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A fundamental task in science is to design experiments that yield valuable insights about the system under study. Mathematically, these insights can be represented as a utility or risk function that shapes the value of conducting each experiment. We present PDBAL, a targeted active learning method that adaptively designs experiments to maximize scientific utility. PDBAL takes a user-specified risk function and combines it with a probabilistic model of the experimental outcomes to choose designs that rapidly converge on a high-utility model. We prove theoretical bounds on the label complexity of PDBAL and provide fast closed-form solutions for designing experiments with common exponential family likelihoods. In simulation studies, PDBAL consistently outperforms standard untargeted approaches that focus on maximizing expected information gain over the design space. Finally, we demonstrate the scientific potential of PDBAL through a study on a large cancer drug screen dataset where PDBAL quickly recovers the most efficacious drugs with a small fraction of the total number of experiments.

We prove a new generalization bound that shows for any class of linear predictors in Gaussian space, the Rademacher complexity of the class and the training error under any continuous loss $\ell$ can control the test error under all Moreau envelopes of the loss $\ell$. We use our finite-sample bound to directly recover the "optimistic rate" of Zhou et al. (2021) for linear regression with the square loss, which is known to be tight for minimal $\ell_2$-norm interpolation, but we also handle more general settings where the label is generated by a potentially misspecified multi-index model. The same argument can analyze noisy interpolation of max-margin classifiers through the squared hinge loss, and establishes consistency results in spiked-covariance settings. More generally, when the loss is only assumed to be Lipschitz, our bound effectively improves Talagrand's well-known contraction lemma by a factor of two, and we prove uniform convergence of interpolators (Koehler et al. 2021) for all smooth, non-negative losses. Finally, we show that application of our generalization bound using localized Gaussian width will generally be sharp for empirical risk minimizers, establishing a non-asymptotic Moreau envelope theory for generalization that applies outside of proportional scaling regimes, handles model misspecification, and complements existing asymptotic Moreau envelope theories for M-estimation.

In this paper, we investigate the Gaussian graphical model inference problem in a novel setting that we call erose measurements, referring to irregularly measured or observed data. For graphs, this results in different node pairs having vastly different sample sizes which frequently arises in data integration, genomics, neuroscience, and sensor networks. Existing works characterize the graph selection performance using the minimum pairwise sample size, which provides little insights for erosely measured data, and no existing inference method is applicable. We aim to fill in this gap by proposing the first inference method that characterizes the different uncertainty levels over the graph caused by the erose measurements, named GI-JOE (Graph Inference when Joint Observations are Erose). Specifically, we develop an edge-wise inference method and an affiliated FDR control procedure, where the variance of each edge depends on the sample sizes associated with corresponding neighbors. We prove statistical validity under erose measurements, thanks to careful localized edge-wise analysis and disentangling the dependencies across the graph. Finally, through simulation studies and a real neuroscience data example, we demonstrate the advantages of our inference methods for graph selection from erosely measured data.

Two-stage randomized experiments are becoming an increasingly popular experimental design for causal inference when the outcome of one unit may be affected by the treatment assignments of other units in the same cluster. In this paper, we provide a methodological framework for general tools of statistical inference and power analysis for two-stage randomized experiments. Under the randomization-based framework, we consider the estimation of a new direct effect of interest as well as the average direct and spillover effects studied in the literature. We provide unbiased estimators of these causal quantities and their conservative variance estimators in a general setting. Using these results, we then develop hypothesis testing procedures and derive sample size formulas. We theoretically compare the two-stage randomized design with the completely randomized and cluster randomized designs, which represent two limiting designs. Finally, we conduct simulation studies to evaluate the empirical performance of our sample size formulas. For empirical illustration, the proposed methodology is applied to the randomized evaluation of the Indian national health insurance program. An open-source software package is available for implementing the proposed methodology.

Bayesian variable selection methods are powerful techniques for fitting and inferring on sparse high-dimensional linear regression models. However, many are computationally intensive or require restrictive prior distributions on model parameters. Likelihood based penalization methods are more computationally friendly, but resource intensive refitting techniques are needed for inference. In this paper, we proposed an efficient and powerful Bayesian approach for sparse high-dimensional linear regression. Minimal prior assumptions on the parameters are required through the use of plug-in empirical Bayes estimates of hyperparameters. Efficient maximum a posteriori probability (MAP) estimation is completed through the use of a partitioned and extended expectation conditional maximization (ECM) algorithm. The result is a PaRtitiOned empirical Bayes Ecm (PROBE) algorithm applied to sparse high-dimensional linear regression. We propose methods to estimate credible and prediction intervals for predictions of future values. We compare the empirical properties of predictions and our predictive inference to comparable approaches with numerous simulation studies and an analysis of cancer cell lines drug response study. The proposed approach is implemented in the R package probe.

High-dimensional matrix-variate time series data are becoming widely available in many scientific fields, such as economics, biology, and meteorology. To achieve significant dimension reduction while preserving the intrinsic matrix structure and temporal dynamics in such data, Wang et al. (2017) proposed a matrix factor model that is shown to provide effective analysis. In this paper, we establish a general framework for incorporating domain or prior knowledge in the matrix factor model through linear constraints. The proposed framework is shown to be useful in achieving parsimonious parameterization, facilitating interpretation of the latent matrix factor, and identifying specific factors of interest. Fully utilizing the prior-knowledge-induced constraints results in more efficient and accurate modeling, inference, dimension reduction as well as a clear and better interpretation of the results. In this paper, constrained, multi-term, and partially constrained factor models for matrix-variate time series are developed, with efficient estimation procedures and their asymptotic properties. We show that the convergence rates of the constrained factor loading matrices are much faster than those of the conventional matrix factor analysis under many situations. Simulation studies are carried out to demonstrate the finite-sample performance of the proposed method and its associated asymptotic properties. We illustrate the proposed model with three applications, where the constrained matrix-factor models outperform their unconstrained counterparts in the power of variance explanation under the out-of-sample 10-fold cross-validation setting.

This paper considers the estimation and inference of the low-rank components in high-dimensional matrix-variate factor models, where each dimension of the matrix-variates ($p \times q$) is comparable to or greater than the number of observations ($T$). We propose an estimation method called $\alpha$-PCA that preserves the matrix structure and aggregates mean and contemporary covariance through a hyper-parameter $\alpha$. We develop an inferential theory, establishing consistency, the rate of convergence, and the limiting distributions, under general conditions that allow for correlations across time, rows, or columns of the noise. We show both theoretical and empirical methods of choosing the best $\alpha$, depending on the use-case criteria. Simulation results demonstrate the adequacy of the asymptotic results in approximating the finite sample properties. The $\alpha$-PCA compares favorably with the existing ones. Finally, we illustrate its applications with a real numeric data set and two real image data sets. In all applications, the proposed estimation procedure outperforms previous methods in the power of variance explanation using out-of-sample 10-fold cross-validation.

In this paper we integrate the isotonic regression with Stone's cross-validation-based method to estimate discrete infinitely supported distribution. We prove that the estimator is strongly consistent, derive its rate of convergence for any underlying distribution, and for one-dimensional case we derive Marshal-type inequality for cumulative distribution function of the estimator. Also, we construct the asymptotically correct conservative global confidence band for the estimator. It is shown that, first, the estimator performs good even for small sized data sets, second, the estimator outperforms in the case of non-monotone underlying distribution, and, third, it performs almost as good as Grenander estimator when the true distribution is isotonic. Therefore, the new estimator provides a trade-off between goodness-of-fit, monotonicity and quality of probabilistic forecast. We apply the estimator to the time-to-onset data of visceral leishmaniasis in Brazil collected from 2007 to 2014.

Convolutional neural networks (CNNs) provide flexible function approximations for a wide variety of applications when the input variables are in the form of images or spatial data. Although CNNs often outperform traditional statistical models in prediction accuracy, statistical inference, such as estimating the effects of covariates and quantifying the prediction uncertainty, is not trivial due to the highly complicated model structure and overparameterization. To address this challenge, we propose a new Bayes approach by embedding CNNs within the generalized linear model (GLM) framework. We use extracted nodes from the last hidden layer of CNN with Monte Carlo dropout as informative covariates in GLM. This improves accuracy in prediction and regression coefficient inference, allowing for the interpretation of coefficient and uncertainty quantification. By fitting ensemble GLMs across multiple realizations from Monte Carlo dropout, we can fully account for uncertainties in model estimation. We apply our methods to simulated and real data examples, including non-Gaussian spatial data, brain tumor image data, and fMRI data. The algorithm can be broadly applicable to image regressions or correlated data analysis by enabling accurate Bayesian inference quickly.

Causal inference is a critical research topic across many domains, such as statistics, computer science, education, public policy and economics, for decades. Nowadays, estimating causal effect from observational data has become an appealing research direction owing to the large amount of available data and low budget requirement, compared with randomized controlled trials. Embraced with the rapidly developed machine learning area, various causal effect estimation methods for observational data have sprung up. In this survey, we provide a comprehensive review of causal inference methods under the potential outcome framework, one of the well known causal inference framework. The methods are divided into two categories depending on whether they require all three assumptions of the potential outcome framework or not. For each category, both the traditional statistical methods and the recent machine learning enhanced methods are discussed and compared. The plausible applications of these methods are also presented, including the applications in advertising, recommendation, medicine and so on. Moreover, the commonly used benchmark datasets as well as the open-source codes are also summarized, which facilitate researchers and practitioners to explore, evaluate and apply the causal inference methods.