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In this work we propose a semiparametric bivariate copula whose density is defined by a picewise constant function on disjoint squares. We obtain the maximum likelihood estimators which reduce to the sample copula under specific conditions. We carry out a full Bayesian analysis of the model and propose a spatial dependent prior distribution for the model parameters. This prior allows the parameters to borrow strength across neighbouring regions to produce smooth posterior estimates. We implement a simulation study and illustrate the performance of our model with a real dataset.

This paper develops a general methodology to conduct statistical inference for observations indexed by multiple sets of entities. We propose a novel multiway empirical likelihood statistic that converges to a chi-square distribution under the non-degenerate case, where corresponding Hoeffding type decomposition is dominated by linear terms. Our methodology is related to the notion of jackknife empirical likelihood but the leave-out pseudo values are constructed by leaving columns or rows. We further develop a modified version of our multiway empirical likelihood statistic, which converges to a chi-square distribution regardless of the degeneracy, and discover its desirable higher-order property compared to the t-ratio by the conventional Eicker-White type variance estimator. The proposed methodology is illustrated by several important statistical problems, such as bipartite network, two-stage sampling, generalized estimating equations, and three-way observations.

Bayesian nonparametric methods are a popular choice for analysing survival data due to their ability to flexibly model the distribution of survival times. These methods typically employ a nonparametric prior on the survival function that is conjugate with respect to right-censored data. Eliciting these priors, particularly in the presence of covariates, can be challenging and inference typically relies on computationally intensive Markov chain Monte Carlo schemes. In this paper, we build on recent work that recasts Bayesian inference as assigning a predictive distribution on the unseen values of a population conditional on the observed samples, thus avoiding the need to specify a complex prior. We describe a copula-based predictive update which admits a scalable sequential importance sampling algorithm to perform inference that properly accounts for right-censoring. We provide theoretical justification through an extension of Doob's consistency theorem and illustrate the method on a number of simulated and real data sets, including an example with covariates. Our approach enables analysts to perform Bayesian nonparametric inference through only the specification of a predictive distribution.

There is a growing interest in the estimation of the number of unseen features, mostly driven by biological applications. A recent work brought out a peculiar property of the popular completely random measures (CRMs) as prior models in Bayesian nonparametric (BNP) inference for the unseen-features problem: for fixed prior's parameters, they all lead to a Poisson posterior distribution for the number of unseen features, which depends on the sampling information only through the sample size. CRMs are thus not a flexible prior model for the unseen-features problem and, while the Poisson posterior distribution may be appealing for analytical tractability and ease of interpretability, its independence from the sampling information makes the BNP approach a questionable oversimplification, with posterior inferences being completely determined by the estimation of unknown prior's parameters. In this paper, we introduce the stable-Beta scaled process (SB-SP) prior, and we show that it allows to enrich the posterior distribution of the number of unseen features arising under CRM priors, while maintaining its analytical tractability and interpretability. That is, the SB-SP prior leads to a negative Binomial posterior distribution, which depends on the sampling information through the sample size and the number of distinct features, with corresponding estimates being simple, linear in the sampling information and computationally efficient. We apply our BNP approach to synthetic data and to real cancer genomic data, showing that: i) it outperforms the most popular parametric and nonparametric competitors in terms of estimation accuracy; ii) it provides improved coverage for the estimation with respect to a BNP approach under CRM priors.

The statistical finite element method (StatFEM) is an emerging probabilistic method that allows observations of a physical system to be synthesised with the numerical solution of a PDE intended to describe it in a coherent statistical framework, to compensate for model error. This work presents a new theoretical analysis of the statistical finite element method demonstrating that it has similar convergence properties to the finite element method on which it is based. Our results constitute a bound on the Wasserstein-2 distance between the ideal prior and posterior and the StatFEM approximation thereof, and show that this distance converges at the same mesh-dependent rate as finite element solutions converge to the true solution. Several numerical examples are presented to demonstrate our theory, including an example which test the robustness of StatFEM when extended to nonlinear quantities of interest.

Ensemble methods based on subsampling, such as random forests, are popular in applications due to their high predictive accuracy. Existing literature views a random forest prediction as an infinite-order incomplete U-statistic to quantify its uncertainty. However, these methods focus on a small subsampling size of each tree, which is theoretically valid but practically limited. This paper develops an unbiased variance estimator based on incomplete U-statistics, which allows the tree size to be comparable with the overall sample size, making statistical inference possible in a broader range of real applications. Simulation results demonstrate that our estimators enjoy lower bias and more accurate confidence interval coverage without additional computational costs. We also propose a local smoothing procedure to reduce the variation of our estimator, which shows improved numerical performance when the number of trees is relatively small. Further, we investigate the ratio consistency of our proposed variance estimator under specific scenarios. In particular, we develop a new "double U-statistic" formulation to analyze the Hoeffding decomposition of the estimator's variance.

Dyadic data is often encountered when quantities of interest are associated with the edges of a network. As such it plays an important role in statistics, econometrics and many other data science disciplines. We consider the problem of uniformly estimating a dyadic Lebesgue density function, focusing on nonparametric kernel-based estimators taking the form of dyadic empirical processes. Our main contributions include the minimax-optimal uniform convergence rate of the dyadic kernel density estimator, along with strong approximation results for the associated standardized and Studentized $t$-processes. A consistent variance estimator enables the construction of valid and feasible uniform confidence bands for the unknown density function. A crucial feature of dyadic distributions is that they may be "degenerate" at certain points in the support of the data, a property making our analysis somewhat delicate. Nonetheless our methods for uniform inference remain robust to the potential presence of such points. For implementation purposes, we discuss procedures based on positive semi-definite covariance estimators, mean squared error optimal bandwidth selectors and robust bias-correction techniques. We illustrate the empirical finite-sample performance of our methods both in simulations and with real-world data. Our technical results concerning strong approximations and maximal inequalities are of potential independent interest.

Though remarkable progress has been achieved in various vision tasks, deep neural networks still suffer obvious performance degradation when tested in out-of-distribution scenarios. We argue that the feature statistics (mean and standard deviation), which carry the domain characteristics of the training data, can be properly manipulated to improve the generalization ability of deep learning models. Common methods often consider the feature statistics as deterministic values measured from the learned features and do not explicitly consider the uncertain statistics discrepancy caused by potential domain shifts during testing. In this paper, we improve the network generalization ability by modeling the uncertainty of domain shifts with synthesized feature statistics during training. Specifically, we hypothesize that the feature statistic, after considering the potential uncertainties, follows a multivariate Gaussian distribution. Hence, each feature statistic is no longer a deterministic value, but a probabilistic point with diverse distribution possibilities. With the uncertain feature statistics, the models can be trained to alleviate the domain perturbations and achieve better robustness against potential domain shifts. Our method can be readily integrated into networks without additional parameters. Extensive experiments demonstrate that our proposed method consistently improves the network generalization ability on multiple vision tasks, including image classification, semantic segmentation, and instance retrieval. The code will be released soon at //github.com/lixiaotong97/DSU.

Data augmentation has been widely used for training deep learning systems for medical image segmentation and plays an important role in obtaining robust and transformation-invariant predictions. However, it has seldom been used at test time for segmentation and not been formulated in a consistent mathematical framework. In this paper, we first propose a theoretical formulation of test-time augmentation for deep learning in image recognition, where the prediction is obtained through estimating its expectation by Monte Carlo simulation with prior distributions of parameters in an image acquisition model that involves image transformations and noise. We then propose a novel uncertainty estimation method based on the formulated test-time augmentation. Experiments with segmentation of fetal brains and brain tumors from 2D and 3D Magnetic Resonance Images (MRI) showed that 1) our test-time augmentation outperforms a single-prediction baseline and dropout-based multiple predictions, and 2) it provides a better uncertainty estimation than calculating the model-based uncertainty alone and helps to reduce overconfident incorrect predictions.

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.