In regression modelling, the primary objective is to approximate the true conditional mean of a response given a set of features. To this end, various statistical models are used to fit a regression function that provides a mean estimate for each single set of features. This function is said to be calibrated if the resulting mean estimates match the true conditional means for almost all features. Aiming for calibration seems not achievable in practice as models are fitted on finite samples of noisy observations. A weaker notion of calibration is auto-calibration (sometimes also called mean-calibration or well-calibration); see, for example, Kr uger-Ziegel [22] and Denuit et al. [7]. This notion goes back to earlier works on the reliability of probabilistic forecasts in meteorology; we refer to Bross [2], Sanders [26] and Murphy-Winkler [23]. It means that when responses are grouped according to their mean estimates, the average of the responses within each group matches this estimate. This property is important in various applications where sums of mean estimates have to match sums of responses at a global and local level. This is, for example, the case in insurance pricing as an auto-calibrated pricing system avoids systematic cross-subsidy between different price cohorts; we refer the reader to Pohle [24], Denuit et al. [6], Fissler et al. [9] and W uthrich-Merz [30].

Gaussian processes with stationary kernels on bounded domains exhibit inflated posterior variance near the boundary. Despite being a long-recognized artifact in geostatistics and a source of over-exploration in Bayesian optimization, the causes and effects of boundary-induced acquisition bias are underexplored. We trace the root cause to a simple geometric mechanism: the truncation of the kernel correlation neighborhood at the domain boundary creates an observation-independent distortion that worsens with dimensionality. We show how this distortion manifests across three acquisition classes: variance maximization concentrates selections at the corners, whereas negative integrated posterior variance and expected predictive information gain move selections inward to axis-aligned interior shells. These patterns arise without reference to any objective function, meaning that acquisition behavior can be dominated by kernel geometry rather than the desired task-specific uncertainty. To quantify this, we introduce a function-free selection-profile diagnostic for arbitrary acquisitions, kernels, and bounded-domain geometries.

Most statistical and machine learning methods for directed interactions focus on pairwise effects among variables. Even existing cyclic models represent feedback primarily through node-level dependencies, making large-scale recurrent organization difficult to estimate and compare. This limitation is particularly acute in biological and neural systems, where interactions are highly recurrent and involve many overlapping cycles. We introduce a variational framework for statistical inference on cyclic interactions. Directed interactions are represented as edge flows on a simplicial complex and evolved under an energy-minimizing dynamical system. The resulting dynamics separate transient interaction components from persistent harmonic flows, yielding a low-dimensional cycle space that captures stable recurrent organization. Rather than enumerating individual cycles, the proposed framework represents cyclic interactions as elements of a Hilbert space, enabling projection, averaging, comparison, and population-level statistical inference. We establish theoretical properties of the harmonic projection, including characterization of the cycle space, variance reduction, and population inference. Simulations demonstrate substantially improved recovery of cyclic structure in dense recurrent systems compared with existing directed-interaction methods. Applied to resting-state fMRI from 400 human subjects, the framework reveals reproducible large-scale cyclic organization that is not detectable through edgewise averaging. These results provide a scalable statistical framework for studying recurrent interactions in high-dimensional dynamical systems.

Bayesian optimization (BO) is a popular and effective approach for tuning expensive, noisy experiments, but requires the formulation of an explicit objective function. Preferential BO (PBO) removes this requirement by learning from pairwise human feedback, yet existing methods struggle to efficiently optimize beyond low- and medium-dimensional problems due to their global search approaches. We address this limitation by developing a family of local PBO methods that transfer key ideas from high-dimensional BO to the preferential setting. In particular, we introduce local PBO methods which adapt trust-region and derivative-informed local search to pairwise preference feedback, where the latter exploits first- and second-order derivatives of the Laplace-approximated GP posterior. Our benchmark on GP sample paths, standard optimization benchmark functions, and policy-search tasks shows that local PBO methods are especially effective in high-dimensional and complex landscapes with steep optima. Compared with global preference-based baselines, they can substantially reduce cumulative regret, making them particularly useful for real-world preference-based optimization tasks such as policy search.

Could humanoid robots be heading for the battlefield? I've come to an industrial space in a tech-heavy area of San Francisco expecting to see a menacing humanoid robot solider doing something combat-like: the future of land-based warfare, perhaps. Instead, the black shiny faceless Phantom robot is engaged in free play, manipulating a bunch of coloured kids blocks. We need data from it just interacting with its environment [and] this is today's menu, explains Sankaet Pathak, co-founder and CEO of two-year-old start-up Foundation Robotics, which is developing Phantom for military and civilian applications. Later he pushes its 80kg steel-covered body around the room to demonstrate its stability and shows me how it walks.

PCWorld examines Apple's revamped Siri, which integrates AI across macOS with a focus on productivity and seamless ecosystem functionality. Apple's approach prioritizes local, private AI processing and practical user benefits, contrasting sharply with Microsoft's fragmented Copilot solutions. The unified Siri experience demonstrates how Apple's strategic AI integration creates a more holistic user experience than Windows' various disconnected tools. Apple's secret is that, like the Queen of England, it is never early, and never late. It is always on time.

Massive twist in JPMorgan'sex slave' case as accuser unveils NEW dossier of wild claims: 'The story is about to change dramatically' Trump's return to New York City for historic NBA Finals game takes dramatic turn as White House reveals sweeping ICE plot When I lost 150lbs men did double-takes in the street. Then I handed my husband sexy photos of my new body and he barely looked, so I moved on. I used to only drink socially but after a bad break-up I started boozing every night and gorging myself on crisps - then I took this £3 pill... now both wine AND junk food taste awful and I have dropped to a size 8 Americans finally snap over'ridiculous' tipping culture as millions slash gratuities... and reveal the trick they are refusing to fall for Luigi Mangione's sister lands prized job at America's most prestigious hospital... as she makes bold public move before murder trial Incredible footage emerges of star-studded celebrity row from Knicks' last appearance in NBA Finals in 1999 Kim Kardashian'steals Monaco Grand Prix winner's TOWEL' on disastrous debut as an F1 WAG to Lewis Hamilton after snubbing iconic TV reporter Explosive new details of accused baby killer's sickening video confession... as glamorous PhD student makes grim bathroom claim after 13-hour interrogation MOLLY CLAYTON: Whispers in the paddock, and some VERY raised eyebrows: Why Kim Kardashian's first appearance with Lewis Hamilton at the Monaco Grand Prix is sending the rumour mill into overdrive... as his mother stays firmly away Kanye West's wife Bianca Censori almost pops out of her nude suit on his 49th birthday after she calls him her '4ever' Idaho murder victim's father reveals new account of chilling moment he came face-to-face with Bryan Kohberger: 'I want to hunt killers' Gilmore Girls star Alexis Bledel became a TV icon in her teens... see her now during rare red carpet appearance at age 44 US tourists are'tricked' into paying 44 euros for two ice creams in Rome Bill Gates's embarrassing secrets revealed by Epstein-linked mistress who Melinda HATED Uber's robotaxis arrive in the UK: Self-driving cars will be available in London this summer READ MORE: Waymo'goes rogue': Self-driving car wakes residents at 4am Uber has unveiled its fleet of self-driving robotaxis, which will soon take to the streets of London . Designed in collaboration with Wayve, the robotaxi is an all-electric Ford Mustang Mach-e, equipped with surround cameras and radar. The high-tech set-up allows Wayve's AI to see the world with full 360-degree visibility around the car at all times.

'Record the faces': Tense moment NBA boss gives VERY honest take on Trump attending Knicks game Outrage as Netanyahu is caught SPYING on Trump's Iran negotiators... as JD Vance reveals a chilling truth about Israel Massive twist in JPMorgan'sex slave' case as accuser unveils NEW dossier of wild claims: 'The story is about to change dramatically' Countless men have a condition that turns sex into agony - but few talk about it. Meghan Markle's As Ever website has had'less than 400,000 US visitors' since January - as Duchess launches collaboration with a lifestyle influencer to plug her products Karmelo Anthony's parents seen leaving the courtroom in tears just before son's defense team pulls shock move No one will admit the sleazy truth about skinny Serena Williams's sudden return to tennis. Call me evil... but I'm exposing her: LIZ JONES Gaming influencer Alex Cimo dies'very suddenly' aged 32 just a month after'refusing to accept his fate' Donald Trump's threat to Knicks fans revealed by lipreader in secret chat with MSG owner James Dolan during Spurs loss Apple just made five popular Apple Watches'obsolete' - and it will leave users without any support if something goes wrong Everyone always said I cleared my throat a lot. But then I developed shoulder pain and doctors discovered the sinister cause... the world's deadliest cancer. Don't leave it too late like I did Why Kate always wears such pale colours to weddings: Princess of Wales wouldn't dream of upstaging the bride, says LAURA CRAIK, after her off-white Roland Mouret dress raised eyebrows at Peter Phillips' wedding Cunning new tactic women are using to cheat.

Caitlyn Jenner biographer and Robin Riker's ex William Hasley found dead on hiking trail at 78 Disgraceful texts'hot' teacher sent boy, 17, who she had illegal sex with where she moaned about her HUSBAND Everyone always said I cleared my throat a lot. But then I developed shoulder pain and doctors discovered the sinister cause... the world's deadliest cancer. Don't leave it too late like I did Urgent recall for 1.1m vehicles over fears they could spontaneously CATCH FIRE even when parked Leaked transcript of UNAIRED 60 Minutes interview exposes REAL reason'callous' CBS star Scott Pelley'deserved to be fired' Karmelo Anthony's parents seen leaving the courtroom in tears just before son's defense team pulls shock move'Great' mom, 32, tried to gas herself and her three young kids to death after inviting them to'popcorn sleepover' in car, prosecutors allege The porn-fuelled fantasy middle-class husbands are desperate to try with their wives... and it almost always ends in divorce: JANA HOCKING The historic steel mill that helped build America was written off for dead. Medical student, 24, died by suicide in his white coat a day after he was suspended for alleged'inappropriate' behavior towards female patient, lawsuit alleges, as his heartbreaking goodbye note to parents is revealed Furious dad films his partner in bed with his 19-year-old son: You've seen the viral video - now all three tell the Daily Mail what REALLY happened in the scandal gripping Australia Woke Vegas school compared boy to racist cross burner over pro-ICE stickers and expelled him... but did not punish pro-migrant students for class walkout, lawsuit alleges Gaming influencer Alex Cimo dies'very suddenly' aged 32 just a month after'refusing to accept his fate' Mother's final words before she was shot dead'by new husband' in front of her two young children All the backstage gossip from Miami Swim Week: Insider exposes'catty' VIP's diva demands... STEALING... and'morbidly embarrassing' celeb moment everyone is whispering about They said their toothpaste was the best for my daughter... then I read the sickening claims I am the type of mom who reads every label before buying a product for my four-year-old daughter. So when I learned about a lawsuit against a toothpaste marketed as safe, natural and free from artificial dyes and sweeteners, I immediately checked the tube sitting in my bathroom.

Karmelo Anthony's sobbing mother pleads with jurors to show'mercy' for her son as they prepare to sentence him for murder of Austin Metcalf, 17, that shocked America Caitlyn Jenner biographer and Robin Riker's ex William Hasley found dead on hiking trail at 78 Karmelo Anthony's mother sobs with shock as son is found guilty of murdering Austin Metcalf, 17, in stabbing that horrified America: Live updates Disgraceful texts'hot' teacher sent boy, 17, who she had illegal sex with where she moaned about her HUSBAND Everyone always said I cleared my throat a lot. But then I developed shoulder pain and doctors discovered the sinister cause... the world's deadliest cancer. Don't leave it too late like I did Leaked transcript of UNAIRED 60 Minutes interview exposes REAL reason'callous' CBS star Scott Pelley'deserved to be fired' Urgent recall for 1.1m vehicles over fears they could spontaneously CATCH FIRE even when parked Disturbing new death scene photos show tech whistleblower's haunting final moments... as forensic report casts doubt on suicide claims: 'Execution angle' 'Great' mom, 32, tried to gas herself and her three young kids to death after inviting them to'popcorn sleepover' in car, prosecutors allege The porn-fuelled fantasy middle-class husbands are desperate to try with their wives... and it almost always ends in divorce: JANA HOCKING Grim-faced former Louisiana mayor Misty Roberts arrives in court for sentencing after being found guilty of having sex with son's teenage friend John Oliver's private panic: Late-night curse spreads and host prepares for worst as insiders reveal his desperate'plan B'... and the industry whispers swirling about his fate Woke Vegas school compared boy to racist cross burner over pro-ICE stickers and expelled him... but did not punish pro-migrant students for class walkout, lawsuit alleges Medical student, 24, died by suicide in his white coat a day after he was suspended for alleged'inappropriate' behavior towards female patient, lawsuit alleges, as his heartbreaking goodbye note to parents is revealed Mother's final words before she was shot dead'by new husband' in front of her two young children All the backstage gossip from Miami Swim Week: Insider exposes'catty' VIP's diva demands... STEALING... and'morbidly embarrassing' celeb moment everyone is whispering about The brewing super El Niño will likely be the strongest ever recorded, new predictions suggest. The latest modelling from the European Centre for Medium-Range Weather Forecasts (ECMWF) shows that sea temperatures will be well above average later this year. Scientists measure the intensity of El Niño using the Niño 3.4 index, which records sea surface temperature anomalies between 5 degrees north and 5 degrees south latitude, and 120 degrees west and 170 degrees west longitude.