Amer. Statist. Assoc. 68 (1973) 117-130) empirical Bayes approach, whereas inversely in proportion to their variances in Berger's (Ann. Statist. 4 (1976) 223-226) minimax estimators. We propose a new minimax estimator, by approximately minimizing the Bayes risk with a normal prior among a class of minimax estimators where the shrinkage direction is o…
We consider the problem of estimating a continuous distribution function $F$, as well as meaningful functions $τ(F)$ under a large class of loss functions. We obtain best invariant estimators and establish their minimaxity for Hölder continuous $τ$'s and strict bowl-shaped losses with a bounded derivative. We also introduce and motivate the use of integrated balanced loss functions which combine the criteria of an integrated distance between a decision $d$ and $F$, with the proximity of $d$ with a target estimator $d_0$. Moreover, we show how the risk analysis of procedures under such an integ…
Let y=Aβ+ε, where y is an N\times1 vector of observations, βis a p\times1 vector of unknown regression coefficients, A is an N\times p design matrix and εis a spherically symmetric error term with unknown scale parameter σ. We consider estimation of βunder general quadratic loss functions, and, in particular, extend the work of Strawderman [J. Amer. Statist. Assoc. 73 (1978) 623-627] and Casella [Ann. Statist. 8 (1980) 1036-1056, J. Amer. Statist. Assoc. 80 (1985) 753-758] by finding adaptive minimax estimators (which are, under the normality assumption, also generalized Bayes) of β, which hav…
For the important classical problem of inference on a sparse high-dimensional normal mean vector, we propose a novel empirical Bayes model that admits a posterior distribution with desirable properties under mild conditions. In particular, our empirical Bayes posterior distribution concentrates on balls, centered at the true mean vector, with squared radius proportional to the minimax rate, and its posterior mean is an asymptotically minimax estimator. We also show that, asymptotically, the support of our empirical Bayes posterior has roughly the same effective dimension as the true sparse mea…
Two major research tasks lie at the heart of high dimensional data analysis: accurate parameter estimation and correct support recovery. The existing literature mostly aims for either the best parameter estimation or the best model selection result, however little has been done to understand the potential interaction between the estimation precision and the selection behavior. In this work, our minimax result shows that an estimator's performance of type I error control directly links with its $L_2$ estimation error rate, and reveals a trade-off phenomenon between the rate of convergence and t…
We address the problem of adaptive minimax density estimation on $\bR^d$ with $\bL_p$--loss on the anisotropic Nikol'skii classes. We fully characterize behavior of the minimax risk for different relationships between regularity parameters and norm indexes in definitions of the functional class and of the risk. In particular, we show that there are four different regimes with respect to the behavior of the minimax risk. We develop a single estimator which is (nearly) optimal in orderover the complete scale of the anisotropic Nikol'skii classes. Our estimation procedure is based on a data-drive…
Proper Bayes minimax multiple shrinkage estimation
研究在平方误差损失下估计多元正态均值问题，通过引入新方法证明适当贝叶斯极小极大多重收缩估计量存在，其能自适应向更有前景目标收缩，解决多目标选择挑战，且构建方式与以往不同。
Minimax Private Estimation of Smooth Optimal-Transport Maps
针对差分隐私约束下光滑最优传输映射的估计问题，提出基于小波的密度估计器和稳定性界，在中心与局部DP模型下达到近极小极大最优率，并建立匹配的极小极大下界。
We study predictive density estimation under Kullback-Leibler loss in $\ell_0$-sparse Gaussian sequence models. We propose proper Bayes predictive density estimates and establish asymptotic minimaxity in sparse models. A surprise is the existence of a phase transition in the future-to-past variance ratio $r$. For $r < r_0 = (\surd 5 - 1)/4$, the natural discrete prior ceases to be asymptotically optimal. Instead, for subcritical $r$, a `bi-grid' prior with a central region of reduced grid spacing recovers asymptotic minimaxity. This phenomenon seems to have no analog in the otherwise parallel …
A new class of minimax Stein-type shrinkage estimators of a multivariate normal mean is studied where the shrinkage factor is based on an l_p norm. The proposed estimators allow some but not all coordinates to be estimated by 0 thereby allow sparsity as well as minimaxity.
In this article, we consider the problem of inverting the exponential Radon transform of a function in the presence of noise. We propose a kernel estimator to estimate the true function, analogous to the one proposed by Korostelëv and Tsybakov in their article `Optimal rates of convergence of estimators in a probabilistic setup of tomography problem', Problems of Information Transmission, 27:73-81,1991. For the estimator proposed in this article, we then show that it converges to the true function at a minimax optimal rate.
Standard approaches to stochastic gradient estimation, with only noisy black-box function evaluations, use the finite-difference method or its variants. While natural, it is open to our knowledge whether their statistical accuracy is the best possible. This paper argues so by showing that central finite-difference is a nearly minimax optimal zeroth-order gradient estimator for a suitable class of objective functions and mean squared risk, among both the class of linear estimators and the much larger class of all (nonlinear) estimators.
We present estimators for a well studied statistical estimation problem: the estimation for the linear regression model with soft sparsity constraints ($\ell_q$ constraint with $0<q\leq1$) in the high-dimensional setting. We first present a family of estimators, called the projected nearest neighbor estimator and show, by using results from Convex Geometry, that such estimator is within a logarithmic factor of the optimal for any design matrix. Then by utilizing a semi-definite programming relaxation technique developed in [SIAM J. Comput. 36 (2007) 1764-1776], we obtain an approximation algor…
Signal-to-noise ratio aware minimaxity and higher-order asymptotics
本文提出信噪比感知的极小极大框架，通过高阶渐近精确近似风险并发现最优估计量，以解决经典极小极大理论在稀疏高斯序列模型中无法解释经验次优性能的问题。
Analytic evaluation of heteroskedasticity consistent covariance matrix estimates (HCCME) is difficult because of the complexity of the formulae currently available. We obtain new analytic formulae for the bias of a class of estimators of the covariance matrix of OLS in a standard linear regression model. These formulae provide substantial insight into the properties and performance characteristics of these estimators. In particular, we find a new estimator which minimizes the maximum possible bias and improves substantially on the standard Eicker-White estimate.
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