The nested sampling (NS) technique has gained widespread attention, particularly in cosmology and astronomy, due to its ability to efficiently explore high-likelihood regions - a feature akin to an implicit likelihood optimization that underlies its success. While the full theoretical derivation of NS is complex and involves several approximations, the central challenge lies in sampling from the likelihood-constrained priors, which is crucial for its performance. This work provides a comprehensive and detailed exposition of NS derivation, clarifying both its theoretical foundations and practical challenges. We provide a thorough description of the NS procedure, emphasizing both its strengths and potential limitations. In doing so, this work seeks to deepen understanding of the method and to foster the development of future enhancements, novel variants, and more efficient implementations across a wide range of scientific applications. Thus, the main contribution of this work is twofold: it serves both as a tutorial for newcomers to the field and as a critical review for experienced practitioners.

In almost all these applications the quality of the embedding is measured by variousaverage case criteria.

Fast, efficient, robust communication during wildfire and other emergency responses is critical. One way to achieve this is by coordinating swarms of autonomous aerial vehicles carrying communications equipment to form an ad-hoc network connecting emergency response personnel to both each other and central command. However, operating in such extreme environments may lead to individual networking agents being damaged or rendered inoperable, which could bring down the network and interrupt communications. To overcome this challenge and enable multi-agent UAV networking in difficult environments, this paper introduces and formalizes the problem of Robust Task Networking Under Attrition (RTNUA), which extends connectivity maintenance in multi-robot systems to explicitly address proactive redundancy and attrition recovery. We introduce Physics-Informed Robust Employment of Multi-Agent Networks ($Φ$IREMAN), a topological algorithm leveraging physics-inspired potential fields to solve this problem. Through simulation across 25 problem configurations, $Φ$IREMAN consistently outperforms the DCCRS baseline, and on large-scale problems with up to 100 tasks and 500 drones, maintains $>99.9\%$ task uptime despite substantial attrition, demonstrating both effectiveness and scalability.

Humans build viewpoint-independent cognitive maps through navigation, enabling intuitive reasoning about object permanence and spatial relations. We argue that multimodal large language models (MLLMs), despite extensive video training, lack this fundamental spatial reasoning capability, a critical limitation for embodied applications. To demonstrate these limitations and drive research, we introduce REM (Reasoning over Embodied Multi-Frame Trajectories), a benchmark using controllable 3D environments for long-horizon embodied spatial reasoning. REM systematically evaluates key aspects like object permanence/distinction, spatial relationships, and numerical tracking across dynamic embodied viewpoints. Our evaluation shows that the best-performing current models exhibit promising overall performance, but become increasingly unreliable at even moderate complexity levels easily handled by humans. These findings highlight challenges MLLMs face in developing robust spatial representations from sequential visual input. Consequently, REM provides targeted metrics and diagnostics to foster improved spatial understanding in future models.

Policy learning algorithms are widely used in areas such as personalized medicine and advertising to develop individualized treatment regimes. However, most methods force a decision even when predictions are uncertain, which is risky in high-stakes settings. We study policy learning with abstention, where a policy may defer to a safe default or an expert. When a policy abstains, it receives a small additive reward on top of the value of a random guess. We propose a two-stage learner that first identifies a set of near-optimal policies and then constructs an abstention rule from their disagreements. We establish fast O(1/n)-type regret guarantees when propensities are known, and extend these guarantees to the unknown-propensity case via a doubly robust (DR) objective. We further show that abstention is a versatile tool with direct applications to other core problems in policy learning: it yields improved guarantees under margin conditions without the common realizability assumption, connects to distributionally robust policy learning by hedging against small data shifts, and supports safe policy improvement by ensuring improvement over a baseline policy with high probability.

The clients' positions may change in complex and unpredictable ways and thus an a priori knowledge

This paper focuses on the privacy risks of disclosing the community structure in an online social network.

As AI rapidly advances, the security risks posed by AI are becoming increasingly severe, especially in critical scenarios, including those posing existential risks. If AI becomes uncontrollable, manipulated, or actively evades safety mechanisms, it could trigger systemic disasters. Existing AI safety approaches-such as model enhancement, value alignment, and human intervention-suffer from fundamental, in-principle limitations when facing AI with extreme motivations and unlimited intelligence, and cannot guarantee security. To address this challenge, we propose a Governable AI (GAI) framework that shifts from traditional internal constraints to externally enforced structural compliance based on cryptographic mechanisms that are computationally infeasible to break, even for future AI, under the defined threat model and well-established cryptographic assumptions.The GAI framework is composed of a simple yet reliable, fully deterministic, powerful, flexible, and general-purpose rule enforcement module (REM); governance rules; and a governable secure super-platform (GSSP) that offers end-to-end protection against compromise or subversion by AI. The decoupling of the governance rules and the technical platform further enables a feasible and generalizable technical pathway for the safety governance of AI. REM enforces the bottom line defined by governance rules, while GSSP ensures non-bypassability, tamper-resistance, and unforgeability to eliminate all identified attack vectors. This paper also presents a rigorous formal proof of the security properties of this mechanism and demonstrates its effectiveness through a prototype implementation evaluated in representative high-stakes scenarios.

In this paper, we introduce a simple and computationally efficient algorithm for the correlation clustering problem with provable privacy guarantees.