This chapter introduces the Bayesian reflex -- an analogy with the autonomic nervous system -- as a unifying framework for online learning in AI. Bayesian online algorithms automatically maintain equilibrium in dynamic environments via three mechanisms: belief maintenance through probabilistic representations, sequential updating via Bayes' theorem, and uncertainty-driven action balancing exploration and exploitation. We survey online Bayesian methods, highlighting two computational principles: the look-up table principle for sequential inference in function space, and the ellipsoidal decomposition framework for nearly exact i.i.d. sampling from arbitrary posteriors. These principles are generalized across dynamic emulation, nonparametric state-space models, circular time series, inverse regression for climate model evaluation, and deep architectures via Recursive Gaussian Processes. Decision-making is explored via Thompson sampling and restless bandits. We extend the framework to assess infinite series convergence (applied to climate dynamics and the Riemann Hypothesis), model prime number distributions leading to the discovery of 184 strong Mersenne prime candidates, detect stationarity, and characterize point processes. The Bayesian reflex provides a foundational infrastructure for adaptive AI that continuously learns in a complex world.

Probabilistic values, including Shapley values and semivalues, provide a model-agnostic framework to attribute the behavior of a black-box model to data points or features, with a wide range of applications including explainable artificial intelligence and data valuation. However, their exact computation requires utility evaluations over exponentially many coalitions, making Monte Carlo approximation essential in modern machine learning applications. Existing estimators are often developed through different identification strategies, including weighted averages, self-normalized weighting, regression adjustment, and weighted least squares. Our key observation is that these seemingly distinct constructions share a common first-order error structure, in which the leading term is an augmented inverse-probability weighted influence term determined by the sampling law and a working surrogate function. This first-order representation yields an explicit expression for the leading mean squared error (MSE), which characterizes how the sampling law and the surrogate jointly determine statistical efficiency. Guided by this criterion, we propose an Efficiency-Aware Surrogate-adjusted Estimator (EASE) that directly chooses the sampling law and surrogate to minimize the first-order MSE. We demonstrate that EASE consistently outperforms state-of-the-art estimators for various probabilistic values.

Things have not been looking good for Spirit Airlines for years now. The budget airline known for its bare-bones approach to the sky filed for bankruptcy in 2024 and then again in 2025. And yet, its demise on Saturday felt sudden and shocking: Spirit said it would go out of business, canceling flights, shuttering its customer service lines, and laying off workers without warning. What does it mean for flyers, and for the busy summer travel season? WIRED spoke to experts to find out.

A federal review of new AI models ahead of their public release is being considered as a possible power for that committee, according to the publication's sources. No clear approach has been decided, but the suggested it could mimic what's currently happening within the UK government, where multiple layers of oversight confirm that AI models meet safety standards. There's also still a chance the entire concept fizzles and comes to nothing. If an oversight group is created, it would mark quite a reversal from the hands-off attitude presented in the White House's previously introduced AI Action Plan. The plan appeared willing to offer the AI companies most of the concessions they wanted, although it did leave a lot of potential to create plenty of new problems .

BirdCast season is here once again. More information Adding us as a Preferred Source in Google by using this link indicates that you would like to see more of our content in Google News results. The species migrates annually from South America to North America to breed, traveling thousands of miles each spring. Breakthroughs, discoveries, and DIY tips sent six days a week. Tonight, there will be more birds in the sky than there are people in the United States.

More information Adding us as a Preferred Source in Google by using this link indicates that you would like to see more of our content in Google News results. It is estimated that fewer than 2,500 mature Speartooth sharks remain in the wild. Breakthroughs, discoveries, and DIY tips sent six days a week. Combining lasers and sharks may sound like a bad idea, but marine ecologists are banking on it to help save some of the planet's most threatened species. By merging optical technology and geochemistry, a group of researchers in Australia are gaining far more accurate information on the snub-nosed speartooth shark's () age, as well as the health of its environment.

GameStop's CEO said he could turn eBay into something worth hundreds of billions of dollars. GameStop's CEO said he could turn eBay into something worth hundreds of billions of dollars. GameStop makes $55.5bn takeover offer for eBay Video game retailer's CEO warns that unsolicited bid could turn hostile if it is rebuffed by resale site's board US video games retailer GameStop has offered to buy eBay for $55.5bn (£41bn) in an unsolicited bid that its boss warned could turn hostile if the proposal is rebuffed by eBay's board. GameStop, which has quietly accumulated a 5% stake in eBay, said it was willing to pay $125 a share, split 50-50 between cash and stock. It is an ambitious move by the games company, which catapulted to fame during the meme-stock craze of 2021 but is worth far less than its takeover target.

The Normalized Transformer, or nGPT (arXiv:2410.01131) achieves impressive training speedups and does not require weight decay or learning rate warmup. However, despite having hyperparameters that explicitly scale with model size, we observe that nGPT does not exhibit learning rate transfer across model dimension and token horizon. To rectify this, we combine numerical experiments with a principled use of alignment exponents (arXiv:2407.05872) to revisit and modify the $μ$P approach to hyperparameter transfer (arXiv:2011.14522). The result is a novel nGPT parameterization we call $ν$GPT. Through extensive empirical validation, we find $ν$GPT exhibits learning rate transfer across width, depth, and token horizon.

Independent sample generation is the prevailing paradigm in modern diffusion-based generative models of AI. We ask a different question: can samples coordinate through shared population statistics to transport probability mass more efficiently? We introduce Mean-Field Path-Integral Diffusion (MF-PID), a framework in which samples are promoted to interacting agents whose drift depends self-consistently on the evolving population density. We identify two analytically tractable regimes: a Linear-Quadratic-Gaussian (LQG) benchmark in which the infinite-dimensional mean-field system reduces to a finite set of Riccati and linear ODEs, and a Gaussian-mixture regime governed by a piecewise-constant protocol that preserves closed-form solvability. For a quadratic interaction potential with schedule βt and zero base drift we prove that the self-consistent MF guidance is the exact linear interpolant between initial and target global means -- a result that holds for arbitrary initial and target densities and any βt. Applied to demand-response control of energy systems, where agents aggregated into an ensemble are energy consumers (e.g. The energy saving is independent of the number of zones per building (d = 1-32 tested), confirming that the linear guidance formula broadcasts a single d-vector with O(d) communication and grows mildly in compute (sub-cubically for d 32, asymptotically O(d3) for d 1). Introduction Generative AI has been transformed by diffusion models, which frame sample generation as a stochastic process steered from noise to data [1-3]. A key structural feature of these models -- shared with other generative models, e.g. Similarly, stochastic optimal transport (SOT) and Schrödinger bridge formulations [6-8] cast distribution matching as an independent-particle path optimization, yielding tractable convolutions of Green functions but discarding inter-particle information; stochastic interpolants [9] construct flexible transport bridges between arbitrary densities via tunable continuous-time stochastic processes, recovering the Schrödinger bridge as a special limit -- again in an independent-particle framework.

Groundwater in the Densu Basin is increasingly threatened by heavy metal contamination, but conventional methods fail to capture the statistical complexity and spatial heterogeneity of pollution indicators. A key challenge is modelling the Heavy Metal Pollution Index (HPI), which is typically skewed and affected by correlated contaminants, leading to biased predictions without transformation. This study develops a predictive framework integrating response transformations with nested cross-validated ensemble machine learning. Three transformations (raw, log, and Gaussian copula) were applied to HPI and evaluated across six learners: support vector regression (SVM), $k$-nearest neighbours (k-NN), CART, Elastic Net, kernel ridge regression, and a stacked Lasso ensemble. Raw-scale models produced deceptively high fits (Elastic Net and stacked ensemble $R^2 \approx 1.0$), suggesting over-optimism. The log transformation stabilised variance (SVM: $R^2 = 0.93$, RMSE $= 0.18$; k-NN: $R^2 = 0.92$, RMSE $= 0.20$). The Gaussian copula gave the most reliable results: stacked ensemble $R^2 = 0.96$ (RMSE $= 0.19$), with other learners maintaining high accuracy. Copula-based models improved residuals and produced spatially plausible maps. DBSCAN clustering revealed Fe and Mn as primary HPI contributors, consistent with regional hydrogeochemistry. Limitations include reliance on random (not spatial) cross-validation and basin-specific scope. Future work should explore spatial validation and other geological settings. Overall, distribution-aware ensembles with clustering diagnostics offer robust, interpretable assessments of groundwater contamination.