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We commonly encounter the problem of identifying an optimally weight-adjusted version of the empirical distribution of observed data, adhering to predefined constraints on the weights. Such constraints often manifest as restrictions on the moments, tail behavior, shapes, number of modes, etc., of the resulting weight-adjusted empirical distribution. In this article, we substantially enhance the flexibility of such a methodology by introducing a nonparametrically imbued distributional constraint on the weights and developing a general framework leveraging the maximum entropy principle and tools from optimal transport. The key idea is to ensure that the maximum entropy weight-adjusted empirical distribution of the observed data is close to a pre-specified probability distribution in terms of the optimal transport metric, while allowing for subtle departures. The proposed scheme for the re-weighting of observations subject to constraints is reminiscent of the empirical likelihood and related ideas, but offers greater flexibility in applications where parametric distribution-guided constraints arise naturally. The versatility of the proposed framework is demonstrated in the context of three disparate applications where data re-weighting is warranted to satisfy side constraints on the optimization problem at the heart of the statistical task—namely, portfolio allocation, semi-parametric inference for complex surveys, and ensuring algorithmic fairness in machine learning algorithms. 

The rise of machine learning-driven decision-making has sparked a growing emphasis on algorithmic fairness. Within the realm of clustering, the notion of balance is utilized as a criterion for attaining fairness, which characterizes a clustering mechanism as fair when the resulting clusters maintain a consistent proportion of observations representing individuals from distinct groups delineated by protected attributes. Building on this idea, the literature has rapidly incorporated a myriad of extensions, devising fair versions of the existing frequentist clustering algorithms, e.g., k-means, k-medioids, etc., that aim at minimizing specific loss functions. These approaches lack uncertainty quantification associated with the optimal clustering configuration and only provide clustering boundaries without quantifying the probabilities associated with each observation belonging to the different clusters. In this article, we intend to offer a novel probabilistic formulation of the fair clustering problem that facilitates valid uncertainty quantification even under mild model misspecifications, without incurring substantial computational overhead. Mixture model-based fair clustering frameworks facilitate automatic uncertainty quantification, but tend to showcase brittleness under model misspecification and involve significant computational challenges. To circumnavigate such issues, we propose a generalized Bayesian fair clustering framework that inherently enjoys decision-theoretic interpretation. Moreover, we devise efficient computational algorithms that crucially leverage techniques from the existing literature on optimal transport and clustering based on loss functions. The gain from the proposed technology is showcased via numerical experiments and real data examples. 

Abstract In the wake of lead‐halide perovskite research, bismuth‐ and antimony‐based perovskite‐inspired semiconducting materials are attracting increasing attention as safer and potentially more robust alternatives to lead‐based archetypes. Of particular interest are the group IB–group VA halide compositions with a generic formula A_xB_yX_x+3y(A⁺ = Cu⁺/Ag⁺; B³⁺ = Bi³⁺/Sb³⁺; X^– = I^–/Br^–), i.e., silver/copper pnictohalides and derivatives thereof. This family of materials forms 3D structures with much higher solar cell efficiencies and greater potential for indoor photovoltaics than the lower‐dimensional bismuth/antimony‐based perovskite‐inspired semiconductors. Furthermore, silver/copper pnictohalides are being investigated for applications beyond photovoltaics, e.g., for photodetection, ionization radiation detection, memristors, and chemical sensors. Such versatility parallels the wide range of possible compositions and synthetic routes, which enable various structural, morphological, and optoelectronic properties. This manuscript surveys the growing research on silver/copper pnictohalides, highlighting their composition–structure–property relationships and the status and prospects of the photovoltaic and optoelectronic devices based thereon. The authors hope that the insights provided herein might accelerate the development of eco‐friendly and stable perovskite‐inspired materials for next‐generation photovoltaics and optoelectronics.