In networks, effective dynamic treatment allocation requires deciding both whom to treat and also when, so as to amplify policy impact through spillovers. An early intervention at a well-connected node can trigger cascades that change which nodes are worth targeting in the next period. Existing treatment strategies under network interference are largely static while dynamic treatment frameworks typically ignore network structure altogether. We integrate these perspectives and propose Q-Ising, a three-stage pipeline that (i) estimates network adoption dynamics via a Bayesian dynamic Ising model from a single observed panel, (ii) augments treatment adoption histories with continuous posterior latent states, and (iii) learns a dynamic policy via offline reinforcement learning. The Bayesian mechanism enables uncertainty quantification over dynamic decisions, yielding posterior ensemble policies with interpretable spillover estimates. We provide a finite-sample regret upper bound that decomposes into standard offline-RL uncertainty, network abstraction error, and first stage error in Ising state estimation. We apply our method to data from Indian village microfinance networks and synthetic stochastic block models under simulated heterogeneous susceptible-infected-susceptible (SIS) dynamics and demonstrate that adaptive targeting outperforms static centrality benchmarks.

This paper proposes a hybrid methodology to improve the approximation of SABR (Stochastic Alpha Beta Rho) implied volatility by combining analytical structure with machine learning. The approach augments the neural-network input representation with geometric features derived from the stochastic differential equations of the SABR model. Unlike approaches that fully replace analytical formulas with black-box models, the proposed framework preserves the analytical backbone of the model. The hybridization operates along two complementary dimensions. First, geometry-aware variables reflecting intrinsic properties of the SABR dynamics are used as structured inputs to the network. Second, the neural network is trained to learn the residual error relative to Hagan's closed-form approximation rather than implied volatility directly. The resulting model acts as a structured residual correction to the analytical formula, retaining interpretability while capturing higher-order effects that are not included in the asymptotic expansion. Numerical experiments conducted over realistic parameter domains, as well as stressed environments, show that the method improves accuracy and robustness compared with both analytical approximations and standard neural-network approaches. Because the correction remains lightweight and structurally consistent with the underlying model, the framework is well suited for real-time pricing and calibration in practical trading environments.

Despite the prevalence of the attention sink phenomenon in Large Language Models (LLMs), where initial tokens disproportionately monopolize attention scores, its structural origins remain elusive. This work provides a \textit{mechanistic explanation} for this phenomenon. First, we trace its root to the value aggregation process inherent in self-attention, which induces a systematic variance discrepancy. We further demonstrate that this discrepancy is drastically amplified by the activation of super neurons within Feed-Forward Network (FFN) layers. Specifically, the channel-sparse down-projections trigger a dimension disparity of the first-token representation, necessitating the formation of attention sinks as a structural anchor. Then, we validate this causal chain through two controlled interventions: (i) isolating the aggregation effect via attention mask modifications and (ii) amplifying the variance of targeted token representations. Both interventions can replicate attention sinks at arbitrary positions. Our mechanistic understanding offers a foundation for the systematic control of sink formation. Finally, as a proof of concept, we propose \textit{head-wise RMSNorm}, an architectural modification that stabilizes value aggregation outputs during pre-training. Our experiments demonstrate that restoring statistical parity across positions significantly accelerates convergence.

Bayesian model calibration is central to digital twins and computer experiments, as it aligns model outputs with field observations by estimating calibration parameters and correcting systematic model bias. Classical Bayesian calibration introduces latent parameters and a discrepancy function to model bias, but suffers from parameter--discrepancy confounding and is typically formulated as an offline procedure under a stationary data-generating assumption. These limitations are restrictive in modern digital twin applications, where systems evolve over time and may exhibit gradual drift and abrupt regime shifts. While data assimilation methods enable sequential updates, they generally do not explicitly model systematic bias and are less effective under abrupt changes. We propose Bayesian Recursive Projected Calibration (BRPC), an online Bayesian calibration framework for streaming data under simulator mismatch and nonstationarity. BRPC extends projected calibration to the online setting by separating a discrepancy-free particle update for calibration parameters from a conditional Gaussian process update for discrepancy, preserving identifiability while enabling bias-aware adaptation under gradual system evolution. To handle abrupt changes, BRPC is integrated with restart mechanisms that detect regime shifts and reset the calibration process. We establish theoretical guarantees for both components, including tracking performance under gradual evolution and false-alarm and detection behavior for restart mechanisms. Empirical studies on synthetic and plant-simulation benchmarks show that BRPC improves calibration accuracy under gradual changes, while restart-augmented BRPC further improves robustness and predictive performance under abrupt regime shifts compared to sliding-window Bayesian calibration and data assimilation baselines.

Megan Rapinoe, in a shock to no one, backs Angel Reese skipping interviews as'taking power back' Here's why the coaches association's 24-team College Football Playoff could ruin the sport Boston Celtics star Jaylen Brown tells ESPN's Stephen A Smith to'be quiet and retire' President Trump on $1,000 World Cup ticket prices: 'I wouldn't pay it either, to be honest' Pirates vs. Diamondbacks betting preview targets the under as both offenses go cold in series Former LSU coach Brian Kelly uses AI to prepare for job interviews, proving he's just like the rest of us Newsom office source responds to planned protest against trans athlete at state playoff girls' track meet US waits for Iran's response on peace proposal Authorities try to'connect the dots' on hantavirus infections Jesse Watters: Spencer Pratt is a'charismatic, common-sense populist' Greg Gutfeld: Dana White laughs off the'toxic masculinity thing' Iranians are fearful of facing the regime's frustration and anger after the war, activist says OutKick White House calls out Newsom as California girls' track and field controversy reignites Spokeswoman called Newsom'a truly sick individual who has no regard for fairness, dignity, and respect' Jurupa Valley High School graduate Hadeel Hazameh responded to the news that the Trump administration has launched a Title IX investigation into her district over an incident involving trans volleyball teammate, which has resulted in her graduating early and leaving her sports career behind. President Donald Trump's White House has officially put California Gov. Gavin Newsom on notice as a controversial girls' track and field postseason is set to begin this weekend. A White House spokesperson called out Newsom in a statement to Fox News Digital as his state continues to allow biological male trans athletes to compete in girls' high school sports. Gavin Newscum is a truly sick individual who has no regard for fairness, dignity, and respect. If he did, he wouldn't allow men to compete in women's sports, limiting women's opportunities and jeopardizing their health and safety.

A man has been charged after allegedly threatening Andrew Mountbatten-Windsor during an incident near his home on the Sandringham Estate in Norfolk. Norfolk Police earlier said a man was arrested shortly after 19:30 BST on Wednesday after officers received a report of a man a behaving in an intimidating manner in Wolferton. The Daily Telegraph reported Mountbatten-Windsor was threatened by a balaclava-clad man while out walking his dogs and fled to his car along with his security. Alex Jenkinson, 39, of Stowmarket, Suffolk, has been remanded in custody and is due to appear at Norwich Magistrates' Court on Friday. Police said he has been charged with two counts of using threatening, abusive or insulting words or behaviour to harass someone or cause alarm or distress and failing to provide a specimen of blood in custody.

Here's why the coaches association's 24-team College Football Playoff could ruin the sport Boston Celtics star Jaylen Brown tells ESPN's Stephen A Smith to'be quiet and retire' President Trump on $1,000 World Cup ticket prices: 'I wouldn't pay it either, to be honest' Pirates vs. Diamondbacks betting preview targets the under as both offenses go cold in series Former LSU coach Brian Kelly uses AI to prepare for job interviews, proving he's just like the rest of us Newsom office source responds to planned protest against trans athlete at state playoff girls' track meet'This can touch anyone': Gorman family speaks following loss of Sheridan'Project Freedom' could soon resume: Report Iranian people are not citizens, but'subjects' of the regime: Middle East expert Vice Admiral Robert Harward weighs in on restarting'Project Freedom' in Strait of Hormuz Largest teachers' union accused of antisemitism in federal civil rights complaint McEnany's URGENT plea: 'Be Spencer Pratt!' WHO doesn't expect large Hantavirus outbreak US blockade keeps stranglehold on Iran's economy What And Who Is Next In The Russini Saga? | OutKick Hot Mic TMZ Sports obtained documents this week showing that New England Patriots coach Mike Vrabel and former NFL reporter Dianna Russini signed a waiver for a private boating trip during her pregnancy in 2021. The report adds that Vrabel and Russini were the only two people on board for the two-to-three-hour excursion. On Thursday, the outlet obtained video and photos of the two on a dock in Putnam County, TN. See TMZ Sports' video below: ARE WE SURE MIKE VRABEL WILL SURVIVE RUSSINI SCANDAL AND COACH PATRIOTS THIS SEASON? According to the report, Russini gave birth to her first child with her husband, Kevin, later that summer. She was about six to seven months pregnant during the boating trip with Vrabel.

Boston Celtics star Jaylen Brown tells ESPN's Stephen A Smith to'be quiet and retire' President Trump on $1,000 World Cup ticket prices: 'I wouldn't pay it either, to be honest' Pirates vs. Diamondbacks betting preview targets the under as both offenses go cold in series Former LSU coach Brian Kelly uses AI to prepare for job interviews, proving he's just like the rest of us Newsom office source responds to planned protest against trans athlete at state playoff girls' track meet'This can touch anyone': Gorman family speaks following loss of Sheridan'Project Freedom' could soon resume: Report Iranian people are not citizens, but'subjects' of the regime: Middle East expert Vice Admiral Robert Harward weighs in on restarting'Project Freedom' in Strait of Hormuz Largest teachers' union accused of antisemitism in federal civil rights complaint McEnany's URGENT plea: 'Be Spencer Pratt!' WHO doesn't expect large Hantavirus outbreak US blockade keeps stranglehold on Iran's economy OutKick Here's why the coaches association's 24-team College Football Playoff could ruin the sport College Football Playoff Eying 24 Team Expansion: Is Bigger Better? The college football playoff could expand to 24 teams -- and a lot of people aren't happy about it. After just moving to 12, this feels like overkill. Is the sport trying to fix something that wasn't broken? When college football first moved to a playoff format, there were just four teams included.

Today on, we're diving into recent reports that the Trump administration is considering an executive order that would establish some sort of federal oversight over new AI models. This week on, the team discusses the surprising reports of the Trump administration seemingly reversing its stance when it comes to AI safety and regulation. We also look into what exactly is going on with the Hantavirus outbreak, and whether you should be worried. Also, we get into the story of how a former federal employee who was ousted by Elon Musk's so-called Department of Government Efficiency is now running for office. Plus, a Spirit Airlines laid off employee shares with us how they experienced the company's shutdown news last weekend and what they'll miss most about the job. A Federal Worker Was Fired for Filming DOGE. Write to us at [email protected] . You can always listen to this week's podcast through the audio player on this page, but if you want to subscribe for free to get every episode, here's how: If you're on an iPhone or iPad, open the app called Podcasts, or just tap this link . And we're going to talk about whether this move actually signals a meaningful shift in future regulation of this technology.

Here's why the coaches association's 24-team College Football Playoff could ruin the sport President Trump on $1,000 World Cup ticket prices: 'I wouldn't pay it either, to be honest' Pirates vs. Diamondbacks betting preview targets the under as both offenses go cold in series Former LSU coach Brian Kelly uses AI to prepare for job interviews, proving he's just like the rest of us Newsom office source responds to planned protest against trans athlete at state playoff girls' track meet Iranians are fearful of facing the regime's frustration and anger after the war, activist says'This can touch anyone': Gorman family speaks following loss of Sheridan'Project Freedom' could soon resume: Report Iranian people are not citizens, but'subjects' of the regime: Middle East expert Vice Admiral Robert Harward weighs in on restarting'Project Freedom' in Strait of Hormuz Largest teachers' union accused of antisemitism in federal civil rights complaint McEnany's URGENT plea: 'Be Spencer Pratt!' WHO doesn't expect large Hantavirus outbreak OutKick Boston Celtics star Jaylen Brown tells ESPN's Stephen A Smith to'be quiet and retire' The viral exchange on X adds Brown to a list of NBA stars, including LeBron James and Kevin Durant, who've feuded with Smith ESPN commentator Stephen A. Smith is no stranger to having beef with NBA stars. It's time to add Celtics guard Jaylen Brown to the list. The latest dust-up started, naturally, on First Take, where Smith took aim at Brown for his comments following Boston's playoff collapse . Brown recently said this was his favorite year, despite the Celtics blowing a 3-1 series lead to the Philadelphia 76ers and getting eliminated in the first round of the NBA playoffs. That didn't sit well with Smith, who made it very clear Thursday that he thought Brown should have kept that to himself.