Circuit representations are becoming the lingua franca to express and reason about tractable generative and discriminative models.

We focus on monotonicity constraints, which are common and require that the function's output increases with increasing values of specific input features.

Bayesian optimization has become widely popular across various experimental sciences due to its favorable attributes: it can handle noisy data, perform well with relatively small datasets, and provide adaptive suggestions for sequential experimentation. While still in its infancy, Bayesian optimization has recently gained traction in bioprocess engineering. However, experimentation with biological systems is highly complex and the resulting experimental uncertainty requires specific extensions to classical Bayesian optimization. Moreover, current literature often targets readers with a strong statistical background, limiting its accessibility for practitioners. In light of these developments, this review has two aims: first, to provide an intuitive and practical introduction to Bayesian optimization; and second, to outline promising application areas and open algorithmic challenges, thereby highlighting opportunities for future research in machine learning.

We present BKP, a user-friendly and extensible R package that implements the Beta Kernel Process (BKP) -- a fully nonparametric and computationally efficient framework for modeling spatially varying binomial probabilities. The BKP model combines localized kernel-weighted likelihoods with conjugate beta priors, resulting in closed-form posterior inference without requiring latent variable augmentation or intensive MCMC sampling. The package supports binary and aggregated binomial responses, allows flexible choices of kernel functions and prior specification, and provides loss-based kernel hyperparameter tuning procedures. In addition, BKP extends naturally to the Dirichlet Kernel Process (DKP) for modeling spatially varying multinomial or compositional data. To our knowledge, this is the first publicly available R package for implementing BKP-based methods. We illustrate the use of BKP through several synthetic and real-world datasets, highlighting its interpretability, accuracy, and scalability. The package aims to facilitate practical application and future methodological development of kernel-based beta modeling in statistics and machine learning.

Scientific progress is tightly coupled to the emergence of new research tools. Today, machine learning (ML)-especially deep learning (DL)-has become a transformative instrument for quantum science and technology. Owing to the intrinsic complexity of quantum systems, DL enables efficient exploration of large parameter spaces, extraction of patterns from experimental data, and data-driven guidance for research directions. These capabilities already support tasks such as refining quantum control protocols and accelerating the discovery of materials with targeted quantum properties, making ML/DL literacy an essential skill for the next generation of quantum scientists. At the same time, DL's power brings risks: models can overfit noisy data, obscure causal structure, and yield results with limited physical interpretability. Recognizing these limitations and deploying mitigation strategies is crucial for scientific rigor. These lecture notes provide a comprehensive, graduate-level introduction to DL for quantum applications, combining conceptual exposition with hands-on examples. Organized as a progressive sequence, they aim to equip readers to decide when and how to apply DL effectively, to understand its practical constraints, and to adapt AI methods responsibly to problems across quantum physics, chemistry, and engineering.

This work demonstrates how the concept of the entropic potential of events -- a parameter quantifying the influence of discrete events on the expected future entropy of a system -- can enhance uncertainty quantification, decision-making, and interpretability in artificial intelligence (AI). Building on its original formulation in physics, the framework is adapted for AI by introducing an event-centric measure that captures how actions, observations, or other discrete occurrences impact uncertainty at future time horizons. Both the original and AI-adjusted definitions of entropic potential are formalized, with the latter emphasizing conditional expectations to account for counterfactual scenarios. Applications are explored in policy evaluation, intrinsic reward design, explainable AI, and anomaly detection, highlighting the metric's potential to unify and strengthen uncertainty modeling in intelligent systems. Conceptual examples illustrate its use in reinforcement learning, Bayesian inference, and anomaly detection, while practical considerations for computation in complex AI models are discussed. The entropic potential framework offers a theoretically grounded, interpretable, and versatile approach to managing uncertainty in AI, bridging principles from thermodynamics, information theory, and machine learning.

Background: Cirrhosis is a progressive liver disease with high mortality and frequent complications, notably acute kidney injury (AKI), which occurs in up to 50% of hospitalized patients and worsens outcomes. AKI stems from complex hemodynamic, inflammatory, and metabolic changes, making early detection essential. Many predictive tools lack accuracy, interpretability, and alignment with intensive care unit (ICU) workflows. This study developed an interpretable machine learning model for early AKI prediction in critically ill patients with cirrhosis. Methods: We conducted a retrospective analysis of the MIMIC-IV v2.2 database, identifying 1240 adult ICU patients with cirrhosis and excluding those with ICU stays under 48 hours or missing key data. Laboratory and physiological variables from the first 48 hours were extracted. The pipeline included preprocessing, missingness filtering, LASSO feature selection, and SMOTE class balancing. Six algorithms-LightGBM, CatBoost, XGBoost, logistic regression, naive Bayes, and neural networks-were trained and evaluated using AUROC, accuracy, F1-score, sensitivity, specificity, and predictive values. Results: LightGBM achieved the best performance (AUROC 0.808, 95% CI 0.741-0.856; accuracy 0.704; NPV 0.911). Key predictors included prolonged partial thromboplastin time, absence of outside-facility 20G placement, low pH, and altered pO2, consistent with known cirrhosis-AKI mechanisms and suggesting actionable targets. Conclusion: The LightGBM-based model enables accurate early AKI risk stratification in ICU patients with cirrhosis using routine clinical variables. Its high negative predictive value supports safe de-escalation for low-risk patients, and interpretability fosters clinician trust and targeted prevention. External validation and integration into electronic health record systems are warranted.

The automatic curation of discussion forums in online courses requires constant updates, making frequent retraining of Large Language Models (LLMs) a resource-intensive process. To circumvent the need for costly fine-tuning, this paper proposes and evaluates the use of Bayesian fusion. The approach combines the multidimensional classification scores of a pre-trained generic LLM with those of a classifier trained on local data. The performance comparison demonstrated that the proposed fusion improves the results compared to each classifier individually, and is competitive with the LLM fine-tuning approach

Instead, we impose a condition on the residual variances that is closely related to previous work on linear models with equal variances.