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Estimating the entropy rate of discrete time series is a challenging problem with important applications in numerous areas including neuroscience, genomics, image processing and natural language processing. A number of approaches have been developed for this task, typically based either on universal data compression algorithms, or on statistical estimators of the underlying process distribution. In this work, we propose a fully-Bayesian approach for entropy estimation. Building on the recently introduced Bayesian Context Trees (BCT) framework for modelling discrete time series as variable-memory Markov chains, we show that it is possible to sample directly from the induced posterior on the entropy rate. This can be used to estimate the entire posterior distribution, providing much richer information than point estimates. We develop theoretical results for the posterior distribution of the entropy rate, including proofs of consistency and asymptotic normality. The practical utility of the method is illustrated on both simulated and real-world data, where it is found to outperform state-of-the-art alternatives.

We revisit the Bayesian Context Trees (BCT) modelling framework for discrete time series, which was recently found to be very effective in numerous tasks including model selection, estimation and prediction. A novel representation of the induced posterior distribution on model space is derived in terms of a simple branching process, and several consequences of this are explored in theory and in practice. First, it is shown that the branching process representation leads to a simple variable-dimensional Monte Carlo sampler for the joint posterior distribution on models and parameters, which can efficiently produce independent samples. This sampler is found to be more efficient than earlier MCMC samplers for the same tasks. Then, the branching process representation is used to establish the asymptotic consistency of the BCT posterior, including the derivation of an almost-sure convergence rate. Finally, an extensive study is carried out on the performance of the induced Bayesian entropy estimator. Its utility is illustrated through both simulation experiments and real-world applications, where it is found to outperform several state-of-the-art methods.

We prove a weak rate of convergence of a fully discrete scheme for stochastic Cahn--Hilliard equation with additive noise, where the spectral Galerkin method is used in space and the backward Euler method is used in time. Compared with the Allen--Cahn type stochastic partial differential equation, the error analysis here is much more sophisticated due to the presence of the unbounded operator in front of the nonlinear term. To address such issues, a novel and direct approach has been exploited which does not rely on a Kolmogorov equation but on the integration by parts formula from Malliavin calculus. To the best of our knowledge, the rates of weak convergence are revealed in the stochastic Cahn--Hilliard equation setting for the first time.

In survival contexts, substantial literature exists on estimating optimal treatment regimes, where treatments are assigned based on personal characteristics for the purpose of maximizing the survival probability. These methods assume that a set of covariates is sufficient to deconfound the treatment-outcome relationship. Nevertheless, the assumption can be limiting in observational studies or randomized trials in which noncompliance occurs. Thus, we advance a novel approach for estimating the optimal treatment regime when certain confounders are not observable and a binary instrumental variable is available. Specifically, via a binary instrumental variable, we propose two semiparametric estimators for the optimal treatment regime, one of which possesses the desirable property of double robustness, by maximizing Kaplan-Meier-like estimators within a pre-defined class of regimes. Because the Kaplan-Meier-like estimators are jagged, we incorporate kernel smoothing methods to enhance their performance. Under appropriate regularity conditions, the asymptotic properties are rigorously established. Furthermore, the finite sample performance is assessed through simulation studies. We exemplify our method using data from the National Cancer Institute's (NCI) prostate, lung, colorectal, and ovarian cancer screening trial.

We consider network games where a large number of agents interact according to a network sampled from a random network model, represented by a graphon. By exploiting previous results on convergence of such large network games to graphon games, we examine a procedure for estimating unknown payoff parameters, from observations of equilibrium actions, without the need for exact network information. We prove smoothness and local convexity of the optimization problem involved in computing the proposed estimator. Additionally, under a notion of graphon parameter identifiability, we show that the optimal estimator is globally unique. We present several examples of identifiable homogeneous and heterogeneous parameters in different classes of linear quadratic network games with numerical simulations to validate the proposed estimator.

The coresets approach, also called subsampling or subset selection, aims to select a subsample as a surrogate for the observed sample. Such an approach has been used pervasively in large-scale data analysis. Existing coresets methods construct the subsample using a subset of rows from the predictor matrix. Such methods can be significantly inefficient when the predictor matrix is sparse or numerically sparse. To overcome the limitation, we develop a novel element-wise subset selection approach, called core-elements, for large-scale least squares estimation in classical linear regression. We provide a deterministic algorithm to construct the core-elements estimator, only requiring an $O(\mbox{nnz}(\mathbf{X})+rp^2)$ computational cost, where $\mathbf{X}$ is an $n\times p$ predictor matrix, $r$ is the number of elements selected from each column of $\mathbf{X}$, and $\mbox{nnz}(\cdot)$ denotes the number of non-zero elements. Theoretically, we show that the proposed estimator is unbiased and approximately minimizes an upper bound of the estimation variance. We also provide an approximation guarantee by deriving a coresets-like finite sample bound for the proposed estimator. To handle potential outliers in the data, we further combine core-elements with the median-of-means procedure, resulting in an efficient and robust estimator with theoretical consistency guarantees. Numerical studies on various synthetic and open-source datasets demonstrate the proposed method's superior performance compared to mainstream competitors.

Learning precise surrogate models of complex computer simulations and physical machines often require long-lasting or expensive experiments. Furthermore, the modeled physical dependencies exhibit nonlinear and nonstationary behavior. Machine learning methods that are used to produce the surrogate model should therefore address these problems by providing a scheme to keep the number of queries small, e.g. by using active learning and be able to capture the nonlinear and nonstationary properties of the system. One way of modeling the nonstationarity is to induce input-partitioning, a principle that has proven to be advantageous in active learning for Gaussian processes. However, these methods either assume a known partitioning, need to introduce complex sampling schemes or rely on very simple geometries. In this work, we present a simple, yet powerful kernel family that incorporates a partitioning that: i) is learnable via gradient-based methods, ii) uses a geometry that is more flexible than previous ones, while still being applicable in the low data regime. Thus, it provides a good prior for active learning procedures. We empirically demonstrate excellent performance on various active learning tasks.

An instrumental variable (IV) is a device that encourages units in a study to be exposed to a treatment. Under a set of key assumptions, a valid instrument allows for consistent estimation of treatment effects for compliers (those who are only exposed to treatment when encouraged to do so) even in the presence of unobserved confounders. Unfortunately, popular IV estimators can be unstable in studies with a small fraction of compliers. Here, we explore post-stratifying the data using variables that predict complier status (and, potentially, the outcome) to yield better estimation and inferential properties. We outline an estimator that is a weighted average of IV estimates within each stratum, weighing the stratum estimates by their estimated proportion of compliers. We then explore the benefits of post-stratification in terms of bias reduction, variance reduction, and improved standard error estimates, providing derivations that identify the direction of bias as a function of the relative means of the compliers and non-compliers. We also provide a finite-sample asymptotic formula for the variance of the post-stratified estimators. We demonstrate the relative performances of different IV approaches in simulations studies and discuss the advantages of our design-based post-stratification approach over incorporating compliance-predictive covariates into two-stage least squares regressions. In the end, we show covariates predictive of outcome can increase precision, but only if one is willing to make a bias-variance trade-off by down-weighting or dropping those strata with few compliers. Our methods are further exemplified in an application.

Selection of covariates is crucial in the estimation of average treatment effects given observational data with high or even ultra-high dimensional pretreatment variables. Existing methods for this problem typically assume sparse linear models for both outcome and univariate treatment, and cannot handle situations with ultra-high dimensional covariates. In this paper, we propose a new covariate selection strategy called double screening prior adaptive lasso (DSPAL) to select confounders and predictors of the outcome for multivariate treatments, which combines the adaptive lasso method with the marginal conditional (in)dependence prior information to select target covariates, in order to eliminate confounding bias and improve statistical efficiency. The distinctive features of our proposal are that it can be applied to high-dimensional or even ultra-high dimensional covariates for multivariate treatments, and can deal with the cases of both parametric and nonparametric outcome models, which makes it more robust compared to other methods. Our theoretical analyses show that the proposed procedure enjoys the sure screening property, the ranking consistency property and the variable selection consistency. Through a simulation study, we demonstrate that the proposed approach selects all confounders and predictors consistently and estimates the multivariate treatment effects with smaller bias and mean squared error compared to several alternatives under various scenarios. In real data analysis, the method is applied to estimate the causal effect of a three-dimensional continuous environmental treatment on cholesterol level and enlightening results are obtained.

We propose a certainty-equivalence scheme for adaptive control of scalar linear systems subject to additive, i.i.d. Gaussian disturbances and bounded control input constraints, without requiring prior knowledge of the bounds of the system parameters, nor the control direction. Assuming that the system is at-worst marginally stable, mean square boundedness of the closed-loop system states is proven. Lastly, numerical examples are presented to illustrate our results.