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Learning the graphical structure of Bayesian networks is key to describing data-generating mechanisms in many complex applications but poses considerable computational challenges. Observational data can only identify the equivalence class of the directed acyclic graph underlying a Bayesian network model, and a variety of methods exist to tackle the problem. Under certain assumptions, the popular PC algorithm can consistently recover the correct equivalence class by reverse-engineering the conditional independence (CI) relationships holding in the variable distribution. The dual PC algorithm is a novel scheme to carry out the CI tests within the PC algorithm by leveraging the inverse relationship between covariance and precision matrices. By exploiting block matrix inversions we can simultaneously perform tests on partial correlations of complementary (or dual) conditioning sets. The multiple CI tests of the dual PC algorithm proceed by first considering marginal and full-order CI relationships and progressively moving to central-order ones. Simulation studies show that the dual PC algorithm outperforms the classic PC algorithm both in terms of run time and in recovering the underlying network structure, even in the presence of deviations from Gaussianity. Additionally, we show that the dual PC algorithm applies for Gaussian copula models, and demonstrate its performance in that setting.

Selecting the appropriate requirements to develop in the next release of an open market software product under evolution, is a compulsory step of each software development project. This selection should be done by maximizing stakeholders' satisfaction and minimizing development costs, while keeping constraints. In this work we investigate what is the requirements interactions impact when searching for solutions of the bi-objective Next Release Problem. In one hand, these interactions are explicitly included in two algorithms: a branch and bound algorithm and an estimation of distribution algorithm (EDA). And on the other, we study the performance of these not previously used solving approaches by applying them in several instances of small, medium and large size data sets. We find that interactions inclusion do enhance the search and when time restrictions exists, as in the case of the bi-objective Next Release Problem, EDAs have proven to be stable and reliable locating a large number of solutions on the reference Pareto front.

In this paper, we employ the thoughts and methodologies of Shannon's information theory to solve the problem of the optimal radar parameter estimation. Based on a general radar system model, the \textit{a posteriori} probability density function of targets' parameters is derived. Range information (RI) and entropy error (EE) are defined to evaluate the performance. It is proved that acquiring 1 bit of the range information is equivalent to reducing estimation deviation by half. The closed-form approximation for the EE is deduced in all signal-to-noise ratio (SNR) regions, which demonstrates that the EE degenerates to the mean square error (MSE) when the SNR is tending to infinity. Parameter estimation theorem is then proved, which claims that the theoretical RI is achievable. The converse claims that there exists no unbiased estimator whose empirical RI is larger than the theoretical RI. Simulation result demonstrates that the theoretical EE is tighter than the commonly used Cram\'er-Rao bound and the ZivZakai bound.

AI advice is becoming increasingly popular, e.g., in investment and medical treatment decisions. As this advice is typically imperfect, decision-makers have to exert discretion as to whether actually follow that advice: they have to "appropriately" rely on correct and turn down incorrect advice. However, current research on appropriate reliance still lacks a common definition as well as an operational measurement concept. Additionally, no in-depth behavioral experiments have been conducted that help understand the factors influencing this behavior. In this paper, we propose Appropriateness of Reliance (AoR) as an underlying, quantifiable two-dimensional measurement concept. We develop a research model that analyzes the effect of providing explanations for AI advice. In an experiment with 200 participants, we demonstrate how these explanations influence the AoR, and, thus, the effectiveness of AI advice. Our work contributes fundamental concepts for the analysis of reliance behavior and the purposeful design of AI advisors.

Conventional clustering methods based on pairwise affinity usually suffer from the concentration effect while processing huge dimensional features yet low sample sizes data, resulting in inaccuracy to encode the sample proximity and suboptimal performance in clustering. To address this issue, we propose a unified tensor clustering method (UTC) that characterizes sample proximity using multiple samples' affinity, thereby supplementing rich spatial sample distributions to boost clustering. Specifically, we find that the triadic tensor affinity can be constructed via the Khari-Rao product of two affinity matrices. Furthermore, our early work shows that the fourth-order tensor affinity is defined by the Kronecker product. Therefore, we utilize arithmetical products, Khatri-Rao and Kronecker products, to mathematically integrate different orders of affinity into a unified tensor clustering framework. Thus, the UTC jointly learns a joint low-dimensional embedding to combine various orders. Finally, a numerical scheme is designed to solve the problem. Experiments on synthetic datasets and real-world datasets demonstrate that 1) the usage of high-order tensor affinity could provide a supplementary characterization of sample proximity to the popular affinity matrix; 2) the proposed method of UTC is affirmed to enhance clustering by exploiting different order affinities when processing high-dimensional data.

The principle of minimum potential and complementary energy are the most important variational principles in solid mechanics. The deep energy method (DEM), which has received much attention, is based on the principle of minimum potential energy and lacks the important form of minimum complementary energy. Thus, we propose the deep energy method based on the principle of minimum complementary energy (DCM). The output function of DCM is the stress function that naturally satisfies the equilibrium equation. We extend the proposed DCM algorithm (DCM-P), adding the terms that naturally satisfy the biharmonic equation in the Airy stress function. We combine operator learning with physical equations and propose a deep complementary energy operator method (DCM-O), including branch net, trunk net, basis net, and particular net. DCM-O first combines existing high-fidelity numerical results to train DCM-O through data. Then the complementary energy is used to train the branch net and trunk net in DCM-O. To analyze DCM performance, we present the numerical result of the most common stress functions, the Prandtl and Airy stress function. The proposed method DCM is used to model the representative mechanical problems with the different types of boundary conditions. We compare DCM with the existing PINNs and DEM algorithms. The result shows the advantage of the proposed DCM is suitable for dealing with problems of dominated displacement boundary conditions, which is reflected in theory and our numerical experiments. DCM-P and DCM-O improve the accuracy of DCM and the speed of calculation convergence. DCM is an essential supplementary energy form of the deep energy method. We believe that operator learning based on the energy method can balance data and physical equations well, giving computational mechanics broad research prospects.

Recent results on optimization and generalization properties of neural networks showed that in a simple two-layer network, the alignment of the labels to the eigenvectors of the corresponding Gram matrix determines the convergence of the optimization during training. Such analyses also provide upper bounds on the generalization error. We experimentally investigate the implications of these results to deeper networks via embeddings. We regard the layers preceding the final hidden layer as producing different representations of the input data which are then fed to the two-layer model. We show that these representations improve both optimization and generalization. In particular, we investigate three kernel representations when fed to the final hidden layer: the Gaussian kernel and its approximation by random Fourier features, kernels designed to imitate representations produced by neural networks and finally an optimal kernel designed to align the data with target labels. The approximated representations induced by these kernels are fed to the neural network and the optimization and generalization properties of the final model are evaluated and compared.

Many models for point process data are defined through a thinning procedure where locations of a base process (often Poisson) are either kept (observed) or discarded (thinned). In this paper, we go back to the fundamentals of the distribution theory for point processes and provide a colouring theorem that characterizes the joint density of thinned and observed locations in any of such models. In practice, the marginal model of observed points is often intractable, but thinned locations can be instantiated from their conditional distribution and typical data augmentation schemes can be employed to circumvent this problem. Such approaches have been employed in recent publications, but conceptual flaws have been introduced in this literature. We concentrate on an example: the so-called sigmoidal Gaussian Cox process. We apply our general theory to resolve what are contradicting viewpoints in the data augmentation step of the inference procedures therein. Finally, we provide a multitype extension to this process and conduct Bayesian inference on data consisting of positions of 2 different species of trees in Lansing Woods, Illinois.

This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.

Modeling multivariate time series has long been a subject that has attracted researchers from a diverse range of fields including economics, finance, and traffic. A basic assumption behind multivariate time series forecasting is that its variables depend on one another but, upon looking closely, it is fair to say that existing methods fail to fully exploit latent spatial dependencies between pairs of variables. In recent years, meanwhile, graph neural networks (GNNs) have shown high capability in handling relational dependencies. GNNs require well-defined graph structures for information propagation which means they cannot be applied directly for multivariate time series where the dependencies are not known in advance. In this paper, we propose a general graph neural network framework designed specifically for multivariate time series data. Our approach automatically extracts the uni-directed relations among variables through a graph learning module, into which external knowledge like variable attributes can be easily integrated. A novel mix-hop propagation layer and a dilated inception layer are further proposed to capture the spatial and temporal dependencies within the time series. The graph learning, graph convolution, and temporal convolution modules are jointly learned in an end-to-end framework. Experimental results show that our proposed model outperforms the state-of-the-art baseline methods on 3 of 4 benchmark datasets and achieves on-par performance with other approaches on two traffic datasets which provide extra structural information.