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Kundu与Schmidt-Hieber对《流形假说的统计探索》的讨论AI 解读

The authors have done an admirable job by combining new methodology, meaningful statistical theory and applications all within one article. The work is both deep and accessible. As a first discussion point, let us relate the proposed latent metric model (LMM) to functional PCA. This comparison can serve as basis to borrow strength and to advance the understanding of both approaches or even to unify them. The latent metric model assumes that we observe where Z1,…,Zn∼μ are…

偏差与方差权衡的下界AI 解读

It is a common phenomenon that for high-dimensional and nonparametric statistical models, rate-optimal estimators balance squared bias and variance. Although this balancing is widely observed, little is known whether methods exist that could avoid the trade-off between bias and variance. We propose a general strategy to obtain lower bounds on the variance of any estimator with bias smaller than a prespecified bound. This shows to which extent the bias-variance trade-off is…

二项模型中参数均未知时总体规模的后验分析及定量纳米成像应用AI 解读

Estimation of the population size n from k i.i.d. binomial observations with unknown success probability p is relevant to a multitude of applications and has a long history. Without additional prior information this is a notoriously difficult task when p becomes small, and the Bayesian approach becomes particularly useful. For a large class of priors, we establish posterior contraction and a Bernstein-von Mises type theorem in a setting where p→0 and n→∞ as k→∞. Furthermore,…

使用 ReLU 深度神经网络的非参数回归AI 解读

Consider the multivariate nonparametric regression model. It is shown that estimators based on sparsely connected deep neural networks with ReLU activation function and properly chosen network architecture achieve the minimax rates of convergence (up to $\log n$-factors) under a general composition assumption on the regression function. The framework includes many well-studied structural constraints such as (generalized) additive models. While there is a lot of flexibility…

支撑边界恢复中复合泊松先验的非参数贝叶斯分析AI 解读

Given data from a Poisson point process with intensity $(x,y)\mapston\mathbf{1}(f(x)\leq y)$, frequentist properties for the Bayesian reconstruction of the support boundary function $f$ are derived. We mainly study compound Poisson process priors with fixed intensity proving that the posterior contracts with nearly optimal rate for monotone support boundaries and adapts to Hölder smooth boundaries. We then derive a limiting shape result for a compound Poisson process prior…

自适应后验集中速率AI 解读

We investigate the problem of deriving posterior concentration rates under different loss functions in nonparametric Bayes. We first provide a lower bound on posterior coverages of shrinking neighbourhoods that relates the metric or loss under which the shrinking neighbourhood is considered, and an intrinsic pre-metric linked to frequentist separation rates. In the Gaussian white noise model, we construct feasible priors based on a spike and slab procedure reminiscent of…

采用稀疏先验的贝叶斯线性回归AI 解读

We study full Bayesian procedures for high-dimensional linear regression under sparsity constraints. The prior is a mixture of point masses at zero and continuous distributions. Under compatibility conditions on the design matrix, the posterior distribution is shown to contract at the optimal rate for recovery of the unknown sparse vector, and to give optimal prediction of the response vector. It is also shown to select the correct sparse model, or at least the coefficients…

分数噪声回归模型的渐近等价性AI 解读

Consider estimation of the regression function based on a model with equidistant design and measurement errors generated from a fractional Gaussian noise process. In previous literature, this model has been heuristically linked to an experiment, where the anti-derivative of the regression function is continuously observed under additive perturbation by a fractional Brownian motion. Based on a reformulation of the problem using reproducing kernel Hilbert spaces, we derive…

反卷积中的形状约束多尺度方法：定性特征的置信陈述AI 解读

We derive multiscale statistics for deconvolution in order to detect qualitative features of the unknown density. An important example covered within this framework is to test for local monotonicity on all scales simultaneously. We investigate the moderately ill-posed setting, where the Fourier transform of the error density in the deconvolution model is of polynomial decay. For multiscale testing, we consider a calibration, motivated by the modulus of continuity of Brownian…