In a range of genomic applications, it is of interest to quantify the evidence that the signal at site~$i$ is active given conditionally independent replicate observations summarized by the sample mean and variance $(\bar Y, s^2)$ at each site. We study the version of the problem in which the signal distribution is sparse, and the error distribution has an unknown site-specific variance so that the null distribution of the standardized statistic is Student-$t$ rather than Gaussian. The main contribution of this paper is a sparse-mixture approximation to the non-null density of the $t$-ratio. This formula demonstrates the effect of low degrees of freedom on the Bayes factor, or the conditional probability that the site is active. We illustrate some differences on a HIV dataset for gene-expression data previously analyzed by Efron (2012).
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We consider nonparametric Bayesian inference in a multidimensional diffusion model with reflecting boundary conditions based on discrete high-frequency observations. We prove a general posterior contraction rate theorem in $L^2$-loss, which is applied to Gaussian priors. The resulting posteriors, as well as their posterior means, are shown to converge to the ground truth at the minimax optimal rate over H\"older smoothness classes in any dimension. Of independent interest and as part of our proofs, we show that certain frequentist penalized least squares estimators are also minimax optimal.
Collecting large quantities of high-quality data is often prohibitively expensive or impractical, and a crucial bottleneck in machine learning. One may instead augment a small set of $n$ data points from the target distribution with data from more accessible sources like public datasets, data collected under different circumstances, or synthesized by generative models. Blurring distinctions, we refer to such data as `surrogate data'. We define a simple scheme for integrating surrogate data into training and use both theoretical models and empirical studies to explore its behavior. Our main findings are: $(i)$ Integrating surrogate data can significantly reduce the test error on the original distribution; $(ii)$ In order to reap this benefit, it is crucial to use optimally weighted empirical risk minimization; $(iii)$ The test error of models trained on mixtures of real and surrogate data is well described by a scaling law. This can be used to predict the optimal weighting and the gain from surrogate data.
We consider the estimation of the cumulative hazard function, and equivalently the distribution function, with censored data under a setup that preserves the privacy of the survival database. This is done through a $\alpha$-locally differentially private mechanism for the failure indicators and by proposing a non-parametric kernel estimator for the cumulative hazard function that remains consistent under the privatization. Under mild conditions, we also prove lowers bounds for the minimax rates of convergence and show that estimator is minimax optimal under a well-chosen bandwidth.
In this paper, we consider the numerical approximation of a time-fractional stochastic Cahn--Hilliard equation driven by an additive fractionally integrated Gaussian noise. The model involves a Caputo fractional derivative in time of order $\alpha\in(0,1)$ and a fractional time-integral noise of order $\gamma\in[0,1]$. The numerical scheme approximates the model by a piecewise linear finite element method in space and a convolution quadrature in time (for both time-fractional operators), along with the $L^2$-projection for the noise. We carefully investigate the spatially semidiscrete and fully discrete schemes, and obtain strong convergence rates by using clever energy arguments. The temporal H\"older continuity property of the solution played a key role in the error analysis. Unlike the stochastic Allen--Cahn equation, the presence of the unbounded elliptic operator in front of the cubic nonlinearity in the underlying model adds complexity and challenges to the error analysis. To overcome these difficulties, several new techniques and error estimates are developed. The study concludes with numerical examples that validate the theoretical findings.
Vecchia approximation has been widely used to accurately scale Gaussian-process (GP) inference to large datasets, by expressing the joint density as a product of conditional densities with small conditioning sets. We study fixed-domain asymptotic properties of Vecchia-based GP inference for a large class of covariance functions (including Mat\'ern covariances) with boundary conditioning. In this setting, we establish that consistency and asymptotic normality of maximum exact-likelihood estimators imply those of maximum Vecchia-likelihood estimators, and that exact GP prediction can be approximated accurately by Vecchia GP prediction, given that the size of conditioning sets grows polylogarithmically with the data size. Hence, Vecchia-based inference with quasilinear complexity is asymptotically equivalent to exact GP inference with cubic complexity. This also provides a general new result on the screening effect. Our findings are illustrated by numerical experiments, which also show that Vecchia approximation can be more accurate than alternative approaches such as covariance tapering and reduced-rank approximations.
Spatially distributed functional data are prevalent in many statistical applications such as meteorology, energy forecasting, census data, disease mapping, and neurological studies. Given their complex and high-dimensional nature, functional data often require dimension reduction methods to extract meaningful information. Inverse regression is one such approach that has become very popular in the past two decades. We study the inverse regression in the framework of functional data observed at irregularly positioned spatial sites. The functional predictor is the sum of a spatially dependent functional effect and a spatially independent functional nugget effect, while the relation between the scalar response and the functional predictor is modeled using the inverse regression framework. For estimation, we consider local linear smoothing with a general weighting scheme, which includes as special cases the schemes under which equal weights are assigned to each observation or to each subject. This framework enables us to present the asymptotic results for different types of sampling plans over time such as non-dense, dense, and ultra-dense. We discuss the domain-expanding infill (DEI) framework for spatial asymptotics, which is a mix of the traditional expanding domain and infill frameworks. The DEI framework overcomes the limitations of traditional spatial asymptotics in the existing literature. Under this unified framework, we develop asymptotic theory and identify conditions that are necessary for the estimated eigen-directions to achieve optimal rates of convergence. Our asymptotic results include pointwise and $L_2$ convergence rates. Simulation studies using synthetic data and an application to a real-world dataset confirm the effectiveness of our methods.
We investigate the multiplicity model with m values of some test statistic independently drawn from a mixture of no effect (null) and positive effect (alternative), where we seek to identify, the alternative test results with a controlled error rate. We are interested in the case where the alternatives are rare. A number of multiple testing procedures filter the set of ordered p-values in order to eliminate the nulls. Such an approach can only work if the p-values originating from the alternatives form one or several identifiable clusters. The Benjamini and Hochberg (BH) method, for example, assumes that this cluster occurs in a small interval $(0,\Delta)$ and filters out all or most of the ordered p-values $p_{(r)}$ above a linear threshold $s \times r$. In repeated applications this filter controls the false discovery rate via the slope s. We propose a new adaptive filter that deletes the p-values from regions of uniform distribution. In cases where a single cluster remains, the p-values in an interval are declared alternatives, with the mid-point and the length of the interval chosen by controlling the data-dependent FDR at a desired level.
We consider a quantum and classical version multi-party function computation problem with $n$ players, where players $2, \dots, n$ need to communicate appropriate information to player 1, so that a "generalized" inner product function with an appropriate promise can be calculated. The communication complexity of a protocol is the total number of bits that need to be communicated. When $n$ is prime and for our chosen function, we exhibit a quantum protocol (with complexity $(n-1) \log n$ bits) and a classical protocol (with complexity $(n-1)^2 (\log n^2$) bits). In the quantum protocol, the players have access to entangled qudits but the communication is still classical. Furthermore, we present an integer linear programming formulation for determining a lower bound on the classical communication complexity. This demonstrates that our quantum protocol is strictly better than classical protocols.
We present a subspace method to solve large-scale trace ratio problems. This method is matrix-free, only needing the action of the two matrices in the trace ratio. At each iteration, a smaller trace ratio problem is addressed in the search subspace. Additionally, our algorithm is endowed with a restarting strategy, that ensures the monotonicity of the trace ratio value throughout the iterations. We also investigate the behavior of the approximate solution from a theoretical viewpoint, extending existing results on Ritz values and vectors, as the angle between the search subspace and the exact solution to the trace ratio approaches zero. In the context of multigroup classification, numerical experiments show that the new subspace method tends to be more efficient than iterative approaches that need a (partial) eigenvalue decomposition in every step.
For a sequence of random structures with $n$-element domains over a relational signature, we define its first order (FO) complexity as a certain subset in the Banach space $\ell^{\infty}/c_0$. The well-known FO zero-one law and FO convergence law correspond to FO complexities equal to $\{0,1\}$ and a subset of $\mathbb{R}$, respectively. We present a hierarchy of FO complexity classes, introduce a stochastic FO reduction that allows to transfer complexity results between different random structures, and deduce using this tool several new logical limit laws for binomial random structures. Finally, we introduce a conditional distribution on graphs, subject to a FO sentence $\varphi$, that generalises certain well-known random graph models, show instances of this distribution for every complexity class, and prove that the set of all $\varphi$ validating 0--1 law is not recursively enumerable.
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