Algorithmic decisions about individuals require predictions that are not only accurate but also fair with respect to sensitive attributes such as gender and race. Causal notions of fairness align with legal requirements, yet many methods assume access to detailed knowledge of the underlying causal graph, which is a demanding assumption in practice. We propose a learning framework that achieves interventional fairness by leveraging a causal graph over \textit{clusters of variables}, which is substantially easier to estimate than a variable-level graph. With possible \textit{adjustment cluster sets} identified from such a cluster causal graph, our framework trains a prediction model by reducing the worst-case discrepancy between interventional distributions across these sets. To this end, we develop a computationally efficient barycenter kernel maximum mean discrepancy (MMD) that scales favorably with the number of sensitive attribute values. Extensive experiments show that our framework strikes a better balance between fairness and accuracy than existing approaches, highlighting its effectiveness under limited causal graph knowledge.

Horrifying next twist in the Alexander brothers case: MAUREEN CALLAHAN exposes an unthinkable perversion that's been hiding in plain sight Alexander brothers' alleged HIGH SCHOOL gang rape video: Classmates speak out on sick'taking turns' footage... as creepy unseen photos are exposed Model Cindy Crawford, 60, mocked for her'out of touch' morning routine: 'Nothing about this is normal' Kentucky mother and daughter turn down $26.5MILLION to sell their farms to secretive tech giant that wants to build data center there Tucker Carlson erupts at Trump adviser as she hurls'SLANDER' claim linking him to synagogue shooting NFL superstar Xavier Worthy spills all on Travis Kelce, the Chiefs' struggles... and having Taylor Swift as his No 1 fan Heartbreaking video shows very elderly DoorDash driver shuffle down customer's driveway with coffee order because he is too poor to retire Amber Valletta, 52, was a '90s Vogue model who made movies with Sandra Bullock and Kate Hudson, see her now Nancy Mace throws herself into Iran warzone as she goes rogue on Middle East rescue mission: 'I AM that person' Hidden toxins in kids' treats EXPOSED: Health guru Jillian Michaels' sit-down with Casey DeSantis reveals dangers lurking in popular foods READ MORE: I communicated with non-human intelligence... and what they told me proves God's existence New research has suggested that UFOs could be clustering around underwater canyons off the US coastline, raising fresh questions about whether mysterious craft could be operating beneath the ocean. An analysis of more than 80,000 reports found concentrated clusters of sightings near steep submarine canyon systems, particularly along the West Coast. The findings stem from an independent study testing the so-called'cryptoterrestrial hypothesis,' which proposes that unidentified aerial phenomena could originate from hidden non-human intelligence on Earth rather than distant planets. Using publicly available UFO sighting data and detailed ocean depth maps, the report identified correlations between reported sightings and deep underwater terrain features. The analysis also uncovered a striking geographical anomaly, with clustering patterns appearing on the West Coast but not on the East or Gulf coasts.

Kentucky mother and daughter turn down $26.5MILLION to sell their farms to secretive tech giant that wants to build data center there Horrifying next twist in the Alexander brothers case: MAUREEN CALLAHAN exposes an unthinkable perversion that's been hiding in plain sight Hollywood icon who starred in Psycho after Hitchcock dubbed her'my new Grace Kelly' looks incredible at 95 Kylie Jenner's total humiliation in Hollywood: Derogatory rumor leaves her boyfriend's peers'laughing at her' behind her back Tucker Carlson erupts at Trump adviser as she hurls'SLANDER' claim linking him to synagogue shooting Ben Affleck'scores $600m deal' with Netflix to sell his AI film start-up Long hair over 45 is ageing and try-hard. I've finally cut mine off. Alexander brothers' alleged HIGH SCHOOL rape video: Classmates speak out on sickening footage... as creepy unseen photos are exposed Heartbreaking video shows very elderly DoorDash driver shuffle down customer's driveway with coffee order because he is too poor to retire Amber Valletta, 52, was a '90s Vogue model who made movies with Sandra Bullock and Kate Hudson, see her now Model Cindy Crawford, 60, mocked for her'out of touch' morning routine: 'Nothing about this is normal' MORE: Who were the Three Wise Men? New research rewrites the mystery of the Bible's magi A Babylonian'time capsule' buried for more than two millennia under the ruins of a ziggurat in modern-day Iraq has revealed never-before-seen details about the biblical king Nebuchadnezzar II. Two cylinders bearing a royal inscription were buried as'foundation deposits' - ritual objects buried under ancient buildings as a divine blessing believed to ensure the structure's longevity. The cylinders, each made of baked clay, were originally unearthed at the ruins of the temple in the ancient city of Kish, one of the most important cities in Mesopotamia.

ROTC students at Old Dominion subdued and killed ISIS-linked gunman who left one dead, two wounded after shouting'Allahu Akbar' and opened fire Horrifying next twist in the Alexander brothers case: MAUREEN CALLAHAN exposes an unthinkable perversion that's been hiding in plain sight Kentucky mother and daughter turn down $26.5MILLION to sell their farms to secretive tech giant that wants to build data center there Hollywood icon who starred in Psycho after Hitchcock dubbed her'my new Grace Kelly' looks incredible at 95 Kylie Jenner's total humiliation in Hollywood: Derogatory rumor leaves her boyfriend's peers'laughing at her' behind her back Tucker Carlson erupts at Trump adviser as she hurls'SLANDER' claim linking him to synagogue shooting Ben Affleck'scores $600m deal' with Netflix to sell his AI film start-up Long hair over 45 is ageing and try-hard. I've finally cut mine off. Alexander brothers' alleged HIGH SCHOOL rape video: Classmates speak out on sickening footage... as creepy unseen photos are exposed Heartbreaking video shows very elderly DoorDash driver shuffle down customer's driveway with coffee order because he is too poor to retire Amber Valletta, 52, was a '90s Vogue model who made movies with Sandra Bullock and Kate Hudson, see her now Model Cindy Crawford, 60, mocked for her'out of touch' morning routine: 'Nothing about this is normal' Revealed: The most cringeworthy words, according to Gen Z - so, are you still using them? Are you down with the kids, or beginning to show your age? The answer may lie in how many of these'cringeworthy' words you're still using.

ROTC students at Old Dominion subdued and killed ISIS-linked gunman who left one dead, two wounded after shouting'Allahu Akbar' and opened fire Horrifying next twist in the Alexander brothers case: MAUREEN CALLAHAN exposes an unthinkable perversion that's been hiding in plain sight Kentucky mother and daughter turn down $26.5MILLION to sell their farms to secretive tech giant that wants to build data center there Hollywood icon who starred in Psycho after Hitchcock dubbed her'my new Grace Kelly' looks incredible at 95 Kylie Jenner's total humiliation in Hollywood: Derogatory rumor leaves her boyfriend's peers'laughing at her' behind her back Tucker Carlson erupts at Trump adviser as she hurls'SLANDER' claim linking him to synagogue shooting Ben Affleck'scores $600m deal' with Netflix to sell his AI film start-up Long hair over 45 is ageing and try-hard. I've finally cut mine off. Alexander brothers' alleged HIGH SCHOOL rape video: Classmates speak out on sickening footage... as creepy unseen photos are exposed Heartbreaking video shows very elderly DoorDash driver shuffle down customer's driveway with coffee order because he is too poor to retire Amber Valletta, 52, was a '90s Vogue model who made movies with Sandra Bullock and Kate Hudson, see her now Model Cindy Crawford, 60, mocked for her'out of touch' morning routine: 'Nothing about this is normal' Are you killing your dog with kindness? A single sausage is the canine equivalent of THREE bags of crisps - while a trendy'puppuccino' is comparable to a McDonald's milkshake, vets warn It can be hard to resist those puppy dog eyes - but treating your pet to a sausage or trendy'puppuccino' could be putting their health at risk, vets have warned. As pet treat trends go viral online, experts say these seemingly fun snacks can pack a significant calorie punch.

Her husband wanted to use ChatGPT to create sustainable housing. Then it took over his life. Kate Fox says Joe Ceccanti was the'most hopeful person' before he started spending 12 hours a day with a chatbot On 7 August, Kate Fox received a phone call that upended her life. A medical examiner said that her husband, Joe Ceccanti - who had been missing for several hours - had jumped from a railway overpass and died. Ceccanti had no history of depression, she said, nor was he suicidal - he was the "most hopeful person" she had ever known. In fact, according to the witness accounts shared with Fox later, just before Ceccanti jumped, he smiled and yelled: "I'm great!" to the rail yard attendants below when they asked him if he was OK.

Kentucky mother and daughter turn down $26.5MILLION to sell their farms to secretive tech giant that wants to build data center there Horrifying next twist in the Alexander brothers case: MAUREEN CALLAHAN exposes an unthinkable perversion that's been hiding in plain sight Hollywood icon who starred in Psycho after Hitchcock dubbed her'my new Grace Kelly' looks incredible at 95 Kylie Jenner's total humiliation in Hollywood: Derogatory rumor leaves her boyfriend's peers'laughing at her' behind her back Tucker Carlson erupts at Trump adviser as she hurls'SLANDER' claim linking him to synagogue shooting Ben Affleck'scores $600m deal' with Netflix to sell his AI film start-up Long hair over 45 is ageing and try-hard. I've finally cut mine off. Alexander brothers' alleged HIGH SCHOOL rape video: Classmates speak out on sickening footage... as creepy unseen photos are exposed Heartbreaking video shows very elderly DoorDash driver shuffle down customer's driveway with coffee order because he is too poor to retire Amber Valletta, 52, was a '90s Vogue model who made movies with Sandra Bullock and Kate Hudson, see her now Model Cindy Crawford, 60, mocked for her'out of touch' morning routine: 'Nothing about this is normal' Do YOU have one gathering dust in your attic? It's time to feel old - as believe it or not, Pokémon turns 30 today. The franchise continues to sell billions of cards every single year, but a rare handful of highly sought-after cards can be worth thousands, if not millions, at auction.

A clump of human brain cells can play the classic computer game . While its performance is not up to par with humans, experts say it brings biological computers a step closer to useful real-world applications, like controlling robot arms. In 2021, the Australian company Cortical Labs used its neuron-powered computer chips to play . The chips consisted of clumps of more than 800,000 living brain cells grown on top of microelectrode arrays that can both send and receive electrical signals. Researchers had to carefully train the chips to control the paddles on either side of the screen.

In this work, we analyze alternative effective sample size (ESS) metrics for importance sampling algorithms, and discuss a possible extended range of applications. We show the relationship between the ESS expressions used in the literature and two entropy families, the Rényi and Tsallis entropy. The Rényi entropy is connected to the Huggins-Roy's ESS family introduced in \cite{Huggins15}. We prove that that all the ESS functions included in the Huggins-Roy's family fulfill all the desirable theoretical conditions. We analyzed and remark the connections with several other fields, such as the Hill numbers introduced in ecology, the Gini inequality coefficient employed in economics, and the Gini impurity index used mainly in machine learning, to name a few. Finally, by numerical simulations, we study the performance of different ESS expressions contained in the previous ESS families in terms of approximation of the theoretical ESS definition, and show the application of ESS formulas in a variable selection problem.

Discrete diffusion models have achieved strong empirical performance in text and other symbolic domains, with masked (absorbing-rate) variants emerging as competitive alternatives to autoregressive models. Among existing samplers, the Euler method remains the standard choice in many applications, and more recently, the First-Hitting Sampler (FHS) has shown considerable promise for masked diffusion models. Despite their practical success, the theoretical understanding of these samplers remains limited. Existing analyses are conducted in Kullback-Leibler (KL) divergence, which often yields loose parameter dependencies and requires strong assumptions on score estimation. Moreover, these guarantees do not cover recently developed high-performance sampler of FHS. In this work, we first develop a direct total-variation (TV) based analysis for the Euler method that overcomes these limitations. Our results relax assumptions on score estimation, improve parameter dependencies, and establish convergence guarantees without requiring any surrogate initialization. Also for this setting, we provide the first convergence lower bound for the Euler sampler, establishing tightness with respect to both the data dimension $d$ and the target accuracy $\varepsilon$. Finally, we analyze the FHS sampler and show that it incurs no sampling error beyond that induced by score estimation, which we show to be tight with a matching lower error bound. Overall, our analysis introduces a direct TV-based error decomposition along the CTMC trajectory and a decoupling-based path-wise analysis for FHS, which may be of independent interest.