亚洲男人的天堂2018av,欧美草比,久久久久久免费视频精选,国色天香在线看免费,久久久久亚洲av成人片仓井空

In the context of principal components analysis (PCA), the bootstrap is commonly applied to solve a variety of inference problems, such as constructing confidence intervals for the eigenvalues of the population covariance matrix $\Sigma$. However, when the data are high-dimensional, there are relatively few theoretical guarantees that quantify the performance of the bootstrap. Our aim in this paper is to analyze how well the bootstrap can approximate the joint distribution of the leading eigenvalues of the sample covariance matrix $\hat\Sigma$, and we establish non-asymptotic rates of approximation with respect to the multivariate Kolmogorov metric. Under certain assumptions, we show that the bootstrap can achieve the dimension-free rate of ${\tt{r}}(\Sigma)/\sqrt n$ up to logarithmic factors, where ${\tt{r}}(\Sigma)$ is the effective rank of $\Sigma$, and $n$ is the sample size. From a methodological standpoint, our work also illustrates that applying a transformation to the eigenvalues of $\hat\Sigma$ before bootstrapping is an important consideration in high-dimensional settings.

We study the problem of robustly estimating the parameter $p$ of an Erd\H{o}s-R\'enyi random graph on $n$ nodes, where a $\gamma$ fraction of nodes may be adversarially corrupted. After showing the deficiencies of canonical estimators, we design a computationally-efficient spectral algorithm which estimates $p$ up to accuracy $\tilde O(\sqrt{p(1-p)}/n + \gamma\sqrt{p(1-p)} /\sqrt{n}+ \gamma/n)$ for $\gamma < 1/60$. Furthermore, we give an inefficient algorithm with similar accuracy for all $\gamma <1/2$, the information-theoretic limit. Finally, we prove a nearly-matching statistical lower bound, showing that the error of our algorithms is optimal up to logarithmic factors.

In this paper we treat statistical inference for an intrinsic wavelet estimator of curves of symmetric positive definite (SPD) matrices in a log-Euclidean manifold. This estimator preserves positive-definiteness and enjoys permutation-equivariance, which is particularly relevant for covariance matrices. Our second-generation wavelet estimator is based on average-interpolation and allows the same powerful properties, including fast algorithms, known from nonparametric curve estimation with wavelets in standard Euclidean set-ups. The core of our work is the proposition of confidence sets for our high-level wavelet estimator in a non-Euclidean geometry. We derive asymptotic normality of this estimator, including explicit expressions of its asymptotic variance. This opens the door for constructing asymptotic confidence regions which we compare with our proposed bootstrap scheme for inference. Detailed numerical simulations confirm the appropriateness of our suggested inference schemes.

This paper provides some first steps in developing empirical process theory for functions taking values in a vector space. Our main results provide bounds on the entropy of classes of smooth functions taking values in a Hilbert space, by leveraging theory from differential calculus of vector-valued functions and fractal dimension theory of metric spaces. We demonstrate how these entropy bounds can be used to show the uniform law of large numbers and asymptotic equicontinuity of the function classes, and also apply it to statistical learning theory in which the output space is a Hilbert space. We conclude with a discussion on the extension of Rademacher complexities to vector-valued function classes.

Human medical data can be challenging to obtain due to data privacy concerns, difficulties conducting certain types of experiments, or prohibitive associated costs. In many settings, data from animal models or in-vitro cell lines are available to help augment our understanding of human data. However, this data is known for having low etiological validity in comparison to human data. In this work, we augment small human medical datasets with in-vitro data and animal models. We use Invariant Risk Minimisation (IRM) to elucidate invariant features by considering cross-organism data as belonging to different data-generating environments. Our models identify genes of relevance to human cancer development. We observe a degree of consistency between varying the amounts of human and mouse data used, however, further work is required to obtain conclusive insights. As a secondary contribution, we enhance existing open source datasets and provide two uniformly processed, cross-organism, homologue gene-matched datasets to the community.

For an AI system to be reliable, the confidence it expresses in its decisions must match its accuracy. To assess the degree of match, examples are typically binned by confidence and the per-bin mean confidence and accuracy are compared. Most research in calibration focuses on techniques to reduce this empirical measure of calibration error, ECE_bin. We instead focus on assessing statistical bias in this empirical measure, and we identify better estimators. We propose a framework through which we can compute the bias of a particular estimator for an evaluation data set of a given size. The framework involves synthesizing model outputs that have the same statistics as common neural architectures on popular data sets. We find that binning-based estimators with bins of equal mass (number of instances) have lower bias than estimators with bins of equal width. Our results indicate two reliable calibration-error estimators: the debiased estimator (Brocker, 2012; Ferro and Fricker, 2012) and a method we propose, ECE_sweep, which uses equal-mass bins and chooses the number of bins to be as large as possible while preserving monotonicity in the calibration function. With these estimators, we observe improvements in the effectiveness of recalibration methods and in the detection of model miscalibration.

As a traditional and widely-adopted mortality rate projection technique, by representing the log mortality rate as a simple bilinear form $\log(m_{x,t})=a_x+b_xk_t$. The Lee-Carter model has been extensively studied throughout the past 30 years, however, the performance of the model in the presence of outliers has been paid little attention, particularly for the parameter estimation of $b_x$. In this paper, we propose a robust estimation method for Lee-Carter model by formulating it as a probabilistic principal component analysis (PPCA) with multivariate $t$-distributions, and an efficient expectation-maximization (EM) algorithm for implementation. The advantages of the method are threefold. It yields significantly more robust estimates of both $b_x$ and $k_t$, preserves the fundamental interpretation for $b_x$ as the first principal component as in the traditional approach and is flexible to be integrated into other existing time series models for $k_t$. The parameter uncertainties are examined by adopting a standard residual bootstrap. A simulation study based on Human Mortality Database shows superior performance of the proposed model compared to other conventional approaches.

Feature attribution is often loosely presented as the process of selecting a subset of relevant features as a rationale of a prediction. This lack of clarity stems from the fact that we usually do not have access to any notion of ground-truth attribution and from a more general debate on what good interpretations are. In this paper we propose to formalise feature selection/attribution based on the concept of relaxed functional dependence. In particular, we extend our notions to the instance-wise setting and derive necessary properties for candidate selection solutions, while leaving room for task-dependence. By computing ground-truth attributions on synthetic datasets, we evaluate many state-of-the-art attribution methods and show that, even when optimised, some fail to verify the proposed properties and provide wrong solutions.

This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.

Discrete random structures are important tools in Bayesian nonparametrics and the resulting models have proven effective in density estimation, clustering, topic modeling and prediction, among others. In this paper, we consider nested processes and study the dependence structures they induce. Dependence ranges between homogeneity, corresponding to full exchangeability, and maximum heterogeneity, corresponding to (unconditional) independence across samples. The popular nested Dirichlet process is shown to degenerate to the fully exchangeable case when there are ties across samples at the observed or latent level. To overcome this drawback, inherent to nesting general discrete random measures, we introduce a novel class of latent nested processes. These are obtained by adding common and group-specific completely random measures and, then, normalising to yield dependent random probability measures. We provide results on the partition distributions induced by latent nested processes, and develop an Markov Chain Monte Carlo sampler for Bayesian inferences. A test for distributional homogeneity across groups is obtained as a by product. The results and their inferential implications are showcased on synthetic and real data.