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The Tsallis $q$-Gaussian distribution is a powerful generalization of the standard Gaussian distribution and is commonly used in various fields, including non-extensive statistical mechanics, financial markets and image processing. It belongs to the $q$-distribution family, which is characterized by a non-additive entropy. Due to their versatility and practicality, $q$-Gaussians are a natural choice for modeling input quantities in measurement models. This paper presents the characteristic function of a linear combination of independent $q$-Gaussian random variables and proposes a numerical method for its inversion. The proposed technique makes it possible to determine the exact probability distribution of the output quantity in linear measurement models, with the input quantities modeled as independent $q$-Gaussian random variables. It provides an alternative computational procedure to the Monte Carlo method for uncertainty analysis through the propagation of distributions.

Prevention is better than cure. This old truth applies not only to the prevention of diseases but also to the prevention of issues with AI models used in medicine. The source of malfunctioning of predictive models often lies not in the training process but reaches the data acquisition phase or design of the experiment phase. In this paper, we analyze in detail a single use case - a Kaggle competition related to the detection of abnormalities in X-ray lung images. We demonstrate how a series of simple tests for data imbalance exposes faults in the data acquisition and annotation process. Complex models are able to learn such artifacts and it is difficult to remove this bias during or after the training. Errors made at the data collection stage make it difficult to validate the model correctly. Based on this use case, we show how to monitor data and model balance (fairness) throughout the life cycle of a predictive model, from data acquisition to parity analysis of model scores.

A posteriori ratemaking in insurance uses a Bayesian credibility model to upgrade the current premiums of a contract by taking into account policyholders' attributes and their claim history. Most data-driven models used for this task are mathematically intractable, and premiums must be then obtained through numerical methods such as simulation such MCMC. However, these methods can be computationally expensive and prohibitive for large portfolios when applied at the policyholder level. Additionally, these computations become ``black-box" procedures as there is no expression showing how the claim history of policyholders is used to upgrade their premiums. To address these challenges, this paper proposes the use of a surrogate modeling approach to inexpensively derive a closed-form expression for computing the Bayesian credibility premiums for any given model. As a part of the methodology, the paper introduces the ``credibility index", which is a summary statistic of a policyholder's claim history that serves as the main input of the surrogate model and that is sufficient for several distribution families, including the exponential dispersion family. As a result, the computational burden of a posteriori ratemaking for large portfolios is therefore reduced through the direct evaluation of the closed-form expression, which additionally can provide a transparent and interpretable way of computing Bayesian premiums.

In this paper, we propose the Ordered Median Tree Location Problem (OMT). The OMT is a single-allocation facility location problem where p facilities must be placed on a network connected by a non-directed tree. The objective is to minimize the sum of the ordered weighted averaged allocation costs plus the sum of the costs of connecting the facilities in the tree. We present different MILP formulations for the OMT based on properties of the minimum spanning tree problem and the ordered median optimization. Given that ordered median hub location problems are rather difficult to solve we have improved the OMT solution performance by introducing covering variables in a valid reformulation plus developing two pre-processing phases to reduce the size of this formulations. In addition, we propose a Benders decomposition algorithm to approach the OMT. We establish an empirical comparison between these new formulations and we also provide enhancements that together with a proper formulation allow to solve medium size instances on general random graphs.

Oceanographers are interested in predicting ocean currents and identifying divergences in a current vector field based on sparse observations of buoy velocities. Since we expect current velocity to be a continuous but highly non-linear function of spatial location, Gaussian processes (GPs) offer an attractive model. But we show that applying a GP with a standard stationary kernel directly to buoy data can struggle at both current prediction and divergence identification -- due to some physically unrealistic prior assumptions. To better reflect known physical properties of currents, we propose to instead put a standard stationary kernel on the divergence and curl-free components of a vector field obtained through a Helmholtz decomposition. We show that, because this decomposition relates to the original vector field just via mixed partial derivatives, we can still perform inference given the original data with only a small constant multiple of additional computational expense. We illustrate the benefits of our method on synthetic and real ocean data.

The Immersed Boundary (IB) method of Peskin (J. Comput. Phys., 1977) is useful for problems involving fluid-structure interactions or complex geometries. By making use of a regular Cartesian grid that is independent of the geometry, the IB framework yields a robust numerical scheme that can efficiently handle immersed deformable structures. Additionally, the IB method has been adapted to problems with prescribed motion and other PDEs with given boundary data. IB methods for these problems traditionally involve penalty forces which only approximately satisfy boundary conditions, or they are formulated as constraint problems. In the latter approach, one must find the unknown forces by solving an equation that corresponds to a poorly conditioned first-kind integral equation. This operation can require a large number of iterations of a Krylov method, and since a time-dependent problem requires this solve at each time step, this method can be prohibitively inefficient without preconditioning. In this work, we introduce a new, well-conditioned IB formulation for boundary value problems, which we call the Immersed Boundary Double Layer (IBDL) method. We present the method as it applies to Poisson and Helmholtz problems to demonstrate its efficiency over the original constraint method. In this double layer formulation, the equation for the unknown boundary distribution corresponds to a well-conditioned second-kind integral equation that can be solved efficiently with a small number of iterations of a Krylov method. Furthermore, the iteration count is independent of both the mesh size and immersed boundary point spacing. The method converges away from the boundary, and when combined with a local interpolation, it converges in the entire PDE domain. Additionally, while the original constraint method applies only to Dirichlet problems, the IBDL formulation can also be used for Neumann conditions.

Many variations of the classical graph coloring model have been intensively studied due to their multiple applications; scheduling problems and aircraft assignments, for instance, motivate the robust coloring problem. This model gets to capture natural constraints of those optimization problems by combining the information provided by two colorings: a vertex coloring of a graph and the induced edge coloring on a subgraph of its complement; the goal is to minimize, among all proper colorings of the graph for a fixed number of colors, the number of edges in the subgraph with the endpoints of the same color. The study of the robust coloring model has been focused on the search for heuristics due to its NP-hard character when using at least three colors, but little progress has been made in other directions. We present a new approach on the problem obtaining the first collection of non-heuristic results for general graphs; among them, we prove that robust coloring is the model that better approaches the equitable partition of the vertex set, even when the graph does not admit a so-called \emph{equitable coloring}. We also show the NP-completeness of its decision problem for the unsolved case of two colors, obtain bounds on the associated robust coloring parameter, and solve a conjecture on paths that illustrates the complexity of studying this coloring model.

Causal discovery and causal reasoning are classically treated as separate and consecutive tasks: one first infers the causal graph, and then uses it to estimate causal effects of interventions. However, such a two-stage approach is uneconomical, especially in terms of actively collected interventional data, since the causal query of interest may not require a fully-specified causal model. From a Bayesian perspective, it is also unnatural, since a causal query (e.g., the causal graph or some causal effect) can be viewed as a latent quantity subject to posterior inference -- other unobserved quantities that are not of direct interest (e.g., the full causal model) ought to be marginalized out in this process and contribute to our epistemic uncertainty. In this work, we propose Active Bayesian Causal Inference (ABCI), a fully-Bayesian active learning framework for integrated causal discovery and reasoning, which jointly infers a posterior over causal models and queries of interest. In our approach to ABCI, we focus on the class of causally-sufficient, nonlinear additive noise models, which we model using Gaussian processes. We sequentially design experiments that are maximally informative about our target causal query, collect the corresponding interventional data, and update our beliefs to choose the next experiment. Through simulations, we demonstrate that our approach is more data-efficient than several baselines that only focus on learning the full causal graph. This allows us to accurately learn downstream causal queries from fewer samples while providing well-calibrated uncertainty estimates for the quantities of interest.

Seeking the equivalent entities among multi-source Knowledge Graphs (KGs) is the pivotal step to KGs integration, also known as \emph{entity alignment} (EA). However, most existing EA methods are inefficient and poor in scalability. A recent summary points out that some of them even require several days to deal with a dataset containing 200,000 nodes (DWY100K). We believe over-complex graph encoder and inefficient negative sampling strategy are the two main reasons. In this paper, we propose a novel KG encoder -- Dual Attention Matching Network (Dual-AMN), which not only models both intra-graph and cross-graph information smartly, but also greatly reduces computational complexity. Furthermore, we propose the Normalized Hard Sample Mining Loss to smoothly select hard negative samples with reduced loss shift. The experimental results on widely used public datasets indicate that our method achieves both high accuracy and high efficiency. On DWY100K, the whole running process of our method could be finished in 1,100 seconds, at least 10* faster than previous work. The performances of our method also outperform previous works across all datasets, where Hits@1 and MRR have been improved from 6% to 13%.

Image segmentation is still an open problem especially when intensities of the interested objects are overlapped due to the presence of intensity inhomogeneity (also known as bias field). To segment images with intensity inhomogeneities, a bias correction embedded level set model is proposed where Inhomogeneities are Estimated by Orthogonal Primary Functions (IEOPF). In the proposed model, the smoothly varying bias is estimated by a linear combination of a given set of orthogonal primary functions. An inhomogeneous intensity clustering energy is then defined and membership functions of the clusters described by the level set function are introduced to rewrite the energy as a data term of the proposed model. Similar to popular level set methods, a regularization term and an arc length term are also included to regularize and smooth the level set function, respectively. The proposed model is then extended to multichannel and multiphase patterns to segment colourful images and images with multiple objects, respectively. It has been extensively tested on both synthetic and real images that are widely used in the literature and public BrainWeb and IBSR datasets. Experimental results and comparison with state-of-the-art methods demonstrate that advantages of the proposed model in terms of bias correction and segmentation accuracy.