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High-performing out-of-distribution (OOD) detection, both anomaly and novel class, is an important prerequisite for the practical use of classification models. In this paper, we focus on the species recognition task in images concerned with large databases, a large number of fine-grained hierarchical classes, severe class imbalance, and varying image quality. We propose a framework for combining individual OOD measures into one combined OOD (COOD) measure using a supervised model. The individual measures are several existing state-of-the-art measures and several novel OOD measures developed with novel class detection and hierarchical class structure in mind. COOD was extensively evaluated on three large-scale (500k+ images) biodiversity datasets in the context of anomaly and novel class detection. We show that COOD outperforms individual, including state-of-the-art, OOD measures by a large margin in terms of TPR@1% FPR in the majority of experiments, e.g., improving detecting ImageNet images (OOD) from 54.3% to 85.4% for the iNaturalist 2018 dataset. SHAP (feature contribution) analysis shows that different individual OOD measures are essential for various tasks, indicating that multiple OOD measures and combinations are needed to generalize. Additionally, we show that explicitly considering ID images that are incorrectly classified for the original (species) recognition task is important for constructing high-performing OOD detection methods and for practical applicability. The framework can easily be extended or adapted to other tasks and media modalities.

Maximum entropy (Maxent) models are a class of statistical models that use the maximum entropy principle to estimate probability distributions from data. Due to the size of modern data sets, Maxent models need efficient optimization algorithms to scale well for big data applications. State-of-the-art algorithms for Maxent models, however, were not originally designed to handle big data sets; these algorithms either rely on technical devices that may yield unreliable numerical results, scale poorly, or require smoothness assumptions that many practical Maxent models lack. In this paper, we present novel optimization algorithms that overcome the shortcomings of state-of-the-art algorithms for training large-scale, non-smooth Maxent models. Our proposed first-order algorithms leverage the Kullback-Leibler divergence to train large-scale and non-smooth Maxent models efficiently. For Maxent models with discrete probability distribution of $n$ elements built from samples, each containing $m$ features, the stepsize parameters estimation and iterations in our algorithms scale on the order of $O(mn)$ operations and can be trivially parallelized. Moreover, the strong $\ell_{1}$ convexity of the Kullback--Leibler divergence allows for larger stepsize parameters, thereby speeding up the convergence rate of our algorithms. To illustrate the efficiency of our novel algorithms, we consider the problem of estimating probabilities of fire occurrences as a function of ecological features in the Western US MTBS-Interagency wildfire data set. Our numerical results show that our algorithms outperform the state of the arts by one order of magnitude and yield results that agree with physical models of wildfire occurrence and previous statistical analyses of wildfire drivers.

In decision-making, maxitive functions are used for worst-case and best-case evaluations. Maxitivity gives rise to a rich structure that is well-studied in the context of the pointwise order. In this article, we investigate maxitivity with respect to general preorders and provide a representation theorem for such functionals. The results are illustrated for different stochastic orders in the literature, including the usual stochastic order, the increasing convex/concave order, and the dispersive order.

We give a short survey of recent results on sparse-grid linear algorithms of approximate recovery and integration of functions possessing a unweighted or weighted Sobolev mixed smoothness based on their sampled values at a certain finite set. Some of them are extended to more general cases.

A higher-order change-of-measure multilevel Monte Carlo (MLMC) method is developed for computing weak approximations of the invariant measures of SDE with drift coefficients that do not satisfy the contractivity condition. This is achieved by introducing a spring term in the pairwise coupling of the MLMC trajectories employing the order 1.5 strong It\^o--Taylor method. Through this, we can recover the contractivity property of the drift coefficient while still retaining the telescoping sum property needed for implementing the MLMC method. We show that the variance of the change-of-measure MLMC method grows linearly in time $T$ for all $T > 0$, and for all sufficiently small timestep size $h > 0$. For a given error tolerance $\epsilon > 0$, we prove that the method achieves a mean-square-error accuracy of $O(\epsilon^2)$ with a computational cost of $O(\epsilon^{-2} \big\vert \log \epsilon \big\vert^{3/2} (\log \big\vert \log \epsilon \big\vert)^{1/2})$ for uniformly Lipschitz continuous payoff functions and $O \big( \epsilon^{-2} \big\vert \log \epsilon \big\vert^{5/3 + \xi} \big)$ for discontinuous payoffs, respectively, where $\xi > 0$. We also observe an improvement in the constant associated with the computational cost of the higher-order change-of-measure MLMC method, marking an improvement over the Milstein change-of-measure method in the aforementioned seminal work by M. Giles and W. Fang. Several numerical tests were performed to verify the theoretical results and assess the robustness of the method.

The sparsity-ranked lasso (SRL) has been developed for model selection and estimation in the presence of interactions and polynomials. The main tenet of the SRL is that an algorithm should be more skeptical of higher-order polynomials and interactions *a priori* compared to main effects, and hence the inclusion of these more complex terms should require a higher level of evidence. In time series, the same idea of ranked prior skepticism can be applied to the possibly seasonal autoregressive (AR) structure of the series during the model fitting process, becoming especially useful in settings with uncertain or multiple modes of seasonality. The SRL can naturally incorporate exogenous variables, with streamlined options for inference and/or feature selection. The fitting process is quick even for large series with a high-dimensional feature set. In this work, we discuss both the formulation of this procedure and the software we have developed for its implementation via the **fastTS** R package. We explore the performance of our SRL-based approach in a novel application involving the autoregressive modeling of hourly emergency room arrivals at the University of Iowa Hospitals and Clinics. We find that the SRL is considerably faster than its competitors, while producing more accurate predictions.

Mendelian randomization is an instrumental variable method that utilizes genetic information to investigate the causal effect of a modifiable exposure on an outcome. In most cases, the exposure changes over time. Understanding the time-varying causal effect of the exposure can yield detailed insights into mechanistic effects and the potential impact of public health interventions. Recently, a growing number of Mendelian randomization studies have attempted to explore time-varying causal effects. However, the proposed approaches oversimplify temporal information and rely on overly restrictive structural assumptions, limiting their reliability in addressing time-varying causal problems. This paper considers a novel approach to estimate time-varying effects through continuous-time modelling by combining functional principal component analysis and weak-instrument-robust techniques. Our method effectively utilizes available data without making strong structural assumptions and can be applied in general settings where the exposure measurements occur at different timepoints for different individuals. We demonstrate through simulations that our proposed method performs well in estimating time-varying effects and provides reliable inference results when the time-varying effect form is correctly specified. The method could theoretically be used to estimate arbitrarily complex time-varying effects. However, there is a trade-off between model complexity and instrument strength. Estimating complex time-varying effects requires instruments that are unrealistically strong. We illustrate the application of this method in a case study examining the time-varying effects of systolic blood pressure on urea levels.

An integrated Equation of State (EOS) and strength/pore-crush/damage model framework is provided for modeling near to source (near-field) ground-shock response, where large deformations and pressures necessitate coupling EOS with pressure-dependent plastic yield and damage. Nonlinear pressure-dependence of strength up to high-pressures is combined with a Modified Cam-Clay-like cap-plasticity model in a way to allow degradation of strength from pore-crush damage, what we call the "Yp-Cap" model. Nonlinear hardening under compaction allows modeling the crush-out of pores in combination with a fully saturated EOS, i.e., for modeling partially saturated ground-shock response, where air-filled voids crush. Attention is given to algorithmic clarity and efficiency of the provided model, and the model is employed in example numerical simulations, including finite element simulations of underground explosions to exemplify its robustness and utility.

Robust Markov Decision Processes (RMDPs) are a widely used framework for sequential decision-making under parameter uncertainty. RMDPs have been extensively studied when the objective is to maximize the discounted return, but little is known for average optimality (optimizing the long-run average of the rewards obtained over time) and Blackwell optimality (remaining discount optimal for all discount factors sufficiently close to 1). In this paper, we prove several foundational results for RMDPs beyond the discounted return. We show that average optimal policies can be chosen stationary and deterministic for sa-rectangular RMDPs but, perhaps surprisingly, that history-dependent (Markovian) policies strictly outperform stationary policies for average optimality in s-rectangular RMDPs. We also study Blackwell optimality for sa-rectangular RMDPs, where we show that {\em approximate} Blackwell optimal policies always exist, although Blackwell optimal policies may not exist. We also provide a sufficient condition for their existence, which encompasses virtually any examples from the literature. We then discuss the connection between average and Blackwell optimality, and we describe several algorithms to compute the optimal average return. Interestingly, our approach leverages the connections between RMDPs and stochastic games.

One of the main challenges for interpreting black-box models is the ability to uniquely decompose square-integrable functions of non-independent random inputs into a sum of functions of every possible subset of variables. However, dealing with dependencies among inputs can be complicated. We propose a novel framework to study this problem, linking three domains of mathematics: probability theory, functional analysis, and combinatorics. We show that, under two reasonable assumptions on the inputs (non-perfect functional dependence and non-degenerate stochastic dependence), it is always possible to decompose such a function uniquely. This generalizes the well-known Hoeffding decomposition. The elements of this decomposition can be expressed using oblique projections and allow for novel interpretability indices for evaluation and variance decomposition purposes. The properties of these novel indices are studied and discussed. This generalization offers a path towards a more precise uncertainty quantification, which can benefit sensitivity analysis and interpretability studies whenever the inputs are dependent. This decomposition is illustrated analytically, and the challenges for adopting these results in practice are discussed.