Gaussian covariance graph model is a popular model in revealing underlying dependency structures among random variables. A Bayesian approach to the estimation of covariance structures uses priors that force zeros on some off-diagonal entries of covariance matrices and put a positive definite constraint on matrices. In this paper, we consider a spike and slab prior on off-diagonal entries, which uses a mixture of point-mass and normal distribution. The point-mass naturally introduces sparsity to covariance structures so that the resulting posterior from this prior renders covariance structure learning. Under this prior, we calculate posterior model probabilities of covariance structures using Laplace approximation. We show that the error due to Laplace approximation becomes asymptotically marginal at some rate depending on the posterior convergence rate of covariance matrix under the Frobenius norm. With the approximated posterior model probabilities, we propose a new framework for estimating a covariance structure. Since the Laplace approximation is done around the mode of conditional posterior of covariance matrix, which cannot be obtained in the closed form, we propose a block coordinate descent algorithm to find the mode and show that the covariance matrix can be estimated using this algorithm once the structure is chosen. Through a simulation study based on five numerical models, we show that the proposed method outperforms graphical lasso and sample covariance matrix in terms of root mean squared error, max norm, spectral norm, specificity, and sensitivity. Also, the advantage of the proposed method is demonstrated in terms of accuracy compared to our competitors when it is applied to linear discriminant analysis (LDA) classification to breast cancer diagnostic dataset.
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Reliable probability estimation is of crucial importance in many real-world applications where there is inherent uncertainty, such as weather forecasting, medical prognosis, or collision avoidance in autonomous vehicles. Probability-estimation models are trained on observed outcomes (e.g. whether it has rained or not, or whether a patient has died or not), because the ground-truth probabilities of the events of interest are typically unknown. The problem is therefore analogous to binary classification, with the important difference that the objective is to estimate probabilities rather than predicting the specific outcome. The goal of this work is to investigate probability estimation from high-dimensional data using deep neural networks. There exist several methods to improve the probabilities generated by these models but they mostly focus on classification problems where the probabilities are related to model uncertainty. In the case of problems with inherent uncertainty, it is challenging to evaluate performance without access to ground-truth probabilities. To address this, we build a synthetic dataset to study and compare different computable metrics. We evaluate existing methods on the synthetic data as well as on three real-world probability estimation tasks, all of which involve inherent uncertainty. We also give a theoretical analysis of a model for high-dimensional probability estimation which reproduces several of the phenomena evinced in our experiments. Finally, we propose a new method for probability estimation using neural networks, which modifies the training process to promote output probabilities that are consistent with empirical probabilities computed from the data. The method outperforms existing approaches on most metrics on the simulated as well as real-world data.
A solution to control for nonresponse bias consists of multiplying the design weights of respondents by the inverse of estimated response probabilities to compensate for the nonrespondents. Maximum likelihood and calibration are two approaches that can be applied to obtain estimated response probabilities. The paper develops asymptotic properties of the resulting estimator when calibration is applied. A logistic regression model for the response probabilities is postulated and missing at random data is supposed. The author shows that the estimators with the response probabilities estimated via calibration are asymptotically equivalent to unbiased estimators and that a gain in efficiency is obtained when estimating the response probabilities via calibration as compared to the estimator with the true response probabilities.
We study ROUND-UFP and ROUND-SAP, two generalizations of the classical BIN PACKING problem that correspond to the unsplittable flow problem on a path (UFP) and the storage allocation problem (SAP), respectively. We are given a path with capacities on its edges and a set of tasks where for each task we are given a demand and a subpath. In ROUND-UFP, the goal is to find a packing of all tasks into a minimum number of copies (rounds) of the given path such that for each copy, the total demand of tasks on any edge does not exceed the capacity of the respective edge. In ROUND-SAP, the tasks are considered to be rectangles and the goal is to find a non-overlapping packing of these rectangles into a minimum number of rounds such that all rectangles lie completely below the capacity profile of the edges. We show that in contrast to BIN PACKING, both the problems do not admit an asymptotic polynomial-time approximation scheme (APTAS), even when all edge capacities are equal. However, for this setting, we obtain asymptotic $(2+\varepsilon)$-approximations for both problems. For the general case, we obtain an $O(\log\log n)$-approximation algorithm and an $O(\log\log\frac{1}{\delta})$-approximation under $(1+\delta)$-resource augmentation for both problems. For the intermediate setting of the no bottleneck assumption (i.e., the maximum task demand is at most the minimum edge capacity), we obtain absolute $12$- and asymptotic $(16+\varepsilon)$-approximation algorithms for ROUND-UFP and ROUND-SAP, respectively.
Missing data is frequently encountered in practice. Propensity score estimation is a popular tool for handling such missingness. The propensity score is often developed using a model for the response probability, which can be subject to model misspecification. In this paper, we consider an alternative approach of estimating the inverse of the propensity scores using the density ratio function. The smoothed density ratio function is obtained by the solution to the information projection onto the space satisfying the moment conditions on the balancing scores. By including the covariates for the outcome regression models only into the density ratio model, we can achieve efficient propensity score estimation. Penalized regression is used to identify important covariates. We further extend the proposed approach to the multivariate missing case. Some limited simulation studies are presented to compare with the existing methods.
We consider the problem of estimating high-dimensional covariance matrices of $K$-populations or classes in the setting where the sample sizes are comparable to the data dimension. We propose estimating each class covariance matrix as a distinct linear combination of all class sample covariance matrices. This approach is shown to reduce the estimation error when the sample sizes are limited, and the true class covariance matrices share a somewhat similar structure. We develop an effective method for estimating the coefficients in the linear combination that minimize the mean squared error under the general assumption that the samples are drawn from (unspecified) elliptically symmetric distributions possessing finite fourth-order moments. To this end, we utilize the spatial sign covariance matrix, which we show (under rather general conditions) to be an asymptotically unbiased estimator of the normalized covariance matrix as the dimension grows to infinity. We also show how the proposed method can be used in choosing the regularization parameters for multiple target matrices in a single class covariance matrix estimation problem. We assess the proposed method via numerical simulation studies including an application in global minimum variance portfolio optimization using real stock data.
We consider an elliptic linear-quadratic parameter estimation problem with a finite number of parameters. A novel a priori bound for the parameter error is proved and, based on this bound, an adaptive finite element method driven by an a posteriori error estimator is presented. Unlike prior results in the literature, our estimator, which is composed of standard energy error residual estimators for the state equation and suitable co-state problems, reflects the faster convergence of the parameter error compared to the (co)-state variables. We show optimal convergence rates of our method; in particular and unlike prior works, we prove that the estimator decreases with a rate that is the sum of the best approximation rates of the state and co-state variables. Experiments confirm that our method matches the convergence rate of the parameter error.
A new approach to $L_2$-consistent estimation of a general density functional using $k$-nearest neighbor distances is proposed, where the functional under consideration is in the form of the expectation of some function $f$ of the densities at each point. The estimator is designed to be asymptotically unbiased, using the convergence of the normalized volume of a $k$-nearest neighbor ball to a Gamma distribution in the large-sample limit, and naturally involves the inverse Laplace transform of a scaled version of the function $f.$ Some instantiations of the proposed estimator recover existing $k$-nearest neighbor based estimators of Shannon and R\'enyi entropies and Kullback--Leibler and R\'enyi divergences, and discover new consistent estimators for many other functionals such as logarithmic entropies and divergences. The $L_2$-consistency of the proposed estimator is established for a broad class of densities for general functionals, and the convergence rate in mean squared error is established as a function of the sample size for smooth, bounded densities.
In this chapter, we discuss recent work on learning sparse approximations to high-dimensional functions on data, where the target functions may be scalar-, vector- or even Hilbert space-valued. Our main objective is to study how the sampling strategy affects the sample complexity -- that is, the number of samples that suffice for accurate and stable recovery -- and to use this insight to obtain optimal or near-optimal sampling procedures. We consider two settings. First, when a target sparse representation is known, in which case we present a near-complete answer based on drawing independent random samples from carefully-designed probability measures. Second, we consider the more challenging scenario when such representation is unknown. In this case, while not giving a full answer, we describe a general construction of sampling measures that improves over standard Monte Carlo sampling. We present examples using algebraic and trigonometric polynomials, and for the former, we also introduce a new procedure for function approximation on irregular (i.e., nontensorial) domains. The effectiveness of this procedure is shown through numerical examples. Finally, we discuss a number of structured sparsity models, and how they may lead to better approximations.
We consider a problem of manifold estimation from noisy observations. Many manifold learning procedures locally approximate a manifold by a weighted average over a small neighborhood. However, in the presence of large noise, the assigned weights become so corrupted that the averaged estimate shows very poor performance. We suggest a structure-adaptive procedure, which simultaneously reconstructs a smooth manifold and estimates projections of the point cloud onto this manifold. The proposed approach iteratively refines the weights on each step, using the structural information obtained at previous steps. After several iterations, we obtain nearly "oracle" weights, so that the final estimates are nearly efficient even in the presence of relatively large noise. In our theoretical study, we establish tight lower and upper bounds proving asymptotic optimality of the method for manifold estimation under the Hausdorff loss, provided that the noise degrades to zero fast enough.
Implicit probabilistic models are models defined naturally in terms of a sampling procedure and often induces a likelihood function that cannot be expressed explicitly. We develop a simple method for estimating parameters in implicit models that does not require knowledge of the form of the likelihood function or any derived quantities, but can be shown to be equivalent to maximizing likelihood under some conditions. Our result holds in the non-asymptotic parametric setting, where both the capacity of the model and the number of data examples are finite. We also demonstrate encouraging experimental results.
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